quantum gravity confusion
Greg Weeks
Every now and then my brain boils over with my confusion regarding the
state of quantum gravity. In general relativity, the metric and (global)
topology of space-time are dynamically determined. So, unless something
really weird is going on, a theory of quantum gravity cannot be based on an
underlying space-time with known global topology.
And yet Weinberg (for example) says it's just fine to quantize gravity just
like any other field theory. The only problem from his point of view is
that it didn't turn out to be renormalizable. If only that could be fixed,
everything would be okay.
To overstate slightly, either he's crazy or I am.
My impression is that the supersymmetric theories also presume an
underlying space-time with known topology. Indeed, my impression is that
every "quantum gravity" theory I've heard about presumes an underlying
space-time with known topology. (This space-time is not necessarily
Minkowski space-time [Kaluza-Klein?], which raises even more questions.)
So, I don't know what the heck people (eg, Hawking) think they're doing.
Perhaps the "quantum gravity" of today is to quantum gravity as the
"quantum mechanics" of 1915 is to quantum field theory. If so, it isn't
clear how seriously to take its conclusions.
Do people agree with this and just not bother to mention it very often?
Or ...?
Greg Weeks
[Mod. Note - I'm not really and expert on this, and so I may get some
things wrong. Perhaps John Baez or one of the other experts on quantum
gravity can point out my errors and provide much more information for you.
There are many approaches to quantizing gravity, as you note. One standard
approach, known as canonical quantization, is to fix an initial 3-geometry
and quantize the evolution from there. In this case, one does not fix the
entire background (although one does restrict oneself to geometries that
are compatible with the initial slice). One then writes down the ADM action
and quantizes as one would for any other field theory for which the action
is known. It turns out that the theory treated in this way is a gauge theory,
essentially due to the fact that there are many ways of choosing time (or
the lapse function, as it is known) that give equivalent evolutions.
Thus, it is the diffeomorphism group that acts as the gauge group. The
standard Fadeev-Popov approach where one introduces ghost fields can be
carried over and the theory quantised. Now, what one gets is a field
theory for a field who's coupling constant carries dimension, and this
leads to the famous non-renormalisability of quantum gravity. However, it
has been pointed out in recent years (see gr-qc/9512024 for an overview) by
a fellow from U.Mass. (Amherst) named Donoghue that this simply means that
we are dealing with an effective field theory, much like the Fermi theory
of electrodynamics. This simply means that there is a more fundamental
underlying theory which you have not been smart enough to figure out, but
as long as you stay in the regime where you effective theory captures the
important aspects, you can use that to do your calculations. In this sense,
gravity is no worse off than so-called renormalisable theories, since they
also have a cut-off scale below which they are replaced by a more
fundamental theory. I know field theorists who basically claim that
renormalisability is an outdated requirement. I don't know if that
clarifies anything, but it's grist for thought!]
Dan Evens
Greg Weeks wrote:
> And yet Weinberg (for example) says it's just fine to quantize gravity just
> like any other field theory. The only problem from his point of view is
> that it didn't turn out to be renormalizable. If only that could be fixed,
> everything would be okay.
When you are talking renormalization, you are talking perturbation. That
is, you start with some solution which is special in some way, and hope
that the real answer is in some sense close to it.
This works pretty well for QED. You start with the free solutions
to the system (ignoring all interactions) which are the electron
and the photon (and their anti-particles). Then you introduce
interactions and hope that their effects are small. You let the
interactions perturb the free solutions. Then you put in higher
and higher orders of calculations.
Renormalizable means that, at each order of calculation, you
can adjust the finite set of initial parameters so that the
observed values are agreed with. Then use that finite set
to predict things other than the measured values for the
parameters. Non-renormalizable means that you will find
new parameters required at each order of calculation. This
makes it basically impossible to predict anything with the
theory.
So, the idea is, you start with some solution of general
relativity (a flat empty space-time for example). Then
you introduce some interactions and work out how that
modifies things. The hope is that the changes will be
in some sense small. Since gravity seems to be a
relatively weak interaction, this ought to have a good
hope of being true.
But at each order of interaction, you need more parameters
to adjust to match the calculation with observed parameters.
For GR, these amount to higher order interaction constants.
People have done some work along this line and gotten some
interesting results at one loop. But, things get difficult
and confusing after that.
So, yes, a background of some definite geometry is assumed
by Weinberg to do what he was doing, which is to use a
perturbation scheme and hope the gravity interaction is
small. Sadly this scheme fails because general relativity
is not renormalizable.
String theories have two thrusts for attacking this.
One is to produce a renormalizable theory of spin 2 particles,
the other is to produce a finite theory based on supersymmetry.
I'm not real sure how succesful either has been.
--
The preceding are my opinions alone and have nothing
whatever to do with my employer. I don't even know what my
employer thinks. I'm not even real sure who the CEO is.
Dan Evens
Matt McIrvin
In article <533fci$13...@pulp.ucs.ualberta.ca>, Greg Weeks
<we...@dtc.hp.com> wrote:
> A theory of quantum gravity should not be based on an underlying
> space-time of known topology.
>
> Agree? Disagree?
Disagree-- it depends on what the theory is for. A *complete* theory of
quantum gravity, valid at the Planck scale, probably should not be based on
an underlying space-time of known topology. For a low-energy effective
theory this presents no problem.
Weinberg's opinion seems to be that quantum gravity in the first sense
isn't even going to stand on its own, that it will have to be included as
part of something like superstring theory, and that the only reasonable way
to quantize GR by itself is in the second sense. I am not sure that he is
right, but I agree that the effective-theory quantization is a reasonable
thing to do.
--
Matt McIrvin <http://world.std.com/~mmcirvin/>
john baez
In <530ue6$u...@pulp.ucs.ualberta.ca> Greg Weeks <we...@dtc.hp.com>
writes:
>Every now and then my brain boils over with my confusion regarding the
>state of quantum gravity.
Consider yourself lucky: my brain is *always* boiling over with
confusion regarding quantum gravity!
You note quite correctly, in my opinion, that treating quantum gravity
perturbatively on flat space just like any other quantum
field theory is a bit schizophrenic. The whole point of general
relativity is that Minkowski space is a very special solution of
Einstein's equation, so privileging it with a special role in
quantum gravity should immediately raise ones suspicions. After all,
if our quantum field theory on Minkowski space is *causal* with
respect to the Minkowski metric, like typical quantum field theories
try to be, then it's impossible for influences to be propagated at
the speed of light with respect to a general sort of *curved* metric,
as in general relativity! You will find this obvious problem simply
NEGLECTED in most work on perturbative quantum gravity.
Of course the perturbative approach is a reasonable thing to play
around with, in the absence of some better idea. Indeed one
might expect to use it to successfully predict corrections, proportional
to Planck's constant hbar, to various calculations that neglect
quantum gravity. But as a fundamental theory it's highly dubious, and
it's probably no accident that when we try to work out all the effects
of order hbar^n, we get utter garbage. (Technically speaking,
the problem here is that perturbative quantum gravity is nonrenormalizable.)
So what do we do? Well, if you come from a particle physics background,
you try to doctor up quantum gravity until it *is* renormalizable.
You invent supergravity, Kaluza-Klein theories, string theories, and
finally superstring theories which take advantage of all these ideas ---
and finally, you luck out and get something renormalizable. Then
later you realize that you still don't understand the nonperturbative
effects and you start thinking about those....
On the other hand, if you come from a general relativity background,
you try to figure out a way to do quantum field theory that doesn't
require a fixed background metric. You get stuck for about 20 years
and finally you invent Ashtekar's new variables and the loop representation
of quantum gravity. But you are faced at every moment with the fact
that none of the standard quantum field theory recipes apply, so you
have to constantly be worrying about very basic conceptual issues....
Anyway, that's what people are doing, now. If nature is kind to
us --- and we need some kindness, now more than ever, if we are
going to figure this out --- it will turn out that string theory
and the loop representation are two sides of the same thing,
some theory that we are only dimly beginning to understand.
(I have argued that this is the case in "Strings, loops, knots
and gauge fields," available as hep-th/9309067.) But only time
will tell... it could be that one or *both* of these approaches is
completely wrong!
I urge you to read stuff on quantum gravity written by general
relativists, because they are more attuned to the conceptual
problems with quantizing gravity "just like any other field" than
the particle physicists are. You will find that a lot of neat
stuff *is* understood. And you will see that there other people
out there, worried about what you are worried about.
Try:
Prima facie questions in quantum gravity, by Chris Isham, lecture at
Bad Honeff, September 1993, preprint available in LaTeX form as
gr-qc/9310031.
and
Conceptual Problems of Quantum Gravity, edited by Abhay Ashtekar and
John Stachel, based on the proceedings of the 1988 Osgood Hill
Conference, 15-19 May 1988, Birkhaueser, Boston, 1991.
Matt McIrvin
In article <325438...@hydro.on.ca>, dan....@hydro.on.ca wrote:
[Mod. Note: description of nonrenormalizability deleted to save
bandwidth. WGA]
> So, yes, a background of some definite geometry is assumed
> by Weinberg to do what he was doing, which is to use a
> perturbation scheme and hope the gravity interaction is
> small. Sadly this scheme fails because general relativity
> is not renormalizable.
Weinberg's whole point (based on what I've heard him say about the subject)
is that renormalizability is not really necessary here, as long as you are
satisfied with an effective theory of what gravity does at energy scales
much smaller than the Planck energy (or, equivalently, larger distances).
In effective field theories, these proliferating nonrenormalizable coupling
terms are divided by increasingly large denominators (in this case, powers
of the Planck energy), and you can at least hold out the hope that this
means you can ignore most of them and truncate the expansion to get
finite-precision results.
Of course, this tells you absolutely nothing about what happens up at the
Planck energy, when the numerators get comparable to the denominators. In
that regime, the effective field theory approach no longer works, and it is
presumably necessary to bite the bullet and figure out what is really going
on (or, at least, discover the next theory up), rather than doing
perturbation theory on background-quantized GR.
In the context of effective field theory, nonrenormalizability is not
really a problem. The field theories that describe the rest of the world
are renormalizable, probably because the nonrenormalizable interactions
tend to become very weak at energies far below the cutoff scale of the
effective theory. In the case of gravity, though, there is no
renormalizable interaction to swamp the nonrenormalizable ones, so all we
are left with in our low-energy world is a really weak interaction, which
is why gravity is such a feeble thing compared to the other forces we know.
If it weren't for the fact that there are humongous quantities of its
"charge" flying around, we probably wouldn't know it was there.
Greg Weeks
Greg Weeks (we...@dtc.hp.com) wrote:
: Every now and then my brain boils over with my confusion regarding the
: state of quantum gravity...
>From the responses so far, it appears that I did not phrase my question
precisely enough. I'll try again:
A theory of quantum gravity should not be based on an underlying
space-time of known topology.
Agree? Disagree?
The well-known theories of quantum gravity are based on an underlying
space-time of known topology.
Agree? Disagree?
Thanks for your patience and input.
Greg
Philip Gibbs
Greg Weeks wrote:
> My impression is that the supersymmetric theories also presume an
> underlying space-time with known topology. Indeed, my impression is that
> every "quantum gravity" theory I've heard about presumes an underlying
> space-time with known topology.
You are almost certainly right about the importance of
topology and topology change in quantum gravity, but
you are wrong to suppose that all quantum gravity
theories assume a background topology.
John Wheeler was one of the first to recognize that
topology might be important at the Planck scale.
Sometimes physicists such as Hawking talk about the
space-time foam.
Although superstring theory is formulated on a fixed
background, it has been recognized that topology change
is part of string theory. See for example
SPACE-TIME TOPOLOGY CHANGE AND STRINGY GEOMETRY.
By P.S. Aspinwall (Princeton, Inst. Advanced Study),
B.R. Greene (Cornell U., LNS), D.R. Morrison (Duke U.). 1995.
J. Math. Phys. 35 (1994) 5321-5337.
It sounds contradictory to say that topology
changes in a theory which is formulated on a
fixed background. I don't know enough about the
subject to explain how this works, but I
imagine that answering the question properly
would be a big step forward in quantum gravity.
--
====================================================
Phil Gibbs p...@pobox.com http://pobox.com/~pg
Steve Carlip
Greg Weeks (we...@dtc.hp.com) wrote:
: >From the responses so far, it appears that I did not phrase my question
: precisely enough. I'll try again:
: A theory of quantum gravity should not be based on an underlying
: space-time of known topology.
: Agree? Disagree?
Don't know. It is completely unclear whether quantum gravity
should fix the spacetime topology or allow it to fluctuate.
Certainly the geometry must not be fixed, but topology is harder.
In 2+1 dimensions, for example, there are consistent quantizations
in which the spacetime topology is fixed; but there are also
(probably) consistent quantizations in which the topology is not
fixed. At the moment, we have no way to know which, if any, of
these approaches are "correct."
(I say "probably" because the approaches that allow topology
change, which are based on path integrals, typically have infinite
factors coming from integrals over zero-modes. These aren't the
usual infinities of quantum field theory, and it is plausible, but
not certain, that they will disappear when one moves from pure
gravity to gravity plus matter.)
: The well-known theories of quantum gravity are based on an underlying
: space-time of known topology.
: Agree? Disagree?
Some are, some aren't. Canonical approaches usually fix the
topology of space S and assume that the overall spacetime topology
is RxS. Path integral approaches can fix topology or can include
sums over topologies, typically with undetermined (although not
completely arbitrary) relative phases. I'm not sure how to even
pose the question in the Ashtekar/loop variable approach, since
the basic states there seem to describe something that isn't much
like a manifold at all; there's been some work, but the passage to
a classical manifold-like spacetime is still very unclear. In
string theory, transitions between topologies can occur, but again
it's not clear what the classical limit means. (For example,
manifolds which are classically different can be described by the
same state in a string theory.)
You're confused because everybody is confused. (Isn't it fun?)
Steve Carlip
car...@dirac.ucdavis.edu
john baez
> A theory of quantum gravity should not be based on an underlying
> space-time of known topology.
>
>Agree? Disagree?
Nobody knows, although different people have different opinions.
> The well-known theories of quantum gravity are based on an underlying
> space-time of known topology.
>
>Agree? Disagree?
As Philip Gibbs noted in a previous post, J. A. Wheeler's idea
of "spacetime foam" amounts to the idea that one should not work
with a spacetime of fixed topology in quantum gravity. His
work, while quite sketchy, was the motivation for lots of work
on quantum gravity that takes the possibility of topology change
seriously.
For example, the Hartle-Hawking path-integral approach to quantum
gravity involves a "sum over topologies" in addition to the
"integral over metrics", so they are clearly not assuming an underlying
spacetime of known topology. Indeed, Hawking gets interesting effects
from assuming topological fluctuations. For example, working in the
Riemannian signature as he does (people call this "Euclidean quantum
gravity" but I can't abide that), a virtual black hole pair appears
as a topological fluctuation in which one forms the connected sum
of some manifold with an S^2 x S^2. See below for more details.
On the other hand, perturbative string theory not only works
with a spacetime of a given topology, but even goes so far
as to fix a particular *geometry*! This is clearly odd for
a fundamental theory of quantum gravity, and string theorists
have not failed to notice this: the most exciting work in
string theory these days involves comparing string theories
in different geometries and topologies; they turn out to be
related by many symmetries ("mirror symmetry", "duality" are
some of the buzzwords here), and are expected to arise as
different manifestations of some more fundamental approach
to string theory ("background-free string field theory" and
"M-theory" are key buzzwords here). Here is a rather well-known
paper that studies topology change in string theory:
Black Hole Condensation and the Unification of String Vacua
Brian R. Greene, David R. Morrison, Andrew Strominger,
hep-th/9504145
It is argued that black hole condensation can occur at conifold singularities in the
moduli space of type II Calabi--Yau string vacua. The condensate signals a
smooth transition to a new Calabi--Yau space with different Euler characteristic
and Hodge numbers. In this manner string theory unifies the moduli spaces of many
or possibly all Calabi--Yau vacua. Elementary string states and black holes are
smoothly interchanged under the transitions, and therefore cannot be invariantly
distinguished. Furthermore, the transitions establish the existence of mirror
symmetry for many or possibly all Calabi--Yau manifolds.
The loop representation of quantum gravity is an approach to
canonical (i.e., Hamiltonian) quantum gravity. As such, it
assumes a fixed topology of the form R x S, where S is
some 3-manifold. However, it does not assume a fixed background
geometry. Ashtekar argues that a conservative approach to
quantum gravity would not assume topology change until it is
somehow forced upon one. There are some mild indications that
perhaps topology change might arise naturally when one gets
the loop representation working, but it is too early to tell.
In short, the question of topology change in quantum gravity is
wide open, and not something one should form premature prejudices about.
From "week67":
2) Stephen W. Hawking, Virtual black holes, preprint available as
hep-th/510029.
Hawking likes the "Euclidean path-integral approach" to quantum gravity.
The word "Euclidean" is a horrible misnomer here, but it seems to have
stuck. It should really read "Riemannian", the idea being to replace
the Lorentzian metric on spacetime by one in which time is on the same
footing as space. One thus attempts to compute answers to quantum
gravity problems by integrating over all Riemannian metrics on some
4-manifold, possibly with some boundary conditions. Of course, this is
tough --- impossible so far --- to make rigorous. But Hawking isn't
scared; he also wants to sum over all 4-manifolds (possibly having a
fixed boundary). Of course, to do this one needs to have some idea of
what "all 4-manifolds" are. Lots of people like to consider wormholes,
which means considering 4-manifolds that aren't simply connected. Here,
however, Hawking argues against wormholes, and concentrates on
simply-connected 4-manifolds. He writes: "Barring some pure
mathematical details, it seems that the topology of simply connected
four-manifolds can be essentially represented by gluing together three
elementary units, which I call bubbles. The three elementary units are
S^2 x S^2, CP^2, and K3. The latter two have orientation reversed
versions, -CP^2 and -K3." S^2 x S^2 is just the product of the
2-dimensional sphere with itself, and he argues that this sort of bubble
corresponds to a virtual black hole pair. He considers the effect on
the Euclidean path integral when you have lots of these around (i.e.,
when you take the connected sum of S^4 with lots of these). He argues
that particles scattering off these lose quantum coherence, i.e., pure
states turn to mixed states. And he argues that this effect is very
small at low energies *except* for scalar fields, leading him to predict
that we may never observe the Higgs particle! Yes, a real honest
particle physics prediction from quantum gravity! As he notes, "unless
quantum gravity can make contact with observation, it will become as
academic as arguments about how many angels can dance on the head of a
pin". I suspect he also realizes that he'll never get a Nobel prize
unless he goes out on a limb like this. :-) He also gives an argument
for why the "theta angle" measuring CP violation by the strong force may
be zero. This parameter sits in front of a term in the Standard Model
Lagrangian; there seems to be no good reason for it to be zero, but
measurements of the neutron electric dipole moment show that it has to
be less than 10^{-9}....
Greg Weeks
Steve Carlip (sjca...@ucdavis.edu) wrote:
... what I found to be a helpful survey. There is a tangential question
I'd like to ask:
: ... Path integral approaches can fix topology or can include
: sums over topologies ...
What do these path integrals mean?
Minkowski path integrals are viewed (initially at least) as transition
amplitudes, eg, <q2 t2 | q1 t1>. But what is the Hilbert space of |q>'s?
And doesn't the space of |q>'s impose a global spatial topology?
Alternatively, Euclidean path integrals may be viewed simply as solutions
to the Euclidean Schwinger-Dyson equations. (The field equations imply
Wightman function equations, and the Wightman function equations
analytically continue to Schwinger function equations.) But what
quantities in general relativity can be continued to some form of Euclidean
space?
So I'm baffled.
Greg
PS: Part of me blames myself for not reading the literature on the subject.
But -- besides the fact that I'm two decades behind the times -- when I was
first exposed to gravitational path integrals, there was no rationale
presented for them. They were simply ascii (well, mostly ascii) strings
that you could write down that had a pleasing resemblence to other ascii
strings that had proved valuable in other contexts.
Greg Weeks
Steve Carlip (sjca...@ucdavis.edu) wrote:
: In 2+1 dimensions, for example, there are consistent quantizations
: in which the spacetime topology is fixed;
I'm EXTREMELY intrigued. What form does the theory take?
The theory can't be a local quantum field theory in the usual sense, since
there is no fixed metric with respect to which local causality can be
defined.
If diffeomorphisms form the gauge group, then there are no local
observables. (So there is no definition of the vacuum sector as the
closure of local observables applied to the vacuum.)
Indeed, the only nontrivial observable that I can imagine is the S-matrix.
This would mean the space-time is asymptotically flat. (And I bet there is
no way to distinguish physical from unphysical states without resorting to
the asymptotic fields.)
Are the above musings at all close to the facts?
Greg
Philip Gibbs
Greg Weeks wrote:
>
> A theory of quantum gravity should not be based on an underlying
> space-time of known topology.
>
> Agree? Disagree?
Agree
There is another paper which adds more confusion to
this question of topology. It is from a lecture by
Witten and as far as I know it is only available as
a pre-print or e-print:
1) PHASE TRANSITIONS IN M THEORY AND F THEORY.
By Edward Witten (Princeton, Inst. Advanced Study).
IASSNS-HEP-96-26, Mar 1996. 27pp.
Published in Nucl.Phys.B471:195-216,1996.
e-Print Archive: hep-th/9603150
Witten discusses the relation of the hypothetical
symmetry group ,str(X), of string theory on a background
manifold X and the diffeomorphism symmetry group diff(X).
If string theories on manifolds with different topology
X and X' are really the same as a result of mirror symmetry
dualities or whatever, then str(X)= str(X'). But
we expect the diffeomorphism groups to be part of the
symmetry in some sense. Perhaps diff(X) is a subgroup
of str(X), or there is a homomorphism from str(X) onto
diff(X). In that case but diff(X) and diff(X') are
embedded inside str(X), but this is puzzling since there
is no obvious way to combine different diffeomorphism
groups which preserves the idea that they act on the
same space.
My account is a bit vague so please read Witten's which
is perhaps slightly less so.
So how is Witten's puzzle resolved? I have made the
suggestion that str(X) contains the symmetric group
S(X) on the manifold, i.e. the group of all 1-1 mappings
ignoring the topology altogether. Then the puzzle is
resolved because diff(X) is a subgroup of S(X) and
S(X) is isomorphic to S(X'). Perhaps there are
possible ways to generalise the concept of symmetry
so that the puzzle can be resolved without abandoning
the ideas of topology change or universal string symmetry.
Any suggestions?
Daryl McCullough
In article <5398iq$1...@charity.ucr.edu>, ba...@math.ucr.edu says...
>The loop representation of quantum gravity is an approach to
>canonical (i.e., Hamiltonian) quantum gravity. As such, it
>assumes a fixed topology of the form R x S, where S is
>some 3-manifold.
Does R x S mean the time evolution of 3-space? If so, is that perfectly
general? Aren't there 4-space solutions to general relativity that cannot
be described that way? In particular, I'm thinking of the formation of
singularities such as black holes, which appear never quite to happen
if you try to describe things in terms of the time evolution of 3-space.
Or is the loop representation mostly concerned with singularity-free evolution?
Daryl McCullough
CoGenTex, Inc.
Ithaca, NY
Dave Pandres, Jr.
Philip Gibbs wrote:
[Moderator's Note: Vast wad of quoted text deleted here -
please, folks, trim the quoted text! - jb]
> So how is Witten's puzzle resolved? I have made the
> suggestion that str(X) contains the symmetric group
> S(X) on the manifold, i.e. the group of all 1-1 mappings
> ignoring the topology altogether. Then the puzzle is
> resolved because diff(X) is a subgroup of S(X) and
> S(X) is isomorphic to S(X'). Perhaps there are
> possible ways to generalise the concept of symmetry
> so that the puzzle can be resolved without abandoning
> the ideas of topology change or universal string symmetry.
> Any suggestions?
I suggest that the "string group" str(X) is the "conservation group,"
which is discussed in my paper "Unified Gravitational and Yang-Mills
Fields," International Journal of Theoretical Physics, Vol.34,
p. 733-759. (May 1995). I have shown that the diffeomorphism group
diff(X) is a subgroup of the conservation group.
The abstract of this paper reads as follows:
We unify the gravitational and Yang-Mills fields by extending the
diffeomorphisms in (N=4+n)-dimensional space-time to a larger group,
called the conservation group. This is the largest group of coordinate
transformations under which conservation laws are covariant statements.
We present two theories that are invariant under the conservation group.
Both theories have field equations that imply the validity of Einstein'
equations for general relativity with the stress-energy tensor of a
non-Abelian Yang-Mills field (with massive quanta) and associated
currents. Both provide a geometrical foundation for string theory and
admit solutions that describe the direct product of a compact
n-dimensional space and flat four-dimensional space-time. One of the
theories requires that the cosmological constant shall vanish. The
conservation group symmetry is so large that there is reason to believe
the theories are finite or renormalizable.
--
Dave Pandres, Jr., Department of Mathematics and Computer Science
North Georgia College, Dahlonega, GA 30597
e-mail dpan...@nugget.ngc.peachnet.edu
(706) 864-1809 FAX (706) 864-1678
Steve Carlip
Greg Weeks (we...@dtc.hp.com) wrote:
: Steve Carlip (sjca...@ucdavis.edu) wrote:
: : In 2+1 dimensions, for example, there are consistent quantizations
: : in which the spacetime topology is fixed;
: I'm EXTREMELY intrigued. What form does the theory take?
: The theory can't be a local quantum field theory in the usual sense, since
: there is no fixed metric with respect to which local causality can be
: defined.
: If diffeomorphisms form the gauge group, then there are no local
: observables. (So there is no definition of the vacuum sector as the
: closure of local observables applied to the vacuum.)
In many ways, (2+1)-dimensional gravity is the ideal playground
for this kind of question. The issues you raise exist, in a
very clear form, classically. The classical vacuum Einstein
equations (with a cosmological constant, say) imply that the
spacetime manifold is one of constant Riemann curvature, so
even classically there are no local degrees of freedom. Never-
theless, when space has a nontrivial topology the classical
evolution can be highly nontrivial. For example, you can take
a closed universe with spatial slices of genus g>0 and slice
it along spatial slices of constant mean (extrinsic) curvature;
the resulting spatial geometry is then described by a set of
moduli that change with time, as well as a volume that increases
or decreases with time.
So already in the classical theory, one can ask how to reconstruct
a rather complicated local geometry from a set of well-defined
nonlocal moduli and their conjugate momenta. Once one has an
answer to this question---and there are several approaches to
the answer, depending on the classical formulation you prefer---
it is (sometimes) not too hard to convert that answer into a
statement about operators in a quantum theory. Depending on the
classical starting point, there are several known possible
quantum theories, which are not all equivalent.
You are absolutely correct in saying that there are no local
observers and no vacuum state.
The kinds of questions you can ask in one of these quantum
theories are the same as those you can ask classically, as
long as you are careful to phrase your classical questions
without implicit reference to a coordinate system. For
example: "If I look at the spatial slice for which the trace
of the extrinsic curvature is -17, and find the two shortest
noncontractible geodesics, what is the probability that the
angle between them is 31 degrees?" This kind of question
seems obscure, but the classical answers really do determine
the geometry, and there is a reasonable correspondence
principle.
If you want details, I have a pretty readable review article on
the gr-qc archives, gr-qc/9503024.
Steve Carlip
car...@dirac.ucdavis.edu
John Baez
Greg Weeks writes, concerning path integrals in quantum gravity:
>What do these path integrals mean?
You'll never figure out the laws of nature if you keep asking
questions like that! Shut up and calculate! :-)
>Minkowski path integrals are viewed (initially at least) as transition
>amplitudes, eg, <q2 t2 | q1 t1>. But what is the Hilbert space of |q>'s?
>And doesn't the space of |q>'s impose a global spatial topology?
The Hilbert space of |q>'s in Lorentzian quantum gravity is often
taken to be L^2(Met(S)), the space of square-integrable wavefunctions
on the space Met(S) of Riemannian metrics on "space", a given 3-dimensional
manifold S. In this approach, yes, we are assuming a fixed topology on
space from the very beginning. There is no obvious reason why it's
bad to do this. One would then hope to compute transition amplitudes
<q'|q> by doing a path integral over the space of Lorentzian metrics
on "spacetime", that is, S x [0,1].
What does the path integral MEAN? I.e., what does the transition
amplitude <q'|q> MEAN? It means "if the universe has some spacelike
slice on which the metric is q, what is the amplitude for it to have
some other spacelike slice in the future on which the metric is q'?"
Note: the reason we are working with the interval [0,1], that is,
with time going from 0 to 1, is that it makes no difference what interval
we use, thanks to the diffeomorphism-invariance of the theory. In other
words, the time t here serves only as an arbitrary coordinate function
on spacetime and we can reparametrize it without changing any of the physics.
If this freaks you out, welcome to the "problem of time"; it's one of the main
things that distinguishes quantum gravity from ordinary quantum field
theory on a spacetime with fixed metric. There is much more to say about
this, and Chris Isham's huge review article on the problem of time is
a good place to start. Carlo Rovelli's work is also very good to read.
The main *technical* problem with all of the above is that, a priori, we
have no idea what we mean by L^2 of an infinite-dimensional space like Met(S),
and we have no idea what we mean by the path integral. In the Ashtekar
approach one takes as ones classical configuration space not Met(S) but
the space Conn(S) of SU(2) connections on S. Thanks to work by people
like Rovelli, Smolin, Ashtekar, Isham, Lewandowski, and Baez, we now
have a way to define L^2(Conn(S)). It's not clear yet that it's the
"right" way, though, and we still don't know how to do the path integral.
In fact, most work along these lines takes the Hamiltonian rather
than the Lagrangian approach, so the problem becomes one of defining the
Hamiltonian constraint as an operator on L^2(Conn(S)).
For more on this you can download
http://math.ucr.edu/home/baez/net.tex
for a LaTeX file of an expository article called "Spin networks in
nonperturbative quantum gravity." This is also available in the
book Interface of Knots and Physics, ed. Louis Kauffman, American
Mathematical Society, Providence, Rhode Island, 1996. For a more
recent summary of the state of the art, you can download LaTeX file
http://math.ucr.edu/home/baez/hamiltonian.tex
of if you just want to eyeball something right away,
try the slightly watered-down html version,
http://vishnu.nirvana.phys.psu.edu/mog/mog8/node7.html
In addition to the approach where we integrate over metrics
on the spacetime S x [0,1], there is an approach where we
try to *sum* over all 4-manifolds going from the t = 0 copy
of S to the t = 1 copy of S. So we can either assume spacetime
has a fixed topology or "all possible" topologies. Steve Carlip has
already described these two alternatives in the context of 2+1-dimensional
gravity, which is simple enough that one can actually work things out
and compare the answers. Since he is the world's expert on this I
will let *him* answer any questions you may have along these lines.
>Alternatively, Euclidean path integrals may be viewed simply as solutions
>to the Euclidean Schwinger-Dyson equations. (The field equations imply
>Wightman function equations, and the Wightman function equations
>analytically continue to Schwinger function equations.) But what
>quantities in general relativity can be continued to some form of Euclidean
>space?
First a terminological aside:
As you know, there are "Minkowskian" and "Euclidean" path integrals
in quantum field theory. Similarly in quantum gravity one can
do path integrals over spaces of Lorentzian or Riemannian metrics.
People sometimes speak of path integrals over spaces of Riemannian metrics
as "Euclidean quantum gravity", but as I've noted before, this is
the result of ones mouth moving before ones brain is properly engaged.
The whole point of gravity is that spacetime is curved, not Euclidean!
"Riemannian quantum gravity" is a perfectly clear description of a
path integral over Riemannian metrics, and similarly for "Lorentzian
quantum gravity". (Luckily people don't usually speak of "Minkowskian
quantum gravity"; they usually call it just "quantum gravity". So please
don't start people talking about "Minkowskian quantum gravity.")
Okay:
In quantum field theory on flat spacetime, there are a bunch of theorems
saying you can analytically continue results from the Euclidean context
over to Minkowski spacetime. This justifies the use of Euclidean quantum
field theory, also known as "imaginary time", to solve real-world problems.
In quantum gravity there are no such theorems. This is not surprising,
because nobody understands quantum gravity. However, there are sharp
differences in opinion over whether one should *expect* such theorems ---
over whether one should expect to be able to study Lorentzian quantum
gravity by Riemannian path integrals. Clearly Hawking expects this,
because almost all he ever does is Riemannian quantum field theory. If
you examine his work you see he has a variety of tricks for translating
from the Riemannian context to the Lorentzian context --- even when
he does not assume a fixed topology for spacetime! Some of these
seem to invoke the idea that spacetime is asymptotically flat like Minkowski
spacetime, and use standard ideas about analytically S-matrix elements.
That makes sense in a rough-and-ready sort of way --- sort of like the
idea that interacting quantum fields are "free at past and future
infinity", which is often invoked in justifying S-matrix computations.
(Greg Weeks knows full well the limitations of this rough-and-ready
approach, as well as its successes.)
On the other hand, Hawking also uses Riemannian path integrals in
studying cosmology, and here one doesn't expect spacetime to be
asympotically flat. Presumably here one needs some other sort of
justification for analytically continuing ones results to get results
about the real world. Here is where I am very confused.
Other people, like Ashtekar, are more suspicious of
the use of Riemannian path integrals. Ashtekar has studied
Riemannian quantum gravity at times, but not with the expectation
that it will necessarily be directly relevant to the Lorentzian
theory.
>So I'm baffled.
Join the club. This is a subject full of wonderful puzzles.
>PS: Part of me blames myself for not reading the literature on the subject.
It's never too late to start. You'll definitely need to read a
wide selection of stuff to get a feel for the differing opinions
on these controversies.
john baez
http://math.ucr.edu/home/baez/net.tex
http://math.ucr.edu/home/baez/hamiltonian.tex
http://vishnu.nirvana.phys.psu.edu/mog/mog8/node7.html
First a terminological aside:
Okay:
because almost all he ever does is Riemannian quantum gravity. If
you examine his work you see he has a variety of tricks for translating
from the Riemannian context to the Lorentzian context --- even when
he does not assume a fixed topology for spacetime! Some of these
seem to invoke the idea that spacetime is asymptotically flat like Minkowski
spacetime, and use standard ideas about analytically continuing
S-matrix elements. That makes sense in a rough-and-ready sort of way ---
sort of like the idea that interacting quantum fields are "free at
past and future infinity", which is often naively invoked in justifying
Ross Tessien
In article <53e9hb$l...@math.ucr.edu>, ba...@math.ucr.edu says...
>
>Greg Weeks writes, concerning path integrals in quantum gravity:
>
>>What do these path integrals mean?
>
>You'll never figure out the laws of nature if you keep asking
>questions like that! Shut up and calculate! :-)
There are two possibilities to the above question. The first is that
there is no meaning. In this case, calculations can tell us about things
we ought to expect. But we will not gain intuition into any meaning or
foundations underlying the functioning of the universe. In this case,
the response is justified.
The second case is that there is some order at a level we have not yet
discovered. In this instance, we may or may not ever discover the
meaning, or the mechanism(s) at work in the phenomena we know as
gravitation. If we were to discover the principles at this level, then
it would indeed be possible for "meaning" to exist.
If the former is true, to attempt to discover meaning by asking questions
about the way things work will be fruitless. But, it will do no harm to
make the attempt.
If the latter is true, then by posing questions, there exists the
possibility that the structures may be discovered, and the "meaning"
found out.
I therefore respectfully submit that the comments above are out of line
in this forum, which purports to be one where serious questions are
welcomed in the effort to research new and old discoveries alike. If
such efforts are discouraged in the manner above, then we may as well be
satisfied with the foundations of physics we currently have.
These structures, by the way, do not give definition to any of the
concepts of time, space, energy, mass, force. Each is instead defined in
the manner they interelate with the others. But no underlying structure
is provided, and the physics used in quantum mechanics fail to yield the
cosmos as a solution. This fact leads many to seek the unification of
the forces into a single cohesive theory uniting gravitation and the
other forces.
I assume that the comment above was intended to be in good humor, but
some people may not understand that and the result is to impede the
presentation of questions that are not among the accepted concepts, even
though they are well presented and well intentioned, and within the
charter of this group.
I tremendously value the responses I have received from this group, and
am bothered by the above comment since I have been told many times to
simply shut up and quit researching the concepts I am working on.
The fact that we currently find no meaning attributable to the path
integral, does not mean that none exists. Thus, one should not be told
to "shut up".
Ross Tessien
[Mod. Note: As Ross points out, the answer which he objects to was made
in jest (in fact, I would say tongue in cheek). I do not want to put words
into John Baez's mouth, but my guess is that the intended humour of the
answer pivots on the fact that John, of all people, agrees whole heartedly
that such questions are not only useful but necessary. That the comment is
not meant to be taken seriously is supposedly made evident by the smiley
appended to it, which in internet circles is the universal indicator
that a joke has just been made (its very existence is a testament to the
fact that computer media are not the most suitable for expressing humour,
especially sarcasm). That Ross is worried about the comment is a testament
to the fact that such universal indicators are, to borrow a phrase, "well
known to those who know them well", and that they may not work for the
entire intended audience. I hope that, apart from a possible explanation of
the answer in question by John, the discussion of this comment will end
here, or, at worst, be taken up by e-mail, and that this will simply serve
as a reminder to all of us that USENET communication is fraught with
difficulty and that a certain amount of concession must be made by both
the writer and reader accordingly. - WGA]
john baez
Daryl McCullough <da...@cogentex.com> writes:
>In article <5398iq$1...@charity.ucr.edu>, ba...@math.ucr.edu says...
>>The loop representation of quantum gravity is an approach to
>>canonical (i.e., Hamiltonian) quantum gravity. As such, it
>>assumes a fixed topology of the form R x S, where S is
>>some 3-manifold.
>Does R x S mean the time evolution of 3-space?
Right.
>If so, is that perfectly general?
No, it's not.
>Aren't there 4-space solutions to general relativity that cannot
>be described that way?
Sure. There are all sorts of examples, some of which are very silly,
others of which are less so. For a very silly example, take
flat Minkowski spacetime and remove a point! The result is
a solution of the vacuum Einstein equations on a manifold that's
not of the form "R x S". This solution is silly because it
can be "completed" to get good old Minkowski spacetime. Nonetheless
one must understand this "completion" issue, and relativists have
studied it extensively.
Another less silly example would be T^4, the 4-torus, with its
flat metric. In other words, take the circle S^1, form the
product S^1 x S^1 x S^1 x S^1, and give it the metric
dt^2 - dx^2 - dy^2 - dz^2. This is also a solution of Einstein's
equations. It's a bit silly because it has timelike loops. But
are timelike loops really silly? (Even in a crazy subject like
quantum gravity?) Answering this question has led otherwise
perfectly physicists into an extensive discussion of time machines
and the like! It may seem far-out, and I hear it gave Kip Thorne
some difficulty getting grant money from the NSF, but after one
learns about the issues, one realizes that it's not obvious
that solutions with timelike loops are absurd.
I can't resist noting that Thorne's popular book,
"Black holes and time warps: Einstein's outrageous legacy"
gives a nice elementary discussion of these issues ---- and also
some amusing anecdotes about how he struggled, not entirely
successfully, to keep his research from being hyped in the press
with headlines like "CRAZY PROFESSOR INVENTS TIME MACHINE".
But I digress! The point is, there are quite a number of
more or less "pathological" solutions of Einstein's equations,
and when quantizing gravity one must decide which of these
are too "pathological" to worry about, and which aren't.
At least, this is what one must do if one QUANTIZES gravity
starting from the classical theory. Probably when we get
a good theory of quantum gravity, it will be based on
its own first principles, and not the result of waving the
magic wand called "quantization" over the classical theory.
Until we get this theory, however, it is natural to start
with what we know, quantum theory and general relativity, and
try to fit them together somehow --- so for this we need
to choose which solutions of general relativity to take
seriously.
Luckily, relativists have put a lot of work into this
over the last few decades. You can read a lot about it
in Hawking and Ellis' book "The large scale structure of
spacetime", and also Geroch's wonderful paper whose title
I forget. There are a lot of pathologies far weirder
than the ones described above.
>In particular, I'm thinking of the formation of
>singularities such as black holes, which appear never quite to happen
>if you try to describe things in terms of the time evolution of 3-space.
>Or is the loop representation mostly concerned with singularity-free
>evolution?
Yes.
The loop representation, like most "canonical" approaches to
quantum gravity --- i.e., *Hamiltonian* rather than *Lagrangian*
approaches --- implicitly concentrates attention on globally hyperbolic
solutions of Einstein's equations. We say a spacetime is
"globally hyperbolic" if it can be sliced into
spacelike slices like R x S, with each slice {t} x S being
a Cauchy surface. What's a Cauchy surface? Well, before
I get into that, let me just note that a solution of general relativity
in which a star collapses and black hole forms *is* globally
hyperbolic; we just need to pick the right slicing. So we
aren't a priori ruling out that sort of thing.
What's a Cauchy surface? Oh dear, a recursive series of definitions
looms: A "Cauchy surface" is a closed achronal set whose domain of
dependence is all of spacetime. A set is "achronal" if it
doesn't have two distinct points p and q in it such that we can
get from p to q by a future-directed timelike curve. (The
concept of "future-directed" assumes our spacetime is
time orientable --- roughly, that there's a global notion
of which timelike vectors are "future-pointing" and which
are "past-pointing.") The "domain of dependence" of a set
X is the set of points p for which every inextendible causal
curve through p intersects X. I won't define "inextendible
causal curve" except to say that it's roughly a curve that
doesn't go faster than light, and can't be extended to a longer
such curve.
IN SHORT, a Cauchy surface is one such that if you
know what's going on there, you should in principle be able
to calculate the whole history of the universe, if no
influences travel faster than the speed of light.
Globally hyperbolic spacetimes are fun! If you are
going to study the initial value problem for general
relativity, and figure out the sense in which the Hamiltonian
generates time evolution (classically), it's nicest to
start by understanding the globally hyperbolic case.
There's a decent explanation of all this stuff in Wald's
reputable tome, "General relativity".
Anyway, the point is this: quantum gravity is hard and
we barely know where to begin. Therefore many people would
feel it's no shame to try to find a quantum theory that'd
be the quantum analog of Einstein's equations *restricted*
to the case of globally hyperbolic spacetimes. Quite possibly
this won't fly. Then perhaps the mathematics gods are telling us
that globally hyperbolic solutions aren't sufficiently general.
(Meanwhile, other people are working on quantum gravity
from other starting-points.)
There is much more to say about this... people have thought
about these issues a lot... but I hope this is enough to
get going.
Ross Tessien
:-)
>That the comment is
>not meant to be taken seriously is supposedly made evident by the smiley
>appended to it, which in internet circles is the universal indicator
>that a joke has just been made (its very existence is a testament to the
>fact that computer media are not the most suitable for expressing
>humour, especially sarcasm). That Ross is worried about the comment is a
>testament to the fact that such universal indicators are, to borrow a phrase,
>"well known to those who know them well", and that they may not work for the
>entire intended audience.
I agree that this ends this particular thread, but wanted to make one
last point. I see now that it was I who was ignorant of the internet
smiley face, and had I known the meaning of the symbol above, (by looking
at it sideways for anyone who still didn't understand it), I would have
understood his comment was fully intended in jest as I had assumed. I
guess this just goes to show how varied in background the audience is.
Similar misunderstandings occur when those using English as a second
language are chided on their gramatical structuring of their sentences as
I have seen occur in the past.
It is amazing how much information about intent we relay in our facial
gestures that cannot be relayed in words in this forum.
In any case, I guess this is just a note to say that I understand and
should not have been bothered by John's comment due to his appropriate
appendage of the smiley face to show the tongue in cheek nature of the
comment. It was my error, but I hope that any one else like myself
learned about this internet character for future communication and
understanding of intent.
Thanks for the clarification.
Sincerely, Ross Tessien
Steve Carlip
: : ... Path integral approaches can fix topology or can include
: : sums over topologies ...
: What do these path integrals mean?
: Minkowski path integrals are viewed (initially at least) as transition
: amplitudes, eg, <q2 t2 | q1 t1>. But what is the Hilbert space of |q>'s?
: And doesn't the space of |q>'s impose a global spatial topology?
As usual in quantum gravity, no one has a very complete
answer. But here are a few pieces that might eventually
go into an answer:
1. Many of the attempts to use path integrals in quantum
gravity involve processes in which the initial and final
spatial topologies are identical, but a sum is taken over
some or all interpolating manifolds. In this case, the
relevant Hilbert space is the one that comes from your
favorite approach to canonical quantization, and the path
integrals can be interpreted as describing the effects of
virtual fluctuations of topology. This is the picture
relevant to many of the wormhole and "baby universe"
calculations in the literature, and to the attempts to
describe the effect of virtual black holes on particle
propagation.
2. The question is harder, obviously, when one looks at
path integrals in which the initial and final topologies,
and therefore (presumably) the initial and final Hilbert
spaces, are different. One place to look for answers is
the axiomatization of topological field theories by Segal,
Atiyah, and others. The axioms typically involve spaces
with multiple boundaries, with different Hilbert spaces
associated to different boundaries and with transition
amplitudes connecting various Hilbert spaces. These models
are not my area of expertise, and (at least as now formulated)
they do not apply directly to quantum gravity, but they are
at least examples of rigorously formulated quantum field
theories that allow topology change.
3. In (2+1)-dimensional gravity, some topology-changing
amplitudes can be computed explicitly by path integration.
Here the trick is that while the Hilbert spaces corresponding
to surfaces of different genus are different, the Hilbert
space for quantum gravity corresponding to a surface of genus
g is contained in that of a surface of genus g'>g. Roughly
speaking, the Hilbert spaces are spaces of L^2 functions on
genus g Teichmuller space, and the connection is related to
the existence of a universal Teichmuller space. In a more
easily visualizable picture, the idea is that you can get a
genus g surface from a genus g' surface by pinching off some
of the handles. It is certainly not obvious that any such
construction exists in higher dimensions (there has been
some speculation by Wheeler, but so far it's not much beyond
wishful thinking yet), but maybe...
4. The lower dimensional cases also raise some problems. One
I worry about a lot is the following. It seems reasonable to
suppose that amplitudes in quantum gravity should be invariant
under all spacetime diffeomorphisms, including the "large"
diffeomorphisms (those not connected to the identity). But
large diffeomorphisms can mix up the geometry of the initial
and final surfaces, and mix them up with the geometry of the
interpolating manifold, it is not at all clear how to take them
into account in a Hilbert space picture. Furthermore, it seems
very difficult to implement this invariance while still allowing
the possibility of splitting up an amplitude into two pieces
and summing over intermediate states. This problem is one of
the main difficulties in string field theory, and has caused
tremendous complications in what would otherwise be a rather
straightforward theory; it might also kill the counting arguments
used in "baby universe" computations.
Does this add sufficiently to the confusion?
Steve Carlip
car...@dirac.ucdavis.edu
Greg Weeks
I hope I'm not seen as looking a gift horse in the mouth.
john baez (ba...@math.ucr.edu) wrote:
: Hawking likes the "Euclidean path-integral approach" to quantum gravity.
: The word "Euclidean" is a horrible misnomer here, but it seems to have
: stuck. It should really read "Riemannian", the idea being to replace
: the Lorentzian metric on spacetime by one in which time is on the same
: footing as space.
It seems to me that applying "Euclidean" and "Minkowski" to quantum gravity
requires only the implicit prefix "locally". "Lorentzian" and "Riemannian"
seem more troublesome. I wouldn't be surprised if almost no one uses
"Lorentzian" for locally Minkowsky GR. And, in GR texts, "Riemannian" is
used for both locally Euclidean and locally Minkowski manifolds.
Hoping to digest the real issues sometime...
Greg
[Moderator's note: Which GR texts? I don't think that my two
favorites (Wald and MTW) commit this particular sin. I for one use
the terminology the same way John seems to, namely
Minkowskian : Euclidean :: Lorentzian : Riemannian.
For instance, a Lorentzian manifold is locally Minkowskian, while a
Riemannian manifold is locally Euclidean. -TB]
Greg Weeks
Greg Weeks (we...@dtc.hp.com) wrote:
: ... And, in GR texts, "Riemannian" is
: used for both locally Euclidean and locally Minkowski manifolds.
: Hoping to digest the real issues sometime...
: Greg
Oops. Given the potential for misunderstandings here in cyberspace, I
should point out that "Hoping to digest the real issues sometime..." was a
humorous, self-deprecating reference to the fact that, with all the physics
that responders had presented me with, I was commenting on an issue of
nomenclature. I was not digging at John for raising the issue in the first
place.
: [Moderator's note: Which GR texts?
Well, I must confess that my text is 45 years old. Still, I'd be surprised
to hear that Einstein's equation now involves "the Lorentzian curvature
tensor". :-)
[Moderator's note: The curvature tensor is still called the Riemann
tensor, even if the manifold over which it is defined is a Lorentzian
manifold (with a Lorentzian metric). In my experience, most textbooks
respect that distinction, at least most of the time. "Pseudo-riemannian"
is a synonym one sometimes sees for "Lorentzian," by the way. -TB]
Matt McIrvin
> And, in GR texts, "Riemannian" is
> used for both locally Euclidean and locally Minkowski manifolds.
I've usually heard the term "pseudo-Riemannian" used to describe the
latter.
--
Matt McIrvin <http://world.std.com/~mmcirvin/>
[Mod. Note: Let me throw my 2 cents in. I also use terminology exactly as
described by John Baez: Minkowski for flat with signature (-,+,+,+) (or
(+,-,-,-), although, being a relativist, I prefer the former), Euclidean for
flat with signature (+,+,+,+), Lorentzian for arbitrary curvature with
signature (-,+,+,+), and Riemannian for arbitrary curvature with signature
(+,+,+,+). Many mathematicians (including me, when I publish in mathematical
journals) use pseudo-Riemmanian rather than Lorentzian, which is not wrong,
but is less precise since it refers to any metric space with arbitrary
signature (mathematicians tend to be more interested in whether a metric is
sign definite or not than in what the particular indefiniteness is). Many
physicists use Euclidean, rather than Lorentzian, which is confusing and
wrong. All these adjectives are used to modify words such as "manifold",
"path integral", "quantum field theory", and (lest we forget the origin of
this thread), the non-flat ones are used for "quantum gravity", in which
case Euclidean is not only wrong, but oxymoronic. On the other hand, things
that live on the manifold, like curvatures, connections, derivatives, etc.,
are named after the same people, regardless of what the signature is. So
the Weyl, Riemann, Ricci, and Einstein curvatures are called just that,
on manifolds of every signature, as are Lie derivatives and Levi-Civita
connections. It is my (probably vain) hope that this little sermon can put
an end to this tangential discussion of nomenclature and return us to the
infinitely more interesting question of physics (mathematics?) of quantising
gravity. - WGA]
Charles Torre
Given this nice discussion of fixed topology vs. changing topology in quantum
gravity, let me throw two more possibilities into the arena. When considering
which classical structures to carry over into a quantization scheme, one can
also consider (1) fixed topology but change in differentiable structure;
(2) fixed topology and differentiable structure but non-zero gravitational
"kink number"*. Possibility (1) has been pretty well studied, mathematically
speaking, in recent years. Possibility (2) is intrinsically a Lorentzian
option and allows (I believe) non-singular spacetimes, which are not globally
hyperbolic, but which can live on, e.g., R^4, T^4, R x S, etc.
Has anybody seen quantum gravity research incorporating either of these
options?
* Finkelstein and Misner, Ann. Phys. 6, pg. 320 (1959).
Charles Torre
Department of Physics
Utah State University
Logan, UT 84322-4415 USA
Greg Weeks
john baez (ba...@math.ucr.edu) wrote:
: implement the diffeomorphism constraint, which is part of projecting onto
: the physical states. The physical observables are diffemorphism-invariant,
: not localized.
Do these observables make sense in classical GR? Does classical GR have
an algebra of observables?
: For every approach to quantum gravity that I know of, I can easily list
: plenty of potentially fatal objections! I would be utterly shocked if any
: approach currently known actually *worked*. We are just fumbling around
: now and we will almost certainly need a bunch of revolutionary new ideas to
: solve the problem.
I have no problem with this. My problem was with the view of quantum
gravity as just another field theory. I am thankful that the many
variations on that theme have NOT panned out. However, the prevalence of
this view in some circles means that I, as a part-time physics hobbyist,
can use some guidance regarding where I spend my time. And this thread has
been helpful.
Despite the occasional reference to nontrivial topologies in string theory,
the approaches closest to the spirit of GR appear to be Hawking's
Riemannian path integrals (which no one in this thread quite understands)
and Ashtekar's loop approach.
Now, I did indeed have a cow about the canonical quantization of the loop
approach. That was hasty.
Greg
john baez
Greg Weeks (we...@dtc.hp.com) wrote:
john baez (ba...@math.ucr.edu) wrote:
>: The physical observables are diffemorphism-invariant,
>: not localized.
By the way, for folks who don't know what "diffeomorphism-invariant"
means, let me say it means something sort of like "independent of
any choice of coordinates". That's not quite right, but it'll
give you a flavor of what it means, and why it's important.
>Do these observables make sense in classical GR? Does classical GR have
>an algebra of observables?
Oh dear, you have jumped right into one of the most knotty issues
of all: the "problem of time". Before I say anything else, let me
say that there has been a lot of controversy concerning this problem.
For anything I say, there will be some smart person who believes
the opposite. Even worse, anything I say *quickly* will be
somewhat inaccurate.
Nonetheless I will plunge in. Take everything with a big grain of
salt, okay?
First note that, even classically, any quantity we can actually measure in the
lab or the cosmos is diffeomorphism-invariant, so it makes decent sense
to demand that our observables be diffeomorphism-invariant. It is
a curious fact that in physics EXCEPT for general relativity and
quantum gravity, we seem NOT demand our observables be
diffeomorphism-invariant. But in classical general relativity,
the predictions are apparently diffeomorphism-invariant. Moreover, if we apply
the Dirac quantization procedure or some variant thereof to quantum
gravity, the final physical observables are *required* to be diffeomorphism-
invariant.
But what are the diffeomorphism-invariant observables in quantum
gravity? Nobody *knows* what they are in quantum gravity are, though
there are some good guesses out there. It's quite possible that
if and when we get them, we will have some trouble relating them to
observables in the classical theory. Part of the problem is that
mathematical structure of the observables in the classical theory are
still rather poorly understood, though we "know them when we see them".
I could write a huge amount more about this, but I think you
mentioned that you haven't quite had time to absorb the enormous
responses your previous questions evoked, so I will spare you.
Let me simply note that you should always keep this problem
in mind as you try to understand quantum gravity,
because it is, in a way, THE problem of quantum gravity: how to
do quantum physics in a diffeomorphism-invariant way. (Or
perhaps a diffeomorphism-covariant way?)
I already suggested that you read Chris Isham's big paper about this:
Canonical Quantum Gravity and the Problem of Time,
Chris J. Isham, 125 pages of LaTeX output, Imperial/TP/91-92/25
Available electronically as gr-qc/9210011
It's an excellent survey of the current viewpoints on the
subject. One viewpoint which he doesn't cover, since
it wasn't around back then, is the viewpoint of n-category
theory.
John Baez
gravity? Nobody *knows* what they are in quantum gravity, though
there are some good guesses out there. It's quite possible that
if and when we get them, we will have some trouble relating them to
observables in the classical theory. Part of the problem is that
mathematical structure of the observables in the classical theory are
still rather poorly understood, though we "know them when we see them".
I could write a huge amount more about this, but I think you
mentioned that you haven't quite had time to absorb the enormous
responses your previous questions evoked, so I will spare you.
Let me simply note that you should always keep this problem
in mind as you try to understand quantum gravity,
because it is, in a way, THE problem of quantum gravity: how to
do quantum physics in a diffeomorphism-invariant way. (Or
perhaps a diffeomorphism-covariant way???)
I already suggested that you read Chris Isham's big paper about this:
Canonical Quantum Gravity and the Problem of Time,
Chris J. Isham, 125 pages of LaTeX output, Imperial/TP/91-92/25
Available electronically as gr-qc/9210011
It's an excellent survey of different viewpoints on the
subject.
Greg Weeks
John Baez (ba...@math.ucr.edu) wrote:
> Oh dear, you have jumped right into one of the most knotty issues
> of all: the "problem of time".
>
> Part of the problem is that mathematical structure of the observables in
> the classical theory are still rather poorly understood, though we "know
> them when we see them".
>
> I already suggested that you read Chris Isham's big paper about this:
I must confess that I don't understand why you call this issue "the problem
of time". (Also, I've read the Isham paper carefully.) There is something
weird going on, and you started to discuss it. I might as well lay my
cards on the table and continue it.
I consider quantum physics to be most elegantly formulated as an abstract
algebra of observables. However, in every case, there is something bogus
about the observables. These observables require that the stuff of the
universe be divided into two categories: First, there is the system under
observation. Second, there is the measurement apparatus. The measurement
apparatus may be used in particular to label points in spacetime. Quite
obviously, the spacetime point (0,0,0,0) has no absolute meaning. To give
it meaning, some physical "apparatus" is required.
In general relativity, the above separation is impossible. There is no way
to create measurement apparatus that is not influenced by the system under
observation. (There is no substance that is gravitationally neutral.)
So all these slightly-bogus observables become decidedly unobservable.
The true observables in general relativity are similar to the observables
of an irregular piece of glazed pottery that someone is describing to you
from another room. You can't ask the color of the glaze at a particular
point, because you have no way of identifying any particular point a
priori. You CAN ask, for example, if there are any points with a
particular scalar curvature, or whether the glaze at the (unique?) point of
greatest curvature is purple. As you can see, the set of observables that
comes to mind is much less rich and regular than the set of obervables in
theories other than general relativity.
There is much more to be said here, and it could be said much more
pedagogically. And I don't feel that I really grok it anyway. Do any
textbooks deal with this issue?
> I could write a huge amount more about this, but I think you mentioned that
> you haven't quite had time to absorb the enormous responses your previous
> questions evoked, so I will spare you.
Ah, but the one topic I _am_ allowing myself to pursue the the issue of
observables in classical GR.
Greg
PS: And another thing: Even if I miraculously DID have a rich set of
classical GR observables, and even if I did manage to quantize it, it might
be that I wouldn't be able to interpret the algebra experimentally. I can
just barely interpret algebras in SR quantum physics. The interpretation
of the GR algebra -- which we are two steps away from having -- might
indeed lead us to a "problem of time".
Ross Tessien
In article <53nf4u$m...@charity.ucr.edu>, ba...@math.ucr.edu says...
>The other route, the route we're talking about now, is to face
>up to the *nonperturbative* quantization of theories *without a
>fixed background metric*. This is vastly different than ordinary
>quantum field theory. I can't impress upon you sufficiently how
>drastically different it is --- all the eternal verities of quantum
>field theory become highly suspect.
1) Is "fixed background metric" another way of saying a "nodal structure
of spacetime", where the latter is meant to mean a structure that
permeates the known universe and provides a reference for the dimensions
of space and of time?
2) If so, are these structures treated as rigid and fixed, or as
deformable?
For example, in a metallic lattice of atoms, there are at times
dislocations. Does the background metric ever get treated in a manner
that would allow for such distortions which I suppose might lead to the
concept of curvature of spacetime in gravitational fields.
john baez
In article <1996Oct14.1...@cc.usu.edu>,
Charles Torre <to...@cc.usu.edu> wrote:
>When considering
>which classical structures to carry over into a quantization scheme, one can
>also consider (1) fixed topology but change in differentiable structure;
>(2) fixed topology and differentiable structure but non-zero gravitational
>"kink number"*. Possibility (1) has been pretty well studied, mathematically
>speaking, in recent years. Possibility (2) is intrinsically a Lorentzian
>option and allows (I believe) non-singular spacetimes, which are not globally
>hyperbolic, but which can live on, e.g., R^4, T^4, R x S, etc.
>Has anybody seen quantum gravity research incorporating either of these
>options?
I don't really understand this "kink" stuff, but at the quantum gravity
workshop in Vienna, Thomas Strobl spoke about "kinks" in various
2-dimensional gravity theories, and also about the quantization
of these theories. I don't know if he has gotten around to quantizing
these kinky solutions, but you might try:
hep-th/9607226
Title: Solutions of Arbitrary Topology and Kinks in 1+1 Gravity
(Classical and Quantum Gravity in 1+1 Dim., Part III)
Authors: T. Kloesch, T. Strobl
Comments: 45 pages, 26 figures, uses AMSTeX
Bruce Scott TOK
John Baez (ba...@math.ucr.edu) wrote:
[...]
: Let me simply note that you should always keep this problem
: in mind as you try to understand quantum gravity,
: because it is, in a way, THE problem of quantum gravity: how to
: do quantum physics in a diffeomorphism-invariant way. (Or
: perhaps a diffeomorphism-covariant way???)
John, doesn't this spring from the fact that QM (or QFT) is always set
up in terms of Hamiltonians, rather than working straight from the
Lagrangians? That is, by using Hamiltonians it is forcing you into the
3+1 space+time split, which you would rather not do in GR.
I've never even seen this addressed in a QM or QFT text; they just jump
right in with the Hamiltonian...
--
Mach's gut!
Bruce Scott, Max-Planck-Institut fuer Plasmaphysik, b...@ipp-garching.mpg.de
Remember John Hron: http://www.nizkor.org/hweb/people/h/hron-john/
[Moderator's Note: This is one of the manifestations of the problem of
time, and I know that it is addressed in many discussions of QG. One can
sort of weasel out of it using path integrals, which use the Lagrangian
and hence treat all coordinates more or less equally. The problem with
that approach is that path integrals for Lorentzian spacetimes encounter
singularities (just as path integrals for QFT on Lorentzian backgrounds
do). In QFT, we overcome this by computing the Euclidean path integral and
then performing the Wick rotation back to the Minkowski domain, hence
giving us the Feynman propogator uniquely. In QG, there is no reason for
believing that the Wick rotation makes any sense. Since the Wick rotation
is basically a rotation from real to imaginary time, we are again
encountering the problem of time, only in a different guise. I bet John
can say a great deal about some of it's other manifestations. -WGA]
Greg Weeks
Bruce Scott TOK (b...@ipp-garching.mpg.de) wrote:
: John Baez (ba...@math.ucr.edu) wrote:
: : THE problem of quantum gravity: how to
: : do quantum physics in a diffeomorphism-invariant way.
: John, doesn't this spring from the fact that QM (or QFT) is always set
: up in terms of Hamiltonians, rather than working straight from the
: Lagrangians? That is, by using Hamiltonians it is forcing you into the
: 3+1 space+time split, which you would rather not do in GR.
: [Moderator's Note: This is one of the manifestations of the problem of
: time, and I know that it is addressed in many discussions of QG. One can
: sort of weasel out of it using path integrals, which use the Lagrangian
: and hence treat all coordinates more or less equally. The problem with
Perhaps I should EXPLICITLY state the point that I believe John has alluded
to:
In special relativity, spacetime is equipped with a fixed metric. The
symmetries of spacetime (+ metric) are the Poincare group. The
observables of the theory are Poincare-covariant. They transform under
the Poincare group in such a way that the laws of nature are
Poincare-INvariant. There are many such observables: eg,
energy-momentum, electic current, electromagnetic field, etc.
In general relativity, space-time is not equipped with a fixed metric.
The symmetries of spacetime are the diffeomorphisms. The observables
of the theory are diffeomorphism invariant. [Not covariant!] Such
observables are awkward and difficult to write down: eg, the global
topology (homology?) of the whole universe, the maximum scalar
curvature of the whole universe, etc. Curvature, energy-momentum,
electric current, and so on are NOT observable.
Why are special and general relativity so radically different, with
spacetime-symmetry COvariant observables in one case and spacetime-symmetry
INvariant observables in the other? I don't have a simple answer for that.
(I provided a shaky one in a previous post.)
Note, by the way, that this distinction has little to do with Hamiltonian
vs Lagrangian or with canonical quantization vs path integral.
Or so it seems to me.
John, do you a-gree?
Greg
[Moderator's Note: Thanks for pointing out that the discussion of the
problem of time is only part of the diffeomorphism invariance problem. I
should have made that clear in the moderator's note quoted above but did
not. Now, a few words of clarification about the distinction between
special and general relativity. There are, as I see it, 2 distinctions to
be made. The first is the distinction between field theories on a fixed
background spacetime and the field theory of the spacetime itself. For
field theories on a fixed background, we distinguish between those on flat
space (or other spaces with timelike killing fields) and those on curved
backgrounds. If I understand you correctly, when you say "special
relativity" you mean quantum field theory on a flat background and when
you say "general relativity" you mean quantisation of general relativity
itself, ie quantisation of the background itself.
If this is the case then it seems to me that worrying about observables is
a bit premature at this time. What we need first are 1) a state space and
2) an operator algebra on that state space. Once we have this we can worry
about which operators correspond to observables (if, indeed, such a concept
even makes sense in the context of a closed system). As I understand it, it
turns out that diffeomorphism invariance is a problem even at the level of
the Hilbert space. In fact, even QFT on a fixed but curved background runs
into the problem of diffeomorphism invariance. The issue there is that
quantisation, by its very nature, treats time and space differently. In
doing so, it is not diffeomorphism invariant. If I quantise, then hit my
spacetime with a diffeomorphism that changes the time coordinate, then
quantise again, I get ineqivalent quantum theories. Unless I have a
preferred time coordinate, I don't know which of these theories to choose.
Thus, quantum field theory on Lorentzian manifolds seems to be, by its
very nature, diffeomorphism variant.
Now let me explain my shaky understanding of how this extends to
quantisation of the manifold itself. In this case, not only do we not have
a preferred time, we have no time at all a priori. The canonical approach
is to fix a boundary three geometry and quantise from the 4-geometry
subject to those boundary conditions. We mod out the gauge degrees of
freedom (diffeomorphisms of the boundary 3 geometry) but this does not fix
the gauge entirely, since there is 4-diffeomorphism freedom that is not
fixed by specifying the 3-gauge. Roughly speaking, these extra degrees of
freedom correspond to a choice of time coordinate. Thus, diffeomorphism
dependence again rears its ugly head, in a more profound way.
Let me finish by saying that I would greatly appreciate one of the local
experts confirming, denying or clarifying this understanding for me. -WGA]
Peter Peldan
Greg Weeks wrote:
>"In standard QM":
> These observables require that the stuff of the
> universe be divided into two categories: First, there is the system under
> observation. Second, there is the measurement apparatus.
>
> In general relativity, the above separation is impossible. There is no way
> to create measurement apparatus that is not influenced by the system under
> observation.
I'm not sure i agree on this. My opinion is that the present form of
quantum mechanics/QFT is constructed to be used in a situation where we
can distinguish between the qm system of interest and the classical
measuring apparatus, and we should therefore first try yo quantize
gravity in this setting: we should study manifolds with boundary
imbedded in a larger manifold. Only the part inside the boundary should
be quantized, and we keep the outside classical. In this way, we get a
reference system: the coordinates on the boundary and we may relate the
observables to that reference system. However, we should still require
all physical observables to be invariant under all "gauge
transformations" which now consists of the set of diffeomorphisms that
leaves the boundary invariant. The rest of the diffeomorphisms (the ones
that transforms the boundary) should be considered as symmetry
transformations and we only have to require unitarity for the
observables under these transformations.
Just my 2 öres worth of thoughts
Peter Peldan
john baez
In article <54350o$12...@pulp.ucs.ualberta.ca>,
Greg Weeks <we...@dtc.hp.com> wrote:
>I must confess that I don't understand why you call this issue "the problem
>of time". (Also, I've read the Isham paper carefully.)
Perhaps Isham gets into the technical aspects a bit too soon, assuming
the reader is already a bit familiar with the basic philosophical
problem. It can be hard to relate the philosophical problem to the
technical aspects!
>There is something weird going on, and you started to discuss it.
>I might as well lay my cards on the table and continue it.
>I consider quantum physics to be most elegantly formulated as an abstract
>algebra of observables. However, in every case, there is something bogus
>about the observables. These observables require that the stuff of the
>universe be divided into two categories: First, there is the system under
>observation. Second, there is the measurement apparatus. The measurement
>apparatus may be used in particular to label points in spacetime. Quite
>obviously, the spacetime point (0,0,0,0) has no absolute meaning. To give
>it meaning, some physical "apparatus" is required.
>In general relativity, the above separation is impossible. There is no way
>to create measurement apparatus that is not influenced by the system under
>observation. (There is no substance that is gravitationally neutral.)
>So all these slightly-bogus observables become decidedly unobservable.
Right!!! This is exactly what people call the "problem of time". Your
description is very nice, and indicates that you understand the problem
perfectly.
A better name might be the "problem of situated observables," because
its essence is that we can only situate the observable we are trying
to measure --- i.e., specify where and when we are trying to measure it ---
relative to other features of the world, features that *depend on the
state of the world*.
However, the name "problem of time" is what people use, because it has most
often been studied in the context of canonical quantization, as in Isham's
paper. In this context people emphasize the following aspect. We often speak
of setting up experimental conditions at t = t0 and performing a measurement
at t = t1. Of course, these times are measured with a watch. An ideal
classical watch might measure the elapsed proper time along its worldline.
This depends on the metric. But in general relativity, the metric depends
on the experimental setup at t = t0. Thus, unlike in special relativity,
we cannot treat the watch as decoupled from the experiment it is timing.
However, there is nothing really special about time here; the measurement
of spatial locations is similarly affected.
(A bit more technically speaking, the problem is that to relate the space of
solutions of Einstein's equation modulo spacetime diffeomorphisms to the
space of initial data, we need to solve the equations of motion. The physical
observables are presumably functions on the former space, but we know
much more about the latter space.)
Note that the above issue shows up already at the classical level. One
needs to deal with it somehow or other in all general relativity problems.
We have experience in doing it, classically. It becomes more
problematic in quantum gravity. Now your watch is subject to the
uncertainty principle, and so is the metric it interacts with! You may
not be able to discount the back-reaction --- the effect of the watch on
the metric! And so on....
>The true observables in general relativity are similar to the observables
>of an irregular piece of glazed pottery that someone is describing to you
>from another room. You can't ask the color of the glaze at a particular
>point, because you have no way of identifying any particular point a
>priori. You CAN ask, for example, if there are any points with a
>particular scalar curvature, or whether the glaze at the (unique?) point of
>greatest curvature is purple.
Exactly! Now just imagine that your piece of pottery is very much a
quantum-mechanical thing, and see how it gets even harder.
What if "curvature" and "purpleness" don't commute? --- and so on.
(I often use what I call the "coke bottle analogy" myself, but
your version is a bit more classy.)
>As you can see, the set of observables that
>comes to mind is much less rich and regular than the set of obervables in
>theories other than general relativity.
I would prefer to say that we have not yet come to terms with
this sort of problem; the mathematics hasn't been developed yet,
so it seems like a huge mess. In the long run it may turn out
that the diffeomorphism-invariant observables are mathematically nicer
when we take quantum mechanics into account. This seems to be happening
in the loop representation (if one is a bit optimistic).
>There is much more to be said here, and it could be said much more
>pedagogically. And I don't feel that I really grok it anyway. Do any
>textbooks deal with this issue?
It's difficult to find nice textbook treatments of important areas of
ignorance. I repeat my earlier suggestions: try the book
Conceptual Problems of Quantum Gravity, edited by Abhay Ashtekar and
John Stachel, based on the proceedings of the 1988 Osgood Hill
Conference, 15-19 May 1988, Birkhaueser, Boston, 1991.
especially the articles
Is there incompatibility between the ways time is treated in general
relativity and in standard quantum mechanics?, by Carlo Rovelli
The problem of time in canonical quantization of relativistic systems,
by Karel V. Kuchar.
and
Space and time in the quantum universe, by Lee Smolin.
I also find Rovelli's other papers on this subject very enlightening.
If anyone groks this problem, he does, in my opinion. You can probably
find references to his work in the paper above, or in Isham's paper, which
has an extensive bibliography.
john baez
In article <54360n$11...@pulp.ucs.ualberta.ca>,
>: Let me simply note that you should always keep this problem
>: in mind as you try to understand quantum gravity,
>: because it is, in a way, THE problem of quantum gravity: how to
>: do quantum physics in a diffeomorphism-invariant way. (Or
>: perhaps a diffeomorphism-covariant way???)
>John, doesn't this spring from the fact that QM (or QFT) is always set
>up in terms of Hamiltonians, rather than working straight from the
>Lagrangians? That is, by using Hamiltonians it is forcing you into the
>3+1 space+time split, which you would rather not do in GR.
Isham told me he has observed that in general relativity and
quantum gravity there are two kinds of people: "three people"
and "four people". The "three people" think of space and time
as very different, and like canonical quantization (also known
as the Hamiltonian approach). The "four people" think of spacetime
as a unified thing, and like path-integral quantization (also known
as the Lagrangian approach). He believes this is a basic Jungian
opposition of archetypes... but I digress.
One might hope that the problem of time in quantum gravity is
only a problem for "three people", and that it'd just disappear
if one used the Lagrangian approach. However it doesn't seem so.
In the Lagrangian approach, naive path integrals are ill-defined,
*apparently* in part because one is integrating over the space of
all metrics on spacetime when one should be integrating over the
space of metrics modulo diffeomorphisms. In other words, one is
overcounting, by treating situations as different when they
differ only by a coordinate transformation.
The same sort of problem happens in the path-integral approach to gauge
theories like Yang-Mills theory. There the culprit is not diffeomorphisms
but gauge-transformations. One should really do integrals over the
space of connections modulo gauge transformations. However, in this
context people have a recipe for getting around the problem, at least
at the level of perturbation theory. They "gauge-fix" by choosing
(or trying to choose) one connection for each gauge equivalence class.
Technically, a popular way to do this is introduce extra fields into
the theory, called ghosts, which implement the gauge-fixing.
In perturbative Yang-Mils theory, with ghosts included, one gets
a renormalizable field theory. One can make physical predictions!
In perturbative quantum gravity one does not. One is stuck!
This is not to say that the path-integral approach is useless.
Indeed, Hawking has used it to come up with all sorts of interesting
stuff. I'm just saying that it's not a panacea, and that most of
the same basic problems appear to show up, in different guises.
At a more fundamental level, one can argue that n-point
Green's functions, the basic output of standard Lagrangian field
theory, are bound to be meaningless in a diffeomorphism-invariant
theory, because all configurations of n distinct points in spacetime
are isotopic to each other (in a connected spacetime).
Greg Weeks
: [Moderator's Note: ...
: If I understand you correctly, when you say "special
: relativity" you mean quantum field theory on a flat background and when
: you say "general relativity" you mean quantisation of general relativity
: itself, ie quantisation of the background itself.
Yes, except that I was not thinking of quantum theories. I guess the
comments apply both classically and "quantumly".
: If this is the case then it seems to me that worrying about observables is
: a bit premature at this time. What we need first are 1) a state space and
: 2) an operator algebra on that state space.
We have different operator algebras.
For example, in QED, the algebra I refer to contains (bounded functions of)
the energy-momentum tensor, the electric current, the electromagnetic
fields, and so on. It does not contain gauge-variant unobservables like
the Psi and A fields. The algebra you refer to does contain these fields.
Indeed, you have an infinite set of algebras to choose from, corresponding
to different methods of gauge-fixing.
For me, the algebra of OBSERVABLES is the fundamental description of the
theory. It corresponds -- with a few layers of interpretation in between
-- to what we actually observe. To me, a classical theory is quantized by
replacing a classical (commutative) set of observables with a quantum
(noncommutative) set of observables (in such a way that the equations of
motion are the same). But, in GR, there doesn't seem to be an appealing
set of classical observables.
The notion that observables need not be considered classically is new to me
and interesting. However, in SR theories, it seems to be possible to take
at least a formal h-->0 limit of the observable algebra, resulting in the
classical (commutative) observable algebra. If the same holds in GR, then
quantum observables imply classical observables. That is my guess. So I
find it natural to consider the classical observables, which HAVE to be
there, first. But maybe I'm mistaken. Hmmm.
Greg
john baez
In article <54609j$10...@pulp.ucs.ualberta.ca>,
Greg Weeks <we...@dtc.hp.com> wrote:
> In general relativity, space-time is not equipped with a fixed metric.
> The symmetries of spacetime are the diffeomorphisms. The observables
> of the theory are diffeomorphism invariant. [Not covariant!] Such
> observables are awkward and difficult to write down: eg, the global
> topology (homology?) of the whole universe, the maximum scalar
> curvature of the whole universe, etc. Curvature, energy-momentum,
> electric current, and so on are NOT observable.
> Or so it seems to me.
> John, do you a-gree?
Basically yeah.
Although a previous post of yours indicated you understood it perfectly
already, let me clarify this point, because it is very important:
In classical general relativity, the value of some field *at a point with
specified coordinates in some coordinate system* is not, technically,
an observable. The reason is that for any solution S of Einstein's equations
coupled to matter, there are lots of other solutions S' differing only by
spacetime diffeomorphisms (i.e., roughly, coordinate changes.) The solutions
S and S' describe the same universe in different coordinates, but the value
of, say, the temperature at the point with coordinates (2,3,-7,1) may
be very different in these two solutions. Therefore we do not want to call
the value of a field at a point with specified coordinates an observable.
On the other hand, the value of a field *at a point whose location is
determined by the values of other fields* is an observable, because
it is diffeomorphism-invariant. Roughly speaking, "the temperature in the
clock tower at U. C. Riverside when the bell tolled noon on October
18th, 1996" is an observable, because we don't need any coordinate system
to find this point and see what the temperature is there.
However, the above example has limitations which show the subtlety
of the problem. The temperature in the clock tower at U. C. Riverside
when the bell tolled noon on October 18th, 1996 may be well-defined
in states of the universe where there *is* a U. C. Riverside, there *is* a
Gregorian calendar, and so on, but in many states of the universe we could
hunt far and wide for this point in spacetime and never find it. (Worse,
there might be two U. C. Riversides with clock towers, or U. C. Riversides
with dubious entities that we might or might not consider a clock tower.)
In short, observables that are situated at locations determined by the
values of other fields have an annoying tendency to be defined only on
part of the state space, sometimes with the exact domain of definition
being a matter of taste.
This is usually no big deal in classical general relativity, where one
is usually only interested in a small part of the state space at a time.
We don't need to compute the precession of the perihelion of Mercury in
all possible universes --- in most of which the very concept of "Mercury" is
highly ambiguous at best! It suffices to do so for our universe, or
states of the universe very similar to ours. E.g., we could compute
the precession of the perihelion of Mercury assuming that Mercury were
10000 times heavier than it actually is and Jupiter 10000 times lighter,
and only a loony or philosopher would complain that this counterfactual
makes it problematic whether the planet we are studying is still "Mercury";
only a forgetful physicist would get mixed up about what we were calling
"Mercury" and what we were calling "Jupiter".
But these issues, which seem like laughable pedantries in the classical case,
start becoming more worrisome in the quantum case. We would like to
have "situated observables" that are mathematically easy to work with and
correspond nicely to familiar quantities in the classical theory. The first
demand is hard enough already, and combining the second demand makes
it still harder.
What if we get wonderful observables in quantum gravity and have no idea
of what they correspond to in the classical theory? Is this not unbearable?
No, actually it's perfectly okay; for example, as Ashtekar has repeatedly
noted, in the study of atomic physics the quantum numbers of an
orbital --- n, l, and m --- are perfectly fine observables,
even though they are very different from the observables we regarded
as fundamental classically (the positions and momenta of the electrons).
However, it took us a while to get used to these new observables;
for a while the Bohr atom helped out by catering to our classical
intuition about orbits, but eventually we learned to think in a new way.
Of course, part of why it works is that we do know how to relate the
quantum mechanics of atoms to classical mechanics; and similarly, we
will need *some* ways to relate quantum gravity to classical general
relativity, even if the best observables in the quantum theory are
very different from the classical quantities we know and love.
>Why are special and general relativity so radically different, with
>spacetime-symmetry COvariant observables in one case and spacetime-symmetry
>INvariant observables in the other? I don't have a simple answer for that.
>(I provided a shaky one in a previous post.)
The snappy answer is that in special relativity we have a fixed background
geometry: a field, the metric, that appears implicitly in the Lagrangian but
is regarded as fixed rather than a variable. This is obviously crazy
because it affects the motion of other particles and fields without in turn
being affected, violating the Newton's third law. (I'm joking here, of course,
but I could turn this joke into perfectly rigorous mathematical physics,
too.) The symmetries of this field give a preferred subgroup of the
diffeomorphism group --- the Poincare group. In general relativity
we have no fixed background geometry, hence no Lie group like this to
work with. Physics with no background geometry is more honest, but
we are not at all used to it yet, particularly in the quantum case.
We are used to the approximation where one field affects but is not
affected by the rest.
A general remark:
There are many potentially valid alternative ways of thinking about
these issues, so as you read about these things you will never find
any two authors who say exactly the same thing, and this can be very
confusing, particularly when both authors seem to be very smart. Often
very different-sounding approaches to these issues turn out, upon
reflection, to be compatible.
Paul Budnik
john baez (ba...@math.ucr.edu) wrote:
: In article <54350o$12...@pulp.ucs.ualberta.ca>,
: Greg Weeks <we...@dtc.hp.com> wrote:
[...]
: >I consider quantum physics to be most elegantly formulated as an abstract
: >algebra of observables. However, in every case, there is something bogus
: >about the observables. These observables require that the stuff of the
: >universe be divided into two categories: First, there is the system under
: >observation. Second, there is the measurement apparatus.
: [this might be called] the "problem of situated observables," because
: its essence is that we can only situate the observable we are trying
: to measure --- i.e., specify where and when we are trying to measure it ---
: relative to other features of the world, features that *depend on the
Why is the distinction between observables and experimental apparatus
so fundamental in QM. Because observables are not preexisting things.
In QM we do not observe what exists, we create an observable by the
act of observation. Yet the experimental apparatus must exist prior
to the observation and we must be able to control and adjust it. In
standard QM this is fine because the apparatus is classical and quantum
effects are not significant. This bit of duplicity is not
possible in quantum gravity where quantum affects influence everything.
The problem may be more fundamental than anything seriously discussed
in mainstream physics. I think we are still hung up in the conceptual
framework of classical physics and that framework can never really work
in the quantum domain. It inevitably forces us in to a false distinction
between observer and observables.
For me there are two starting point for addressing this issue.
First is the principle point of EPR, quantities that are conserved
absolutely must correspond to some objective element of reality.
Second is the fact that observables are created or at least evolve from
the act of observation. These imply that there is a process completely
outside of existing theory that evolves particular observations and
enforces absolute conservation laws. This in turn suggests
there is an objective state of the universe but this state *cannot be
described in terms of classical observables*.
Perhaps Einstein was that extreme rarity among geniuses who not only
created a revolution but had the vision to see beyond the revolution
that he created.
I consider it quite possible that physics cannot be based on the
field concept, i. e., on continuous structures. In that case
*nothing* remains of my entire castle in the air gravitation
theory included, [and of] the rest of modern physics.
-- Einstein in a 1954 letter to Besso, quoted from:
_Subtle is the Lord_, Abraham Pais, page 467.
On my web site I go into some detail on how such an approach may explain
quantum gravity and everything else. Of course this is all extremely
speculative and a long way from a developed scientific theory, but it
is unlikely that a real solution to this problem is going to come into
the world as a full blown theory. It will start with some vague intuition
that has the potential to become a theory.
--
Paul Budnik
pa...@mtnmath.com, http://www.mtnmath.com