Loop Quantum Gravity vs. M-Theory II.

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Josh Willis

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Jun 22, 2001, 12:05:22 AM6/22/01
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Readers are forewarned: this post is long!

Lubos Motl and others have written much in recent days, and it seems
to me that there are a few main points to be discussed, as well as a
host of other lesser points in comparing String/M-theory to Loop
Quantum Gravity. I shall therefore enumerate what seem to me to be the
general points, and discuss these, and then also have something to say
about some other more specific things later on.

There are two categories here which it seems evident to me are being
confused with one another:

I. What is known rigorously about a theory, be it SMT or LQG or
anything else, as opposed to suggestive and partially heuristic
results. This is a matter of theorems, not opinions, and the goal here
is, as stated in the preface to Streater & Wightman's *PCT, Spin and
Statistics, and All That*, the elimination of all theorems whose
proofs are non-existent.

II. In this category belong many other things: partial or heuristic
results, intuition about what is an important criterion in guiding the
progress of a field, etc. I am not denying that things in this
category are suggestive and may well be a guide to results that
eventually belong in category I above. Indeed, I cannot imagine how
to make progress in any field of science without these. But I am
insisting that we not confuse things in this category with the one
above, and that in comparing different theories we not compare apples
to oranges by comparing the absence of results in category I above in
one theory with the presence of results in category II in another.

Also, I think another general area that has been the source of some
confusion is the generic non-uniqueness of a quantization of a given
theory for which we have a classical description. So I shall say
something about this as well.

So, first a discussion of categories I and II above, as they relate to
SMT and LQG.

String theory's first big success and a driving force behind it is the
absence of ultraviolet divergences. In other words, string theory
lets us calculate a perturbative expansion of the S-matrix and know
that at each order in perturbation theory, the individual terms are
finite. Since this is based heavily on the similar approach to quantum
field theory, it is useful to recall what is known about perturbation
theory in general and the S-matrix in particular in QFT, and to
compare this both to ordinary QM and SMT.

First, on perturbation theory in general. A first approach to putting
one's theory into category I above might be:

1. *Define* your theory by its perturbation expansion.

This is well and good---if your perturbation theory converges. I
don't know anyone who thinks this likely for string theory, and
certainly no *proof* that it converges.

But an immediate obejection to this might be:

2. Perturbation series often do not converge, even in ordinary QM, and
nobody complains about that. Instead, we believe, usually without
rigorous proof, that the perturbation series is asymptotic to the
quantity it purports to calculate.

There is, however, an important difference between QM and QFT in this
respect: in QM, we have nonperturbative proofs that all of the things
we are calculating really do exist; we know that we have a Hilbert
space with appropriate self-adjoint operators and eigenstates
thereof. In QFT, in general, we have none of this. That is what the
whole enterprise of constructive quantum field theory is about:
proving that the heurisitc expressions one learns in a standard QFT
course really do have mathematical meaning. Thus far, this has been
done for interacting theories in two and three dimensions, and only
for free fields in dimension four. See:

J. Glimm and A. Jaffe, *Quantum Physics: A Functional Integral
Point of View*

and references therein for more details.

One might still respond with:

3. Fine, maybe I don't have a mathematical proof that my perturbation
series is asymptotic, but I can have a good enough experimental
proof. QED is the canonical example: we just plug in the numbers and
find unprecedentedly excellent agreement with experiment. Let those
who insist on rigor look for it themselves in their ivory towers,
which after all are usually atop the mathematics department rather
than the physics department anyway.

The problem with this approach is that you may well find that
experimentally important things which your theory should explain are
described nonperturbatively. In QED this rarely, if ever, happens,
but as Matthew Nobes has already indicated in QCD it does. Thus,
because of asymptotic freedom and factorization we can calculate
cross sections for many things of interest in QCD, but we still have
no proof that QCD has a stable eigenstate of a Hamiltonian that looks
like the proton, with the correct mass, etc. The same thing will
happen in gravity, as I explain in more detail below, and indeed
already happens at the classical level: there are interesting and
important solutions of Einstein's equation which cannot be described
as a perturbation about some other, simpler solution. The
Schwarzschild solution is not a perturbation of the Minkowski
solution, for example, simply because the manifolds are not
diffeomorphic to one another.

Finally, we should say something about the S-matrix. Everything we
have said thus far pertains to perturbation theory in general; I can
of course use perturbation theory to calculate many things other than
S-matrix elements, such as wavefunctions or eigenvalues. In QFT, when
we try to calculate transition amplitudes for interactions, we run
immediately into the problem that we have, as indicated following 2)
above, no real Hilbert space whose states we could try to take
expectation amplitudes between. So we argue instead that if we
restrict attention to scattering processes, then when we look
infinitely far back into the past and forward into the future, the
particles are well separated and hence we can treat them as
noninteracting. And noninteracting fields we do have a well defined
theory for, as mentioned above; in particular, we have a Hilbert
space: good old Fock space that we have all come to know and love
since first we learned QFT at our mother's knee. So we speak of "in"
states---elements of Fock space in the infinitely far past---and "out"
states, which are elements of Fock space in the infinitely far
future. Now our definition of the S-matrix is that it calculates
the expectation value between an element of Fock space that represents
the incoming state, and another element of Fock space that represents
the outgoing state. We can do this precisely because the separation
of the particles well outside of the collision region means that the
incoming and outgoing states in the full, interacting theory are
unitarily equivalent to appropriate states in Fock space.

Except that they aren't. Physically, the reason for this is that in
any theory with self interaction, that interaction can never be turned
off because even when particles are very far away from each other,
they are interacting with themselves. Mathematically, this is seen,
for example, in van Hove's paradox, where he shows that the vacuum
state of a particular interacting theory is *orthogonal* to all of the
eigenstates of the corresponding non-interacting theory; there is a
discussion of this in, for example, Chapter V of:

I. E. Segal, *Mathematical Problems of Relativistic Physics*

More generally, there is the result known as Haag's theorem: that for
an interacting quantum field, the corresponding representations of the
canonical commutation relations are unitarily inequivalent and
therefore do not lie in "the same" Hilbert space (rather
unsurprisingly, a good place to read about this is in:

R. Haag, *Local Quantum Physics: Fields, Particles, Algebras*

See in particular the discussion of Section II.1 of the revised
edition). This situation is in stark contrast to ordinary QM, where,
for a given dimension for space and choice of particle type (e.g.,
spin zero), all theories have the same Hilbert space: L_2(R^n,d^n x).

Also, note that in string theory the situation is even worse, because
you do not have the analogue of Fock space---i.e., a Hilbert space of
non-interacting strings---because the theory inherently includes
interactions.

So, what bearing does all of this have on SMT? You have a
perturbative expansion of an S-matrix, but no proof that the
perturbation series converges, or that anything exists to which it
might be asymptotic. On top of that you inherit from QFT the problems
above in the definition of an S-matrix. Simply put, having a
perturbative expansion of an S-matrix, each of whose terms is finite,
does not a full quantum theory make. This is related to the
discussion going on in another thread about causal quantum theory.

You claim that String/M-theory has *already* been shown to be a
quantum theory of gravity. For that claim to be true, you would have
had to have shown the following:

1. That String/M-theory is in fact a quantum theory in the first
place. What is the Hilbert Space? If it is L_2(X,d\mu) for some space
X and measure \mu, what are they? What are the operators corresponding
to observables? Can you prove they are self-adjoint? What are their
spectra?

2. That GR is in fact the semiclassical limit of SMT. To do this,
you should prove that for any classical solution of Einstein's
equation, there is a corresponding state in your Hilbert space that
has, for all of the relevant observables, minimum uncertainty
distributed about the correct, corresponding values of the classical
solution. You should prove some kind of Ehrenfest properties. This
is what is done in ordinary QM; this is what people are now attempting
to do in LQG. It has not been done even for ordinary QFT in an
interacting theory.

The arguments you cite on this second point don't come close to doing
this. I don't have a copy of Green, Schwarz, and Witten, but let's
look at Section 3.7 of Polchinski, page 108. There he argues, as you
have said, that string theory has GR as a classical limit because one
can treat strings moving in a curved background. The immediate
objection to using this as an argument that GR is the limit of string
theory is that it only treats string theory interacting with GR; it
does not show that GR in fact arises from string theory. Polchinski's
answer to this, which you repeat, is that if we consider small
perturbations about a flat metric, the first order term in the action
that we get is the vertex operator for a graviton and hence he
concludes that "[t]he action ... can be thought of as describing a
coherent state of gravitons by exponentiating the graviton vertex
operator." But for this claim to be proven, one would have to know
that the perturbation series converges, which one does not know (and
doubts).

Moreover, one must face a deep conceptual problem: how do I obtain
in string theory a state, from a coherent state of gravitons, that
describes some classical solution to Einsteins equation that is not
diffeomorphic to Minkowski space (if I choose Minkowski space as the
manifold that I perturb around)? At each order of perturbation theory
I am superposing strings on a flat background, and so at each order of
perturbation theory I have a solution that is diffeomorphic to R^4.
Hence such states for solutions whose manifolds are *not*
diffeomorphic to R^4 must be non-perturbative. Again, as I stated above
GR, like QCD, has important physics that is non-perturbative, and
hence a perturbation theory alone cannot define the theory. Yet you
present only perturbative arguments.

Moreover, you say, as does Polchinski, that conformal invariance
requires for consistency Einstein's equation. Yet all he derives, on
pages 111--112, is the *linearized* Einstein equation. Yes, this can
be repeated for any background other than Minkowski, but all this shows
is that you can linearize about a solution to Einstein's equation, not
that the solution itself is well approximated by some state in your
quantum theory, namely an equivalence class of rays in a Hilbert space
(or positive linear functional on a C* algebra, if you prefer). This
is what is needed to show that GR is the semiclassical limit of string
theory (or any other theory, for that matter). You further claim that
string theory is background independent. To show that, you must
provide a formulation that is manifestly background independent. What
is done in Section 3.7 of Polchinski certainly is not that. Instead,
he shows that we can treat strings moving on a curved spacetime
background, and then shows that conformal invariance requires that
this metric satisfy the linearized Einstein equation. But this does
*not* show that GR "comes out of" string theory, because this metric
is *non-dynamical*. It is not varied in the Lagrangian approach, and
it is not integrated over in the path integral approach. The argument
that this is implicit in treating strings on some fixed backgraound
because the graviton is a state of the string would, as I have
emphasized above, require a proof that the perturbation series
converges, which is lacking thus far.

Again, I emphasize that this does not discredit string theory or show
that it is unworthy of attention; it is simply that you claim more has
been shown in string theory than really has been. If I were working
in string theory I'm sure I would find it encouraging that the
linearized Einstein equation (or higher orders, if I carried the
calculation further) was a consistency requirement for my theory; it
is very good evidence in category II above. It just does not yet
belong in category I.

In fact, I think I've now covered all I intended to on why I think
your claims about string theory are overstated; I next wanted to say
some things more specifically about what intuitions are important in
quantum gravity, or in short why I'm more interested in pursuing LQG
than SMT. I don't mind if you have different intuitions, just as long
as you're clear that that's what they are. You seem in your comments
to suggest that LQG people suffer from some kind of psychological
preconceptions about what is needed in quantum gravity, but that
string theorists are above all that and bring no preconceptions of
their own to the table; their work alone is just hard science. That
attitude is simply wrong, for all of the reasons I have outlined
above.

What do I think is attractive about LQG? That it attempts, and seems
close to achieving, a non-perturbative quantization of a
background-independent quantum theory. This is something that has not
yet been accomplished for any interacting quantum theory in four
dimensions, and so LQG will be significant if only for that reason, if
it achieves its goal. String theory, on the other hand, has as its
main strong point the fact that the individual terms in its
perturbation theory are finite; this is closely analagous to
renormalizablility in ordinary QFT. Yet in ordinary QFT we know that
in the cases where we can construct an interacting QFT (namely, less
than four spacetime dimensions) that there are non-renormalizable
interacting QFT's. Moreover, as I mentioned in my earlier post we now
think we understand why we observe only renormalizable QFT's at
presently obervable energies. To quote from Section 12.1 of Peskin
and Schroeder (certainly a well known and well regarded modern
treatment of QFT):

Wilson's analysis takes just the opposite point of view [to the
view that it seems very fortunate that we observe only
renormalizable QFT's], that any quantum field theory is defined
fundamentally with a cutoff \Lambda that has some physical
significance. In statistical mechanical applications, this momentum
scale is the inverse interatomic spacing. In QED and other quantum
field theories appropriate to elementary particle physics, the
cutoff would have to be associated with some fundamental graininess
of spacetime, perhaps a result of quantum fluctuations in gravity.
...But whatever this scale is, it lies far beyond the reach of
present-day experiments. The argument we have just given shows
that this circumstance *explains* the renormalizability of QED and
other quantum field theories of particle interactions. Whatever
the Lagrangian of QED was at its fundamental scale, as long as its
couplings are sufficiently weak, it must be described at the
energies of our experiments by a renormalizable effective
Lagrangian. (pp. 402--403, fifth (corrected) printing)

In light of this understanding, it is unpersausive to me to place too
much emphasis on the finiteness or renormalizablility of terms in a
perturbation theory; such behavior is unlikely to be the underlying
characteristic of the true quantum theory. Especially if that
finiteness comes at the cost of extra particles and dimensions that
are not yet observed.

Moreover, the argument usually advanced in favor of SMT is its
'tightness': there is but one free parameter and everything, from the
spectra of particles to the dimension of spacetime, is dictated by the
theory. You also have repeated this argument. Yet when I and others
have pointed out the wide variety of ways that supersymmetry can be
broken or extra dimensions compactified, resulting in a plethora of
possible 'string theories', you complained that it is unreasonable to
demand of the theory that it predict these things. Which is it? Am I
to believe in the theory because of its unicity, or is unicity an
unreasonable expectation of a theory? If it is unreasonable, then why
is it a criticism of LQG to complain that it works in other spacetime
dimensions or does not unify all of the forces? You see, you are
simply choosing which things you would like to have your theory tell
you, and which must be undetermined inputs to it. But I emphasize
that this is a choice, not a deduction. There is no proof you can
offer that your choice is somehow "better" than another.

Finally, you label the emphasis that LQG places on diffeomorphism
invariance as "arrogant" and some kind of religious obsession. This
again is simply your personal taste; you have no proof of either of
these views because they are the kinds of statements that are
inherently unprovable---i.e., opinions. For me, it seems very
important to take as much from Einstein's theory as we can; it was one
of the breakthrough's of twentieth century physics and it seems
"arrogant" to me to simply ignore it. Note that the analagous thing
to what you suggest is *not* done with special relativity: both the
rigorous treatments of QFTs that I referred to above and the heuristic
treatment of a first graduate course both require *exact* Poincare
invariance of the theory. Why is it somehow unreasonable to try the
same thing with diffeomorphism invariance?

Next, I also promised some general statements about the non-uniqueness
of quantizations of a classical theory, and why I think this is
important to work on quantum gravity. In the early days of quantum
theory, after Dirac's work connecting the quantum commutator with the
Poisson brackets, people had the hope that for every (or at least
many) classical theories given by a Hamiltonian description one could
find a unique quantum theory for which all classical Poisson brackets
went over to quantum commutators. This idea is natural and beautiful,
but has the distinct disadvantage of being provably
impossible. Instead, the best one can do is hope that some classical
observables preserve this property, and you can easily have
inequivalent quantum theories which all have the same classical limit;
i.e., the limit you obtain when hbar goes to zero. This is known to
happen in many cases; in particular, as Steve Carlip has pointed out,
it is known to happen for 2+1 dimensional gravity. This is a generic
feature of "quantization", a very misleading term since it implies
that one can take any given classical theory and somehow uniquely
construct a corresponding quantum theory. But in general the
association is one to many. This is why your claim that it is
sufficient to establish one quantum theory of gravity and then dare
others to find different ones, and that until they do you are
justified in claiming that your theory is unique, is erroneous. You
must do much more: you must provide some kind of impossibility result
to show why 3+1 dimensional quantum gravity is an exception to the
general rule that classical theories have more than one quantization.


So much for general statements; a few specific replies below.


>> Very easy? Name *one* such theory. Aside from loop quantum gravity.
>
>Roger Penrose's theory of twistors is claimed to be another example
>(although none of us is sure why he thinks that it is a physical theory at
>all). Lee Smolin's new book "Three Roads to Quantum Gravity" presents yet
>another example (as is clear already from the title), some information
>networks or something. ;-)
>
>I am ready not only name such a theory but I can construct it in real
>time. Let's call it Membrane Quantum Gravity (MQG, also stands for Motl's
>Quantum Gravity). :-) The Hilbert space is defined to contain all the
>2-dimensional surfaces with arbitrary junctions of 3 surfaces (and no
>boundaries); spin networks are replaced by spin networks of membranes. At
>every minimal 2D area, there is a single "J" operator transforming as an
>SU(2) vector. The length of a line is defined as the sum of sqrt[j(j+1)]
>over all the intersections of the line with the 2-dimensional surfaces and
>j(j+1) is the eigenvalue of J^2 living at this portion of the
>2-dimensional surface, to make the analogy with LQG more obvious.

>Well, this construction obviously looks "dual" to LQG in some sense but it
>is probably not equivalent. But it is equally covariant. Why do the people
>study LQG and not MQG, for example? Only because LQG is related to the
>gauge formulation of the initial problem in GR? The low-energy limit has
>not been found for either of them. I would be happy to hear why MQG is
>really a worse direction of research than LQG. ;-)

None of the theories you cite comes close to being a well formulated
quantum theory, as I have indicated above.

>Unlike string theory
>where all the questions about the number of dimensions etc. are uniquely
>fixed, there are almost no constraints in LQG-like attempts and all LQG,
>MQG or similar theories are comparably arbitrary and unjustified. Anything
>goes.

Nonsense. Again, why should a theory of quantum gravity have to
explain the dimensionality of spacetime? And the fact that LQG does
not hardly justifies your assertion that it is "arbitrary and
unjustified. Anything goes." That is just a nonsequitur.
...

>> As Steve Carlip has pointed out on this thread, things are known to be
>> complicated in 2+1 dimensional gravity, where there are several
>> inequivalent quantizations. It could be the case that 3+1 dimensional
>> gravity, unlike its lower dimensional counterpart, for some reason has
>> an essentially unique quantization. If so, it's not clear that we

> This is what I call "religion in physics". There is no reason known why
> "canonical" quantum gravity in 3+1 dimensions should be better than in
> other dimensions.

No one claims that it is; again, why should it have to be?
...

>> I think you should either back up your claims about the beliefs of
>> workers in LQG with some hard evidence, or be a little more cautious...

>I prefer to back up my claims. Take as an example Rovelli's review of LQG
> http://arXiv.org/abs/gr-qc/9710008

That is fine if you prefer it, but in that case you should do so.
What you cite certainly does not back up your original claim that
people working on LQG "suffer from an unjustified religion." Are there
places in developing the theory where we have to be optimistic that
some things which we do not yet understand will eventually be worked
out? Of course. You have the same in string theory: you must be
optimistic that the universe has at least ten more dimensions and
twice as many particles as we have any experimental evidence for. So
would you therfore suggest that string theorists suffer from an
unjustified religion?

I think also that the specific problem you mention has since been
resolved. But this is based only on my recollection of a talk Thomas
Thiemann gave here three years ago. Perhaps someone else knows more
details.

> And even if LQG succeeded with
> its "modest" task, how it could explain the other forces etc.?

Again, why should it have to? It is an interesting hypothesis that a
correct description of gravity will require us to unify it with the
other forces, but not a logical necessity. Here again as I indicated
in my earlier post, string theorists are banking that the course of
quantum gravity can be accurately inferred from what has been done in
the other forces. People in LQG are banking that the same course can
be inferred from general relativity. One or neither may be correct;
both of these fall under the category of intuition, not rigorous proof.

>> in those claims. No one I know in the loop quantum gravity community
>> argues that LQG "necessarily" has general relativity as a
>> semiclassical limit, simply because it uses the Einstein-Hilbert
>> action as a starting point. (Actually, to be technically
>> correct, it doesn't even do that, at least according to the way the
>> terminology is normally used. Einstein-Hilbert usually refers to the
>> action written in terms of metric, rather than connection variables.)
>
> I do not understand why the same action should be given different names if
> you use different variables.

The reason is that while the solutions to the equations of motion will
always be equivalent, the space of histories (from which you select
the solution by the requirement that it extremize the action) are
different. And this can have an impact on the quantum theories,
because you then perform a path integral over this space of
histories. Hence quantizing theories written in different variables
could easily lead to inequivalent theories; as I indicated above, this
is a pretty normal thing. I don't think it has been shown explicitly
in this case, because I don't think anyone has constructed any kind of
quantum theory of gravity in 3+1 dimensions based on a metric
formulation.
...
>> I don't know why you would say this. Certainly there are consistency
>> checks performed within LQG: the closing of the constraint algebra,
>> and indeed the first-class nature of the constraints come immediately
>> to mind. Of course the checks are not the *same* as in M-theory,
>
> I mean, the closure is mostly a check that you wrote your formulae
> properly; of course, the canonical quantization of general relativity (in
> any variables, as long as they are equivalent) must pass these trivial
> classical checks
...
> I know how to derive the spin network basis of the Hilbert space from the
> loop variables, this relation is quite straightforward. Maybe it looks
> nontrivial to you but it does not look too nontrivial to me.

You arbitrarily label as "trivial" anything that you do not like, and
as deep anything that you do. What I don't understand is why you are
then surprised that others are unimpressed by such sophistry.
...
> Yes, this is a sociological problem of LQG - that the people are
> *relativists* only. To quantize something like field theory, you should
> know quantum field theory.

Your assertion that people working in LQG do not know quantum field
theory is patently absurd. From what I wrote at the beginning of this
post, it seems to me rather that *you* are not as well versed in
quantum field theory as you should be.

I think that my responses to the rest of what you write have already
been dealt with at the beginning of this post.

Josh


[Moderator's note: This thread is becoming somewhat heated--
please confine remarks to the physics rather than personal
baiting... -MM]

Lubos Motl

unread,
Jun 17, 2001, 4:53:13 AM6/17/01
to
On Sat, 16 Jun 2001, Josh Willis wrote (in a <also> long post...):

> I. What is known rigorously about a theory, be it SMT or LQG or
> anything else, as opposed to suggestive and partially heuristic
> results. This is a matter of theorems, not opinions, and the goal here
> is, as stated in the preface to Streater & Wightman's *PCT, Spin and
> Statistics, and All That*, the elimination of all theorems whose
> proofs are non-existent.

I absolutely disagree with this proposal. And I even believe that people
from LQG will disagree with it, too. This program has nothing to do with
string theory, LQG or the discussions between these two groups. I am a
physicist - just like most of string theorists and most of LQG
theoreticians (although in both communities one can find true
mathematicians) - and therefore my job is not to "prove theorems" as Josh
suggests. Physicists should use the standard physical conventions and
methods to calculate, check hypotheses, decide whether something has been
argued convincingly etc. They are not mathematicians and they should not
try to be ones.

Good physicists were always able to use not only the language of
mathematics, but also the common sense, intuition and they could improve
their definitions of something if necessary. Physics is not (just)
mathematics. Physics is not a closed subject and a too specific
mathematical definition, directly applied to the world of physics, will
probably become inappropriate physically at some moment. Frankly speaking,
I do not know the book and I do not think that the authors are that
important for physics so that one should even follow their preface. In
fact, both of them are mathematicians working on Axiomatic Quantum Field
Theory. In my opinion, this field became separated from physics almost
completely. AQFT is a mathematical game with some simple objects (such as
the CPT conjugation in C*-algebras) that are not powerful enough to deal
with fascinating conceptual breakthroughs of modern physics, especially
physics that has showed that there is something beyond QFT.

Just remember how many centuries did it take to define the rigorous
mathematical tools to deal with Calculus (with the infinitesimal
quantities etc.), using the epsilon-delta gymnastics. Did the definitions
change anything about the Newton's equations and the predictions made by
physicists? I do not think so. The axiomatic game with the laws of physics
is an activity of mathematicians who work on something that physicists
understood well enough decades or centuries ago. Maybe I would even often
say a "recreational activity". When string theorists say that something is
reliable, sometimes they mean that it is a mathematical theorem, but more
often they mean that it is as reliable as the calculations derived from
Newton's equations before the invention of the axiomatic calculus. And I
believe that the same is true for LQG, as well as other areas.

In fact, I think that most of the axiomatic effort related to physics is
irrelevant for physics itself. For instance, mathematicians were able to
prove that the Feynman's path integral does not exist (in Minkowski
space); there is no measure on spaces of functions. Great. I am not sure
whether such results are useful. I could enumerate more drastic examples
that show that some of the most incorrect claims in current physics
literature are claimed to be "theorems". For example, a guy called Rehren
proved his "theorem" that a 4-dimensional local field theory on the
boundary is not equivalent to the full string theory in the bulk, but
rather to a *local* field theory in the AdS bulk. If someone is interested
in the reasons why his claims are wrong, maybe I could send him or her our
correspondence with Lee Smolin if he agreed. Anyway, it is very easy to
find a counterexample as Juan Maldacena did in a couple of minutes.

> String theory's first big success and a driving force behind it is the
> absence of ultraviolet divergences. In other words, string theory

Historically speaking, this is an incorrect claim, too. String theory was
supposed to be a theory of the strong interaction and the driving force
behind it was first the so-called worldsheet duality (summing over the
s-channel can give the same result as summing over the intermediate
particles in the t-channel). This led Veneziano to his amplitude (the
Euler Beta function) which also reproduced some properties of hadrons.
Later is was realized by Susskind and others that this formula can be
explained if particles are really strings. When QCD became obviously the
correct description of the strong force, string theory almost died. At
this era, there was no "big success" and no "driving force". Schwarz and
Scherk realized that string theory contained gravity and it probably
became their biggest "driving force". The UV finiteness is nice but in the
QCD era it was rather a handicap because string theory predicted a softer
UV behavior (exponentially decreasing) than was observed (=than predicted
by QCD, a power law).

Then, in the early 80s, string theory was viewed as a difficult game with
oscillators that gave apparently as much results as you put in. At least,
this is what many people thought. It did not contain a gauge symmetry. But
in 1984, Green and Schwarz found the anomaly cancellation. It was a very
complicated calculation. However, people could understand it after some
time and it showed something really nontrivial. Gravity could be suddenly
unified with the gauge theories etc. and hundreds of people started to
work on string theory. [Thanks to J. Distler for discussions on the first
revolution.] ;-)

> 1. *Define* your theory by its perturbation expansion.
>
> This is well and good---if your perturbation theory converges. I
> don't know anyone who thinks this likely for string theory, and
> certainly no *proof* that it converges.

This is like you wrote it in the early 70s. Today we know very well how
the expansion behaves (diverges) asymptotically (Steve Shenker), we know
what are the nonperturbative effects and we know that the asymptotic
expansion represents a nonperturbatively existing set of amplitudes. In
fact, for a couple of years we have several nonperturbative definitions of
string theory in various backgrounds.

> There is, however, an important difference between QM and QFT in this
> respect: in QM, we have nonperturbative proofs that all of the things
> we are calculating really do exist; we know that we have a Hilbert

You are mostly ignoring Wilson's remarkable discoveries of the
renormalization group that really showed what QFT is, how you should look
at it, what you should compute from it and how you should interpret it.
And I am afraid that most of the people in AQFT ignore them, too. I find
your descriptions of QFT rather unpractical and unpragmatic. Many of the
"problems" you describe are not real problems and it seems that you have
almost no idea what the truly subtle problems are. For instance, in
quantum gravity, the S-matrix is the only gauge invariant observable we
know of and the finite-time transition amplitudes will probably never have
an accurate definition in any theory of quantum gravity.

> Also, note that in string theory the situation is even worse, because
> you do not have the analogue of Fock space---i.e., a Hilbert space of
> non-interacting strings---because the theory inherently includes
> interactions.

This claim is incorrect, too. Perturbative string theory has a perfectly
well-defined Fock space of free, non-interacting and arbitrarily excited
strings. And then you compute the interactions perturbatively, loop by
loop, just like in QFT. There is a free spectrum (determined by the
quadratic piece of the action in QFT - or by a direct second quantization
of a spectrum of one string) and the vertices of Feynman diagrams (in the
case of string theory, the pants diagram replaces the vertex). A g-loop
Feynman diagram in string theory is really a genus g Riemann surface. It
behaves better in the ultraviolet than in QFT, there is only one diagram
at each order (in the case of closed oriented strings) but otherwise the
situation is similar to that in QFT.

> So, what bearing does all of this have on SMT? You have a
> perturbative expansion of an S-matrix, but no proof that the
> perturbation series converges, or that anything exists to which it

We have quite a reliable proof that it diverges just like in QFT (except
for the quantities that are protected and have a finite number of
contributions only). Take lambda.phi^4 theory and identify lambda with
g_string. The divergence in the powers of g_string looks essentially the
same like the divergence in lambda. And we have also evidence that it
converges to something because we have nonperturbative definitions of the
theory. In terms of g_closed=g_open^2, the divergence looks "twice"
faster. It is g_open in which the divergences are really like those in
g_YangMills in field theory.

> might be asymptotic. On top of that you inherit from QFT the problems
> above in the definition of an S-matrix. Simply put, having a
> perturbative expansion of an S-matrix, each of whose terms is finite,
> does not a full quantum theory make. This is related to the
> discussion going on in another thread about causal quantum theory.

Note that your "criticism" can be applied to any theory that should
reproduce physics of QFT in some limit. In fact it can be applied against
any scientific effort to understand anything in reality. :-)

> 1. That String/M-theory is in fact a quantum theory in the first
> place. What is the Hilbert Space? If it is L_2(X,d\mu) for some space

Yes, string theory is a completely quantum theory and it does not modify
anything about the basic principles of quantum theory. The Hilbert space
is H(SMT) :-) and it is isomorphic to the Hilbert space of the harmonic
oscillator - because all the Hilbert spaces are isomorphic to each other
as Hilbert spaces. :-)

> X and measure \mu, what are they?

It is certainly not (in any useful sense) of the form L_2(X,d\mu) for a
finite dimensional X, this would return us to the 1920s (a single-particle
quantum mechanics). ;-) Your arguments are similar to some arguments of a
hypothetical self-confident Newtonian physicists who also comes to the
21st century and argues that string theory is wrong because it cannot
answer his question what is the potential between two point-like particles
with well-defined positions and momenta. ;-)

> What are the operators corresponding to observables? Can you prove
> they are self-adjoint? What are their spectra?

Good questions! Generally, in any theory of quantum gravity you will
probably never be able to define any local operators in the bulk (they
cannot be gauge-invariant etc.). This is related to the general covariance
and holography. In asymptotically flat spaces, you can however define an
S-matrix. In AdS, you can define the local operators in the CFT at the
boundary. Maybe one can also define approximate operators that are
approximately local. And in the light cone gauge, you can define an exact
Hilbert space, a Hermitean Hamiltonian and compute its spectrum.

The S-matrix is not self-adjoint, it is unitary. ;-) Yes, one can show
that it is unitary - the easiest way is to use a physical light-cone gauge
formulation that even contains a (manifestly Hermitean) Hamiltonian. One
can also show that the light-cone gauge formulation is equivalent to the
manifestly covariant RNS formalism. The spectrum of the S-matrix is the
unit circle in the complex plane. ;-) Beyond those operators and
observables, maybe we will find new ones one day - but there is no proof
that there exists a rigorous and physically meaningful observable beyond
the S-matrix.

I hope that this answer is complete enough.

> 2. That GR is in fact the semiclassical limit of SMT. To do this,
> you should prove that for any classical solution of Einstein's

This is easy to prove and I have already written a lot of posts about it.

> equation, there is a corresponding state in your Hilbert space that
> has, for all of the relevant observables, minimum uncertainty
> distributed about the correct, corresponding values of the classical

Nope, you do not need to do anything from your proposals to show that
string theory has the correct classical limit. Why do you think that we
should use these medieval tools to do physics?

> The arguments you cite on this second point don't come close to doing
> this. I don't have a copy of Green, Schwarz, and Witten, but let's
> look at Section 3.7 of Polchinski, page 108. There he argues, as you
> have said, that string theory has GR as a classical limit because one
> can treat strings moving in a curved background. The immediate
> objection to using this as an argument that GR is the limit of string
> theory is that it only treats string theory interacting with GR; it
> does not show that GR in fact arises from string theory.

You misunderstood the argument completely. Polchinski's book is written in
a very comprehensible way and if he was not able to explain it to you, I
think that maybe, I should not even try to do better. String theory is not
coupled to some "other" theory! String theory cannot be coupled to a
different theory living in the same spacetime. Once you decide that you do
string theory, there is absolutely no freedom left. There is no consistent
way how to add "new" fields to your spacetime beyond those predicted by
string theory. The background is a coherent state of strings themselves -
as you can see easily because its infinitesimal change has the same effect
as inserting a string vibrating in the appropriate (graviton) vibration
pattern. Only one type of stringy interactions is possible - joining and
splitting - and the whole structure is determined uniquely (once you
choose whether you are in type IIA, IIB etc.).

> operator." But for this claim to be proven, one would have to know
> that the perturbation series converges, which one does not know (and
> doubts).

Yes, the Taylor series for exp(x) converges. I am not sure whether you are
serious. If you have doubts about it, you should ask people in different
newsgroups. This questions seems a bit inappropriate at
sci.physics.research. Did I misunderstand your question?

Or did you mean the perturbation series in "g" for an amplitude? What does
it have to do with background independence of physics? It would be like
claiming "hydrogen atom does not exist because the QED series diverge",
"the magnet does not attract because the QED series diverge", "physics
depends on gauge because the series diverge"... Everything is wakalixes,
as Feynman said when he rated a bad textbook. ;-) You must see that such
claims are irrational and unphysical!

> perturbation theory I have a solution that is diffeomorphic to R^4.
> Hence such states for solutions whose manifolds are *not*
> diffeomorphic to R^4 must be non-perturbative. Again, as I stated above

Yes, perturbatively you cannot get S^3 x R^1 from R^4 as a finite
excitation, for example. In infinite spacetime, states with different
asymptotic behavior belong to different superselection sectors. Why do you
think that it should be possible to get from one space into another in
this cheap way? If you ask about the change of topology, it has been
described a few days ago, too. Flop transition is a purely perturbative
effect (in g_string) while the conifold transition is nonperturbative. It
involves a condensate of wrapped D3-branes. Any state can be reached from
any starting point. However if one talks about the Hilbert space, one
usually means the Hilbert space of states with fixed asymptotic behavior
of spacetime only.

> GR, like QCD, has important physics that is non-perturbative, and
> hence a perturbation theory alone cannot define the theory. Yet you
> present only perturbative arguments.

Nope, string theory has a more interesting nonperturbative physics than
QCD and its structure has been being revealed since 1995 in thousands of
papers. Most of our arguments today are nonperturbative. This is what most
people worked on for last 6 years or so. A critic who wants to look at
least partially serious should learn that something has changed in the
90s. ;-)

> Moreover, you say, as does Polchinski, that conformal invariance
> requires for consistency Einstein's equation. Yet all he derives, on
> pages 111--112, is the *linearized* Einstein equation. Yes, this can

Did you buy a wrong copy of the book? ;-) Look at (3.7.14a), for example,
you find the full nonlinear Ricci tensor with everything that should be
there. In my copy of Joe's book, I have a comment written by my pen:
"checked", which means that I can guarantee that you derive the correct
full nonlinear result, too. ;-) The calculation is easy to do on a
completely general curved background and gives you the exact result.

If Joe's book is too fast for you, a detailed calculation of the same sort
can be found in the section 3.4 of Green, Schwarz and Witten, volume 1.

> theory (or any other theory, for that matter). You further claim that
> string theory is background independent. To show that, you must
> provide a formulation that is manifestly background independent. What

No, this is not the case. Physics of string theory is
background-independent (i.e. the theory defined around different
backgrounds is still the same theory) and one does not need to have a
manifest background-independent formulation to prove it (although we would
be happy to have it, too). Do you think that the Fermat's Last Theorem
cannot be claimed to be correct because we do not have a proof that is
manifest?

> is done in Section 3.7 of Polchinski certainly is not that. Instead,
> he shows that we can treat strings moving on a curved spacetime
> background, and then shows that conformal invariance requires that
> this metric satisfy the linearized Einstein equation.

Full equations.

> But this does
> *not* show that GR "comes out of" string theory, because this metric
> is *non-dynamical*. It is not varied in the Lagrangian approach, and

On the contrary, everything in string theory is completely dynamical. For
example, while QFTs can contain dimensionless parameters, all such
parameters (such as the coupling constant) become inevitably dynamical
fields (in the case of the coupling, dilaton). Gravity in string theory is
dynamical, of course, this is the main dynamics that one can compute. It
is hard to interpret your claims in such a way that they make sense.

You know, I find it a bit frustrating. For people who want to do
theoretical physics properly, Polchinski's book represents just a basic
part of their knowledge. They must master and check many other papers
before they can do some relevant research. And then one can meet
physicists who have problems already with an elementary material in
chapter 3 of Polchinski's textbook. So they just decide that the textbook
is wrong. And one should be even afraid to say that such an approach is
not appropriate for a physics student. Is not it sad?

> In light of this understanding, it is unpersausive to me to place too
> much emphasis on the finiteness or renormalizablility of terms in a
> perturbation theory; such behavior is unlikely to be the underlying
> characteristic of the true quantum theory. Especially if that
> finiteness comes at the cost of extra particles and dimensions that
> are not yet observed.

You may *dislike* some predicted particles - such as W-bosons and Z-bosons
(as well as superpartners etc. that are likely to be observed in 5 years).
But there is probably nothing else that you can do about it. ;-) Or can
one sue Nature?

You can also say that infinities do not matter. But if you do not have a
prescription to get finite and meaningful predictions, you do not have a
theory.

> broken or extra dimensions compactified, resulting in a plethora of
> possible 'string theories', you complained that it is unreasonable to

These are not different theories as I have been explaining many times.
Particle sitting at two local minima of a potential also does not imply
that there are two theories.

> inherently unprovable---i.e., opinions. For me, it seems very
> important to take as much from Einstein's theory as we can; it was one
> of the breakthrough's of twentieth century physics and it seems

It was just *one* of the breakthroughs and I have been explaining why
other gauge invariances are equally fundamental - and in fact, they can
transmute into general covariance (in Kaluza-Klein theories etc.).

> "arrogant" to me to simply ignore it. Note that the analagous thing
> to what you suggest is *not* done with special relativity: both the
> rigorous treatments of QFTs that I referred to above and the heuristic
> treatment of a first graduate course both require *exact* Poincare
> invariance of the theory. Why is it somehow unreasonable to try the
> same thing with diffeomorphism invariance?

It is fine to count general covariance as an exact concept as long as you
work with gravity classically. At quantum level, its nature is modified at
Planckian distances. The corresponding statement concerning Poincare group
is the following: you can consider the affine Galilean group to be an
exact symmetry as long as you deal with speeds much smaller than c; then
you must switch to the Poincare group.

In other words, the Poincare group is already the "deformed" or "resolved"
version of the affine Galilean group. Therefore there is no reason to
"deform" it further (in theories without a dynamical geometry). In special
relativity, the fundamental constant is "c". The Newtonian concepts break
down for speeds comparable to "c" and the Galilean symmetry must be
replaced by the Lorentz symmetry. In quantum theory, the essential
constant is "hbar": the classical notions break down and classical physics
must be replaced by Hilbert spaces etc. In general relativity, the
constant is "G": the laws of gravity break down if the radius is
comparable to 2GM/c^2. In quantum gravity you combine "hbar" and "G" as
well as "c" and the classical concepts of geometry break down just like
the classical notions of position and speed broke down in QM; it occurs
near the fundamental Planck scale.

> >Well, this construction obviously looks "dual" to LQG in some sense but it
> >is probably not equivalent. But it is equally covariant. Why do the people
> >study LQG and not MQG, for example? Only because LQG is related to the
> >gauge formulation of the initial problem in GR? The low-energy limit has
> >not been found for either of them. I would be happy to hear why MQG is
> >really a worse direction of research than LQG. ;-)

> None of the theories you cite comes close to being a well formulated
> quantum theory, as I have indicated above.

This claim of yours is probably wrong, I just wanted to check you. The
theory I formulated has most likely the same level of quantum consistency
as LQG; in fact, it may be completely equivalent to LQG (compare with dual
graphs in combinatorics). I just wanted to check whether you are able to
say an incorrect claim if you need to do anything to defend your position.
Unfortunately, you are able to say anything. Of course that you do not
seem to have a clue whether there should be anything wrong about MQG. What
could you answer if I asked you why you think that MQG is not a
well-formulated quantum theory? ;-) That I am not Ashtekar? :-)

> Nonsense. Again, why should a theory of quantum gravity have to
> explain the dimensionality of spacetime?

Because a gravitational theory is by a definition dynamics of geometry.
One of the classical notions describing a geometry is the dimension. A
quantum version of gravity has nonzero amplitudes to get from one state to
another, unless they belong to different superselection sectors. For
finite systems (finite volume etc., the entropy in a given region is
finite by Bekenstein bounds i.e. by holography) there cannot be
any superselection sectors. Consequently, theory of quantum gravity is
able to connect (in finite volume) all the possible geometries and
therefore it must be able to predict which geometries can exist and which
cannot. This is a completely physical question. You can prepare an
experiment with a lot of energy and ask whether you can create a (small)
region that looks 5-dimensional, for example. In string theory, you can
in principle derive the answer to all such questions.

> Of course. You have the same in string theory: you must be
> optimistic that the universe has at least ten more dimensions and

No, the Universe does not have "ten more dimensions" unless you are an
instanton. ;-)

> twice as many particles as we have any experimental evidence for.

Yes, this is called "prediction". This was a frequent sin ;-) that
physicists did before they became axiomatic mathematicians who like to
write formulae that imply no predictions at all. Today, with the discovery
of AQFT and LQG, people are much happier. They do not need to be
optimistic because they know that nothing new (such as SUSY) will be ever
discovered - and if it will, LQG has all the rights to ignore such an
unimportant piece of physics. ;-)

> So would you therfore suggest that string theorists suffer from an
> unjustified religion?

No, on the contrary, I would say that string theorists are sometimes doing
predictions. ;-)

> I think also that the specific problem you mention has since been
> resolved. But this is based only on my recollection of a talk Thomas
> Thiemann gave here three years ago. Perhaps someone else knows more
> details.

:-) Believe and your belief will solve all your problems.

> > And even if LQG succeeded with
> > its "modest" task, how it could explain the other forces etc.?

> Again, why should it have to? It is an interesting hypothesis that a
> correct description of gravity will require us to unify it with the
> other forces, but not a logical necessity.

Well, it is amusing that you try to prove that LQG is not a religion - but
then you repeat a verse from the Holy Scripture by Rovelli gr-qc/9710008
with a couple of words permuted. ;-) To compare, Rovelli said:

"A common criticism to loop quantum gravity is that it does not unify all
interactions. But the idea that quantum gravity can be understood only
in conjunctions with other fields is an interesting hypothesis, not an
established truth."

In this case, I was not talking about the interesting hypothesis :-) that
you can explain quantum gravity (that we know at long distances only but
it has quantum effects at the Planck scale) without knowing anything else
that happens in between (such unimportant things like the strong, weak and
electromagnetic force, chemistry, life etc.). ;-) I was talking about the
future. If LQG succeeds (whatever it means), will you be satisfied that
you found a theory of everything, although your theory predicts that
electromagnetism, the weak and the strong force do not exist? ;-) I just
say that even in this hypothetical case of LQG's success, physicists would
finally have to return to the question of unification, calculation of the
remaining constants i.e. to (something like) string theory.

The goal of LQG seems to be to avoid anything new. For instance, Newton's
coevals were also saying: "The claim that the falling apple must be
descibed in conjunction with the motion of other objects (such as planets)
is an interesting hypothesis, not an established fact." Note that their
position was much more justified than the claim of LQG today because the
three forces certainly do appear between the two relevant scales where we
observe and study gravity. In fact, in the history of fundamental physics,
it never happened that a new explanation of existing questions was given,
without unifying some concepts that were independent before or explaining
some number previously unexplained. The opposite case would in fact mean
that the theory is not scientifical because it does not predict anything.

This seems to be inherently the case of LQG. When it predicts one number -
the entropy of the black hole per unit area of the horizon - it requires
to fine tune one parameter. Furthermore - and I think that this is a very
serious objection - any experiment that disagrees with LQG could be
claimed to be "due to some nongravitational forces that are not included
in LQG". What I want to say is that there can be defined absolutely no
limit in which you could "decouple" quantum gravity from other forces. You
cannot say that the limit is "long distances" because at long distances
you do not need gravity to be quantum. You also cannot say that it is the
limit of short distances because the other forces have certainly a nonzero
effect even at Planck scale. So what is LQG really supposed to describe in
the *real world*?

> Here again as I indicated in my earlier post, string theorists are
> banking that the course of quantum gravity can be accurately inferred
> from what has been done in the other forces.

This is completely unfair. String theory has changed quite dramatically
our understanding what are the fundamental degrees of freedom. Fundamental
point-like particles are gone. This is much more radical change of QFT
than LQG did; LQG is at the end a local quantum field theory with a finite
number of fields. I do not want to say that string theory wants to discard
old wisdoms; on the contrary, it is as conservative as possible. ;-)

> People in LQG are banking that the same course can be inferred from
> general relativity. One or neither may be correct; both of these fall
> under the category of intuition, not rigorous proof.

These sentences try to pretend that LQG people understand general
relativity better than string theorists. This is unfair, too. Strominger,
Horowitz and many others are not worse even in classical GR than the
leaders of LQG. But they realize that it is not reasonable to try to
consider some insights more important just because you know them better
than others.

> > I do not understand why the same action should be given different names if
> > you use different variables.

> The reason is that while the solutions to the equations of motion will
> always be equivalent, the space of histories (from which you select
> the solution by the requirement that it extremize the action) are
> different. And this can have an impact on the quantum theories,
> because you then perform a path integral over this space of
> histories. Hence quantizing theories written in different variables
> could easily lead to inequivalent theories; as I indicated above, this
> is a pretty normal thing. I don't think it has been shown explicitly
> in this case, because I don't think anyone has constructed any kind of
> quantum theory of gravity in 3+1 dimensions based on a metric
> formulation.

Yes, I understand in principle. But the idea that one works on a theory
where it is conceivable that there are in fact many inequivalent ways how
to quantize it, depending on the choice of variables (!!!), is really
scary. If I had a feeling that the construction of string theory can be
changed in several ways at some point, I would probably give it up. What
should I think about a theory that seems to allow modifications,
corrections and other possibilities almost at every step? What is the
probability that such a theory can be *the* correct theory, whatever it
means? Or should we believe that there is no *the* theory? The results of
experiments are real and can be measured - and this, in principle, decides
between any pair of theories that are inequivalent (if they are able to
predict at all; but if they are not, too bad).

> You arbitrarily label as "trivial" anything that you do not like, and
> as deep anything that you do. What I don't understand is why you are
> then surprised that others are unimpressed by such sophistry.

On the contrary, I use the term "trivial" for things that are easy to
understand. "Trivial" does not mean that I like it or I dislike it. There
are both trivial claims that I like and dislike, as well as nontrivial
claims that I like and dislike. ;-) John Baez made a good presentation of
LQG. For me it is a system of ideas as large as say D-branes on orbifolds;
it also fits the number of papers etc. I do not see a reason why should
one consider LQG to be more relevant than D-branes on orbifolds, only
because its proponents claim that it is a competition of the whole
building of M-theory. Pride is not enough. When Fidel Castro says that
Cuba will defeat the whole imperialistic United States of America, it does
not mean that Cuba becomes much more important than Florida. ;-)

> Your assertion that people working in LQG do not know quantum field
> theory is patently absurd. From what I wrote at the beginning of this
> post, it seems to me rather that *you* are not as well versed in
> quantum field theory as you should be.

If people in LQG understand QFT as *well* as you do ;-), I would be a
little bit more sceptic about its future than I was so far...

Best wishes
Lubos
______________________________________________________________________________
E-mail: lu...@matfyz.cz Web: http://www.matfyz.cz/lumo tel.+1-805/893-5025
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Superstring/M-theory is the language in which God wrote the world.

Aaron Bergman

unread,
Jun 25, 2001, 12:01:21 AM6/25/01
to
In article <Pine.SOL.4.05.101061...@athena.phys.psu.edu>,
Josh Willis wrote:
>
>There are two categories here which it seems evident to me are being
>confused with one another:
>
>I. What is known rigorously about a theory, be it SMT or LQG or
>anything else, as opposed to suggestive and partially heuristic
>results. This is a matter of theorems, not opinions, and the goal here
>is, as stated in the preface to Streater & Wightman's *PCT, Spin and
>Statistics, and All That*, the elimination of all theorems whose
>proofs are non-existent.

It's worth noting that nothing interesting in high energy physics
falls into this category. I don't think I'm being hyperbolic here, either.

>II. In this category belong many other things: partial or heuristic
>results, intuition about what is an important criterion in guiding the
>progress of a field, etc. I am not denying that things in this
>category are suggestive and may well be a guide to results that
>eventually belong in category I above. Indeed, I cannot imagine how
>to make progress in any field of science without these. But I am
>insisting that we not confuse things in this category with the one
>above, and that in comparing different theories we not compare apples
>to oranges by comparing the absence of results in category I above in
>one theory with the presence of results in category II in another.

I would argue that essentially all the progress in high energy
physics in the last forty years or so has been in this category.

[...]

>First, on perturbation theory in general. A first approach to putting
>one's theory into category I above might be:
>
>1. *Define* your theory by its perturbation expansion.

Nobody I know argues this.


>
>This is well and good---if your perturbation theory converges.

I would think that some sort of Borel resummability would be OK.

>I
>don't know anyone who thinks this likely for string theory, and
>certainly no *proof* that it converges.

Periwal and Gross have argued that it doesn't converge.


>
>But an immediate obejection to this might be:

The stuff I've snipped is essentially a synopsis of all the
things that go wrong in quantizing an interacting field theory.
My personal prejudices are that the reason for all these things
is that we have no idea what a quantum field theory is, but
that's just me.

>So, what bearing does all of this have on SMT? You have a
>perturbative expansion of an S-matrix, but no proof that the
>perturbation series converges, or that anything exists to which it
>might be asymptotic. On top of that you inherit from QFT the problems
>above in the definition of an S-matrix. Simply put, having a
>perturbative expansion of an S-matrix, each of whose terms is finite,
>does not a full quantum theory make. This is related to the
>discussion going on in another thread about causal quantum theory.

No one has argued this.

>You claim that String/M-theory has *already* been shown to be a
>quantum theory of gravity. For that claim to be true, you would have
>had to have shown the following:
>
> 1. That String/M-theory is in fact a quantum theory in the first
>place. What is the Hilbert Space? If it is L_2(X,d\mu) for some space
>X and measure \mu, what are they? What are the operators corresponding
>to observables? Can you prove they are self-adjoint? What are their
>spectra?

I'm not even convinced that a good definition for QFTs should
include Hilbert spaces.

> 2. That GR is in fact the semiclassical limit of SMT. To do this,
>you should prove that for any classical solution of Einstein's
>equation, there is a corresponding state in your Hilbert space that
>has, for all of the relevant observables, minimum uncertainty
>distributed about the correct, corresponding values of the classical
>solution. You should prove some kind of Ehrenfest properties. This
>is what is done in ordinary QM; this is what people are now attempting
>to do in LQG. It has not been done even for ordinary QFT in an
>interacting theory.

Which is why you're asking an absurd amount. What has been done
in string theory is essentially what has been done in interacting
QFT.

>The arguments you cite on this second point don't come close to doing
>this. I don't have a copy of Green, Schwarz, and Witten, but let's
>look at Section 3.7 of Polchinski, page 108. There he argues, as you
>have said, that string theory has GR as a classical limit because one
>can treat strings moving in a curved background.

This is not the entirety of the argument.

>The immediate
>objection to using this as an argument that GR is the limit of string
>theory is that it only treats string theory interacting with GR; it
>does not show that GR in fact arises from string theory.

Not at all. String theory contains a transverse traceless
spin-two particle. This is just the perturbations in the metric.
The (perturbative) dynamics of GR arise from the string theory.
Consistency conditions for the theory require that the background
metric satisfies Einstein's equations. This is the best you're
going to get from a perturbative theory.

[...]

>Moreover, one must face a deep conceptual problem: how do I obtain
>in string theory a state, from a coherent state of gravitons, that
>describes some classical solution to Einsteins equation that is not
>diffeomorphic to Minkowski space (if I choose Minkowski space as the
>manifold that I perturb around)? At each order of perturbation theory
>I am superposing strings on a flat background, and so at each order of
>perturbation theory I have a solution that is diffeomorphic to R^4.

You can (theoretically) do perturbation theory around any
background that satisfies Einstein's equations.

>Hence such states for solutions whose manifolds are *not*
>diffeomorphic to R^4 must be non-perturbative. Again, as I stated above
>GR, like QCD, has important physics that is non-perturbative, and
>hence a perturbation theory alone cannot define the theory.

This is a straw man that I wish you would stop repeating.

>Yet you present only perturbative arguments.

No one's denying that the thing we have the best control over in
string theory is the perurbation expansion.


>
>Moreover, you say, as does Polchinski, that conformal invariance
>requires for consistency Einstein's equation. Yet all he derives, on
>pages 111--112, is the *linearized* Einstein equation.

As I remember it, the beta functions contain the Ricci scalar,
and are identified with the effective spacetime action, so
there's no linearization involved. I don't have Polchinski on me,
so I can't check the pages you reference to see if you're
referring to the correct pages.

[...]

The main point I want to make now is that you seem to be
requiring of string theory things that haven't even been
accomplished for the simplest interacting QFTs. I think it's a
bit unfair.

>What do I think is attractive about LQG? That it attempts, and seems
>close to achieving, a non-perturbative quantization of a
>background-independent quantum theory. This is something that has not
>yet been accomplished for any interacting quantum theory in four
>dimensions, and so LQG will be significant if only for that reason, if
>it achieves its goal. String theory, on the other hand, has as its
>main strong point the fact that the individual terms in its
>perturbation theory are finite; this is closely analagous to
>renormalizablility in ordinary QFT. Yet in ordinary QFT we know that
>in the cases where we can construct an interacting QFT (namely, less
>than four spacetime dimensions) that there are non-renormalizable
>interacting QFT's. Moreover, as I mentioned in my earlier post we now
>think we understand why we observe only renormalizable QFT's at
>presently obervable energies. To quote from Section 12.1 of Peskin
>and Schroeder (certainly a well known and well regarded modern
>treatment of QFT):

The existence of UV fixed points is almost universally known
within the string theory community. I'm not sure what you're
trying to communicate here. You're arguing against things people
might have said 20 years ago.

[...]


>
>In light of this understanding, it is unpersausive to me to place too
>much emphasis on the finiteness or renormalizablility of terms in a
>perturbation theory; such behavior is unlikely to be the underlying
>characteristic of the true quantum theory.

The point is that we can actually calculate things in string
theory. This is a good thing. One can argue all one wants about
the possibility of UV fixed points, but it's all just handwaving
until you actually show some evidence. Some theories really don't
exist, and thus, showing explicitly that UV divergences are
regularized is a nice thing. It's also nice because string
theories seem to be anomaly free.

[...]

And now, since editorializing seems to be the name of the game, I
suppose I shall engage in some of it myself. First of all, all
too many lqg people seem to argue with string theory as if it had
not changed in the last 15 years. Worse, there is a lot of
complete mischaracterization of the people who work in the field.
For example, the idea of defining the theory through its
perturbation theory died out years ago if it ever even existed.
Or, take the intimation that string theorists don't care about
background independence or even understand "the lessons of GR"
(which sounds frightfully condescending, I might add). I hear
string theorists talk about background independence all the time.
I can also assure you that vast chunks of the community know GR
and understand its philosophy and lessons. One thing that is
worth noting, however, is that its lessons might be wrong.
Background independence is really pretty, but that doesn't mean
it is necessarily correct. Maybe life isn't background
independent. It's worth thinking about, don't you think?

I've also talked to some string theorists who know something
about lqg, and maybe I can communicate their objections. To many
people, lqg feels like constructing something. You decide what
your variables should be and then you go about making a Hilbert
space. Well, we need an inner product, so lets do some
complicated math and define that. Etc. etc. It all feels very
artifical. Many people would rather feel that one is discovering
things rather than creating them. Whatever the Hilbert space may
be, it should be something natural rather than something someone
has to go through extensive work on in order to define. For all
that string theory is horribly undefined, the developments mostly
seem to come out of the theory whatever it may be, rather than
being put in. There are also some technical objections.
Apparently, quantization of gauge theories in terms of loops has
been tried in the past and there are known (but not known to me)
issues in this. One person who I talked to saw a talk and said
that this issue (maybe something to do with what happens when
Wilson lines cross?) led Gross and Witten to crucify the speaker.
But I know next to nothing about lqg, so I don't know the
details. Just don't fall into the trap of believing that string
theorists are just rejecting lqg because they're provincial and
arrogant or somesuch.

Aaron
--
Aaron Bergman
<http://www.princeton.edu/~abergman/>

zirkus

unread,
Jun 25, 2001, 12:39:30 AM6/25/01
to
Lubos Motl <mo...@physics.rutgers.edu> wrote:

> AQFT is a mathematical game with some simple objects (such as
> the CPT conjugation in C*-algebras) that are not powerful enough to deal
> with fascinating conceptual breakthroughs of modern physics, especially
> physics that has showed that there is something beyond QFT.

Here I would like to quote a criticism of AQFT by S. Majid and also
mention some things about string theory (ST). Regarding AFQT, "one
still assumes an underlying classical spacetime and classical Poincare
group etc., on which the operator fields live. Yet if the real world
is quantum then phase space and hence probably spacetime itself should
be 'fuzzy' and only approximately modeled by classical geometrical
conepts. Why then should one take classical geometrical concepts
inside the functional integral except other than as an effective
theory or approximate model tailored to the desired classical geometry
that we hope to come out. This can be useful but it cannot possibly be
the fundamental 'theory of evrything' if it is built in such an
illogical manner. There is simply no evidence for the assumption of
nice smooth manifolds other than now-discredited classical mechanics.
And in certain domains such as, but not only, in Planck scale physics
or quantum gravity, it will certainly be unjustified even as an
approximation." From page 11 of [1].

Perhaps two nice things about ST is that perturbative ST does not obey
the axioms of local QFT, and that the usual kind of idea of a
spacetime manifold does not have to be fundamental to ST. I clarify
this second statement some by quoting from page 3 of [2] by C.M. Hull:

"The heterotic string on M x T^4 is equivalent to the type IIA string
on M x K3, even though T^4 and K3 are very different spaces with
different properties (e.g. they have different topologies and
different curvatures) and there is no invariant answer to the
question: what is the spacetime manifold? In the same way that
spacetimes related by diffeomorphisms are regarded as equivalent, so
too must spacetimes related by dualities, and the concept of spacetime
manifold should be replaced by duality equivalence classes of
spacetimes (or, more generally, duality equivalence classes of string
or M-theory solutions)."

(BTW, this paper suggests that the geometry, topology, signature and
dimension of spacetime are relative rather than absolute concepts
which would depend on the values of certain parameters or couplings.
But the correct relativity principle that might underlie M-theory does
not appear to have been discovered yet.)

> [ST] does not modify


> anything about the basic principles of quantum theory.

It is not certain that this statement will remain true. Please see my
other post which refers to sources concerning black holes and QM in
M-theory.

The Hilbert space
> is H(SMT) :-) and it is isomorphic to the Hilbert space of the harmonic
> oscillator - because all the Hilbert spaces are isomorphic to each other
> as Hilbert spaces. :-)

All seperable Hilbert spaces are isomorphic which means that their
algebras of operators are also all isomorphic. I wonder if this might
imply anything about the possibility of finding observables other than
the S-matrix.

Regarding the issue of QG and dimensions, the Beckenstein-Hawking
formula seems to imply that 4 dimensional Minkowski spacetimes have
more states at a given asymptotic energy level than higher dimensional
spaces. Is it possible (even in principle) that LQG could tell us
something about why 4 dimensions might be special (maybe somehow
privileged) ?


[1] http://arxiv.org/abs/hep-th/0006167

[2] http://arxiv.org/abs/hep-th/9911080

Josh Willis

unread,
Jun 22, 2001, 2:52:23 PM6/22/01
to

I am making responses out of the original order, to clarify
presentation.

There is an essential question of in what sense, if any, it is true
that GR is the semiclassical limit of string theory. You write:

Lubos Motl writes:

> Josh Willis wrote:

> > *not* show that GR "comes out of" string theory, because this metric
> > is *non-dynamical*. It is not varied in the Lagrangian approach, and

> On the contrary, everything in string theory is completely dynamical. For
> example, while QFTs can contain dimensionless parameters, all such
> parameters (such as the coupling constant) become inevitably dynamical
> fields (in the case of the coupling, dilaton). Gravity in string theory is
> dynamical, of course, this is the main dynamics that one can
> compute.

There are two obvious approaches to making the "metric" dynamical
in string theory:

(1) Explicitly include an integral over all possible spacetime metrics
in your path integral. This means you would have to do a nonlinear
sigma model where you integrate over "all possible" nonlinear
interactions. Alternatively, you could:

(2) Fix some particular background metric. You would then argue that
as one considers the interactions of the theory you generate all other
metrics dynamically.

Of course, (1) is impossible, as far as we know. If we could do this
we could just as well define quantum gravity as a path integral over
all metrics.

Instead, what is always done is (2). The argument that we do actually
obtain all metrics then hinges on expressing an arbitrary metric as a sum
of the background metric and a perturbation, and then "Taylor
expanding" the action in this perturbation, noting that the first
order term gives us the vertex operator for the graviton, which is
essentially a creation operator for an "asymptotic" graviton state
that we are now therefore adding to our interaction. So does this
"Taylor expansion" converge? You claim:

> > operator." But for this claim to be proven, one would have to know
> > that the perturbation series converges, which one does not know (and
> > doubts).

> Yes, the Taylor series for exp(x) converges.

but while this is true in general for a complex number or finite
dimensional square matrix, it is very far from true for an arbitrary
operator on an infinite dimensional Hilbert (or Banach) space. It
would be sufficent to know that your operator is bounded and
self-adjoint, but since your operator is a creation operator, neither
is true. In fact, a simpler example of this problem can be found
already in the quantum theory of the finite dimensional harmonic
oscillator: if our creation operator a* and annihilation operator a
are normalized so that [a,a*] = 1, as usual, then the action of a* is
not defined on an arbitrary state, let alone the exponentiation of
that operator.

Since you seem to have disdain for rigorous arguments, note that we
can also see the same thing in ordinary quantum field theory. If it
were true that "convergence" of the Taylor series for exp(x)
guaranteed us that we could Taylor expand a quantity of the form
exp[-\int (H0 + HI)] as exp[-\int H0] [ 1 - \int HI + ...], then it
would have to be true that every perturbation series in QFT would
converge, since it is precisely by this argument that these
perturbative expansions are "derived". Note I don't mean simply that
the perturbation series would be renormalizable; more than that, it
would have to converge. But we know perfectly well that this is not
true.

Also observe that you do not solve this problem simply by working in
the path integral framework; you merely relocate it to the proper
definition of the path integral.

In general, even to know if you have an S-matrix, you must know what
the asymptotic states of your full, "second-quantized" theory are,
since it is these states that must be evolved, and evolved according
to the "time" translation generated by the physical Hamiltonian. In
short, you must know what your Hilbert space is, and you do not. Yes,


I know that you write:

> > 1. That String/M-theory is in fact a quantum theory in the first
> > place. What is the Hilbert Space? If it is L_2(X,d\mu) for some space

> Yes, string theory is a completely quantum theory and it does not modify
> anything about the basic principles of quantum theory. The Hilbert space


> is H(SMT) :-) and it is isomorphic to the Hilbert space of the harmonic
> oscillator - because all the Hilbert spaces are isomorphic to each other
> as Hilbert spaces. :-)

but the fact that all infinite dimensional Hilbert spaces are
isomorphic to one another is irrelevant. For Hilbert spaces, the
meaningful question is whether or not they are unitarily equivalent;
you must preserve not only the algebraic vector space structure but
also the inner product structure in order to regard two Hilbert spaces
as "the same". And a Hilbert space for string theory is certainly not
going to be unitarily equivalent to Fock space. Moreover, you don't
prove the existence of a Hilbert space for your theory just by noting
that it would be bad news for your theory if it doesn't have one.

Indeed as I noted in my earlier post, one can see at once (at least a
relativist can) that you cannot get an arbitrary solution to
Einstein's equation by superposing gravitons on Minkowski (or any
other) space, simply because the underlying manifolds are not
diffeomorphic. In response you have claimed several times that string
theory has a known nonperturbative definition, but I have heard
several prominent string theorists (Edward Witten, Michael Green, and
Michael Duff) make exactly the opposite claim, and I have much more
confidence in their assessment of that point than yours.

A larger question is to what extent rigor (i.e., mathematical
consistency) is important in a physical theory. You say that it is
not. If you mean by this that it is not needed for progress in a
field; i.e., that we can work on a theory not knowing how rigoroous it
is but trusting that this can later be put right, then that is a
defensible position. If you mean that it is unimportant always, and
that we can be satisfied if we never have a good mathematical
foundation for our theory, then I couldn't disagree more. There is
nothing in any theory attractive enough to justify the sacrifice of
the idea that nature be mathematically consistent.

But even the first viewpoint is much harder to defend when it comes to
a theory of quantum gravity, for two reasons:

(1) We do not have experiment to guide us. Early pioneers in QED, for
example, could argue that the phenomenal agreement of their theory was
argument enough that there was something "essentially right" about
what they were doing. In quantum gravity we do not have that; there
are no experiments to be checking predictions against, as yet.

(2) There is also a deeper reason. We now have, thanks to the
renormalization group, some understanding of just why earlier QFT's,
based on renormalization, were "essentially right." As indicated in
the quote from Peskin and Schroeder in my earlier post, we now know
that any QFT has a fundamental length scale associated to it: the
scale at which the quantum nature of spacetime, whatever that turns
out to be, becomes important. As long as we are only probing energies
far away from that scale, our theories are only effective theories in
which nonrenormalizable terms in a Lagrangian become irrelevant.
Hence, in retrospect, we could almost have demanded that any theory at
these scales be renormalizable.

But when we come to a theory of quantum gravity things change: we are
now attempting to describe physics at that fundamental scale, rather
than far away from it. Hence nonrenormalizable theories must be
considered on equal footing with renormalizable or super-
renormalizable ones; good UV behavior of the perturbation expansion of
a theory, though certainly not bad, is also not "good": it doesn't
tell us one way or the other about whether or not that theory is,
nonperturbatively, a consistent quantum theory. And now we cannot
count on shoving off mathematically dubious manipulations in our
theory on physics that we don't understand and aren't modeling at some
higher energy scale: we are claiming that we are now at that "higher
energy scale." Of course, it is still a logical possibility that
there *is* some other physics at a higher scale and that theories at
the Planck scale really are just effective theories, but I doubt
anyone will want to devote much effort to that hypothesis without a
much clearer indication that it is necessary.

Note that perturbative nonrenormalizability of any such fundamental
theory does not, as you suppose, mean that

> ... infinities do not matter. But if you do not have a


> prescription to get finite and meaningful predictions, you do not have a
> theory.

Rather, it simply means that we cannot use perturbation theory to
calculate things. And indeed, in LQG we generally use other tools (I
say "generally" because Pullin and Gambini have recently made some
calculations that are perturbations in the inverse cosmological
constant). Because we start with a well defined Hilbert space and
self adjoint operators on it, we know that all of these quantities are
well defined. In string theory you do not.

You further argue that supersymmetric partners and extra dimensions
are predictions, and therefore perfectly acceptable in string theory
and do not in any way indicate that it is a "religion", wheras for LQG
to proceed from exact diffeomorphism invariance, or not to include
other forces, means that they practice an "unjustified religion". Yet
the "predictions" that string theory can make are precisely those things
that have not been seen experimentally; when it comes to predicting
parameters of the Standard Model it has nothing to say because of the
uncertainties of compactification and super-symmetry breaking. In
other words, it does not predict precisely the things that we do
observe, but does predict what we don't observe. Perhaps this will
change in the future, but at the moment there are no "predictions" in
string theory to commend it, per se.

LQG, in contrast, proceeds from diffeomorphism invariance, which at
all scales we have thus probed we do observe, and from assuming that
we can quantize gravity without including the other forces. Note that
we do not assume it is impossible to include other forces, just
unnecessary. In fact, as I mentioned in my last post, if LQG succeeds
we will have a completely rigorous quantum theory of an interacting
field in four dimensions---the first time that will have been
achieved. There is good reason to suppose that the techniques we
develop in this process will help us in then tackling other fields
rigorously, especially gauge fields, since the kinematics of the
theories are so similar. Indeed, in two spacetime dimensions this has
already been shown to be the case, see:

A. Ashtekar, J. Lewandowski, D. Marolf, J. Mourao, and T. Thiemann,
"Closed Formula for Wilson Loops for $SU(N)$ Quantum Yang-Mills
Theory in Two Dimensions." J. Math. Phys 38 (1997) 5453--5482.
Available as: hep-th/9605128.

Again, you seem to suppose that it has somehow been already
established that a successful quantization of gravity must of
necessity include a unification of gravity with other forces:

> In this case, I was not talking about the interesting hypothesis :-) that
> you can explain quantum gravity (that we know at long distances only but
> it has quantum effects at the Planck scale) without knowing anything else
> that happens in between (such unimportant things like the strong, weak and
> electromagnetic force, chemistry, life etc.). ;-) I was talking about the
> future. If LQG succeeds (whatever it means), will you be satisfied that
> you found a theory of everything, although your theory predicts that
> electromagnetism, the weak and the strong force do not exist? ;-) I just
> say that even in this hypothetical case of LQG's success, physicists would
> finally have to return to the question of unification, calculation of the
> remaining constants i.e. to (something like) string theory.

I will of course not be satisfied that I found a theory of everything,
but then I have stated all along that I am not looking for one. If
LQG succeeds we will then return to the *incorporation* of other
interactions---which may or may not involve their unification with
gravity. And people have already started addressing this
incorporation; see for example:

gr-qc/9705021 and hep-th/9703112.

Furthermore, LQG does not "ignore" everything in between the Planck
scale and cosmological, it simply proceeds from the assumption that
they are not necessary ingredients in a quantization of gravity. We
are perfectly well aware that this is an assumption; any progress in
theoretical physics---especially in an area where you have no
experimental input---requires some assumption. String theory is no
exception: you require the assumption that spacetime has more
dimensions and particles than we observe, and that a correct quantum
description of gravity requires that it be unified with the other
forces. None of these assumptions has experimental support. I know
you like to view all of these things as predictions, but nonetheless
they are logically assumptions because you cannot construct your
theory without them. I could just as well claim diffeomorphism
invariance as a prediction of LQG---and my prediction *has*
experimental support, at every scale for which we have any
experimental result.

Indeed, you seem particulary confused on this point; you write:

> > Again, why should it have to? It is an interesting hypothesis that a
> > correct description of gravity will require us to unify it with the
> > other forces, but not a logical necessity.

> Well, it is amusing that you try to prove that LQG is not a religion - but
> then you repeat a verse from the Holy Scripture by Rovelli gr-qc/9710008
> with a couple of words permuted. ;-) To compare, Rovelli said:
>
> "A common criticism to loop quantum gravity is that it does not unify all
> interactions. But the idea that quantum gravity can be understood only
> in conjunctions with other fields is an interesting hypothesis, not an
> established truth."

Of course I say essentially the same thing as Rovelli---because his
point is quite valid. You label his article "Holy Scripture" but this
is just sophistry on your part: you use labels where you should instead
provide arguments. In short, you are unable or unwilling to recognize
that the unification of gravity with other forces is just a
hypothesis, and you thereby become guilty of the very thing you claim
to criticize in others: "religion" in physics. It is never good
science to be unaware of what your hypotheses are, because you may
well find that experiment later forces you to reject them.

Another example where you confuse hypothesis with conclusion:

> > Note that the analagous thing
> > to what you suggest is *not* done with special relativity: both the
> > rigorous treatments of QFTs that I referred to above and the heuristic
> > treatment of a first graduate course both require *exact* Poincare
> > invariance of the theory. Why is it somehow unreasonable to try the
> > same thing with diffeomorphism invariance?
>
> It is fine to count general covariance as an exact concept as long as you
> work with gravity classically. At quantum level, its nature is modified at
> Planckian distances. The corresponding statement concerning Poincare group
> is the following: you can consider the affine Galilean group to be an
> exact symmetry as long as you deal with speeds much smaller than c; then
> you must switch to the Poincare group.
>
> In other words, the Poincare group is already the "deformed" or "resolved"
> version of the affine Galilean group. Therefore there is no reason to
> "deform" it further (in theories without a dynamical geometry). In special

Your analogy could only be appropriate if you already knew that
diffeomorphism invariance was not an exact symmetry. But you do not
know that; you would have to know already that something like string
theory is a correct theory of Nature, and we do not know that. Your
argument is therefore circular.

My analogy, in contrast, is that it is worth investigating whether or
not diffeomorphism invariance is exact: SR (and other symmetries as
well, of course) show us that classical symmetries can carry over
exactly to a quantum theory. We don't know whether or not this is
true for diffeomorphism invariance; part of the goal of LQG is to find
out.

I have said all along that I expect there are many quantum theories of
gravity, simply because of the evidence of 2+1 dimensional gravity,
and the fact that it is a generic fact of life that many inequivalent
quantum theories can have the same classical limit. You may not like
that, but it cannot be avoided. The goal of loop quantum gravity is
to attempt the construction of one such theory, not because we know in
advance that it is necessarily the right one or the only one, but just
because it has not been done yet and therefore is an important step in
our understanding of quantum gravity. And by construct I mean
*rigorously* construct---I have explained above why this is important
for any theory that would be the fundamental description of spacetime
at even the smallest scales. String theory does not provide, as yet,
such a rigorous construction, and in exchange for sacrificing rigor it
offers perturbative UV finiteness (which is not needed) and
predictions that are not observed. I therefore don't find this to be an
appealing research avenue, but if you do, fine. As long as you
recognize that in choosing this you are making a heuristic guess, not
a deduction, as to what is needed for quantum gravity, and have an
open enough mind to realize that your theory may well turn out to be
wrong. I realize that LQG could, but I've just summarized the reasons
why I find it a more promising avenue of research than string theory.
Of course I don't know if either of these approaches will in the end
turn out to be correct: to know that, we would have had to have
already solved the problem.

----------------------------------------------------------------------

Now a few other random comments:

> > String theory's first big success and a driving force behind it is the
> > absence of ultraviolet divergences. In other words, string theory

> Historically speaking, this is an incorrect claim, too. String theory was
> supposed to be a theory of the strong interaction and the driving force

> behind it was first the so-called worldsheet duality ...

I was speaking, of course, about the successes that lead string theory
to be considered a candidate quantum theory of gravity, which is what
we are talking about on this thread.

> > So, what bearing does all of this have on SMT? You have a
> > perturbative expansion of an S-matrix, but no proof that the
> > perturbation series converges, or that anything exists to which it

> We have quite a reliable proof that it diverges just like in QFT (except
> for the quantities that are protected and have a finite number of
> contributions only).

That is my point, and the whole problem, as explained both in my
earlier post and again above.



> > might be asymptotic. On top of that you inherit from QFT the problems
> > above in the definition of an S-matrix. Simply put, having a
> > perturbative expansion of an S-matrix, each of whose terms is finite,
> > does not a full quantum theory make. This is related to the
> > discussion going on in another thread about causal quantum theory.

> Note that your "criticism" can be applied to any theory that should
> reproduce physics of QFT in some limit. In fact it can be applied against
> any scientific effort to understand anything in reality. :-)

Huh? The criticism applies to any theory claiming to be fundamental,
for the reasons I have outlined above. It does not apply to effective
theories, precisely because of renormalization group theory, again as
explained above, And how you get from there to "any scientific effort
to understand anything in reality" is beyond me.

> > X and measure \mu, what are they?

> It is certainly not (in any useful sense) of the form L_2(X,d\mu) for a
> finite dimensional X, this would return us to the 1920s (a single-particle
> quantum mechanics).

I never said it should be finite dimensional. In LQG, or for that
matter those QFTs that have been rigorously constructed, the space is
of course infinite dimensional, and not single particle quantum
mechanics.

> Your arguments are similar to some arguments of a
> hypothetical self-confident Newtonian physicists who also comes to the
> 21st century and argues that string theory is wrong because it cannot
> answer his question what is the potential between two point-like particles
> with well-defined positions and momenta. ;-)

My arguments are not anything like that.

> > >Well, this construction obviously looks "dual" to LQG in some sense but it
> > >is probably not equivalent. But it is equally covariant. Why do the people
> > >study LQG and not MQG, for example? Only because LQG is related to the
> > >gauge formulation of the initial problem in GR? The low-energy limit has
> > >not been found for either of them. I would be happy to hear why MQG is
> > >really a worse direction of research than LQG. ;-)

> > None of the theories you cite comes close to being a well formulated
> > quantum theory, as I have indicated above.

> This claim of yours is probably wrong, I just wanted to check you. The
> theory I formulated has most likely the same level of quantum consistency
> as LQG; in fact, it may be completely equivalent to LQG (compare with dual
> graphs in combinatorics). I just wanted to check whether you are able to
> say an incorrect claim if you need to do anything to defend your position.
> Unfortunately, you are able to say anything. Of course that you do not
> seem to have a clue whether there should be anything wrong about MQG. What
> could you answer if I asked you why you think that MQG is not a
> well-formulated quantum theory? ;-) That I am not Ashtekar? :-)

If you want more specific problems with your theory, fine. Note to
begin with, however, that it is not my job or any one else's to prove
that your theory is not well formulated; it is again your job to prove
that it is. In particular, if you wish to make it a rigorous theory
of quantum gravity instead of hand-waving verbiage, then you should
answer the following questions:

(1) What is your classical configuration space? In LQG we take the
space of connections modulo gauge transformations, A/G. You seem to
want to have surfaces correspond to some kind of functionals on this
space; if it is A/G, how will you do this? The connections are one
forms and therefore naturally integrated along one-dimensional
objects; you will need a two form to integrate along two dimensional
surfaces. Which two form will you use? What is the canonically
conjugate variable? What are the constraints in terms of thes
variables? Are they polynomial? If not, how will you implement them
in your quantum theory? Are they first-class? Does the constrain
algebra close? If it does not, what secondary constraint must you add?
Are any of these second-class? If they are, are your contraints still
polynomial (if they ever were) after you have solved for all
second-class constraints?

(2) What is your quantum configuration space? For a system with
infinitely many degrees of freedom we must enlarge beyond the
classical space of smooth solutions. In LQG we use \bar{A/G}; a
compact Hausdorff space for which we have three definitions, shown,
nonperturbatively and rigorously, to be equivalent. What space will
you use, and how will you define it? Do you have any measures on this
space, so that you can construct a Hilbert space of square integrable
functions on it to serve as your space of quantum states?

(3) What operators will you have? You have proposed an area operator,
but given no justification for it. In LQG we derive our expression by
starting with the formula for that operator in terms of the classical
canonical variables, regularizing it, and then showing that the limit
when the regularizer is removed exists and is independent of the
choice of regularizer. Have you done this for your expression? It is
in general not easy, in part because the operators corresponding to
the canonical variables do not exist as operators on the whole Hilbert
space, so there is no guarantee that an expression formed from them
can be even densley defined. Is your operator densely defined?

You have previously labelled all of the analgous calculations in LQG
to be "trivial", so I assume this will not take you too long.


> > Nonsense. Again, why should a theory of quantum gravity have to

> > explain the dimensionality of spacetime? And the fact that LQG does

> Because a gravitational theory is by a definition dynamics of geometry.
> One of the classical notions describing a geometry is the
> dimension.

Again, you are picking and choosing what elements of classical
geometry a quantum theory should explain based precisely on what it is
string theory attempts to explain. I could just as well ask you to
explain, from string theory, why spacetime should (classically) be a
manifold. The only thing that must be explained by a quantum theory
of geometry is the metric, because that is the field that is dynamical
in GR. The dimensionality of spacetime is not.



>>> I do not understand why the same action should be given
>>> different names if
>>> you use different variables.

> > The reason is that while the solutions to the equations of motion will
> > always be equivalent, the space of histories (from which you select
> > the solution by the requirement that it extremize the action) are
> > different. And this can have an impact on the quantum theories,
> > because you then perform a path integral over this space of
> > histories. Hence quantizing theories written in different variables
> > could easily lead to inequivalent theories; as I indicated above, this
> > is a pretty normal thing. I don't think it has been shown explicitly
> > in this case, because I don't think anyone has constructed any kind of
> > quantum theory of gravity in 3+1 dimensions based on a metric
> > formulation.

> Yes, I understand in principle. But the idea that one works on a theory
> where it is conceivable that there are in fact many inequivalent ways how
> to quantize it, depending on the choice of variables (!!!), is really
> scary.

That would be any quantum theory. You are making an enormous number of
choices in simply choosing to work in on string theory. It is, at
best, *a* quantum theory of gravity. There is *no way* to
mathematically determine that a particular quantum theory which has a
given classical theory as its semiclassical limit is the "correct"
quantization of that theory. As I have emphasized repeatedly, it is a
generic feature of quantization that classical theories have multiple,
inequivalent quantizations. The *only* way to tell which of these is
correct is to find some experimental prediction on which they
disagree, and then go and do the experiment and see if any of them
gave the right answer.

Josh


Lubos Motl

unread,
Jun 26, 2001, 9:40:22 PM6/26/01
to
On Mon, 25 Jun 2001, zirkus wrote:

> Here I would like to quote a criticism of AQFT by S. Majid and also

> mention some things about string theory (ST). Regarding AFQT, "one...

Your comments are interesting and sound reasonable to me. Majid's claims
seem to be confirmed by the developments in ST.

> Perhaps two nice things about ST is that perturbative ST does not obey
> the axioms of local QFT, and that the usual kind of idea of a
> spacetime manifold does not have to be fundamental to ST. I clarify
> this second statement some by quoting from page 3 of [2] by C.M. Hull:

Yes, that's fine, but there is also the opposite characteristics of string
theory. The amplitudes computed from string theory seem to satisfy almost
all the possible principles of local field theories, just like if we
derived them from a local field theory. The S-matrix amplitudes have
simple poles only etc. The violations of locality in string theory have a
very subtle character (they are implied by the nonlocal character of the
fundamental object - string - but the local dynamics of strings and their
interactions is still "local at the worldsheet"). These nonlocalities are
probably responsible for solving the Hawking's information loss paradox
(in the case of black holes), but otherwise they do not damage the usual
properties that allow string theory to agree with the logic of local
quantum field theories.

String field theory is essentially the attempt to trivialize those
observations and properties. It is intended to be a formulation of string
theory that is as similar to local quantum field theories as possible.
String field theory is "just" a quantum field theory with infinitely many
fundamental fields, corresponding to various excitations of a single
string, whose masses and interactions are determined from the usual string
theory.

Maybe there is another equivalent and local formulation of string theory -
as Lenny Susskind believes. I personally believe that it has become clear
that spacetime geometry makes sense only if it becomes large enough - if a
new "decompactification limit" appears. And string theory has taught us
that the moduli spaces have usually many different decompactification
limits which require a completely different geometrical interpretation, as
well as the choice of the theory living in the spacetime. The M-theory in
the "bulk" of the moduli space, a generic state, is a fuzzy and funny
monster. There is no priviliged spacetime interpretation of such a generic
state.

> It is not certain that this statement will remain true. Please see my
> other post which refers to sources concerning black holes and QM in
> M-theory.

I will try to look at it if I have time.

> All seperable Hilbert spaces are isomorphic which means that their
> algebras of operators are also all isomorphic. I wonder if this might
> imply anything about the possibility of finding observables other than
> the S-matrix.

Well, the whole question of dynamics depends on the choice of "time" or
"geometry" or at least "asymptotic boundary conditions". Without these
geometric choices, we have the Hilbert space only and its structure is
completely universal and empty. The whole question is then "which" choices
of geometry, time (and its dual Hamiltonian) are allowed etc. But once you
specify some boundary conditions, you put a lot of structure to your
theory and the S-matrix tells you a lot.

Kevin A. Scaldeferri

unread,
Jun 26, 2001, 9:41:05 PM6/26/01
to
In article <slrn9j5n44....@cardinal0.Stanford.EDU>,

Aaron Bergman <aber...@princeton.edu> wrote:
>In article <Pine.SOL.4.05.101061...@athena.phys.psu.edu>,
>Josh Willis wrote:
>>
>>There are two categories here which it seems evident to me are being
>>confused with one another:
>>
>>I. What is known rigorously about a theory, be it SMT or LQG or
>>anything else, as opposed to suggestive and partially heuristic
>>results. This is a matter of theorems, not opinions, and the goal here
>>is, as stated in the preface to Streater & Wightman's *PCT, Spin and
>>Statistics, and All That*, the elimination of all theorems whose
>>proofs are non-existent.
>
>It's worth noting that nothing interesting in high energy physics
>falls into this category. I don't think I'm being hyperbolic here, either.

Well, to be obvious, the CPT theorem and the Spin-Statistics theorem.
These are interesting in the sense that an experimental signature that
violated either one would turn the theoretical community on it's head.

Although I've never been clear quite how rigorous it is, the
Coleman-Mandula theorem might also qualify given how much work it has
inspired.


>The main point I want to make now is that you seem to be
>requiring of string theory things that haven't even been
>accomplished for the simplest interacting QFTs. I think it's a
>bit unfair.

There is an issue here, which I have heard JB state several times, and
I suspect is a relatively commonly held feeling in the LGQ camp, which
is that for theories where we have experimental evidence we can afford
to be a little sloppy. When we don't have experiments, we should be a
little more cautious. It's not clear that the conventional wisdom of
QFT is applicable to the quantization of extended objects. And we
know from lower-dimensional QG theories that funny stuff can happen.


--
======================================================================
Kevin Scaldeferri Calif. Institute of Technology
The INTJ's Prayer:
Lord keep me open to others' ideas, WRONG though they may be.

Jacques Distler

unread,
Jun 26, 2001, 9:41:34 PM6/26/01
to
References:
<Pine.SOL.4.05.101061...@athena.phys.psu.edu>
<slrn9j5n44....@cardinal0.Stanford.EDU>

In article <slrn9j5n44....@cardinal0.Stanford.EDU>,
aber...@princeton.edu (Aaron Bergman) wrote:

> One person who I talked to saw a talk and said
>that this issue (maybe something to do with what happens when
>Wilson lines cross?) led Gross and Witten to crucify the speaker.

The speaker was Carlo Rovelli, and the funniest moment in the seminar
occurred when Witten, who was arguing some point in functional
analysis with the speaker turned around -- for added moral support --
and said, "Isn't that so, Arthur?" Arthur Wightman just smiled . . .

Jacques

Aaron Bergman

unread,
Jun 27, 2001, 3:56:44 PM6/27/01
to
In article <Pine.SOL.4.05.101062...@athena.phys.psu.edu>, Josh
Willis wrote:

>There are two obvious approaches to making the "metric" dynamical
>in string theory:
>
>(1) Explicitly include an integral over all possible spacetime metrics
>in your path integral.

Path integral? What path integral?

[...snip a whole lot...]

>In response you have claimed several times that string
>theory has a known nonperturbative definition, but I have heard
>several prominent string theorists (Edward Witten, Michael Green, and
>Michael Duff) make exactly the opposite claim, and I have much more
>confidence in their assessment of that point than yours.

While this is a nice appeal to authority, the state of affairs
is, as usual, more complicated. There exists a conjectured
nonperturbative definition of string theory in spacetimes that
are asymptotically AdS. There all exists a conjectured
nonperturbative defintion of string theory "in the infinite
momentum frame". Now, it may be that these conjectures only cover
that small region of moduli space, but, then again, maybe they're
part of something more general. The point is that, in those
classes of backgrounds, we can make nonperturbative statements
about string theory.

[...more snip...]

>(2) There is also a deeper reason. We now have, thanks to the
>renormalization group, some understanding of just why earlier QFT's,
>based on renormalization, were "essentially right." As indicated in
>the quote from Peskin and Schroeder in my earlier post, we now know
>that any QFT has a fundamental length scale associated to it: the
>scale at which the quantum nature of spacetime, whatever that turns
>out to be, becomes important.

So you don't think any QFT nonperturbatively exists? What about
QCD? For that matter, what about a CFT? There's no length scale there.

[...snip to end...]

In a lot of what I snipped, you seem to have expressed a fairly
strong idea of what a quantum theory of gravity should be. The
idea seems to be based on the formalisms that have so far
completely failed to define any interesting QFT in >3 dimensions.
Have you considered that these formalisms might be, well, wrong?

zirkus

unread,
Jun 27, 2001, 4:25:29 PM6/27/01
to
Lubos Motl <mo...@physics.rutgers.edu> wrote:

> The whole question is then "which" choices
> of geometry, time (and its dual Hamiltonian) are allowed etc.

Some progress is being made on this kind of issue. For instance, it
was discovered that a constant background B field for open strings or
D-branes implies noncommuting coordinates of spacetime. It is expected
that this will generalize to the case of a non-constant B field (with
a resultant space that is noncommuting and curved).

Majid suggests, though, that theorists should consider an algebra with
a richer and more geometric algebraic structure than the CCR algebra
(at least a variation, say, of the CCR in order, e.g., to preserve
more of the geometric structure of phase space in the quantum case).
The CCR is basically the usual algebra of observables B(H) of all
bounded (self-adjoint) operators on a Hilbert space. But this is the
same abstract algebra in all cases, i.e., it doesn't know about
certain things a quantum system might know about such as the choice of
polarization on the phase space or the choice of the Hamiltonian, etc.
He discusses this issue in his paper [1]. See, for instance, pages 8
and 14.

Somewhat related to what he says is the question of trying to
understand the appearance of Berry phase in quantum theory. In this
regard, I wonder what readers of s.p.r. think of page 1 of [2]. (You
needn't read beyond page 1 because, here, I am not asking anyone to
accept the Consistent Histories interpretation of QM).


[1] http://arxiv.org/abs/hep-th/0006167

[2] http://arxiv.org/abs/quant-ph/0007093

Lubos Motl

unread,
Jun 27, 2001, 2:34:40 PM6/27/01
to
On Wed, 27 Jun 2001, Jacques Distler wrote:

> The speaker was Carlo Rovelli, and the funniest moment in the seminar
> occurred when Witten, who was arguing some point in functional
> analysis with the speaker turned around -- for added moral support --
> and said, "Isn't that so, Arthur?" Arthur Wightman just smiled . . .

That's amusing. Was A.W. the right person to ask for moral support? I have
heard from several sources that Carlo Rovelli is probably the most
militant person in the LQG community. And one is advised even not to play
any card games with him. :-)

Concerning the Wilson loops, I think that something singular must happen
if two Wilson loops cross. At the moment, when their linking number in
three dimensions jumps. Usually this is associated with some UV
divergences of their correlators etc. Well, Carlo Rovelli claims (see his
review, for example) that LQG avoids all these problems known from the
loop quantization of gauge theories, because there is general covariance
underlying everything and it has a magic power to cure almost every
problem (I do not quote him exactly).

But this is just a technicality. What is a more disappointing aspect of
theories similar to LQG is that people are really inventing and
constructing them. This is not what physicists should do. Physicists
should be rather discovering something that objectively exists. Either in
the world of natural phenomena, or (now, when we have almost no
experiments beyond the Standard Model) in the world of important
mathematical (and physically relevant) concepts.

String theory is a typical example of a structure whose complete and
universal definition is clearly missing, but whose new insights are always
unambiguously derived from something that objectively exists. Whether
string theory describes the real world, we don't know for sure. But its
theoretical rigidity is an important characteristic that implies that we
are not just playing with one of thousands of slightly or completely
different collections of formulae; so that the probability that we are on
the right track would converge to zero. We are not pouring sand into the
Atlantic ocean to build one of the possible lands where we can live; we
are discovering a new continent of mathematics. And even in the
hypothetical case in which Nature did not use string theory but something
worse, string theory will dominate *mathematics* of the 3rd millenium, as
Witten said.

Often, I find it frustrating if some people (usually young people) are
almost ready to abandon most of the important insights that people have
learned in the field. There are many things that we do not know yet and
several things which we "know" incorrectly today. But there are also very
many important lessons that we have already learned and we will probably
never "unlearn" them. They will just become parts of a more general
structure.

Aaron Bergman

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Jun 27, 2001, 4:00:05 PM6/27/01
to
In article <9h8vrl$6...@gap.cco.caltech.edu>, Kevin A. Scaldeferri wrote:

>In article <slrn9j5n44....@cardinal0.Stanford.EDU>,
>Aaron Bergman <aber...@princeton.edu> wrote:

>>In article <Pine.SOL.4.05.101061...@athena.phys.psu.edu>,
>>Josh Willis wrote:

>>>There are two categories here which it seems evident to me are being
>>>confused with one another:
>>>
>>>I. What is known rigorously about a theory, be it SMT or LQG or
>>>anything else, as opposed to suggestive and partially heuristic
>>>results. This is a matter of theorems, not opinions, and the goal here
>>>is, as stated in the preface to Streater & Wightman's *PCT, Spin and
>>>Statistics, and All That*, the elimination of all theorems whose
>>>proofs are non-existent.

>>It's worth noting that nothing interesting in high energy physics
>>falls into this category. I don't think I'm being hyperbolic here, either.

>Well, to be obvious, the CPT theorem and the Spin-Statistics theorem.
>These are interesting in the sense that an experimental signature that
>violated either one would turn the theoretical community on it's head.

Those are theorems that have been proved about theories that
satisfy the Wightman axioms IIRC. Has any interesting theory been
shown to satisfy the axioms?

>Although I've never been clear quite how rigorous it is, the
>Coleman-Mandula theorem might also qualify given how much work it has
>inspired.

I'm not sure about the rigor; it mainly seems to be a statement
about scattering amplitudes.

[...]

>I suspect is a relatively commonly held feeling in the LGQ camp, which
>is that for theories where we have experimental evidence we can afford
>to be a little sloppy. When we don't have experiments, we should be a
>little more cautious.

It's a belief, but I don't really see the point behind it. If you
want to make something rigorous, do it to something we already
know about. That would be really cool.

Lubos Motl

unread,
Jun 26, 2001, 8:53:16 PM6/26/01
to
On Fri, 22 Jun 2001, Josh Willis wrote:

> There is an essential question of in what sense, if any, it is true
> that GR is the semiclassical limit of string theory. You write:

General relativity (coupled to few other fields in various contexts, such
as Yang-Mills and their superpartners) is not only the (easily-derived)
low-energy "semiclassical" limit of string theory, but I know that many
readers have understood technically why is it so. If you have not, we
might argue whether I am a bad teacher or you are a bad student, but we
cannot do anything more, it seems. ;-)

[Moderator's note: another possibility is that Josh Willis is
demanding that string theory reduce to GR in a strong sense than
it actually does, or has rigorously been shown to do. - jb]

> There are two obvious approaches to making the "metric" dynamical
> in string theory:

First of all, there is no way to "make" metric *non*-dynamical in full
string theory. And there is only one way in which the metric is dynamical.
This is the way that you can learn from string theory. All the other ways
are either equivalent or incorrect.

> (1) Explicitly include an integral over all possible spacetime metrics
> in your path integral. This means you would have to do a nonlinear
> sigma model where you integrate over "all possible" nonlinear
> interactions. Alternatively, you could:

At the level of the nonlinear sigma model, this is absolutely incorrect
because the spacetime interpretation of the sigma model is the S-matrix
and the S-matrix is not defined without a specific background (asymptotic
boundary conditions).

> Instead, what is always done is (2). The argument that we do actually
> obtain all metrics then hinges on expressing an arbitrary metric as a sum
> of the background metric and a perturbation, and then "Taylor
> expanding" the action in this perturbation, noting that the first
> order term gives us the vertex operator for the graviton, which is
> essentially a creation operator for an "asymptotic" graviton state
> that we are now therefore adding to our interaction. So does this
> "Taylor expansion" converge? You claim:

> > Yes, the Taylor series for exp(x) converges.

I believed that there had to be a subtlety and you were talking about the
convergence of something else. If you are really talking about the
convergence of exp(x), it is disappointing. The series converge exactly
iff they should converge. If your perturbation of the metric is large so
that it modifies the boundary conditions, then of course the calculation
is incorrect - because you violated the assumption about the asymptotic
conditions which threw you out of the superselection sector. And such a
"jump" should be incorrect, so everything is fine.

> but while this is true in general for a complex number or finite
> dimensional square matrix, it is very far from true for an arbitrary
> operator on an infinite dimensional Hilbert (or Banach) space. It

I have a feeling that the only thing that you want us to do in physics is
that someone gives us an operator and we mechanically evaluate its
exponential, without a clue about its physical interpretation. This is
the way how physics should become a collection of theorems. Fortunately,
this approach does not lead anywhere in physics.

> In fact, a simpler example of this problem can be found already in the
> quantum theory of the finite dimensional harmonic oscillator: if our
> creation operator a* and annihilation operator a are normalized so
> that [a,a*] = 1, as usual, then the action of a* is not defined on an
> arbitrary state, let alone the exponentiation of that operator.

Beautiful. Great. If you have a state c_n|n> and c_n behaves like 1/n, for
example, than its norm is finite - sum of 1/n^2 -, but the norm of a|n> is
not finite; it is a sum of 1/n (logarithmically diverges), because "a"
acts like sqrt(n+1)|n+1>, roughly. Great. The state a|n> is therefore
outside the axiomatic "Hilbert space". In any theory with an
infinite-dimensional Hilbert space, you can act with some operators to get
non-normalizable states from normalizable ones.

But what does this trivial observation have to do with the fact that the
metric is dynamical in string theory? This claim is equally solid as the
claim that for every state |psi> in QM on a line, there is a state where
the whole wave function of the particle is moved by x0. This state is
gotten as exp(i.p.x0)|psi>. Everyone understands why is it so. The case of
metric in string theory is just a little bit more technical.

But the essence is similar and has nothing to do with the fact that you
can get non-normalizable states by operators acting on some normalizable
ones.

> Since you seem to have disdain for rigorous arguments, note that we
> can also see the same thing in ordinary quantum field theory.

First of all, I want to avoid *irrelevant* arguments. For instance, the
argument about normalizability of a*|psi> is absolutely irrelevant for the
question if metric is dynamical in string theory. The stringy vacua with
some fixed vevs of various fields are analogues of the coherent states
|z>=exp(za)|0>.exp(-|z|^2/2). And these states certainly do exist and are
normalizable. So why do you talk about completely different states in
different systems that have absolutely nothing to do with the question? It
seems that you invest whole your energy to find some bugs in string theory
- even at places where even you must know that you won't find anything.
But there aren't any. You're wasting your times. There are many things
that we do not understand yet and we need smart and original people to
find better ways to study the theory. But there are no bugs.

> If it were true that "convergence" of the Taylor series for exp(x)
> guaranteed us that we could Taylor expand a quantity of the form
> exp[-\int (H0 + HI)] as exp[-\int H0] [ 1 - \int HI + ...], then it
> would have to be true that every perturbation series in QFT would
> converge, since it is precisely by this argument that these
> perturbative expansions are "derived".

This argument is also mostly wrong. Some quantum field theories are
nonrenormalizable but this failure cannot be reduced to the "divergent
Taylor series for the exponential". The exponent HI itself cannot be given
a correct definition in the Hilbert space of physical states at nonzero
coupling; the Hilbert space itself is ill-defined. For free theories (when
HI=0), the Gaussian path integral is always fine. The nonrenormalizable
theories however do not make sense for nonzero coupling constants.

Note that we must use the Hilbert space at finite coupling - and this
Hilbert space is different from the Hilbert space of the free theory. And
in the case of nonrenormalizable theories, even HI itself is ill-defined
(otherwise you could evolve the system for an infinitesimal time).

It is really easy to see that this certainly does not happen in the case
of metric in string theory. The analogue of the "nonzero coupling" (where
you want to get by Taylor expanding) is just "string theory at different
background, solving the correct eqns." - and it is equally well-defined as
the original one. By adding the coherent states of the gravitons, you are
just moving between various backgrounds and all of them are equally
consistent.

> In general, even to know if you have an S-matrix, you must know what
> the asymptotic states of your full, "second-quantized" theory are,
> since it is these states that must be evolved, and evolved according
> to the "time" translation generated by the physical Hamiltonian. In
> short, you must know what your Hilbert space is, and you do not. Yes,
> I know that you write:

Why do you think that I do not know the Hilbert space? These asymptotic
states are obtained perturbatively from the free asymptotic states which
are just the states of the second-quantized string theory that everyone
who has learned string theory for 1 day at least knows very well: an
arbitrary collection of strings vibrating in various ways (determined by
the first quantized picture), satisfying the correct (anti)symmetry. What
we do not know is the "Hamiltonian" (at least in temporal gauge; we know
the light cone gauge Hamiltonian in many cases). And it is good that we do
not know it because it should not exist in a generally covariant theory.

> but the fact that all infinite dimensional Hilbert spaces are
> isomorphic to one another is irrelevant. For Hilbert spaces, the

Yes, I agree that it is irrelevant for this discussion. Note that it was
you, not me, who asked this question.

> meaningful question is whether or not they are unitarily equivalent;

Are you kidding? Every two infinite-dimensional (separable) Hilbert spaces
are unitarily isomorphic, see for example

http://www.ias.ac.in/pramana/fm2001/QT10.htm

> you must preserve not only the algebraic vector space structure but
> also the inner product structure in order to regard two Hilbert spaces
> as "the same". And a Hilbert space for string theory is certainly not
> going to be unitarily equivalent to Fock space. Moreover, you don't
> prove the existence of a Hilbert space for your theory just by noting
> that it would be bad news for your theory if it doesn't have one.

This discussion goes really down the hill. The Hilbert space of states in
string theory with some asymptotic boundary conditions (a superselection
sector) in a flat spacetime, for example, is a separable Hilbert space,
and therefore it is unitarily equivalent to Fock space or to the Hilbert
space of harmonic oscillator or any other infinite-dimensional separable
Hilbert space. Do we really have to argue about such things?

> Indeed as I noted in my earlier post, one can see at once (at least a
> relativist can) that you cannot get an arbitrary solution to
> Einstein's equation by superposing gravitons on Minkowski (or any
> other) space, simply because the underlying manifolds are not
> diffeomorphic. In response you have claimed several times that string
> theory has a known nonperturbative definition, but I have heard
> several prominent string theorists (Edward Witten, Michael Green, and
> Michael Duff) make exactly the opposite claim, and I have much more
> confidence in their assessment of that point than yours.

String theory has known nonperturbative definitions *on some backgrounds*.
A general, non-perturbative and background independent formulation of
string theory is one of its greatest dreams. And this dream has not become
reality yet, of course. If it is too difficult to note all such words, I
advice you to study questions easier than string theory first. String
theory also implies that the topology of space can change. I did not want
to get solutions of GR by "superposing" gravitons on flat space. How could
you get a solution of a nonlinear, interacting theory by superposing
something? Your classical ideas about the geometry of spacetime are
not sufficient to understand questions in quantum gravity.

> A larger question is to what extent rigor (i.e., mathematical
> consistency) is important in a physical theory. You say that it is

Rigor is not the same as mathematical consistency! A true rigor would
require us to behave like un-thinking calculators; something that you
would like us to do. Mathematical consistency is something completely
different. A theory is either mathematically consistent, or it is
mathematically inconsistent. But one does not have to behave as a
calculator to show with a high level of reliability whether a theory is
consistent or not!

> If you mean by this that it is not needed for progress in a field;

I say not only that a rigorous behavior of all theoretical physicists (in
the mathematical sense of the word) is not needed for progress. I even say
that such a behavior would stop the progress in physics almost completely.
Most things in physics were clearly known a long time before someone
converted them into a rigorous and boring collection of mathematical
definitions and theorems. There are a few exceptions.

> But even the first viewpoint is much harder to defend when it comes to
> a theory of quantum gravity, for two reasons:
>
> (1) We do not have experiment to guide us. Early pioneers in QED, for
> example, could argue that the phenomenal agreement of their theory was
> argument enough that there was something "essentially right" about
> what they were doing. In quantum gravity we do not have that; there
> are no experiments to be checking predictions against, as yet.

This implies that we must rely on theoretical considerations more heavily.
But it certainly does not imply that we should become mathematicians and
adopt their language and way of thinking.

> (2) There is also a deeper reason. We now have, thanks to the
> renormalization group, some understanding of just why earlier QFT's,
> based on renormalization, were "essentially right." As indicated in
> the quote from Peskin and Schroeder in my earlier post, we now know
> that any QFT has a fundamental length scale associated to it: the

This is also incorrect. Conformal QFTs do not have any fundamental length
scales associated to them. Most QFTs have several fundamental scales. You
probably misunderstand the role of the scale in the renormalization group
approach. What Wilson taught us is that a field theory is not only a
collection of some fields, constants and interactions, but that all those
objects must be expressed relatively to a scale. This scale is *not*
uniquely associated with a given theory as you say incorrectly. On the
contrary, you can express the same physics relatively to different scales
and these descriptions are related by renormalization group flows.

> scale at which the quantum nature of spacetime, whatever that turns
> out to be, becomes important. As long as we are only probing energies
> far away from that scale, our theories are only effective theories in
> which nonrenormalizable terms in a Lagrangian become irrelevant.
> Hence, in retrospect, we could almost have demanded that any theory at
> these scales be renormalizable.

OK, pretty fair.

> But when we come to a theory of quantum gravity things change: we are
> now attempting to describe physics at that fundamental scale, rather
> than far away from it. Hence nonrenormalizable theories must be
> considered on equal footing with renormalizable or super-
> renormalizable ones; good UV behavior of the perturbation expansion of
> a theory, though certainly not bad, is also not "good": it doesn't
> tell us one way or the other about whether or not that theory is,
> nonperturbatively, a consistent quantum theory. And now we cannot

I agree that in the case of quantum gravity, the usual tools of
renormalization group etc. break down. But the reason is that the *whole*
description by a field theory breaks down. I do not understand your claims
that we should study nonrenormalizable theories. It means that you want us
to study nonsensical theories that give us infinite answers to
well-defined questions? Or how do you want to extract anything interesting
from nonrenormalizable theories? I really do not understand what you want
to do. The correct solution is that we must study theories that are *not*
just local quantum field theories. And we have been working on it for more
than 30 years and a lot of insights have been obtained. This industry is
called "string theory".

> count on shoving off mathematically dubious manipulations in our
> theory on physics that we don't understand and aren't modeling at some
> higher energy scale: we are claiming that we are now at that "higher
> energy scale." Of course, it is still a logical possibility that
> there *is* some other physics at a higher scale and that theories at
> the Planck scale really are just effective theories, but I doubt
> anyone will want to devote much effort to that hypothesis without a
> much clearer indication that it is necessary.

I agree. Furthermore we can argue quite reliably that the concept of
geometry almost certainly breaks down at 10^{-35} meters or sooner (in the
case of large dimensions). Accelerators with transPlanckian energies
certainly produce black holes which are bigger than the Planck length and
cannot be therefore used to study subPlanckian distances.

> Rather, it simply means that we cannot use perturbation theory to
> calculate things. And indeed, in LQG we generally use other tools (I

> say "generally" because Pullin and Gambini have recently made some...

Yes, the belief that a given theory is consistent nonperturbatively, even
though you see some inconsistencies already at the perturbative level, is
a possible belief that I am not able to disprove rigorously right now. I
just consider this belief hopeless and irrational. Nonperturbative
consistency is much more constraining than perturbative consistency.
Whatever quantum gravity is, it should allow you to compute the S-matrix
elements for gravitons at the end. And you can always ask what happens if
you try to Taylor-expand them. ;-) In the case of string theory, the
S-matrix makes sense and the asymptotic perturbative expansions can be
calculated.

> calculations that are perturbations in the inverse cosmological
> constant). Because we start with a well defined Hilbert space and
> self adjoint operators on it, we know that all of these quantities are
> well defined. In string theory you do not.

Quantum Gravity cannot have a clear collection of localizable (in
spacetime) exactly gauge invariant operators, because the theorems about
gauge invariance in GR prove it. Consequently, a theory that has them is
either inconsistent, or these operators are irrelevant for the low-energy
description. In both cases, such a theory is wrong. Fortunately, string
theory does not suffer from this failure because it does not allow the
existence of any physical operators localized in spacetime; it can be used
to compute the S-matrix (or perhaps the light-cone Hamiltonians etc.). If
string theory allowed for such local operators, it would be inevitably
wrong, too (as a theory of quantum gravity).

> You further argue that supersymmetric partners and extra dimensions
> are predictions, and therefore perfectly acceptable in string theory
> and do not in any way indicate that it is a "religion", wheras for LQG
> to proceed from exact diffeomorphism invariance, or not to include
> other forces, means that they practice an "unjustified religion".

This cannot be explained without mathematics and data. But the situations
are simply not equal. Superpartners are completely meaningful - and
preferred by high-precision experiments, by the way. But unchanged general
covariance at Planckian distances, without any corrections or extensions,
is just a plain nonsense.

> Yet the "predictions" that string theory can make are precisely those
> things that have not been seen experimentally; when it comes to
> predicting parameters of the Standard Model it has nothing to say
> because of the uncertainties of compactification and super-symmetry
> breaking.

No. This is also incorrect. The most general predictions of string theory
have been verified experimentally. Namely the existence of gravity and
Yang-Mills fields at low energies. String theory is the only theory known
that predicts these forces (as known in the Standard Model plus General
Relativity) at low energies. You probably wanted to say that string theory
has not offered a *new* experimental prediction that is not contained in
the simpler theories but has been observed. But this cannot be understood
as a criticism against string theory because there are no experimentally
verified phenomena going beyond GR a SM (plus masses of neutrinos) at all.
And henceforth *every* theory would suffer from this handicap.

> LQG, in contrast, proceeds from diffeomorphism invariance, which at
> all scales we have thus probed we do observe, and from assuming that
> we can quantize gravity without including the other forces. Note that
> we do not assume it is impossible to include other forces, just
> unnecessary.

This understanding of LQG cannot be more than just a simplified toy-model.
In the real world, such a model fails essentially for every individual
prediction. For instance, the Hawking radiation is a typical phenomenon
that a theory of quantum gravity must be able to describe. String theory
can do it. But the Hawking radiation is mostly made of photons, some
gravitons, sometimes electrons etc. - and therefore a theory that does not
contain the electromagnetism (such as your stupid version of LQG) gives
us, of course, completely wrong predictions concerning the Hawking
radiation in the real world.

Don't think that it is just a question of Hawking's radiation. There is no
phenomenon in the real world for which you can neglect the other forces
completely (only for astronomical applications it is fine to consider
gravity only, but these effects have been explained by classical GR).
There is no "decoupling limit". For example, the characteristic size of
the electromagnetic interaction at Planckian distances is even bigger than
at low energies and its coupling constant (unified with other forces) is
of order one (or 1/24). Therefore a prediction about any Planckian
phenomenon (a typical phenomenon of Quantum Gravity) will come out wrong
by an error of order 100%, if you neglect the other forces.

Not only that this primitive version of LQG gives inevitably wrong results
for anything in the real world. It is also unscientific, because even if
the disagreement of your theory with the observed data is very large, you
can always say that "it is because of the other forces that we neglected".
The simplified, gravitational-only version of LQG is irrelevant for the
real world, predicts nonsensical results or is completely non-scientific,
depending on your interpretation of the wrong predictions. In the best
case, it can be a toy model for some meaningful theory. In my opinion, it
is a wrong toy model.

The same holds for every theory that would like to ignore the other
forces. No such theory can be relevant for physics of the *real* world.

> In fact, as I mentioned in my last post, if LQG succeeds
> we will have a completely rigorous quantum theory of an interacting
> field in four dimensions---the first time that will have been
> achieved.

Yes, this is why LQG must fail, because there is no purely 4-dimensional
consistent quantum theory of gravity.

> Again, you seem to suppose that it has somehow been already
> established that a successful quantization of gravity must of
> necessity include a unification of gravity with other forces:

I hope that I have explained this fact - and what I mean by it - clearly
enough.

> I will of course not be satisfied that I found a theory of everything,
> but then I have stated all along that I am not looking for one. If
> LQG succeeds we will then return to the *incorporation* of other
> interactions---which may or may not involve their unification with
> gravity. And people have already started addressing this
> incorporation; see for example:

"Incorporation" is not a true scientific progress. A good theory must
_explain_ the other forces (and their properties) and not just add them by
hand.

> experimental input---requires some assumption. String theory is no
> exception: you require the assumption that spacetime has more
> dimensions and particles than we observe, and that a correct quantum

String theory is not based on the assumption that the spacetime has more
(large or small) dimensions. It is a *prediction* of string theory, if you
are able to understand it, because you can *derive* it from more
fundamental principles, including their number. It is not clear to me how
someone can misunderstand things so seriously that he cannot tell apart
"assumptions" and "predictions".

> description of gravity requires that it be unified with the other
> forces. None of these assumptions has experimental support. I know

I have explained the logical support why a theory that completely ignores
the other forces cannot be relevant for Planckian physics in the real
world at all.

> you like to view all of these things as predictions, but nonetheless
> they are logically assumptions because you cannot construct your
> theory without them. I could just as well claim diffeomorphism

It is like if you say that the possibility of an atomic bomb is not a
prediction of nuclear physics but rather an assumption because you cannot
create the nuclear theory without it. Or it is like if anyone says any
other silly comment of this sort. You are simply unable to distinguish
predictions and assumtions, truth and nonsense.

> invariance as a prediction of LQG---and my prediction *has*
> experimental support, at every scale for which we have any
> experimental result.

If you think that the tested general covariance at astronomical scales
implies that it makes sense at Planckian scales, you can also believe that
because you do not die if you jump from the 1st floor, you can also safely
jump from the 20th floor. Actually this experimental verification might be
useful for physics. ;-)

> Of course I say essentially the same thing as Rovelli---because his
> point is quite valid. You label his article "Holy Scripture" but this

His point is invalid just like yours - simply because they are the same -
and I hope that everyone else but you has already understood why.

> is just sophistry on your part: you use labels where you should instead
> provide arguments. In short, you are unable or unwilling to recognize
> that the unification of gravity with other forces is just a
> hypothesis, and you thereby become guilty of the very thing you claim

Because it is *not* just a hypothesis. It is a scientific requirement
for a satisfactory theory - and furthermore taking the other forces into
account is absolutely necessary for every theory at microscopic scales
that should be able to match the experiments.

> Another example where you confuse hypothesis with conclusion:
>

> Your analogy could only be appropriate if you already knew that
> diffeomorphism invariance was not an exact symmetry. But you do not

Of course that we know from string theory that diffeomorphism invariance
is not an exact symmetry at the ultrashort scales. More precisely, it
mixes with infinitely many symmetries of this sort, corresponding to
excited strings etc. Theoretical considerations *force you* to all those
conclusions and whoever has studied the questions intensely enough, knows
how it works and why it must work in this way. You want to reject all this
knowledge without having a single argument. String theory is a large
system of interconnected insights where things are derived unambiguously
from each other and which matches qualitatively everything that we know
from the real world. This is why we tak it more seriously than an isolated
and unsupported claim that a specific conclusion of string theory is
wrong.

> know that; you would have to know already that something like string
> theory is a correct theory of Nature, and we do not know that. Your
> argument is therefore circular.

It is not completely circular; if you view it as a mathematical reasoning,
it starts with the existence of gravity and quantum mechanics
(confirmed!), then derives that this imply stringy physics at short
distances - and then it derives the conclusions about the changes that
general covariance undergoes.

Otherwise, your line of arguments could be used to show that anything in
science is "circular". And in fact, many colleagues of yours use precisely
your strategy to "prove" that Darwin's evolution theory is circular,
unsupported, unscientifical and all things like that. Of course, a good
theory contains a lot of facts which support each other and you can travel
among them on loops. But this is a *good* property of a theory that makes
it look better than a collection of ideas where many things can be
modified.

> My analogy, in contrast, is that it is worth investigating whether or
> not diffeomorphism invariance is exact: SR (and other symmetries as

It has been investigated and it has been answered. Diffeomorphism
invariance - and classical geometry at all - breaks down at Planckian
distances.

> well, of course) show us that classical symmetries can carry over
> exactly to a quantum theory. We don't know whether or not this is
> true for diffeomorphism invariance; part of the goal of LQG is to find
> out.

Maybe you do not know. We do know. The goal of LQG is not to "find out".
LQG already assumes (incorrectly) that geometry works at Planckian
distances, too, and therefore it cannot be used to derive any correct
results.

> I have said all along that I expect there are many quantum theories of
> gravity, simply because of the evidence of 2+1 dimensional gravity,
> and the fact that it is a generic fact of life that many inequivalent
> quantum theories can have the same classical limit. You may not like
> that, but it cannot be avoided. The goal of loop quantum gravity is

I completely agree that different quantum theories can have identical
classical limits at low energies.

> to attempt the construction of one such theory, not because we know in
> advance that it is necessarily the right one or the only one, but just
> because it has not been done yet and therefore is an important step in
> our understanding of quantum gravity. And by construct I mean

Which quantum gravity you talk about? Why do you call it "quantum gravity"
if it is just one of "many" different proposals? In my usage of the word,
there is only *one* collection of correct answers to quantum gravity that
agree with the real world - and theoretically, it seems, there is only one
consistent quantum gravity at all.

> *rigorously* construct---I have explained above why this is important
> for any theory that would be the fundamental description of spacetime
> at even the smallest scales. String theory does not provide, as yet,

How it can be important if it is manifestly wrong? Just compute, via
simple scaling, the quantum fluctuations of the metric and you will find
out that at Planckian distances it is of order one and the notion of
smooth geometry certainly breaks down. In fact it breaks down in LQG, too.
LQG just assumes the standard equations, describing a smooth Einsteinian
geometry, but finally it derives that the Planckian geometry is not
smooth, it is described by spin networks, which in fact contradicts the
assumption. Spin networks are just a very simple toy model showing *that*
the concepts of smooth geometry break down at Planckian scales. But it is
not a well-justified one because it is in fact derived from the assumption
that the concepts of geometry make sense everywhere.

> That is my point, and the whole problem, as explained both in my
> earlier post and again above.

There is no problem, all important things related to the degree of
divergence of the asymptotic series have been understood.

> > Note that your "criticism" can be applied to any theory that should
> > reproduce physics of QFT in some limit. In fact it can be applied against
> > any scientific effort to understand anything in reality. :-)
>
> Huh? The criticism applies to any theory claiming to be fundamental,
> for the reasons I have outlined above. It does not apply to effective

OK, so if your criticism applies to any theory that should be fundamental,
I think that it would be more honest not to be paid by a department whose
role should be to understand fundamental physics, and not to try to
explain why is it impossible.

> I never said it should be finite dimensional. In LQG, or for that
> matter those QFTs that have been rigorously constructed, the space is
> of course infinite dimensional, and not single particle quantum
> mechanics.

The Hilbert space in string theory is unitarily equivalent, because all
separable infinite-dimensional Hilbert spaces are unitarily equivalent, as
you do not want to believe me either.

> > Your arguments are similar to some arguments of a
> > hypothetical self-confident Newtonian physicists who also comes to the
> > 21st century and argues that string theory is wrong because it cannot
> > answer his question what is the potential between two point-like particles
> > with well-defined positions and momenta. ;-)
>
> My arguments are not anything like that.

They are almost exactly like that. You say that you want the notions of
geometry to make sense at Planckian distances, and because string theory
clearly shows that the concepts of geometry do not make any sense at
subPlanckian distances, you think that it is a fault of string theory. But
it is your fault. The problem is that your point of view is too primitive.

> If you want more specific problems with your theory, fine. Note to
> begin with, however, that it is not my job or any one else's to prove
> that your theory is not well formulated; it is again your job to prove
> that it is.

OK, but before anyone shows that MQG is worse than LQG, I expect MQG to be
treated as an equivalent competitor of LQG. If some people think that
Ashtekar's theory must be better than Motl's theory, because Ashtekar is a
king, they can think so. I just think that this way of thinking and this
kind of arguments has nothing to do with science.

And I also strongly disagree with the legal approach "it is your job". It
is absolutely wrong. It is a job of every theoretical physicist to look at
*all* reasonable proposals and judge them completely honestly, without any
prejudices. I certainly do that and I never say that "it is job of Mr. XY"
to convince me.

If you envision physics as an arena for war, where people are fighting to
spread some "truth" that has been decided already at the beginning, and
they do not need to look at the arguments suggesting the opposite (because
"it is not their job"), I really think that science is not the right place
for you.

> In particular, if you wish to make it a rigorous theory
> of quantum gravity instead of hand-waving verbiage, then you should
> answer the following questions:
>
> (1) What is your classical configuration space? In LQG we take the
> space of connections modulo gauge transformations, A/G. You seem to

The full quantum theory does not have any priviliged classical
configuration space. The full M-theory has clearly very many classical
limits in which the "classical degrees of freedom" look completely
differently. Every theory or description that has a fixed classical
configuration space necessarily ignores S-dualities (that can mix "very
classical" with "very quantum") and therefore is not too attractive. The
final formulation of the full (string) theory will have to be quantum from
the beginning, but none knows how to formulate the theory in this way; I
assume that it exists, maybe it does not. Physics can be studied without
such a formulation, too.

> want to have surfaces correspond to some kind of functionals on this
> space; if it is A/G, how will you do this? The connections are one
> forms and therefore naturally integrated along one-dimensional
> objects; you will need a two form to integrate along two dimensional
> surfaces. Which two form will you use?

Your description is just perturbative. It is a good surprise if you were
able to figure out the existence of the B-field; this is the name of the
two-form gauge potential that couples to strings. It is, in a sense, the
antisymmetric part of the "generalized asymmetric" metric. B-field is a
massless field in string theory, just like graviton, dilaton, photon and
their superpartners. In unoriented string theories it goes away.

> What is the canonically conjugate variable?

The canonically dual momentum to B-field is of course its field strength,
the three-form H. In the QFT language, schematically,

[B_{ij},H_{0kl}] = i delta(x-y) . delta_{ik}delta_{jl} - antisym.

> What are the constraints in terms of thes variables? Are they
> polynomial? If not, how will you implement them in your quantum
> theory? Are they first-class? Does the constrain algebra close?

All this junk is automatically solved by string theory. You do not need to
do all those things by hand. String theory allows you to calculate the
S-matrix elements for the quanta of B-field (as well as gravitons etc.)
and all those requirements, physical constraints etc. are automatically
taken into account. String theory is not a kind of dirty job where you
must do all those things by hand, and if it does not work, you must modify
something. String theory is correct, already at the beginning and
guarantees a consistent S-matrix satisfying all the requirements.

> If it does not, what secondary constraint must you add? Are any of
> these second-class? If they are, are your contraints still polynomial
> (if they ever were) after you have solved for all second-class
> constraints?

Yes, this is a good example what is the difference between a theory
invented (and glued together) by Men and the theory used by God that
people just uncover. You can of course study the results of string theory
using the effective quantum field theory approach, where you must answer
all your questions. But in string theory, you do *not* need to do anything
of it.

In the QFT language, the constraints come from the gauge invariance delta
B = d Lambda, where Lambda is a one-form, it is a straighforward
generalization of the electromagnetism, but with 1 more index. So this
also answers whether I know 2nd class constraints etc. But once again, you
can follow the standard string procedures to derive S-matrix elements
(say: path integral over Riemann surfaces with vertex operators inserted)
that satisfy all the properties that you would otherwise have to calculate
by the complicated sequence of nonsense that you noted and that you must
do in LQG.

> (2) What is your quantum configuration space? For a system with
> infinitely many degrees of freedom we must enlarge beyond the
> classical space of smooth solutions. In LQG we use \bar{A/G}; a
> compact Hausdorff space for which we have three definitions, shown,
> nonperturbatively and rigorously, to be equivalent. What space will
> you use, and how will you define it? Do you have any measures on this
> space, so that you can construct a Hilbert space of square integrable
> functions on it to serve as your space of quantum states?

This question has nothing to do with string theory. You assume a lot of
completely incorrect things about the theory. There is no unique
"classical configuration space" in the full theory, and because you
assumed the opposite, you made a lot of incorrect conclusions.

> (3) What operators will you have? You have proposed an area operator,
> but given no justification for it. In LQG we derive our expression by

I have already explained many times that quantum gravity does not allow
for any local gauge invariant operators. The rest of this question is
therefore also completely irrelevant.

> You have previously labelled all of the analgous calculations in LQG
> to be "trivial", so I assume this will not take you too long.

I took me five minutes.

> > Because a gravitational theory is by a definition dynamics of geometry.
> > One of the classical notions describing a geometry is the
> > dimension.
>
> Again, you are picking and choosing what elements of classical
> geometry a quantum theory should explain based precisely on what it is
> string theory attempts to explain. I could just as well ask you to

No, string theory does not *attempt* to explain those things. All of these
things are automatically predicted.

> explain, from string theory, why spacetime should (classically) be a
> manifold. The only thing that must be explained by a quantum theory
> of geometry is the metric, because that is the field that is dynamical
> in GR. The dimensionality of spacetime is not.

You can just see how many completely incorrect assumptions you are making
- and how hopeless the search for the correct theory of quantum gravity
would be if people did not discover string theory (by an accident, in the
late 60s) that directs the research essentially uniquely. You assume that
geometry makes sense at Planckian distances. Wrong. You assume that
topology is fixed. Wrong. You assume that the dimension of spacetime is
four and can be kept as the universal truth. Wrong. You assume that
Einstein's equations work at Planckian distances without corrections.
Wrong. You assume that you can describe quantum gravity in the real world
if you ignore the other forces. Wrong.

You assume millions of things and almost all of them are wrong. In string
theory, we did not assume anything from that. We did not assume the
opposite either. We were forced to *derive* all those properties from a
completely rigid structure. People had to *derive* that the dimension of
the spacetime, including all dimensions, is 26 and then 10. At the end, we
are uncovering a structure that has all the necessary features to be a
consistent theory describing our world, but at the same moment shows that
many of the assumptions that one would like to do (sometimes) are just
incorrect prejudices that the true theory does not satisfy.

We did not want to incorporate gravity into string theory. String theory
was meant as the theory of strong interactions. People however derived
that it *had* to contain gravity, even though they wanted to avoid it at
the beginning. We did not want to assume that there are other forces.
String theory forced us to *derive* that there must be also other forces
and particles. We did not assume anything about the dimension, about the
dynamics of topology. All those things were *derived* without knowing the
answer already at the beginning. This is a completely different way of
doing science. We are not putting hundreds of arbitrary (and mostly
incorrect) assumptions in, so that we must wait for an experiment after
every individual step, without doing real predictions. We study an
objective structure that contains the answers to all those questions.

> That would be any quantum theory. You are making an enormous number of
> choices in simply choosing to work in on string theory. It is, at

We are not doing any choices at all. There is only one string theory and
it contains everything that people found in the thousands of papers
(except for the wrong ones haha).

> quantization of that theory. As I have emphasized repeatedly, it is a
> generic feature of quantization that classical theories have multiple,
> inequivalent quantizations. The *only* way to tell which of these is

But string theory is not a classical theory. There is only one correct
quantization of string theory. This is simply a fact. The theory has many
solutions, vacua etc., but it is one theory. And clearly, it is more
specific and concrete than you would like to.

> correct is to find some experimental prediction on which they
> disagree, and then go and do the experiment and see if any of them
> gave the right answer.

Yes, and the experimentalists will be hopefully able to do it in near
future.

Greetings
Lubos Motl

Lubos Motl

unread,
Jun 27, 2001, 10:02:54 PM6/27/01
to
I agree with essentially everything that Aaron wrote in this thread. But
let me add a few comments.

On Mon, 25 Jun 2001, Aaron Bergman wrote:

> It's worth noting that nothing interesting in high energy physics
> falls into this category. I don't think I'm being hyperbolic here, either.

Someone mentioned the CPT-theorem and the spin-statistics relation etc.
But the CPT theorem for QFTs was proven by Schwinger, Luders, Pauli and
others in the 1950s. Fifty years ago. Physics has changed a lot since
then. There were big hopes in the axiomatic field theories etc. But we
should already conclude that this direction of research was not
successful; it was too abstract and almost everything thas has happened in
high-energy theory was based on very specific Lagrangians. For example,
the program of bootstrap (self-consistent determination of the equations
of motion) was successful in the case of some special conformal field
theories in 2 dimensions only. The belief that a few criteria of
consistency determine the correct field theory uniquely has been ruled out
a long time ago. We can construct many asymptotically free and completely
consistent quantum field theories today. I simply agree with Aaron. BTW
concerning CPT-violation, while my personal guess is that it is an exact
symmetry of Nature (including string theory), Alan Kostelecky wrote about
50 papers that discuss a possible CPT violation and its stringy origin
etc.

http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+a+kostelecky+and+title+cpt
http://www.slac.stanford.edu/spi
res/find/hep/www?rawcmd=FIND+a+kostelecky+and+title+cpt

Someone also argued that the Coleman-Mandula theorem was inspired by the
axiomatic field theory etc. I do not want to argue about that because I
know very little about the history of this theorem. What I know much
better is that this theorem had a serious and very important loophole and
this loophole (called supersymmetry) was first understood (apart from some
Russian mathematical physicists) by Ramond who (successfully) tried to
incorporate fermions into string theory (1971).

> >1. *Define* your theory by its perturbation expansion.

This is really a little bit obsolete approach. One would be happier if the
critics of string theory upgraded their 15 years old software to something
more up-to-date. ;-) Especially because string theorists study the
nonperturbative phenomena most of the time, I would say. And this is the
case already since 1995 at least.

> >This is well and good---if your perturbation theory converges.
> I would think that some sort of Borel resummability would be OK.

Yes, this is an old paper by Gross and Periwal (1988) - by the way, today
these two guys have another paper. ;-) Fine that you mentioned it but
the work of Shenker that shows the (2g)! character of the divergence is
certainly also important. Do I understand incorrectly that there are many
different ways how to resum such diverging series - and these different
choices differ essentially in the way how you treat the nonperturbative
corrections? For example, in field theories, you must compute the
contribution of (a gas of) instantons. An instanton and an instanton can
destroy each other - and the distance how close you allow them to go to
each other must be correlated with the coefficients in the resummation
algorithm; this "number" determines the boundary between perturbative
excitations and a collection of nonperturbative instantons. Is the case of
string theory different?

> >I don't know anyone who thinks this likely for string theory, and
> >certainly no *proof* that it converges.

> Periwal and Gross have argued that it doesn't converge.

In this respect, Shenker is much more quantitative and clear, I think. And
his calculation of the volume of the moduli spaces of Riemann surfaces
also implies the coefficient of the order of (2genus)! in front of
g^{2genus}. If you try to sum the series, the minimal term is of the order
exp(-C/g) and this is where you should stop. Therefore he also got an idea
about the size of the nonperturbative effects. In field theory, the
instantons have exp(-C/g^2) effects. In string theory, you have a larger
contribution exp(-C/g) corresponding to D-branes which are lighter than
NS5-branes - a stringy counterpart of instantons and monopoles in field
theory whose mass goes like 1/g_closed^2. Of course, in terms of g_open =
sqrt(g_closed), string theory's divergences look just like in field theory
and the D-branes have tensions of the order 1/g_open^2.

> The stuff I've snipped is essentially a synopsis of all the
> things that go wrong in quantizing an interacting field theory.
> My personal prejudices are that the reason for all these things
> is that we have no idea what a quantum field theory is, but
> that's just me.

Maybe. ;-) Do you think that there is anything unclear in defining QCD,
for example?

> I'm not even convinced that a good definition for QFTs should
> include Hilbert spaces.

I am personally always happier if a theory does include it. ;-)

> Background independence is really pretty, but that doesn't mean
> it is necessarily correct. Maybe life isn't background
> independent. It's worth thinking about, don't you think?

Maybe that this is just an issue of terminology, but in what sense you
think that background independence can be incorrect? There are many issues
concerning background independence: a manifestly background independent
formulation (which we do not have, generally speaking) and a background
independent physics. I think that physics is clearly background
independent. My feeling is that a "background dependent physics" means
that the quantities in your theory computed for one background have no
relations of equivalence to the results computed with a different
background; so in fact, you should say that you have infinitely many
different theories. My understanding implies that there are no doubts that
*physics* of string theory is, in this sense, background independent. For
example, even if you keep your asymptotic conditions at infinity fixed,
you can create a large region in the Universe which looks like a Universe
with different asymptotic conditions, to an arbitrary accuracy.

And even if we left string theory in 2080 or so, a new theory would have
to have background independent dynamics in the same way as string theory
does, I think.

Kevin wrote:

> There is an issue here, which I have heard JB state several times, and

> I suspect is a relatively commonly held feeling in the LGQ camp, which
> is that for theories where we have experimental evidence we can afford
> to be a little sloppy. When we don't have experiments, we should be a

> little more cautious. It's not clear that the conventional wisdom of
> QFT is applicable to the quantization of extended objects. And we
> know from lower-dimensional QG theories that funny stuff can happen.

I would say that string theorists are approximately equally rigorous (or
equally sloppy) as the people who worked on the Standard Model before
W,Z-bosons were observed etc. Maybe string theorists are little bit more
mathematical and rigorous. But there are many similarities with history:
people computing loop diagrams in QED in the 50s really did not understand
what field theory was. String theorists also know the correct perturbative
(and now many nonperturbative) formulae, but they are still not satisfied
with their answers to the question "What is string theory?". The subject
of theoretical physics is larger than it was but the scientific logic has
not changed much. But what I want to say is that string theory has a lot
of things that have almost the same effect as experiments. People do a lot
of things that could be called "theoretical experiments". They do some
calculations. Sometimes they have a clue what the result should be,
sometimes they have no clue. But in the middle of their calculations they
cannot see any clear reason that would imply that the result would work.
There are thousands of things that can go wrong. If one constructs a
random theory with similar spacetime fields and objects as we can find in
string theory, there is 99.999999...% probability that something must go
wrong.

But the result is always consistent and furthermore either completely
correct or it gives a clear unexpected result that usually shows us
comprehensibly which assumption caused that we had incorrect expectations.
Or it gives us a new result. Even the people who perform relatively
mechanical calculations are finally able to get results that exhibit a
deep conceptual idea behind. An idea that even an original thinker could
have problems to invent without doing a calculation. Who could know, for
example, that a wrapped D3-brane can shrink to zero and such an object
that looks like a black hole melts into an elementary oscillating string,
while the topology of spacetime is changed and the Euler character jumps
by exactly the correct amount? Who could guess that a fivebrane can be
absorbed by a domain wall and melt into an instanton in the gauge field?
Who could have predicted mirror symmetry or T-dualities before the
mathematics was understood?

This is what people mean if they say that string theory is smarter than we
are.

Best wishes
Lubos

John Baez

unread,
Jun 28, 2001, 9:03:09 PM6/28/01
to
I think I'll jump into this discussion. I may make some mistakes
in my technical claims about string theory, since I'm not an expert
on that. But at least I'll learn something, so here goes....

In article <Pine.SOL.4.05.101062...@athena.phys.psu.edu>,
Josh Willis <jwi...@phys.psu.edu> wrote:

>Lubos Motl writes:

>> On the contrary, everything in string theory is completely dynamical.

It's really not clear to say that "everything" is completely
dynamical in some theory. Do even the equations themselves
jump up and dance around? Hopefully not. So, the question
becomes: what is regarded as fixed, and what is treated as a
variable whose values not only affect but are affected by other
variables? I call the former "background" structures and the
latter "dynamical" structures.

So: what are the background structures and what are the dynamical
structures in string theory? It would be a lot of fun to study
this question. But it's not so easy! First, we'd have to agree
on a precise definition of string theory.

This is tough, given the rapidly changing nature of the field. Do
we mean one of the 5 perturbative superstring theories in 10 dimensions?
Or do we mean "M-theory"? If the latter... what the heck is "M-theory",
exactly? Nobody really knows, though there are some interesting
conjectures. Certainly we can't point to some equations and get
lots of people to agree that "these are the fundamental equations of
M-theory". A lot of people believe that the 5 superstring theories
and 11d supergravity are all aspects of some deeper picture, but
nobody has pinned down this picture precisely enough to say
what the background structure and dynamical structures are.

>> Gravity in string theory is
>> dynamical, of course, this is the main dynamics that one can
>> compute.

Just to get specific about this question, let's say we're
talking about one of the 5 perturbative superstring theories
in 10 dimensions. Then gravity becomes "half-dynamical,
half-background", because one STARTS by fixing a solution of
the classical equations of supergravity, including metric g_0,
and THEN does string theory on this background, interpreting
some of the string degrees of freedom as a graviton field h.
One can attempt to define a Lorentzian metric

g = g_0 + h

where g_0 is the "background" part (a classical field)
and h is the "dynamical" part (a quantum field). This is
what I mean by "half-dynamical, half-background": we
START by fixing a background structure - a solution of the
classical supergravity equation - and THEN we are in the
position to get dynamics - strings romping around on this
fixed structure.

Of course, it's not really as nice as I'm pretending, because if
h is large there's no guarantee that g really is a Lorentzian metric!
You can't add a large symmetric tensor to a given Lorentzian
metric and get another Lorentzian metric. I have no idea if
any string theorists care about this issue. They might just say
that when h get large, perturbing about g_0 was dumb in the first
place. However, it's not so easy to restrict attention to "small"
quantum fields h.

>There are two obvious approaches to making the "metric" dynamical
>in string theory:
>
>(1) Explicitly include an integral over all possible spacetime metrics
>in your path integral.

>(2) Fix some particular background metric. You would then argue that


>as one considers the interactions of the theory you generate all other
>metrics dynamically.

>Of course, (1) is impossible, as far as we know.

Instead of saying it's "impossible", I'd prefer to say that we
haven't the foggiest clue as to how to do it, so most people have
given up, at least for now.

Perhaps more to the point, nobody even tries to write string theory as a
path integral involving a metric (on 10d spacetime).

>Instead, what is always done is (2). The argument that we do actually
>obtain all metrics then hinges on expressing an arbitrary metric as a sum
>of the background metric and a perturbation, and then "Taylor
>expanding" the action in this perturbation, noting that the first
>order term gives us the vertex operator for the graviton, which is
>essentially a creation operator for an "asymptotic" graviton state
>that we are now therefore adding to our interaction. So does this
>"Taylor expansion" converge?

I'm a bit confused about this. Are you talking about string
theory here? It sounds more like a description of perturbative
quantum gravity. In perturbative quantum gravity we can write
the Einstein action as

S(g) = S(g_0 + h)

and start Taylor-expanding stuff. But in string theory the action
is not a function of the spacetime metric g.

>> Yes, the Taylor series for exp(x) converges.

>but while this is true in general for a complex number or finite
>dimensional square matrix, it is very far from true for an arbitrary
>operator on an infinite dimensional Hilbert (or Banach) space.

That's true. But I'm not quite sure why you guys switched from a
discussion of path integrals to a discussion of the convergence
of exp(x). Perhaps I wasn't paying enough attention. Maybe I can
guess....

>Since you seem to have disdain for rigorous arguments, note that we
>can also see the same thing in ordinary quantum field theory. If it
>were true that "convergence" of the Taylor series for exp(x)
>guaranteed us that we could Taylor expand a quantity of the form
>exp[-\int (H0 + HI)] as exp[-\int H0] [ 1 - \int HI + ...], then it
>would have to be true that every perturbation series in QFT would
>converge, since it is precisely by this argument that these
>perturbative expansions are "derived". Note I don't mean simply that
>the perturbation series would be renormalizable; more than that, it
>would have to converge. But we know perfectly well that this is not
>true.

Okay, so it seems you're now discussing the Hamiltonian approach
to perturbation theory. I'm not sure this is relevant to how
gravity shows up in string theory, but this won't prevent me
from tossing in my 2 cents:

1) In ordinary quantum mechanics it's easy to find situations
where H0 and H1 are unbounded self-adjoint operators but H0 + H1 is not.
For example, when H0 is the free particle Hamiltonian and H1 is a
potential that goes to negative infinity rapidly as we go to spatial
infinity. In such situations it would be sheer folly to try to make
sense of exp(i(H0+H1)t)) as a unitary operator, much less expand it
perturbatively and hope to get a convergent series.

2) In quantum field theory, where H0 is a free field Hamiltonian
and H1 is an typical interaction Hamiltonian, the situation is
even worse. H0 is a self-adjoint operator on Fock space, but
in most cases H1 is not even a well-defined operator!!! This is
one way to see that one is begging for trouble in perturbative
quantum field theory.

>For Hilbert spaces, the
>meaningful question is whether or not they are unitarily equivalent;
>you must preserve not only the algebraic vector space structure but
>also the inner product structure in order to regard two Hilbert spaces
>as "the same". And a Hilbert space for string theory is certainly not
>going to be unitarily equivalent to Fock space.

Eh? All Hilbert spaces of the same dimension are unitarily
equivalent, and the only dimensions physicists are likely to see
are 0,1,2,3,... up to aleph_0, which is math jargon for the
cardinality of the set of natural numbers. The Hilbert space
for the harmonic oscillator, and all the Fock spaces that show
up in ordinary quantum field theory, have dimension aleph_0.
So if string theory has a well-defined Hilbert space, it's probably
the countable-dimensional one.

The more interesting question is to ask for an explicit description
of the Hilbert space for perturbative string theory on a fixed
(static) background metric. I could probably answer this myself
with a little work, but it'll be fun to see how many string theorists
can do it.



>Moreover, you don't
>prove the existence of a Hilbert space for your theory just by noting
>that it would be bad news for your theory if it doesn't have one.

Oh, really? Then a lot of physicists are in trouble. :-)

>Indeed as I noted in my earlier post, one can see at once (at least a
>relativist can) that you cannot get an arbitrary solution to
>Einstein's equation by superposing gravitons on Minkowski (or any
>other) space, simply because the underlying manifolds are not
>diffeomorphic.

That's certainly true if by "graviton" you mean something
like a solution of the linearized Einstein equations. I don't
think any string theorists claim you can pull a stunt like this.

>In response you have claimed several times that string
>theory has a known nonperturbative definition, but I have heard
>several prominent string theorists (Edward Witten, Michael Green, and
>Michael Duff) make exactly the opposite claim, and I have much more
>confidence in their assessment of that point than yours.

Well, if anyone out there DOES think string theory has a
clearly formulated nonperturbative definition, they should
point me to it!

Hmm, I have to go set the table - it's dinner time.

Greg Kuperberg

unread,
Jun 28, 2001, 11:47:08 PM6/28/01
to
>I've also talked to some string theorists who know something
>about lqg, and maybe I can communicate their objections.

I can believe, for purely external reasons, that the string theorists
are right in their intuition that loop quantum gravity is at best an
"elegant cul-de-sac". (John Baez suggested in the New York Times that
this is what string theory could be.) It would be interesting to see a
critical review in hep-th that explains this point of view.

In 1966 the topologist John Stallings wrote a paper with a famous title,
"How not to prove the Poincare conjecture". I think that "How not to
quantize gravity" would be an even better title for you or Lubos Motl
or another colleague.

(In fact Stallings was writing only about his own failed attempt
and not a review, see http://www.math.berkeley.edu/~stall/ .)
--
/\ Greg Kuperberg (UC Davis)
/ \
\ / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/
\/ * All the math that's fit to e-print *

Josh Willis

unread,
Jun 28, 2001, 11:49:41 PM6/28/01
to
In article <slrn9jkejr....@cardinal0.Stanford.EDU>,

aber...@princeton.edu (Aaron Bergman) writes:
>In article <Pine.SOL.4.05.101062...@athena.phys.psu.edu>, Josh
>Willis wrote:
>
>>In response you have claimed several times that string
>>theory has a known nonperturbative definition, but I have heard
>>several prominent string theorists (Edward Witten, Michael Green, and
>>Michael Duff) make exactly the opposite claim, and I have much more
>>confidence in their assessment of that point than yours.
>
>While this is a nice appeal to authority, the state of affairs
>is, as usual, more complicated. There exists a conjectured
>nonperturbative definition of string theory in spacetimes that
>are asymptotically AdS. There all exists a conjectured
>nonperturbative defintion of string theory "in the infinite
>momentum frame". Now, it may be that these conjectures only cover
>that small region of moduli space, but, then again, maybe they're
>part of something more general. The point is that, in those
>classes of backgrounds, we can make nonperturbative statements
>about string theory.

Well, my main point is the use of the word "conjecture": I feel Lubos has been
leaving it out of his posts at rather crucial junctures. I don't think there's
anything wrong per se with conjecture, and find it difficult to imagine in
progress in most scientific fields without conjecture. But Lubos seems to me on
a number of occasions to have argued that it is a problem that LQG has
conjectures of its own, and indeed it is perfectly valid for him to be
pessimistic about the outcome of any conjectures driving LQG research, while at
the same time treating what are in fact conjectures in string theory as
established facts, and thus that these somehow back up his belief in string
theory. And I don't have any problem with him or any one else believing in
string theory, as long as they are fair in their comparisons to other theories.

[Later on you write]

>>(2) There is also a deeper reason. We now have, thanks to the
>>renormalization group, some understanding of just why earlier QFT's,
>>based on renormalization, were "essentially right." As indicated in
>>the quote from Peskin and Schroeder in my earlier post, we now know
>>that any QFT has a fundamental length scale associated to it: the
>>scale at which the quantum nature of spacetime, whatever that turns
>>out to be, becomes important.
>
>So you don't think any QFT nonperturbatively exists? What about
>QCD? For that matter, what about a CFT? There's no length scale there.

[and also]

>
>[...snip to end...]
>
>In a lot of what I snipped, you seem to have expressed a fairly
>strong idea of what a quantum theory of gravity should be. The
>idea seems to be based on the formalisms that have so far
>completely failed to define any interesting QFT in >3 dimensions.
>Have you considered that these formalisms might be, well, wrong?

I must admit I'm puzzled by your remarks here, since I feel I've been arguing
fairly strongly against both of the viewpoints you ascribe to me. I don't think
I've ever said that I don't think QFTs nonperturbatively exist (indeed, I've
made reference a few times to the fact that we know they do exist in lower
dimensions). Rather, I was in the first point you reference explaining why
perturbative renormalizablility can be a useful guide at the energy scales of
effective field theories (because nonrenormalizable terms in the Lagrangian
become irrelevant), and yet not a useful guide at Planck length scales. I know
you have in an earlier post said that many string theorists (and others)
do not equate nonrenormalizability with the failure of a theory to be well
defined, but again, Lubos has said otherwise, and that is what I am arguing
against. I don't believe perturbative renormalizability, or indeed the
perturbative behavior of a theory such as string theory which is not, per se, a
field theory, is a reliable guide either for or against the belief that the
perturbative theory has an underlying, consistent nonperturbative definition[*].

As to your second point above, the only thing I feel I have expressed a belief
in is ensuring that the theory you are proposing for a quantum theory of gravity
is well defined. In LQG we try to do this by working from the outset with a
nonperturbative Hilbert space, self-adjoint operators, etc. Certainly neither
LQG nor, I would expect, any successful quantum theory of gravity can satisfy
any of the usual axioms of a QFT in constructive or axiomatic QFT, if only
because these schemes all make reference to a pre-existing causal structure of
spacetime induced by the Minkowski metric.

It is of course a possibility, which some have considered, that the structure of
quantum theory---as distinct from quantum field theory---may require some
modification in order to successfully incorporate gravity. LQG proceeds from
the assumption that it will not, but again, I think that as long as we are aware
of this assumption LQG gives us a good tool, in its attempted construction,
to analyze what features of quantum mechanics might need to be modified in order
to accomodate gravity. In other words, I think that if you pay careful
attention to rigor in the construction, you will be in a better position to
evaluate "what went wrong" if something does. It seems to me to be harder to do
this in string theory so long as one lacks a rigorous nonperturbative definition
of the theory, because I think it will be harder to ferret out if a difficulty
is only apparent and results from some lack of rigor, or fundamental and
points to a need to reevaluate the formulation of quantum mechanics.

Josh


Note:
[*] What I mean by this statement, somewhat more carefully: We know in lower
dimensions that they are theories that are nonrenormalizeable yet exist
nonperturbatively. Hence the blanket statement "a theory is not well defined if
it is nonrenormalizeable" cannot be true. A modification of it, for instance,
restricting to four dimensions, might be true as far as we know right now, but I
think we would need a more careful demonstration of this, and not just our own
inability to come up with such a theory thus far.

The converse statement would be something like "if a theory is renormalizable,
then it is nonperturbatively well defined". I don't think anything like this has
been proven, yet at the same time I don't know of any rigorously constructed
counter examples (though I admit I haven't looked very hard). But I do think it
is likely, on physical grounds, that such counterexamples exist. Consider phi^3
theory in 3 dimensions; this is superrenormalizable. Yet the classical
Hamiltonian is unbounded below, so it is difficult to see how a nonperturbative
quantum theory could have a stable ground state. Of course, this is not a proof.

I also hasten to add that I'm not implying that Aaron in specific or the string
theory community in general holds the belief that a theory is
consistent iff it is renormalizable, but I am trying to make clear why I think
the perturbative behavior of string theory, good though it may be, does not seem
to me to be a "selling point" for the theory. That may not be the only selling
point its adherents propose, but it certainly was the earliest and still gets
repeated. See, for example, p. 222 of Polchinski, vol II: "This [black hole
entropy] is a remarkable result, and another indication, *beyond perturbative
finiteness*, that string theory defines a sensible theory of quantum gravity"
(emphasis added).

John Baez

unread,
Jun 29, 2001, 2:58:00 AM6/29/01
to
In article <Pine.SOL.4.10.101062...@strings.rutgers.edu>,
Lubos Motl <mo...@physics.rutgers.edu> wrote:

>On Mon, 25 Jun 2001, zirkus wrote:

>> Perhaps two nice things about ST is that perturbative ST does not obey

>> the axioms of local QFT, and [...]

>Yes, that's fine, but there is also the opposite characteristics of string
>theory. The amplitudes computed from string theory seem to satisfy almost
>all the possible principles of local field theories, just like if we
>derived them from a local field theory. The S-matrix amplitudes have
>simple poles only etc. The violations of locality in string theory have a
>very subtle character (they are implied by the nonlocal character of the
>fundamental object - string - but the local dynamics of strings and their
>interactions is still "local at the worldsheet"). These nonlocalities are
>probably responsible for solving the Hawking's information loss paradox
>(in the case of black holes), but otherwise they do not damage the usual
>properties that allow string theory to agree with the logic of local
>quantum field theories.

So: am I right in guessing that only way perturbative string theory
does not obey the axioms of local QFT is that it's just perturbative???
Or did zirkus have something more interesting in mind?

Obviously string theory does not satisfy something like the Garding-
Wightman axioms, which are formulated in terms of "fields at a point"
like phi(x). But most serious work on axiomatic quantum field theory no
longer uses those axioms anyway - people use things like the Haag-Kastler
axioms, which are formulated in terms of "nets of C*-algebras" describing
the observables that can be measured inside various open subsets of
spacetime.

A while ago these axioms were generalized to handle perturbative quantum
field theories, basically by replacing numbers everywhere by formal power
series in a coupling constant. If I remember correctly, all the perturbative
QFTs we know and love have been shown to satisfy these generalized axioms.
So I wonder: can perturbative string theory also be shown to satisfy
these? Or does the fact that strings are extended objects somehow
prevent this? I don't see why it should.

I don't expect anyone here to be able to answer this question, since
the string theory community and the axiomatic QFT community seem to be a
loggerheads for some silly reason. If anyone knows the answer, it's
probably Detlev Buchholz, but he doesn't read this newsgroup.


John Baez

unread,
Jun 29, 2001, 3:45:30 AM6/29/01
to

>I have
>heard from several sources that Carlo Rovelli is probably the most
>militant person in the LQG community. And one is advised even not to play
>any card games with him. :-)

He is a very nice guy who let me stay in his apartment in Verona.
He has a charming accent and he loves to drink espresso and talk
about physics. He also likes poetry, philosophy, and music. Even
you would like him if you somehow managed to discuss something other
than "loop quantum gravity versus M-theory". Please don't demonize
the poor fellow.

>Concerning the Wilson loops, I think that something singular must happen
>if two Wilson loops cross. At the moment, when their linking number in
>three dimensions jumps. Usually this is associated with some UV
>divergences of their correlators etc. Well, Carlo Rovelli claims (see his
>review, for example) that LQG avoids all these problems known from the
>loop quantization of gauge theories, because there is general covariance
>underlying everything and it has a magic power to cure almost every
>problem (I do not quote him exactly).

Well, you certainly don't quote him exactly here. One certainly
should *worry* whether the singular behavior of crossing Wilson loops
in ordinary QFTs like Yang-Mills or Maxwell theory is an indication
that loop quantum gravity cannot work. I have spent plenty of time
worrying about this. However, there are lots of reasons to believe
this issue is not a real problem. The simplest one is that all we
need is a theory that approximates semiclassical quantum gravity at
large distance scales: we don't need a theory whose Wilson loop
expectation values have the exact same (nasty) ultraviolet behavior
that one would expect from perturbative calculations.

The issue of general covariance is important here, because in
loop quantum gravity can't use a background metric to measure
how badly (or not) things blow up at short distance scales, the
way one can in ordinary perturbative QFT or string theory. In
loop quantum gravity, the state itself provides the geometry of
space. This means one has to be very careful in comparing loop
quantum gravity to theories you may be more familiar with.

>But this is just a technicality. What is a more disappointing aspect of
>theories similar to LQG is that people are really inventing and
>constructing them. This is not what physicists should do.

I know... they should all be working on string theory.

>Physicists should be rather discovering something that objectively
>exists.

Oh, whoops. I must have gotten mixed up! Forget string theory:
they should all be doing experiments! :-)

>Either in
>the world of natural phenomena, or (now, when we have almost no
>experiments beyond the Standard Model) in the world of important
>mathematical (and physically relevant) concepts.

Oh, okay, so math counts as objective reality too. That's fine; as
a closet Platonist I agree with you here. But believe it or not,
loop quantum gravity involves a lot of important mathematical (and
physically relevant) concepts. I'm a mathematician who works on
loop quantum gravity. I should know!

You may not like loop quantum gravity as much as string theory, but
I think it's odd that you'd find it "disappointing" that a small
number of people wish to study alternatives to string theory.
If string theory is wrong or even less than perfect, it may come
in handy to have some people with different ideas thinking about
the same problems from a different angle. Even if string theory is
perfect in every respect, the wasted energy will really not have
been so great. Besides, you'll get to gloat!

Let's face it: if loop quantum gravity did not exist, you would
need to invent it, just to have something to argue about on s.p.r..


John Baez

unread,
Jun 29, 2001, 10:38:58 PM6/29/01
to
Aaron Bergman recently commented on a post of Josh Willis. I'll
now add some further comments, starting with a bunch of basic stuff
that both Bergman and Willis undoubtedly know. I hope this will
help nonexperts follow the discussion, while not annoying the
experts too much with my plodding pedantry, peppered with
oversimplifications (some made for the sake of smooth exposition,
others out of sheer stupidity).

I guess Aaron is saying this because none of the quantum field
theories on which high energy physics is founded have been rigorously
shown to be consistent, except

1) perturbatively

or

2) with ultraviolet and/or infrared cutoffs

In case 1), the answers to physical questions are expressed as
infinite series, but we are unable to prove that the sums of
these series exist in any reasonable sense, so we typically add
up a few of the first terms, cross our fingers, and hope we've
gotten something close to the "right answer".

In case 2) we truncate the theory we're really interested in
by ignoring all effects that occur beneath some short distance
scale and/or above some long distance scale. A good example is
lattice gauge theory, where we replace 4d spacetime by a hypercubic
lattice consisting of finitely many points. We are unable to
determine how much this truncation affects the answers to the
questions we're interested in, so we again cross our fingers and
hope we've gotten something close to the "right answer".

I put "right answer" in quotes because the whole point is that
we are doing an approximate calculation of a mathematical quantity
that may not even exist. The only justifications for this maneuver
are that:

a) we don't know anything better to do and

b) the results we get are often in agreement with experiment.

In short, there is an excellent pragmatic justification for
this strategy, but the mathematically minded among us are
bound to be unsatisfied.

Now let me list a couple of ways in which Aaron was exaggerating
when he said "nothing interesting" about high energy physics
falls into the category of rigorous mathematics.

First of all, there are interesting rigorous results about both
perturbative quantum field theory and quantum field theory with
cutoffs, e.g. lattice gauge theory. Some random examples include
the proof of the renormalizability of Yang-Mills theory, and Dirk
Kreimer's recent work on renormalization theory and the Hopf algebra
of trees. The former had a great impact on practical particle
physics; the latter has not yet, but may in time.

Secondly, there are lots of interesting results in axiomatic quantum
field theory: the CPT theorem, the spin-statistics theorem, and many
subtler and less well-known results. Of course, the force of these
results is greatly weakened by the fact that none of the quantum
field theories used in high-energy physics have yet been shown to
satisfy the axioms used in these theorems! Rigorous results that
don't actually apply to any examples are not interesting. Nonetheless,
axiomatic quantum field theory has done a great job of clarifying our
ideas about what these theories are like if they indeed exist.

>> 1. That String/M-theory is in fact a quantum theory in the first
>>place. What is the Hilbert Space? If it is L_2(X,d\mu) for some space
>>X and measure \mu, what are they? What are the operators corresponding
>>to observables? Can you prove they are self-adjoint? What are their
>>spectra?

>I'm not even convinced that a good definition for QFTs should
>include Hilbert spaces.

I'm not exactly sure which doubts Aaron is alluding to here, but
it's perhaps mildly relevant to note that the modern approach to
axiomatic quantum field theory radically de-emphasizes the role
of Hilbert spaces.

More generally, I agree with Aaron's point that quantum field
theory is still a mysterious thing, and that we should not jump
to conclusions about what our ultimate understandings of it will
be. This is already true for traditional quantum field theory
on Minkowski spacetime (e.g. the Standard Model). But it's
infinitely more true about theories of quantum gravity (e.g.
string theory or loop quantum gravity), since these haven't
even come close to achieving their final form.

For example, I think I could describe the Fock-like Hilbert space
for perturbative string theory on a fixed 10d manifold M with static
background fields. But no string theorist these days claims that
this is what string theory is ultimately all about. Instead, string
theory is now supposed to be some mysterious unified thing from which
all these perturbative 10d string theories for different M stick up like
tips of an enormous iceberg... along with theories of various other-
dimensional supermembranes in various other-dimensional spacetimes.

There's no terribly good reason to think this mysterious unified thing
is going to involve a single Hilbert space of states. Already for
simpler theories, like topological quantum field theories, we've had
to wave goodbye to that notion.



>> 2. That GR is in fact the semiclassical limit of SMT. To do this,
>>you should prove that for any classical solution of Einstein's
>>equation, there is a corresponding state in your Hilbert space that
>>has, for all of the relevant observables, minimum uncertainty
>>distributed about the correct, corresponding values of the classical
>>solution. You should prove some kind of Ehrenfest properties. This
>>is what is done in ordinary QM; this is what people are now attempting
>>to do in LQG. It has not been done even for ordinary QFT in an
>>interacting theory.

>Which is why you're asking an absurd amount. What has been done
>in string theory is essentially what has been done in interacting
>QFT.

I agree with Aaron here, but note that people in loop quantum gravity
are asking this "absurd amount" of themselves and doing their damnedest
to come up with it.

>>The arguments you cite on this second point don't come close to doing
>>this. I don't have a copy of Green, Schwarz, and Witten, but let's
>>look at Section 3.7 of Polchinski, page 108. There he argues, as you
>>have said, that string theory has GR as a classical limit because one
>>can treat strings moving in a curved background.

>This is not the entirety of the argument.

>>The immediate
>>objection to using this as an argument that GR is the limit of string
>>theory is that it only treats string theory interacting with GR; it
>>does not show that GR in fact arises from string theory.

>Not at all. String theory contains a transverse traceless
>spin-two particle. This is just the perturbations in the metric.
>The (perturbative) dynamics of GR arise from the string theory.
>Consistency conditions for the theory require that the background
>metric satisfies Einstein's equations. This is the best you're
>going to get from a perturbative theory.

I agree with Aaron here - that's the best you're gonna get.

I have repeatedly pointed out my dissatisfaction with the
perturbative approach to string theory. Back in the 1980s,
when I first learned a bit about this stuff, I got really
annoyed about how little attention was being given to the
limitations of this approach. String theorists were in a
triumphant mood at the time, and people like Hawking were
saying that physics might be completely wrapped up by the
end of the century. This seemed crazy to me, given that
no-one yet had a clue as to the correct background-free
formulation of string theory (or any other quantum theory
including gravity). As a righteous young fellow, I decided
it was my duty to point out these problems to everyone in
the universe.

Now that string theorists are acutely aware of this issue,
it seems a bit pointless to keep beating them over the head
about it - especially when there are so many other worse
problems with string theory! Indeed, it was Aaron who helped
convince me of this a couple of years ago.

However, for the sake of *nonexperts*, I think it's still worth
emphasizing that a background-free formulation of a theory
including quantum mechanics and gravity remains one of the
unattained "holy grails" of physics.

>>Moreover, one must face a deep conceptual problem: how do I obtain
>>in string theory a state, from a coherent state of gravitons, that
>>describes some classical solution to Einsteins equation that is not
>>diffeomorphic to Minkowski space (if I choose Minkowski space as the
>>manifold that I perturb around)? At each order of perturbation theory
>>I am superposing strings on a flat background, and so at each order of
>>perturbation theory I have a solution that is diffeomorphic to R^4.
>
>You can (theoretically) do perturbation theory around any
>background that satisfies Einstein's equations.

Yep. But I suspect that string theory will be nonunitary
unless this background is static. After all, this problem
afflicts even free quantum field theories on curved spacetime,
and I don't know why string theories should be better behaved
in this respect. If someone has shown otherwise, I'd like
to know - this is the sort of thing Charles Torre may have
thought about.

>No one's denying that the thing we have the best control over in

>string theory is the perturbation expansion.

Okay, we have that in print now. :-)

>>Moreover, you say, as does Polchinski, that conformal invariance
>>requires for consistency Einstein's equation. Yet all he derives, on
>>pages 111--112, is the *linearized* Einstein equation.

>As I remember it, the beta functions contain the Ricci scalar,
>and are identified with the effective spacetime action, so
>there's no linearization involved. I don't have Polchinski on me,
>so I can't check the pages you reference to see if you're
>referring to the correct pages.

I don't know about Polchinski's book, but I think Aaron is
right: conformal invariance of string theory imposes the
condition that the background fields satisfy the same equations
as obtained by extremizing a slightly souped-up version of
the Einstein-Hilbert action, together with higher-order stringy
correction terms.

(By "slightly souped-up", I'm referring to stuff like the
Kalb-Ramond field.)

>The main point I want to make now is that you seem to be
>requiring of string theory things that haven't even been
>accomplished for the simplest interacting QFTs. I think it's
>a bit unfair.

How unfair it is depends on how the tone of the discussion.
In converation with someone like yourself, I feel no great
urge to lambast string theory for its defects. But when
Lubos Motl goes around proclaiming that string theory "smells
of god" and it's tough to resist pointing out the more unpleasant
odors that still surround certain corners of the subject.

>And now, since editorializing seems to be the name of the game, I
>suppose I shall engage in some of it myself.

Oh good. Now I can stop being polite and start throwing spitballs
at you. :-)

>First of all, all
>too many lqg people seem to argue with string theory as if it had
>not changed in the last 15 years.

The main problem here is that some loop quantum gravity people
haven't had time to keep up with string theory. They may learn
a bit from Green, Schwarz and Witten, and a bit from second-hand
reports by their string-unsympathetic colleagues. Clearly this is
not enough. Really keeping up with string theory is a fulltime job.
It's far from my first priority, but I try these days to be about 5
years out of date, rather than 15, and I urge all other workers in
loop quantum gravity to do at least this much. Having arguments on
sci.physics.research is a good way to keep in shape.

I would also urge more outreach on the part of string theorists -
not just to those irritating loop quantum gravity heretics, who
are doomed anyway, but also to the vast unwashed masses of physicists
who would really like to understand a bit about strings.

>Worse, there is a lot of
>complete mischaracterization of the people who work in the field.
>For example, the idea of defining the theory through its
>perturbation theory died out years ago if it ever even existed.

I associate this idea more with traditional quantum field theory
than with string theory. It's a standard way for lecturers to get
students to shut up and stop asking annoying questions about what
path integrals really mean.

>Or, take the intimation that string theorists don't care about
>background independence or even understand "the lessons of GR"
>(which sounds frightfully condescending, I might add). I hear
>string theorists talk about background independence all the time.

Us old fogies remember the bad old days before this was true,
and are glad to see them gone.

>One thing that is worth noting, however, is that its lessons might be
>wrong. Background independence is really pretty, but that doesn't mean
>it is necessarily correct. Maybe life isn't background
>independent. It's worth thinking about, don't you think?

Yes, but if someone presents me with a theory that is not
background-independent, I can torture it with lots of questions
and make it jump through lots of hoops. So far I've never
seen a theory of this sort that didn't have big bad problems.

>I've also talked to some string theorists who know something
>about lqg, and maybe I can communicate their objections. To many
>people, lqg feels like constructing something. You decide what
>your variables should be and then you go about making a Hilbert
>space. Well, we need an inner product, so lets do some
>complicated math and define that. Etc. etc.

Interesting. You're complaining that loop quantum gravity is too
complicated, while Lubos Motl is complaining that it is too simple!
I think I'll pretend I'm in a kung fu movie and two guys are rushing
at me with drawn swords from opposite directions. I'll step out of the
way.

>It all feels very artifical.

No, no - it smells of god, honest!

>Many people would rather feel that one is discovering
>things rather than creating them. Whatever the Hilbert space may
>be, it should be something natural rather than something someone
>has to go through extensive work on in order to define.

I'll enjoy watching you define the Hilbert space of heterotic
string theory without extensive work. :-)

Seriously....

I know the feeling that one is discovering rather than creating
things. This is the usual feeling of doing mathematics. I have
no doubt that string theory is wonderful mathematics. The big
question is whether it is the right way to describe our universe.
The mathematical elegance of string theory does not settle this
question.

Crudely speaking, string theory describes a meta-universe of all
possible consistent universes in which quantized supersymmetric
membranes move around in background fields. This is a very pretty,
very roomy meta-universe with a compelling internal logic of its own.
There is, unfortunately, no experimental evidence that our actual
world lies somewhere in this meta-universe. Even if it did, we
have no way, currently, of finding where in this meta-universe our
actual world lies.

So: the old approach of building theories to fit the experimental
data may still have some life left in it. Experiments show us
gravity and other forces; they show us quantum mechanics; it is
not such a bad idea to try to build a theory that includes these
things that we see - and not lots of things we DON'T see, like extra
dimensions and supersymmetry.

>Apparently, quantization of gauge theories in terms of loops has
>been tried in the past and there are known (but not known to me)
>issues in this. One person who I talked to saw a talk and said
>that this issue (maybe something to do with what happens when
>Wilson lines cross?) led Gross and Witten to crucify the speaker.
>But I know next to nothing about lqg, so I don't know the
>details.

I can't argue against this objection, since I don't know what
it is. I'll note, however, that lattice gauge theory is a
form of loop quantization (since the basic variables are
holonomies along edges), and it seems to do a good job of dealing
with the gauge theories that show up in the Standard Model. So,
there is no a priori reason why loop quantization of gauge
theories is a bad idea.

Indeed, the only really workable approach to nonperturbative QCD
is lattice gauge theory. Of course lattice gauge theory has a
built-in ultraviolet cutoff... but so does loop quantum gravity,
in a sense!


Whew... what a long post.


John Baez

unread,
Jun 29, 2001, 11:29:46 PM6/29/01
to

>I agree with essentially everything that Aaron wrote in this thread. But
>let me add a few comments.

>On Mon, 25 Jun 2001, Aaron Bergman wrote:

>> It's worth noting that nothing interesting in high energy physics
>> falls into this category. I don't think I'm being hyperbolic here, either.

>Someone mentioned the CPT-theorem and the spin-statistics relation etc.
>But the CPT theorem for QFTs was proven by Schwinger, Luders, Pauli and
>others in the 1950s. Fifty years ago.

They gave some rough-and-ready arguments that work for specific models.
The current proofs rely only on very general principles. They also show,
for example, why the spin-statistics theorem needs to be modified in
2d and 3d spacetime - where exotic statistics are possible.

>Physics has changed a lot since
>then. There were big hopes in the axiomatic field theories etc. But we
>should already conclude that this direction of research was not
>successful; it was too abstract and almost everything thas has happened in
>high-energy theory was based on very specific Lagrangians.

I don't see why you say it was unsuccessful. It has been very
successful in doing what it set out to do - which is not to compute
scattering amplitudes or discover the unique consistent Theory of
Everything, but to study the logical structure of quantum field theory.
You may not be interested in this, but it's a fascinating subject
when you get into it. At least, that's what I've found. Of course
I never complain that anything is 'too abstract' - for a mathematician,
this is like complaining that the ocean is 'too wet'.

Urs Schreiber

unread,
Jun 30, 2001, 10:30:34 AM6/30/01
to
John Baez wrote (regarding loop quantum gravity):

> So: the old approach of building theories to fit the experimental
> data may still have some life left in it. Experiments show us
> gravity and other forces; they show us quantum mechanics; it is
> not such a bad idea to try to build a theory that includes these
> things that we see - and not lots of things we DON'T see, like extra
> dimensions and supersymmetry.

LQG does not require supersymmetry. But can it nevertheless be compatible
with susy? Could both apply to nature? Can one apply loop quantization to
supergravity? (Or is this not a sensible question to ask?)

Aaron Bergman

unread,
Jun 28, 2001, 9:29:59 PM6/28/01
to
In article <Pine.SOL.4.10.101062...@strings.rutgers.edu>,
Lubos Motl wrote:

>And even in the
>hypothetical case in which Nature did not use string theory but something
>worse, string theory will dominate *mathematics* of the 3rd millenium, as
>Witten said.

I figured out why this is true the other day. It seems obvious in
retrospect. It's quite simple:

Witten is a time traveler from the future.

He's also said that string theory is a bit of 21st century physics
that fell into the 20th century. How does he know that? I think
the answer is clear.

A couple of us are planning on breaking into his office and
stealing the 2007 folder. That stuff ought to lead to a good
postdoc.

Hunh. Maybe I shouldn't be posting all my plans online. Forget you
read that last paragraph.

Aaron Bergman

unread,
Jun 29, 2001, 11:16:56 PM6/29/01
to
In article <9hje42$s1k$1...@glue.ucr.edu>, John Baez wrote:

>In article <slrn9j5n44....@cardinal0.Stanford.EDU>,
>Aaron Bergman <aber...@princeton.edu> wrote:

>>I would argue that essentially all the progress in high energy
>>physics in the last forty years or so has been in this category.

[This category essentially meaning "nonrigorous".]

>I guess Aaron is saying this because none of the quantum field
>theories on which high energy physics is founded have been rigorously
>shown to be consistent, except
>
>1) perturbatively
>or
>2) with ultraviolet and/or infrared cutoffs

Just to be a bit more provocative, I'd go so far as to say that
we really haven't learned much physics at all from the rigorous
pursuit of field theory.

I do want to note that I don't think this means that the rigorous
pursuit of QFT isn't a worthwhile endeavor. In fact, I think it's
an incredibly worthwhile endeavor. I just think that we're going
about it all wrong, but that's really just the gut instinct of a
semi-informed graduate student. I also think that when we do
understand what quantum field theory is, a lot of things will
become more clear. I'm just not sure what those things are yet.

>In case 1), the answers to physical questions are expressed as
>infinite series, but we are unable to prove that the sums of
>these series exist in any reasonable sense, so we typically add
>up a few of the first terms, cross our fingers, and hope we've
>gotten something close to the "right answer".

As you're well aware, there are all sorts of things we know to be
present in field theory, like RG flow and instantons to name the
obvious examples, that will never show up in perturbation
theory.

>In case 2) we truncate the theory we're really interested in
>by ignoring all effects that occur beneath some short distance
>scale and/or above some long distance scale. A good example is
>lattice gauge theory, where we replace 4d spacetime by a hypercubic
>lattice consisting of finitely many points. We are unable to
>determine how much this truncation affects the answers to the
>questions we're interested in, so we again cross our fingers and
>hope we've gotten something close to the "right answer".
>
>I put "right answer" in quotes because the whole point is that
>we are doing an approximate calculation of a mathematical quantity
>that may not even exist. The only justifications for this maneuver
>are that:
>
>a) we don't know anything better to do and
>b) the results we get are often in agreement with experiment.

Which is really the rub of the whole thing. It just works so
well.

>In short, there is an excellent pragmatic justification for
>this strategy, but the mathematically minded among us are
>bound to be unsatisfied.
>
>Now let me list a couple of ways in which Aaron was exaggerating
>when he said "nothing interesting" about high energy physics
>falls into the category of rigorous mathematics.

That's not really what I said. All sorts of rigorous mathematics
can show up in physics. You can even give good rigorous
definitions of the path integral in some cases. I'm just saying
that the axiomatic and/or constructive field theory endeavor just
hasn't led too much. Again, by this, I'm not saying that it isn't
worthwhile.

But you are right. I was exaggerating. Unless you interpret my
statement narrowly, the Coleman-Mandula theorem is a very good
example (that Kevin mentioned) of some useful and interesting
quasi-rigorous results iN QFT.

>>> 1. That String/M-theory is in fact a quantum theory in the first
>>>place. What is the Hilbert Space? If it is L_2(X,d\mu) for some space
>>>X and measure \mu, what are they? What are the operators corresponding
>>>to observables? Can you prove they are self-adjoint? What are their
>>>spectra?

>>I'm not even convinced that a good definition for QFTs should
>>include Hilbert spaces.

>I'm not exactly sure which doubts Aaron is alluding to here, but
>it's perhaps mildly relevant to note that the modern approach to
>axiomatic quantum field theory radically de-emphasizes the role
>of Hilbert spaces.

I'm just trying to be a bit provocative. One thing that bugs me
about QFT is that it's often just presented as relativistic
quantum mechanics in a new guise. I don't really think that's
true.

Of course, this is really far out on a limb. CFTs, for example,
have nice Hilbert spaces.

[...snip stuff I agree with...]


>
>>Which is why you're asking an absurd amount. What has been done
>>in string theory is essentially what has been done in interacting
>>QFT.
>
>I agree with Aaron here, but note that people in loop quantum gravity
>are asking this "absurd amount" of themselves and doing their damnedest
>to come up with it.

I happen to think that it's a good thing that there are people
working on something other that string theory. String theory
could be wrong after all. ...pauses for shocked exclamations... I
even wish, as I've stated elsewhere, that there was more
communication between the "camps".

[...]


>
>Now that string theorists are acutely aware of this issue,
>it seems a bit pointless to keep beating them over the head
>about it - especially when there are so many other worse
>problems with string theory! Indeed, it was Aaron who helped
>convince me of this a couple of years ago.

Er, thanx, I think.

[...]

>Yep. But I suspect that string theory will be nonunitary
>unless this background is static. After all, this problem
>afflicts even free quantum field theories on curved spacetime,
>and I don't know why string theories should be better behaved
>in this respect. If someone has shown otherwise, I'd like
>to know - this is the sort of thing Charles Torre may have
>thought about.

String theory is, in a sense, manifestly unitary and I don't see
any reason off the top of my head why the target space of the CFT
would need a global timelike killing vector. Unfortunately,
though, I haven't spent too much time learning about the sigma
model approach, so maybe Lubos or someone else can give a better
answer.

[...]

>>And now, since editorializing seems to be the name of the game, I
>>suppose I shall engage in some of it myself.
>
>Oh good. Now I can stop being polite and start throwing spitballs
>at you. :-)
>
>>First of all, all
>>too many lqg people seem to argue with string theory as if it had
>>not changed in the last 15 years.
>
>The main problem here is that some loop quantum gravity people
>haven't had time to keep up with string theory. They may learn
>a bit from Green, Schwarz and Witten, and a bit from second-hand
>reports by their string-unsympathetic colleagues. Clearly this is
>not enough. Really keeping up with string theory is a fulltime job.

You don't have to tell me that. But we're having a slow few
months now. All the time you need, right? :)

>It's far from my first priority, but I try these days to be about 5
>years out of date, rather than 15, and I urge all other workers in
>loop quantum gravity to do at least this much. Having arguments on
>sci.physics.research is a good way to keep in shape.
>
>I would also urge more outreach on the part of string theorists -
>not just to those irritating loop quantum gravity heretics, who
>are doomed anyway, but also to the vast unwashed masses of physicists
>who would really like to understand a bit about strings.

I don't know if string theory needs more press that it already
has. I would agree that a more circumspect tone is warranted
quite often, though. One can bring up black hole entropy a lot
(and god knows I do), but string theory still hasn't predicted
anything that anyone has seen yet.

[...]


>
>>I've also talked to some string theorists who know something
>>about lqg, and maybe I can communicate their objections. To many
>>people, lqg feels like constructing something. You decide what
>>your variables should be and then you go about making a Hilbert
>>space. Well, we need an inner product, so lets do some
>>complicated math and define that. Etc. etc.
>
>Interesting. You're complaining that loop quantum gravity is too
>complicated, while Lubos Motl is complaining that it is too simple!
>I think I'll pretend I'm in a kung fu movie and two guys are rushing
>at me with drawn swords from opposite directions. I'll step out of the
>way.

Maybe the point can be made more clear this way. In the process
of workin on string theory, Polchinski discovered that there were
these dynamic objects that carried RR charge. Certainly, no one
put these in when they were trying to explain the strong force.
Thus, in a sense, they were discovered. Or, take the web if
dualities -- these weren't there in the beginning either. Thus,
there's really a sense of discovery rather than creation if that
makes any sense. A lot of string theorists are suspicious when
people start putting things into their theories rather than taking
things out. Whether or not you agree with that, the feeling is
there. This, I think, is what Lubos means when he says that
string theory smells of god rather than of man.

[snip more stuff I agree with essentially]

Josh Willis

unread,
Jun 29, 2001, 7:04:33 PM6/29/01
to
In article <9hgk4d$2cn$1...@glue.ucr.edu>,
ba...@galaxy.ucr.edu (John Baez) writes:

[stuff deleted]

>Josh Willis, though forgetting to cite himself here, was secretly
>the one who wrote:

>>There are two obvious approaches to making the "metric" dynamical
>>in string theory:
>>
>>(1) Explicitly include an integral over all possible spacetime metrics
>>in your path integral.
>>
>>(2) Fix some particular background metric. You would then argue that
>>as one considers the interactions of the theory you generate all other
>>metrics dynamically.
>>
>>Of course, (1) is impossible, as far as we know.

>Instead of saying it's "impossible", I'd prefer to say that we
>haven't the foggiest clue as to how to do it, so most people have
>given up, at least for now.

Well, I thought the "as far as we know" was sufficient caveat; I
usually take that to mean "the preceding could be entirely wrong if we
learn otherwise." :)

>Perhaps more to the point, nobody even tries to write string theory as a
>path integral involving a metric (on 10d spacetime).

Again, I think that's exactly what I say next...

>>Instead, what is always done is (2). The argument that we do actually
>>obtain all metrics then hinges on expressing an arbitrary metric as a sum
>>of the background metric and a perturbation, and then "Taylor
>>expanding" the action in this perturbation, noting that the first
>>order term gives us the vertex operator for the graviton, which is
>>essentially a creation operator for an "asymptotic" graviton state
>>that we are now therefore adding to our interaction. So does this
>>"Taylor expansion" converge?

>I'm a bit confused about this. Are you talking about string
>theory here? It sounds more like a description of perturbative
>quantum gravity. In perturbative quantum gravity we can write
>the Einstein action as
>
>S(g) = S(g_0 + h)
>
>and start Taylor-expanding stuff. But in string theory the action
>is not a function of the spacetime metric g.

Okay, so let me try to sum up. We were starting from the description
on page 108 of Polchinski, vol I, of string theory in a curved
spacetime. There, he starts by replacing the Polyakov action (please
excuse the ascii math):


1 / 2 1/2 ab u v
S_P = ------- | d \sigma g g n (X) d X d X
4 \pi a' / M uv a b

with the action:

1 / 2 1/2 ab u v
S_sigma = ------- | d \sigma g g G (X) d X d X .
4 \pi a' / M uv a b

In these expressions, n is "eta"; i.e., the Minkowski metric, and G is
some arbitrary (for now) curved spacetime metric. The metric g is the
worldsheet metric and the indices a and b run from 0 to 1; the indices
u and v run from 0 to D-1, where D is the dimension of spacetime. In
other words, the only change we've made in going from S_P to S_sigma
is to change the spacetime metric from the Minkowski metric to
something arbitrary. One's first reaction, as Polchinski notes, is
that this "doesn't seem stringy enough". We expect gravity to be
generated dynamically by string theory, and it hardly seems that we do
this by putting a curved spacetime metric in the Polyakov action. The
argument that we do comes from noting that in the path integral, what
we are interested in is exp[-S_sigma]. So now suppose that we can
write our spacetime metric as some small perturbation h about
Minkowski space; i.e., that:

G = n + h
uv uv uv

We now argue that we can expand the exponential of S_sigma as:

/ 1 / 2 1/2 ab u v \
(1) exp(-S_sigma) = exp(-S_P) | 1 - ------- | d \sigma g g h (X) d X d X + ... |
\ 4 \pi a' / M uv a b /

Then one says that the order h term is precisely an integral over all
ways of inserting on the string worldsheet a vertex operator for a
graviton, where that operator is identified through:

ik*X
h (X) = -4\pi g e s
uv c uv

The vertex operator creates an asymptotic state of some particular
mode for a string; ours is a graviton because s_uv is symmetric, since
h_uv is symmetric.

So this is how we came to be discussing the convergence of exp(x).
The basic argument is that this Taylor expansion shows that our curved
metric G can be regarded as the flat metric n with a superposition of
gravitons (obviously this argument generalizes to expansions about
metrics other than the Minkowski metric). What one does next is show
that in string theory, this metric G_uv cannot be arbitrary, but must
satisfy some modified version of the Einstein equations (modified in
part because of supersymmetry, but I'm trying not to muddy the waters
too much with that at the moment. It does, however, mean that we need
to add some other terms to the action S_sigma above). The claim is
therefore that this consistency requirement (that G_uv satisfy
something like the Einstein equations) coupled with the expansion of
exp(-S_sigma) in terms of exp(-S_P) and graviton vertex operators
means that we have established that GR is a low energy limit of string
theory, and gravity is dynamical in GR. And my claim is that it's not
so simple.

First, I think I should mention off the bat one thing that I mentioned
in my previous post: just because we're using a path integral doesn't
mean we can go about applying statements that hold for c-numbers to
everything inside the path integral. The simplest and best known
example of the difficulty is operator ordering. A naive look at a
path integral would seem to imply that operator order doesn't matter,
because we're just dealing with complex numbers and not operators
inside the path integral. But closer inspection shows that different
ways of defining the measure in the path integral correspond to
different operator orderings.

Our situation here is not exactly analagous, but I still think it's
possible to see that something subtle is going on. Namely, on the
left hand side of Eq.(1), we have an action that we'd like to insert
in the path integral when we calculate something for strings
propagating on a given, curved background. On the RHS, we are told
that we may compute this by calculating the same thing on a flat
background, then on a flat background with one asymptotic graviton
added to the mix, then with two, etc., and adding all of these up with
the appropriate coefficients. The point is that each term in the path
integral on the RHS has different boundary conditions from the others
because it has a different number of asymptotic states. So we are
still dealing with an operator, and I think therefore we do need to be
careful when claiming to "Taylor expand" things inside the path
integral.

This is how we came to a discussion of the perturbation series in
ordinary QFT. There we are dealing with Hamiltonians, not actions,
but the argument is the same: we claim that we can Taylor expand
exp(i \int H0 + H1) and then get a series of iterated integrals. But
we know that this leads us to trouble in QFT; the convergence of
exp(x) for real numbers doesn't at all guarantee the convergence of
the perturbation series we get by this method in QFT. And I'm
arguing that we likewise can't conclude that we have really shown that
the curved spacetime metric in the action is equivalent to the
background metric we start with plus a superposition of gravitons.

Even if I agree to suspend this disbelief, it seems to me that at best
what one has shown is that (possibly) we can do small perturbations
about some given curved background, if I assume that all of the
problems I raised above are really just technicalities that could be
put right with a little more care. But then it still seems to me that
one has not shown that gravity is fully dynamical in string theory,
because not only am I splitting up my metric into a "backgound" piece
and a "dynamical" piece, but I can't let my dynamical piece get "very
big". Hence I have to consider a whole class of background metrics,
and to me this loses most of what one would hope for in a theory of
dynamical gravity.

Now, I agree that the appearence of Einstein's equations out of Weyl
invariance is a very suggestive thing, and it very well may be that in
a nonperturbative version of string theory one will see how all of
these different "backgrounds" are in fact states of the full theory.
Certainly I would think that's one of the things string theorists are
hoping will come out of M-theory. But to me this just emphasizes how
important it is to verify the classical limit of your theory
nonperturbatively, and I certainly don't see how string theory is in a
substantially better position than LQG on that front.

>>For Hilbert spaces, the
>>meaningful question is whether or not they are unitarily equivalent;
>>you must preserve not only the algebraic vector space structure but
>>also the inner product structure in order to regard two Hilbert spaces
>>as "the same". And a Hilbert space for string theory is certainly not
>>going to be unitarily equivalent to Fock space.

>Eh? All Hilbert spaces of the same dimension are unitarily
>equivalent, and the only dimensions physicists are likely to see
>are 0,1,2,3,... up to aleph_0, which is math jargon for the
>cardinality of the set of natural numbers. The Hilbert space
>for the harmonic oscillator, and all the Fock spaces that show
>up in ordinary quantum field theory, have dimension aleph_0.
>So if string theory has a well-defined Hilbert space, it's probably
>the countable-dimensional one.

Well, how do I rectify this with Haag's theorem, which says that for
different values of the coupling constant in QFT I must have
inequivalent representations of the CCRs? Surely a unitary equivalence
between the Hilbert spaces will also give me a unitary evquivalence
between the operators representing the CCRs. The only loophole I see
here is that the operators (\phi(x) and \pi(x) for a scalar field,
say) are not really operators. But then it seems to me I'm still
stuck as to a good understanding of my quantum theory: a Hilbert space
with no operators isn't much of a quantum theory, because there's a
lot of physics in them thar operators. I think this is another aspect
of the difficulty you referenced in your post: if my perturbation is
not a really an operator, then even when I act with it on something
that really is a state, what I end up with isn't necessarily a
state. So the Hilbert space structure of my theory is still rather
murky, it seems to me.

Now, I'm not arguing that a Hilbert space is the only way I can have a
well defined quantum theory, and indeed I'm well aware that lots of
people (maybe all) working in axiomatic QFT think it is an outdated
approach to quantum theory and one should instead study algebras of
observables. I could well believe this, but I just don't see how any
use can be made of it at present in quantum gravity. For one thing,
we have no metric lying around to tell us about the locality of our
observables, but worse than that, we don't really know what those
observables should be in QG, local or otherwise. Hopefully when we
have a theory of QG we'll understand this a lot better, but right now
it seems to me that we don't, and certainly I'm not aware of any
understanding in string theory which would let one argue that we know
a lot about it's nonperturbative structure in terms of an algebra
of observables.

>That's certainly true if by "graviton" you mean something
>like a solution of the linearized Einstein equations. I don't
>think any string theorists claim you can pull a stunt like this.

Well, I don't know how else to interpret the following that Lubos
wrote elsewhere on this thread (or what used to be this thread):

(I don't know the message ID; I got this off the spr archives where
it's message 0033596)

>Sure. Read Chapter 3 of Green+Schwarz+Witten's book or Chapter 3 or Joe's
>textbook of string. The physics of perturbative string theory can be
>proved to be background independent as follows:
>
>What we really want to prove is that a different background can be
>understood also in terms of the original background (as a state) and that
>the prediction of physical observables will be identical, regardless of
>the choice of backgrounds.
>
>Well, let us work with the bosonic string to avoid a lot of indices and
>fermions. The worldsheet Lagrangian contains g_mn(X^a).d X^m dbar X^n,
>where I multiply the background metric g_mn (with the worldsheet fields
>X^a(sigma,tau) substituted) by the holomorphic and antiholomorphic
>derivatives of X^a(sigma,tau).
>
>We want to show the equivalence of two backgrounds. So let us divide the
>path between them to infinitesimal pieces and without a loss of generality
>assume that we want to show the equivalence of two infinitesimally
>separated backgrounds. Both of them must satisfy the equations of motion,
>of course.
>
>But this is easy do. Roughly speaking, by expanding the action
>exp(-integral g_mn(X^a).d X^m dbar X^n), if we substitute g_mn + delta
>g_mn instead of g_mn, there will appear an extra factor of
>exp(-integral delta g_mn(X^a).d X^m dbar X^n). The exponential can be
>expanded by Taylor series. Well, let us take only the zeroth and the first
>term, being infinitesimal. The first term is
>
> -integral delta g_mn(X^a).d X^m dbar X^n
>
>This is nothing else than the perturbative vertex operator for a graviton
>on the background g_mn, whose wave function is delta g_mn, so to say. We
>are adding this graviton vertex operator into the scattering. So the
>phenomena computed at one background are equivalent to the phenomena
>computed on the other background, with an extra on-shell particle inserted
>to the process. This shows clearly that the gravitons are not just "some"
>spin 2 particles but their effect is exactly identical to a change of the
>background metric. Therefore the choice of the background metric is a
>technical thing only and does not imply that general covariance (in the
>region where it should hold) is spoiled in any way by string theory.
>Physics of string theory of course preserves all the beauties of the
>Einstein's theory. Maybe just the available formulations do not show it
>manifestly enough. A future expected final formulation is believed to
>contain not only general covariance, but a fancy stringy extension of
>it... Maybe we will never find such a background-independent formulation
>that implies everything else. Who knows. This is partly a question of
>aesthetics and religion. Physics is however clear: a fair analysis shows
>that it is background-independent.

In particular, it seems to me that there is a great loss of generality
in considering "infinitesimal" perturbations like this. We know that it's not
so simple in ordinary QFT; why should it be so simple in string theory?

Josh


Jacques Distler

unread,
Jun 29, 2001, 3:22:34 AM6/29/01
to
In article <9hgjbh$24...@r02n01.cac.psu.edu>,
jwi...@phys.psu.edu (Josh Willis) wrote:

>I also hasten to add that I'm not implying that Aaron in specific or the
>string theory community in general holds the belief that a theory is
>consistent iff it is renormalizable, but I am trying to make clear why I
>think the perturbative behavior of string theory, good though it may be,
>does not seem to me to be a "selling point" for the theory. That may not
>be the only selling
>point its adherents propose, but it certainly was the earliest and still gets
>repeated. See, for example, p. 222 of Polchinski, vol II: "This [black hole
>entropy] is a remarkable result, and another indication, *beyond perturbative
>finiteness*, that string theory defines a sensible theory of quantum gravity"
>(emphasis added).

Sigh! Renormalizability by power counting has little to do with the issue.

We (ie, anyone schooled in modern QFT) are perfectly happy with
"nonrenormalizable" effective field theories, so long as the
ultraviolet behaviour is (ultimately) controlled by *something*. In
asymptotically-free theories, the ultraviolet physics is
weakly-coupled, and we can use perturbation theory to define it.
Other theories are controlled by a nontrivial UV fixed-point. Yet
other theories have their UV behaviour controlled by a string theory.

Any string theorist knows this perfectly well. Indeed, string theory
calculations have shown the existence of nontrivial interacting QFT's
in 5 and 6 dimensions (the UV behaviour is controlled by a UV fixed
point whose existence one can deduce from stringy considerations).

The trouble with trying to quantize "pure" 4d gravity is that there
*is* no UV fixed point (to be more precised, there is not a shred of
evidence for the existence of one, and a good deal of evidence to the
contrary).

Indeed, there is no known way to control the UV behaviour of gravity
in any dimension d>3 (or d=3 with matter) EXCEPT via string theory.

Mind you, there are a zillion ways to define a lattice-regularized
theory of quantum "gravity" (Regge calculus, dynamic triangulations,
LQG, MQG, . . .). People like Ambjorn, Hamber, Gross and others have
spent years doing Monte Carlo simulations in various of these lattice
approaches. The upshot is the NONE of these approaches have YET been
seen to have a "continuum" (large-volume spacetime) limit. They ALWAYS
turn out to be in some horrible crumpled phase where the dominant
spactimes have Planckian curvatures and Planckian sizes. That is to
say, there are lots of theories of quantum "gravity", none of which
turns out to be a theory of quantum gravity.

Maybe LQC (or MQC or whatever) will prove to be different. If so, I
will be VERY interested.

But life, alas, is short. Part of doing science is choosing what the
most promising avenues are.

Anyone who's been to the horse races will understand when I say that
I am glad that someone is betting on the longshot and will be
sincerely congratulatory if, by some chance, his horse does come in.

JD

Greg Kuperberg

unread,
Jun 29, 2001, 2:18:33 PM6/29/01
to
In article <Pine.SOL.4.10.101062...@strings.rutgers.edu>,
Lubos Motl <mo...@physics.rutgers.edu> wrote:

>And even in the hypothetical case in which Nature did not use string
>theory but something worse, string theory will dominate *mathematics*
>of the 3rd millenium, as Witten said.

In other words, all you convex geometers, probabilists, logicians, PDE
analysts, complexity theorists, etc., better drop what you're doing,
because here comes string theory. I have to agree with the skeptics
(not being one myself) that it does sound arrogant.

Moreover, it almost contradicts another thing that you said, that string
theorists aren't and shouldn't be mathematicians. If we mathematicians
are going to be chewing on your ideas for the next millenium, how are
you too good for us? Your argue that mathematical rigor would slow you
down too much. But that doesn't hold water, because consistency checks
in string theory, which are considered crucial, are a kind of rigor.
You have given some examples of ineffectual rigor, but that doesn't
preclude the development of more useful forms. (E.g. among working
mathematicians epsilon-delta rituals have largely been replaced by more
efficient O(f(x)) notation.)

Note that my own math department has two noted string theorists (see
http://math.ucdavis.edu/profiles/). One reason that they are in a math
department is that many physics departments aren't interested in string
theory, fundamental though it may be. (And many math departments feel the
same way about logic and category theory, which are our fundamental core.)


--
/\ Greg Kuperberg (UC Davis)
/ \
\ / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/

\/ * All the math that's fit to e-print *

Aaron Bergman

unread,
Jul 1, 2001, 9:27:02 PM7/1/01
to
In article <Pine.SOL.4.10.101062...@strings.rutgers.edu>,
Lubos Motl wrote:

[...]


>
>Someone also argued that the Coleman-Mandula theorem was inspired by the
>axiomatic field theory etc. I do not want to argue about that because I
>know very little about the history of this theorem. What I know much
>better is that this theorem had a serious and very important loophole and
>this loophole (called supersymmetry) was first understood (apart from some
>Russian mathematical physicists) by Ramond who (successfully) tried to
>incorporate fermions into string theory (1971).

There's another loophole which you're well acquainted with. Note
the existance of the brane central charges on the right hand side
of the SUSY algebra. (Well, technically this is a violation of the
Haag-Loupaszanski-Sohniuys theorem, but close enough, I think.)

>> >This is well and good---if your perturbation theory converges.
>> I would think that some sort of Borel resummability would be OK.
>
>Yes, this is an old paper by Gross and Periwal (1988) - by the way, today
>these two guys have another paper. ;-) Fine that you mentioned it but
>the work of Shenker that shows the (2g)! character of the divergence is
>certainly also important.

I haven't read the Shenker stuff -- I haven't even read Gross and
Periwal, either, to be honest. What's the ref?

>Do I understand incorrectly that there are many
>different ways how to resum such diverging series - and these different
>choices differ essentially in the way how you treat the nonperturbative
>corrections?

The reason I mentioned Borel resummability is that, if I remember
from my quantum class correctly, you get a Borel resummable series
when doing perturbation series on the anharmonic oscillator.
Basically, I think Borel resummability assumes a certain
analyticity property that I forget. There's certainly no reason to
believe that string theory would obey this off the top of my head.
It's was mostly just an example of how a divergent perturbation
expansion is not the end of the world. In the real world, of
course, there's all sorts of O(1/g) corrections.

[...]


>
>> The stuff I've snipped is essentially a synopsis of all the
>> things that go wrong in quantizing an interacting field theory.
>> My personal prejudices are that the reason for all these things
>> is that we have no idea what a quantum field theory is, but
>> that's just me.
>
>Maybe. ;-) Do you think that there is anything unclear in defining QCD,
>for example?

Well, yes, actually. The best bet there is is lattice
calculations, but, as I understand it, there are still problems
there. Otherwise, all we can really do is perturbation theory
around a free fixed point. The only reason we actually know stuff
like chiral symmetry breaking is from educated guessing and
lattice calculations. Or atleast that's my understading of it --
I'm far from an expert on QCD.

>> I'm not even convinced that a good definition for QFTs should
>> include Hilbert spaces.
>
>I am personally always happier if a theory does include it. ;-)

Hilbert spaces seem like relics of the quantum mechanical
approximation to QFT.

(I'm being a bit provocative here....)

>> Background independence is really pretty, but that doesn't mean
>> it is necessarily correct. Maybe life isn't background
>> independent. It's worth thinking about, don't you think?
>
>Maybe that this is just an issue of terminology, but in what sense you
>think that background independence can be incorrect?

It's not unreasonable to think that a theory could explicitly
depend on a background. I'm not saying that string theory does; I
just think that assumptions are always good to question.

Aaron Bergman

unread,
Jul 1, 2001, 9:27:15 PM7/1/01
to
In article <9hgk4d$2cn$1...@glue.ucr.edu>, John Baez wrote:
>
>The more interesting question is to ask for an explicit description
>of the Hilbert space for perturbative string theory on a fixed
>(static) background metric. I could probably answer this myself
>with a little work, but it'll be fun to see how many string th