Group averaging vs Gupta-Bleuler
Urs Schreiber
I was trying to figure out what exactly it is in Th. Thiemanns quantization
hep-th/0401172 of what he calls the 'LQG-string' that makes it so different
from the usual quantization. I now believe that the crucial issue is how to
impose the constraints.
Let there be a physical theory with constraints C_I, I in some set. Then
requiring
<psi|C_I|psi> = 0
is what, in the case where the C_I are the Virasoro constraints, leads to
the usual quantum string.
On the other hand the group averaging method used by Thiemann imposes (see
below his eq. (5.4))
<psi|exp(C_I)|psi> = 0 .
This may seem like essentially the same thing, but the crucial issue is
apparently that the latter form allows to deal quite differently with
operator ordering, which completely changes the quantization. In particular,
it seems to allow Thiemann, in this case, to have no operator re-ordering at
all, which is the basis for him not finding an anomaly, hence no tachyon and
no critical dimension.
(More details of what I am referring to can be found at
http://golem.ph.utexas.edu/string/archives/000299.html .)
If this is true and Group averaging on the one hand and Gupta-Bleuler
quantization on the other hand are two inequivalent consistent quantizations
for the same constrained classical system I would like to understand if they
are related in any sense.
Urs Schreiber
"Urs Schreiber" <Urs.Sc...@uni-essen.de> schrieb im Newsbeitrag
news:4016923b$1...@news.sentex.net...
> <psi|exp(C_I)|psi> = 0 .
Er, of course this should have been
<psi|exp(C_I)|psi'> = <psi|psi'>.
Lubos Motl
On 27 Jan 2004, Urs Schreiber wrote:
> I was trying to figure out what exactly it is in Th. Thiemanns
> quantization hep-th/0401172 of what he calls the 'LQG-string' that
> makes it so different from the usual quantization. I now believe that
> the crucial issue is how to impose the constraints.
Exactly. If physics is done properly, the (Virasoro) constraints are not
arbitrary constraints that are added by hand. They are really Einstein's
equations, derived as the equations of motion from the action if it is
varied with respect to the metric - in this case the worldsheet metric.
The term R_{ab}-R.g_{ab}/2 vanishes identically in two dimensions, and
T_{ab}=0 is the only term in the equation that imposes the constraint. The
constraints are really Einstein's equations, once again.
Moreover, because the (correct) theory is conformal, the trace
T_{ab}g^{ab} vanishes indentically, too, and therefore the three
components of the symmetric tensor T_{ab} actually reduce to two
components, and those two components impose the so-called Virasoro
constraints (which are easiest to be parameterized in the conformal gauge
where the metric is the standard flat metric rescaled by a
spacetime-dependent factor). For closed strings, there are independent
holomorphic and independent antiholomorphic generators - and they become
left-moving and right-moving observables on the Minkowski worldsheet
after we Wick-rotate.
Thomas Thiemann does not appreciate the logic behind all these things, and
he wants to work directly with the (obsolete) Nambu-Goto action to avoid
conformal field theory that he finds too difficult. Of course, the
Nambu-Goto action has no worldsheet metric, and therefore one is not
allowed to impose any further constraints. They simply don't follow and
can't follow from anything such as the equations of motion.
Thiemann does not give up, and imposes "the two" constraints by hand. It
is obvious from his paper that he thinks that one can add any constraints
he likes. Of course, there are no "the two" constraints. If he has no
worldsheet metric, the stress energy tensor has three components, and
there is no way to reduce them to two. Regardless of the effort one makes,
two tensor constraints in a general covariant nonconformal theory can
never transform properly as a tensor - because a symmetric tensor simply
has three components - and therefore his constraints won't close upon an
algebra. His equations are manifestly general non-covariant, in contrast
with his claims.
Equivalently, because he obtained these constraints by artificially
imposing them, they won't behave as conserved currents. (In a general
covariant theory without the worldsheet metric, we can't even say what
does it mean for a current to be conserved, because the conservation law
nabla_a T^{ab} requires a metric to define the covariant derivative.) If
they don't behave as conserved currents, they don't commute with the
Hamiltonian, and imposing these constraints at t=0 will violate them at
nonzero "t" anyway (the constraint is not conserved).
If one summarizes the situation, these constraints simply contradict the
equations of motion. It is not surprising. We are only allowed to derive
*one* equation of motion for each degree of freedom i.e. each component of
X, and this equation was derived from the action. Any further constraint
is inconsistent with such equations unless we add new degrees of freedom.
I hope that this point is absolutely clear. The equations of motion don't
allow any new arbitrarily added constraints unless it is possible to
derive them from extra terms in the action (that can contain Lagrange
multipliers). The Lagrange multipliers for the Virasoro constraints *are*
the components of worldsheet metric, and omitting one component of g_{ab}
makes his theory explicitly non-covariant (even if Thiemann tries to
obscure the situation by using the letters C,D for the two components of
the metric in eqn. (3.1)).
The conformal symmetry is absolutely paramount in the process of solving
the theory and identifying the Virasoro algebra - isolating the two
generators T_{zz} and T_{zBAR zBAR} per point from the general symmetric
tensor. Conformal/Virasoro transformations are those that fix the
conformal gauge - i.e. the requirement that the metric is given by the
unit matrix up to an overall rescaling. Conformal theories give us T_{z
zBAR} (the trace) equal to zero, and this is necessary to decouple T_{zz}
and T_{z zBAR}. In two dimensions, the conformal transformations -
equivalently the maps preserving the angles - are the holomorphic maps
(with possible poles), and the holomorphic automorphisms of a closed
string's worldsheet are generated by two sets of the Virasoro generators.
This material - why it is necessary to go from the Nambu-Goto action to
the Polyakov action and to conformal field theory in order to solve the
relativistic string and quantize it - is a basic material of chapter 1 or
chapter 2 of all elementary books about string theory and conformal field
theory. I think that a careful student should first try to understand this
basic stuff, before he or she decides to write "bombastic" papers boldly
claiming the discovery of new string theories and invalidity of all the
constraints (such as the critical dimension) that we have ever found.
In fact, I think that a careful student should first try to go through the
whole textbook first, before he publishes a paper on a related topic.
Thomas Thiemann is extremely far from being able to understand the chapter
3 about the BRST quantization, for example.
Thiemann's theory has very little to do with string theory, and very
little to do with real physics, and unlike string theory, it is
inconsistent and misled. String theory is a very robust and unique theory
and there is no way to "deform it" from its stringiness, certainly not in
these naive ways.
> This may seem like essentially the same thing, but the crucial issue is
> apparently that the latter form allows to deal quite differently with
> operator ordering, which completely changes the quantization. In particular,
> it seems to allow Thiemann, in this case, to have no operator re-ordering at
> all, which is the basis for him not finding an anomaly, hence no tachyon and
> no critical dimension.
A problem is that you don't know what you're averaging over because his
"group" is not a real symmetry of the dynamics.
By the way, if you want to define physical spectrum by a
Gupta-Bleuler-like method, you must have a rule for a state itself that
decides whether the state is physical or not. In Gupta-Bleuler old
quantization of the string, "L_0 - a" and "L_m" for m>0 are required
to annihilate the physical states. This implies that the matrix element of
any L_n is zero (or "a" for n=0) because the negative ones annihilate the
bra-vector.
It is important that we could have defined the physical spectrum using a
condition that involves the single state only. If you decided to define
the physical spectrum by saying that all matrix elements of an operator
(or many operators) between the physical states must vanish, you might
obtain many solutions of this self-contained condition. For example, you
could switch the roles of L_7 and L_{-7}. However all consistent solutions
would give you an equivalent Hilbert space to the standard one.
The modern BRST quantization allows us to impose the conditions in a
stronger way. All these subtle things - such as the b,c system carrying
the central charge c=-26 - are extremely important for a correct
treatment of the strings, and they can be derived unambiguously.
> If this is true and Group averaging on the one hand and Gupta-Bleuler
> quantization on the other hand are two inequivalent consistent quantizations
> for the same constrained classical system I would like to understand if they
> are related in any sense.
No, they are not. What is called here the "group averaging" is a naive
classical operation that does not allow one any sort of quantization. You
can simply look that at his statements - such as one below eqn. (5.2) -
that in his treatment, the "anomaly" (central charge) in the commutation
relations (of the Virasoro algebra, for example) vanishes, are never
justified by anything. They are only justified by their simple intuition
that things should be simple. This incorrect result is then spread
everywhere, much like many other incorrect results. It is equally wrong as
simply saying that we have constructed a different representation of
quantum mechanics where the operators "x" and "p" commute with one
another.
The central charge - the c-number that appears on the right hand side of
the Virasoro algebra - is absolutely real and unique determined by the
type of field theory that we study (and the theory must be conformal,
otherwise it is not possible to talk about the Virasoro algebra). It can
be calculated in many ways and any treatment that claims that the Virasoro
generators constructed out of X don't carry any central charge is simply
wrong.
There is absolutely no ambiguity in quantization of the perturbative
string. Knowing the background is equivalent to knowing the full theory,
its spectrum, and its interactions. There is no doubt that Thiemann's
paper - one with the big claims about the "ambiguities" of the
quantization of the string - is plain wrong, and exhibits not one, but a
plenty of elementary misunderstanding by the author about the role of
constraints, symmetries, anomalies, and commutators in physics.
Let me summarize a small part of his fundamental errors again. He believes
many very incorrect ideas, for example that
* artificially chosen constraints can be freely imposed on your Hilbert
space, without ruining the theory and contradicting the equations of motion
* two constraints in 2 dimensions can transform as a general symmetric
tensor, and having a tensor with a wrong number of components does not
spoil the general covariance
* he also thinks that the Virasoro generators have nothing to do with the
conformal symmetry and they have the same form in any 2D theory
* in other words, he believes that you can isolate the Virasoro generators
without going to a conformal gauge
* classical Poisson brackets and classical reasoning is enough to
determine the commutators in the corresponding quantum theory
* anomalies in symmetries, carried by various degrees of freedom,
can be ignored or hand-waved away
* there is an ambiguity in defining a representation of the algebra of
creation and annihilation operators
* the calculation of the conformal anomaly does not have to be treated
seriously
* the tools of the so-called axiomatic quantum field theory are useful
in treating two-dimensional <conformal> field theories related to
perturbative string theory
* if a set of formulae looks well enough to him, it must be OK and the
consistent stringy interactions and everything else must follow
Once again, all these things are wrong, much like nearly all of his
conclusions (and completely all "new" conclusions).
Thiemann himself admits that this is the same type of "methods" that they
have also applied to four-dimensional gravity. Well, probably. My research
of the papers on loop quantum gravity confirms it with a high degree of
reliability. Every time one can calculate something that gives them an
interesting but inconvenient result, they claim that in fact we don't need
to calculate it, and it might be ambiguous, and so on. No, this is not
what we can call science. In science, including string theory, we have
pretty well-defined rules how to calculate some class of observables, and
all things calculated according to these rules must be treated seriously.
If a single thing disagrees, the theory must be rejected.
The inevitability of conformal symmetry for a controlled quantization of
the relativistic string - and for isolation (in fact, the definition) of
the Virasoro generators - is real. The theorems of CFT about its being
uniquely determined by certain data are also real. The conformal anomalies
of certain fields are also real. The two-loop divergent diagrams in
ordinary GR are also real. We know how to compute and prove all these
things, and propagating fog and mist can only obscure these
well-established facts from those who don't want to see the truth.
I guess that this paper will demonstrate to most theoretical physicists -
even those who have not been interested in these "alternative" fields -
how bad the situation in the loop quantum gravity community has become.
There are hundreds of people who understand the quantization of a free
string very well, and they can judge whether Thiemann's paper is
reasonable or not and whether funding of this "new kind of science"
should continue.
All the best
Lubos
______________________________________________________________________________
E-mail: lu...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Urs Schreiber
Lubos and I had a little exchange about my original question over at
http://golem.ph.utexas.edu/string/archives/000299.html#c000504 .
It turns out that a crucial point for talking about the methods used in
Thiemann's paper (hep-th/0401172) is that the ordinary lore of quantum field
theory does not (or is not supposed to) apply in that framework.
For instance, in the ordinary version of quantum field theory the operator
:exp(-V):
is not the inverse of
:exp(V):,
due to the normal ordering, which is indicated by the colons. But in
Thiemann's paper (and, as far as I understand, in similar LQG papers) it is
used that there is a representation of QFT operators, obtained by means of
the GNS construction, which satisfy
pi(a) pi(b) = pi(ab),
where a and b are classical observables and pi(a) is the operator
representation of the observable a. This would imply that
pi( exp(-V) )
is indeed the inverse of
pi( exp(V) )
and I believe that this is a relation which is used heavily in Thiemann's
paper. For instance this seems to be the basis for the claim in the 3rd
paragraph of p. 20 that
alpha(W(Y_+-)) = W(alpha(Y_+-)) ,
where Y^mu_+- = p^mu +- X'^mu are essentially what is usually written as
partial X and bar partial X,
i.e. the left- and right-moving bosonic fields on the worldsheet,
W(Y) is the exponentiation of Y
and
alpha is the action of the exponetiated Virasoro constraints.
It is pretty obvious that if this is true then no anomaly does appear, since
the elements generated by exponentiating the operator constraints behave
exactly as those generated by exponentiating the classical constraints.
Is it hence true that we can alternatively have QFTs that have neither
normal ordering issues, nor anomalies, nor non-trivial OPEs, etc? If not, is
there something wrong about Thiemann's assumptions? What is going on here?
Thomas Larsson
"Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote in message news:<4017...@news.sentex.net>...
> the GNS construction, which satisfy
>
> pi(a) pi(b) = pi(ab),
>
> where a and b are classical observables and pi(a) is the operator
> representation of the observable a. This would imply that
Does this mean that you can take a and b to be rotations and pi to
be the spinor rep, and hope to get +1 rather than -1 after a 2\pi
rotation?
Urs Schreiber
news:24a23f36.04012...@posting.google.com...
Good point. I don't know. Yes, this might be a simple example of the same
strange phenomenon that occurs in Thiemann's paper and which essentially
removes all quantum signatures from the quantization! I am hoping that
somebody familiar with the techniques used by Thiemann will help us
understand this.
Meanwhile, I have a suspicion what precisely it is in Thiemann's
construction (and hence in LQG in general, I assume) that allows him to get
rid of all the usual quantum effects. I think the point is that he is
working on a non-seperable Hilbert space.
I may be wrong, but the following is what I seem to understand from
hep-th/0401172 (comments and corrections are welcome). I will try to capture
the basic idea of Thiemanns construction by condensing a lot of details into
the following construction:
Consider observables called W(I), where I is any oriented Borel subset of
S^1, which should be nothing but a finite collection of closed intervals.
Assume that the W satisfy an algebra of the form
W(I)W(J) = (phase factor)W(F(I,J)),
where F is some operation on these sets of which I here only need the
property that it returns the empty set iff I=-J as oriented subsets of S^1.
With the operation
(W(I)^)* = W(-I)
the W generate a unital *-algebra A with unit 1 = W({}).
By means of the GNS construction, if I understand correctly, this algebra
can now be regarded as a vector space and this vector space can be equipped
with a scalar product <.|.> which is induced by some 'state' (a positive
linear functional) omega on A:
<a|b> := omega(a*b) .
Thiemann chooses
omega(W(I)) = 1 if I = {} and = 0 otherwise.
Now, unless I am missing something (which is well possible) this should
imply that the state associated with W(I) is orthogonal to all states W(J)
with I not equal to J, because
<W(I) | W(J)> = omega( W(F(-I,J)) )
is non-zero if and only if I=J, by the above.
So every Borel subset of S^1 corresponds to a linearly independent state.
But surely the number of Borel subsets of S^1 is not countable. Take for
instance the subset
{ [0,x] | x in [1,2] }
of Borel subsets of S^1, which by itself already has the cardinality of the
real numbers.
If all this is right then it would imply that also a basis of the above
Hilbert space is not countable and hence this Hilbert space is
non-seperable.
This wouldn't be a total surprise, since also the (kinematical) Hilbert
space of spin netweorks used in LQG is non-seperable. Indeed, it is this
non-seperability of the Hilbert space which is, after all the technical dust
has settled, the crucial ingredient that Martin Bojowald takes from full LQG
and inserts into the quantization of the Wheelde-deWit equation. By
representing the scale factor of the universe on a non-seperable Hilbert
space he turns it technically into an operator with _discrete_ spectrum
(since it's eigenvalues are normalizable - by construction!!) even though
the range of eigenvalues is still the real numbers. By pulling another trick
(which I haven't understood when I was told about it) it then follows that
the quantum cosmology described by this theory describes a universe which
grows by finite steps in size. There is a growing literature on this
approach and the claim is that the discrete evolution improves the
theoretical fit to the WMAP data at very low values of the parameter l.
Be that as it may, it seems that the use of non-seperable Hilbert spaces
(either for quantizing the WdW equation or for quantizing the relativistic
string) drastically alters the physics as compared to ordinary quantization.
While for loop quantum cosmology the claim is that the physical
interpretation improves, it is not clear to me that the same can be said
about Thiemann's quantization of the string.
The non-seperability (if I am right about this) of the Hilbert space in
Thiemann's construction would intuitively explain why he does not find any
OPEs, i.e. no relation between nearby operators as
W( {x,y})
and
W( {y+epsilon,y+1} )
for small positive epsilon. In the usual quantization normal ordering
effects (singularities in OPEs) in the product of these two operators yield
precisely the effects which lead to the anomaly, the critical dimension,
etc. All this is missing in Thiemann's quantization.
My hunch is that this is because Thiemann's Hilbert space is so enormously
large that there is so much room that the operator
W( {x,y})
can sit happily next to
W( {y+epsilon,y+1} )
without noticing. They simply commute, like classical observables. No
quantum fluctuations are present (expressed by normal ordering or OPEs) that
let these two "vertices" communicate.
Currently I therefore tend to believe that Thiemann's construction, while
mathematically consistent, is not physicall viable. Do we know that quantum
theory should also be "true" (as a physical description of the world) for
non-seperable Hilbert spaces? Are there any simple examples? (Even Bojowald
somehow get's rid of the large non-seperable Hilbert space in favour of an
ordinary one in the end, so this is not really an example.) For instance,
can we even find a path-integral formulation for a quantum theory which has
a non-seperable Hilbert space?
Robert C. Helling
Lubos,
I don't want to defend Thiemann's paper (I cannot as I haven't yet
seriously looked at it), but let me mention one thing that always
comes up in discussions with people under Pohlmeyer's influence: What
we usually do to quantize the string is to go to conformal gauge and
then quantize that free theory giving us Virasoro constraints, level
matching etc. But that's dangerous in a quantum theory because fixing
a symmetry and quantization do not commutein general. So there might
be a point. You could of course say that string theory is defined to
be the theory in conformal gauge, but at least, that's not what GSW
do, they start with Nambu-Goto.
> Let me summarize a small part of his fundamental errors again. He believes
> many very incorrect ideas, for example that
[...]
> * the tools of the so-called axiomatic quantum field theory are useful
> in treating two-dimensional <conformal> field theories related to
> perturbative string theory
Why do you say that? Actually the formalism of BPZ is very close to
the algebraic formalism and many people have made the connection
explicit. AFAIK rational conformal field theories are the only known
examples of interacting theories in the algebraic formalism. See for
example
FROM PATH REPRESENTATIONS TO GLOBAL MORPHISMS FOR A CLASS OF MINIMAL
MODELS.
By Andreas Recknagel (Zurich, ETH),. ETH-TH-96-44, Dec 1996. 50pp.
Published in Commun.Math.Phys.201:365-409,1999
e-Print Archive: hep-th/9612178
Robert
--
.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO
Robert C. Helling Department of Applied Mathematics and Theoretical Physics
University of Cambridge
print "Just another Phone: +44/1223/766870
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling
Urs Schreiber
Newsbeitrag news:bvalkg$q580a$1...@ID-40416.news.uni-berlin.de...
> one thing that always
> comes up in discussions with people under Pohlmeyer's influence: What
> we usually do to quantize the string is to go to conformal gauge and
> then quantize that free theory giving us Virasoro constraints, level
> matching etc. But that's dangerous in a quantum theory because fixing
> a symmetry and quantization do not commutein general. So there might
> be a point.
It might be noteworthy that one does not have to fix conformal gauge in
order to derive the Virasoro constraints. You can either get them from the
Nambu-Goto action or indeed from the Polyakov action with no metric gauge
fixed by using the ADM method on the worldsheet, see e.g. pp. 154 of
http://www-stud.uni-essen.de/~sb0264/sqm.pdf or the lecture notes
author = {M. Henneaux},
title = {Lectures on String Theory, with Emphasis on Hamiltonian and {BRST}
Methods},
booktitle = {Principles of String Theory},
editor = {L. Brink and C. Teitelboim},
pages = {93-292},
publisher = {Plenum Press},
year = {1988} .
Several quantization techniques of the worldsheet theory with different
order of constraining and quantization all yield the same result in the
critical dimension. For instance the lightcone quantization essentially
fixes a symmetry and solves the constraints classically, quantizing only the
result.
Therefore I am wondering if not Pohlmeyer's approach should give the same
result, too. I have only recently introduced myself to Pohlmeyer's work, so
I may be missing something, but if I understand correctly the idea is to
find a complete set of classical observables that Poisson-commute with all
the Virasoro constraints and then to find a quantum representation of the
Poisson algebra of these 'invariant charges'. Is this correct?
If it is I would suggest not to use the invariants that Pohlmeyer finds (by
taking Taylor coefficients of certain Wilson lines along the string) but to
instead use (what must be a linear recombination of them) the classical
version of DDF operators as invariants
http://golem.ph.utexas.edu/string/archives/000301.html. These Poisson
commute with all the constraints, are complete, have a very nice Poisson
algebra whose quantization is known and they immediately carry over to the
superstring. This way of quantizing the string by the Pohlmeyer program but
with DDF invariants hence immediately gives the correct massive spectrum of
the (super-)string. A little more work is needed to find the correct
properties of the worldsheet groundstate, which follows from finding a
Hilbert space representation of the quantized DDF invariants.
Well, or so I think. Maybe something is wrong with this idea, because
otherwise it would have been noticed in the literature? :-)
Lubos Motl
> I don't want to defend Thiemann's paper (I cannot as I haven't yet
> seriously looked at it), but let me mention one thing that always
> comes up in discussions with people under Pohlmeyer's influence: What
> we usually do to quantize the string is to go to conformal gauge and
> then quantize that free theory giving us Virasoro constraints, level
> matching etc. But that's dangerous in a quantum theory because fixing
> a symmetry and quantization do not commutein general.
Robert, this is the whole point of anomalies. If a symmetry survives
quantum mechanically, then the symmetry works and you can use it to reduce
(gauge symmetry) or organize (global symmetry) the quantum mechanical
spectrum. If a symmetry does not survive quantum mechanically, you are not
allowed to impose it in the quantum mechanical theory in any way. You must
leave your quantum theory to decide whether a classical symmetry exists
quantum mechanically or not!
For example, if we consider the worldsheet reparameterization symmetry
(or its conformal group), everyone who has studied at least the first
chapters of the relevant textbooks is able to show that this symmetry will
be destroyed by quantum anomalies unless you work in the critical
dimension. So we are *not* allowed to impose this symmetry in theories
with a wrong field content (whose total central charge is nonzero).
If you talk about the "commutative diagram" involving the symmetries and
quantization, there are only two approaches: the correct approach and the
wrong approach. The correct approach - which is applied by particle
physicists in quantum field theories and string theory - is to quantize
the theory first, and then ask whether it has all the symmetries that we
expected classically. Some symmetries will be preserved, some symmetries
will be broken or anomalous, and anomalous *gauge* symmetries imply that
the theory is eliminated by mathematical inconsistency (e.g. the bosonic
string in a wrong dimension).
The wrong approach is to require the symmetries dogmatically, for
whichever system you deal with, and then to try to find a "quantization"
(meaning a "representation of an algebra") that preserves these
symmetries, regardless of the unphysical character of these spaces. The
correct approach above clearly shows that some theories simply don't have
various classically expected symmetries. This is an important fact, it is
an important lesson of quantum mechanics that constrains the allowed
theories, and if you try to prove the reverse, you obviously construct a
totally unphysical theory - probably with a non-separable Hilbert space as
Urs Schreiber has argued.
In a theory without anomalies, the procedures of quantization and of
imposing the constraints commute with one another, and the bosonic string
in the critical dimension is an example.
> So there might be a point. You could of course say that string theory
> is defined to be the theory in conformal gauge, but at least, that's
> not what GSW do, they start with Nambu-Goto.
Nambu-Goto is certainly our moral starting point, but what you do with it
afterwards (and how you do it) definitely matters. The only mathematically
satisfactory covariant way to quantize the bosonic string is to rewrite
the system using the Polyakov action and add the Faddeev-Popov ghosts. The
anomalies exactly cancel and various constraints can be imposed
"classically". Even if you use a non-covariant quantization, such as the
light-cone gauge, or a mathematically less reliable treatment, such as the
old covariant quantization, the physical results will, of course, agree,
as long as you don't make any errors.
> Why do you say that? Actually the formalism of BPZ is very close to
> the algebraic formalism and many people have made the connection...
I used the word "axiomatic", not "algebraic", and this subtle difference
might have some impact on the conclusions. BPZ is an honest CFT approach,
and Thiemann apparently does not like it. By "axiomatic" QFT I meant an
approach that claims that the operators can have rigorous definition,
leading to an algebra that is independent of the renormalization scheme,
and perhaps even without the singularities in the OPEs. This is not how
the bosonic string can work.
> explicit. AFAIK rational conformal field theories are the only known
> examples of interacting theories in the algebraic formalism.
Yes, I would agree with this formulation, as far as you talk about the
rational CFTs. This observation is not directly related to the question of
the validity of Thiemann's procedures, is it? Moreover, Recknagel's paper
is about the minimal models, and many important things just can't be
extrapolated to the bosonic string. You know that the bosonic string CFT
has a lot of renormalization-scheme-dependent definitions of the operators
at higher order, and so on. This means that the UV divergences on the
worldsheet *do* matter and they include important physical information.
All the best
Lubos
______________________________________________________________________________
E-mail: lu...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Only two things are infinite, the Universe and human stupidity,
and I'm not sure about the former. - Albert Einstein
Thomas Larsson
This is an expanded version of a post to the string coffee table,
http://golem.ph.utexas.edu/string/archives/000300.html . It is in
response to a post by K-H Rehren, who pointed out the crucial
algebraic difference between LQG representations and lowest-energy
representations. This explains the absense of anomalies in Thiemann's
approach and IMO settles the status of LQG as a quantum theory.
K-H Rehren:
>Dorothea Bahns has shown in her diploma thesis, that if one quantizes
>classical invariant observables (Pohlmeyer charges) by embedding them
>into the oscillator algebra via normal odering (N.O.), then the N.O.
>invariants fail to commute with N.O. Virasoro constraints, and
>commutators of N.O. invariants among themselves yield other N.O.
>invariants plus "quantum corrections" which are #not# quantized
>classical invariants. Thus, the quantum algebra has not only
>relations differing from the classical ones by hbar corrections
>(which everybody expects), but it would have #more generators# than
>the classical algebra. This feature ("breakdown of the principle of
>correspondence") is worse than a central extension, because the
>latter is a multiple of one, and as such #is# a quantized classical
>observable, suppressed by hbar. This feature is a property of the
>quantization, i.e., the very choice of the quantum algebra by
>replacing classical invariants by N.O. ones. One may or may not
>appreciate the oscillator quantization with features like this.
The correspondence principle is not necessarily violated. To
construct extensions of the diffeomorphism algebra in more than 1D,
one must first expand all fields in a Taylor series around some point
q. There are no conceptual problems to express classical physics in
terms of Taylor data (q and the Taylor coefficients) rather than in
terms of fields, although there may be problems with convergence.
The reason why such a trivial reformulation is necessary is that the
higher-dimensional generalizations of the Virasoro cocycle (there are
two of them) depend on the expansion point q. The relevant extensions
are thus non-linear functions of data already present classically,
which seems consistent with the correspondence principle.
>A no-go theorem (V. Kac) states that a unitary positive-energy (L0>0)
>representation of Diff without central charge must be trivial
>(one-dimensional). Put otherwise: c=0 is only possible if one
>abandons the positive-definite Hilbert space metric (ghosts), or
>positive energy, or unitarity.
Hence we can not maintain non-triviality, anomaly freedom, positive
energy, unitarity and ghost freedom at the same time. It seems to me
that giving up anomaly freedom makes least damage, especially since
we know that anomalous conformal symmetry is important in 2D
statistical models, such as the Ising and tricritical Ising models.
It is important to realize that such models have been realized
experimentally (e.g. in a monolayer of argon atoms on a graphite
substrate) and that the non-zero conformal anomaly has been measured
(perhaps only in computer experiments). Hence anomalous conformal
symmetry is not intrinsically inconsistent.
A clarification is in order at this point. The multi-dimensional
Virasoro algebra is a kind of gravitational anomaly, but no such
anomalies exist in 4D within a field theory framework. However, it
turns out that the phrase "within a field theory framework" is a
critical assumption. As I explained above, the relevant cocycles
depend on the expansion point q, and thus they can not be expressed
in terms of the fields which are independent of q.
>Thiemann seeks for the quantum algebra within an LQG type auxiliary
>algebra which has a unitary representation of Diff. The latter is
>#not# subject to the positive-energy condition.
The passage to lowest-energy representations is the crucial
quantization step. If such representations do not appear in LQG, then
it cannot IMO be a genuine quantum theory. If LQG is essentially
classical it explains why there is no anomaly.
Let me elaborate on the difference between lowest-energy (LE) reps
and the LQG type of rep, as described in section 2.6 of gr-qc/9910079
by Gaul and Rovelli. The canonical variables are A(x) and E(x) and
the CCR read
[E(x), A(y)] = delta(x-y), [E(x),E(y)] = [A(x),A(y)] = 0.
In the paper, the operators are smeared over suitable submanifolds
and have some index decorations, but that is irrelevant for the
argument. The canonical variables act of functionals Psi(A), with
E(x) = d/dA(x), and the constant functional 1 can be identified with
a vacuum |vac> by
E(x) |vac> = 0 for all x.
Bilinears of the form
A(x)E(y)
generate a gl(infinity), and we can embed diffeomorphism and current
algebras into this gl(infinity). The key point to observe is that
these bilinears are already normal ordered w.r.t. the vacuum |vac>.
Normal ordering means by definition that things that annihilate the
vacuum are moved to the right, and E(y) does annihilate the vacuum.
Since no further normal ordering is necessary, no anomalies arise.
However, this is not what I would call a LE rep. Rather, I would call
it a "lowest-A-number" rep; the A-number operator \int dx A(x)E(x) is
always positive. This space is essentially classical in nature, so it
is not so surprising that there is no anomaly.
Let us contrast this type of rep with LE reps. For simplicity, let us
assume that x and y are points in 1D; the higher-dimensional case
requires a passage to jet space which complicates things, although
not in an essential way. We can now expand A(x) and E(x) in a Fourier
series, and the Fourier components A_m and E_m satisfy the CCR
[E_m, A_n] = delta_m+n,0 , [E_m, E_n] = [A_m, A_n] = 0.
The LQG vacuum satisfies
E_m |vac> = 0 for all m.
The LE vacuum |0>, OTOH, is defined by
E_-m |0> = A_-m |0> = 0 for all -m < 0.
In other words, it is the modes of negative frequency, i.e. those
that travel backwards in time, that annihilate this vacuum.
The bilinears that generate gl(infinity),
A_m E_n ,
are normal ordered w.r.t. the LQG vacuum |vac> but not w.r.t. the
LE vacuum |0>. To normal order w.r.t. the latter, we need to move
negative-frequency modes to the right:
:A_m E_n: = A_m E_n m >= n
E_n A_m m < n
This is the main algebraic difference between the LQG "lowest-A-number"
reps and LE reps. This is an essential difference; in infinite
dimension, there is no Stone-von Neumann theorem that makes different
vacua equivalent.
Urs Schreiber
> Let me elaborate on the difference between lowest-energy (LE) reps
> and the LQG type of rep, as described in section 2.6 of gr-qc/9910079
> by Gaul and Rovelli. The canonical variables are A(x) and E(x) and
> the CCR read
>
> [E(x), A(y)] = delta(x-y), [E(x),E(y)] = [A(x),A(y)] = 0.
>
> In the paper, the operators are smeared over suitable submanifolds
> and have some index decorations, but that is irrelevant for the
> argument. The canonical variables act of functionals Psi(A), with
> E(x) = d/dA(x), and the constant functional 1 can be identified with
> a vacuum |vac> by
>
> E(x) |vac> = 0 for all x.
>
> Bilinears of the form
>
> A(x)E(y)
>
> generate a gl(infinity), and we can embed diffeomorphism and current
> algebras into this gl(infinity). The key point to observe is that
> these bilinears are already normal ordered w.r.t. the vacuum |vac>.
> Normal ordering means by definition that things that annihilate the
> vacuum are moved to the right, and E(y) does annihilate the vacuum.
> Since no further normal ordering is necessary, no anomalies arise.
Sorry, but this is not true.
I made the same mistake here:
http://golem.ph.utexas.edu/string/archives/000299.html#c000527
was corrected by Jacques Distler here:
http://golem.ph.utexas.edu/string/archives/000299.html#c000529
and later posted the correct details here:
http://golem.ph.utexas.edu/string/archives/000299.html#c000572 .
The point is that there is _no_ quantization whatsoever of the Virasoro
algebra which avoids the m^3/12 term of the anomaly, no matter which
ordering you choose.
Furthermore note that this is not how Thomas Thiemann claims to get rid of
the anomaly, according to himself:
http://golem.ph.utexas.edu/string/archives/000299.html#c000577 .
Also note that in LQG (e.g. gr-qc/9910079) we do _not_ have canonical
operators A(x) and E(x). That's because the classical A(x) is represented as
an operator _only_ in its exponentiated form of a holonomy U[A,\gamma].
(e.g. equation (36)).
The analog holds true in Thomas Thiemann's 'LQG-string': Neither canonical
coordinates nor canonical momenta of the string are represented on his
Hilbert space:
http://golem.ph.utexas.edu/string/archives/000299.html#c000549 .
It follows that hence the Virasoro algebra is _not_ reprented (and not
claimed to be represented) on the 'LQG-string' Hilbert space, with or
without anomaly. The quantization method in LQG is _not_ canonical in the
usual strict sense, i.e. first class constraints of the classical theory are
_not_ represented as operators on some Hilbert space. This is made very
clear by Thomas Thiemann here:
http://golem.ph.utexas.edu/string/archives/000299.html#c000554
http://golem.ph.utexas.edu/string/archives/000299.html#c000588 .
The fact that not both of canonical coordinates and momenta are represented
as operators on a Hilbert space in LQG-like quantizations is also emphasized
in
A. Ashtekar, S. Fairhurst & J. Willis, Quantum gravity, shadow states and
quantum mechanics, gr-qc/0207106.
I have summarized some of the aspects of this paper with an eye on the
'LQG-string' here:
http://golem.ph.utexas.edu/string/archives/000299.html#c000647 .
The bottom line seems to be the following, roughly:
LQG-like quantization searches operator representations for the group, not
the algebra.
As shown by the various references mentioned, the result is in general
radically different from what I would call canonical quantization and
furthermore ambiguous, even more ambiguous than ordinary first quantization,
that is.
To make this point quite clear consider the toy version of the 'LQG-string',
the 'LQG Klein-Gordon particle'
http://golem.ph.utexas.edu/~distler/blog/archives/000307.html#c000637 .
The 1+0 dimensional Nambu-Goto action of the KG particle has a single
classical constraint which hence generates the group R, or U(1) if you wish.
So according to the prescription "represent the group, not the generators",
every operator rep of U(1) would be an 'LQG-like' quantization of the KG
particle.
While this sounds extreme, I don't think that it misses the point. When one
look at the equation right above (IV.5) in Ashtekar,Fairhurst&Willis
gr-qc/0207106 one will see that the crucial factor exp(-\alpha^2/2) which is
encluded to ensure at least some similarity with the ordinary quantization
of the 1d nonrelativistic particle, is chosen completely arbitraryly. One
could pick any other value and in particular use a vanishing exponent.
The latter would copy the _classical_ relations to the operator algebra and
is in fact precisely what Thomas Thiemann is doing in the 'LQG-string'.
There an operator representation of the _classical_ conformal group is
constructed by fiat. The same is true for the diffeomorphism constraints of
3+1 dimensional gravity:
http://golem.ph.utexas.edu/string/archives/000299.html#c000559
Maybe it makes sense to technically call such an approach an 'alternative
quantization' as for instance K.-H. Rehren argues in
http://golem.ph.utexas.edu/string/archives/000300.html#c000649
http://golem.ph.utexas.edu/string/archives/000300.html#c000674 .
Thomas Thiemann says that only experiment can show if maybe this
'alternative quantization' is the one followed by nature:
http://golem.ph.utexas.edu/string/archives/000299.html#c000588 .
Of course nobody really knows what might happen at the Planck scale. But one
should be aware of the following points:
- LQG-like quantization in the above sense is not canonical in the usual
sense
- applied to systems which we can test experimentally, like the 1d
nonrelativistic particle or the free electromagnetic field, it produces
results drastically different from the standard ones.
- there seem to be no arguments why this modification of the quantum
principle should be the one used for quantum gravity excetp that 'only
experiment will show'
At least that's what I have learned from the discussion concerning the
'LQG-string'. Admittedly, I am a little surprised to find myself left with
such a rather drastic conclusion, but it seems to be confirmed by many
sources.
Thomas Larsson
Urs Schreiber <Urs.Sc...@uni-essen.de> skrev i
diskussionsgruppsmeddelandet:4039e031$1...@news.sentex.net...
> > Bilinears of the form
> >
> > A(x)E(y)
> >
> > generate a gl(infinity), and we can embed diffeomorphism and current
> > algebras into this gl(infinity).
...
> The point is that there is _no_ quantization whatsoever of the Virasoro
> algebra which avoids the m^3/12 term of the anomaly, no matter which
> ordering you choose.
This is true if you start with a single set of oscillators a_m
satisfying [a_m, a_n] = m delta_m+n. However, I have two sets
A_m, E_m which are canonically conjugate, [E_m, A_n] =
delta_m+n. In this case, you can even add an extra term which
modifies the central charge with standard order.
Set
L_m = sum_k (k - lambda m) A_k E_m-k.
Then one has
[L_m, A_n] = ((1-lambda)m + n) A_m+n
[L_m, E_n] = (lambda m + n) E_m+n
[L_m, L_n] = (n-m) L_m+n
so there is no extension. The primary field A transforms with
weight lambda and E with weight (1-lambda). Since E_n|vac> =
0, we also have
L_m |vac> = 0 for all m.
The L_m's act less trivially on states of the form f(A)|vac>.
One can compute the central charge by considering
[L_m, L_-m] |vac> = 0 = (-2m*0 - c(m^3-m)) |vac>,
and thus c = 0.
If you use standard ordering, you instead get
c = 2(1 - 6 lambda + 6 lambda^2). The same formulas hold if
you let A and E be fermions, except that c -> -c. One can
construct the celebrated minimal models with c < 1 by
factoring out singular vectors in these fermionic Fock
modules; note that the maximal value c = 1 is attained for
lambda = 1/2.
When you order terms and define the vacuum like I did above,
it is not really a quantization, although it superficially
looks like one. Let us define abstractly what axioms a Poisson
bracket and a commutator must satisfy:
1. An associative product.
2. A Lie bracket.
3. The bracket is a derivation of the product.
4c. ab - ba = 0
4q. ab - ba = [a,b]
Axioms 1-4c define a Poisson bracket, 1-4q define a
commutator. However, in my computation with the alternative
order, I only used axioms 1-3. We are therefore free to append
either 4c or 4q, and we can thus regard the bracket as either
a Poisson bracket or as a commutator. The analogous
computation with standard order explicitly uses axiom 4q, so
there we have no choice.
But this is maybe not related to what Thiemann does. Or maybe
it is. He writes on p 4 that "[In string theory] the
constraints are implemented only weakly rather than strongly
so that one does not perform an honest Dirac quantization".
This could mean that he requires
L_m |vac> = 0 for all m, positive and negative,
which is what I got above.
>
> Furthermore note that this is not how Thomas Thiemann claims to get rid of
> the anomaly, according to himself:
>
> http://golem.ph.utexas.edu/string/archives/000299.html#c000577 .
>
> Also note that in LQG (e.g. gr-qc/9910079) we do _not_ have canonical
> operators A(x) and E(x). That's because the classical A(x) is represented as
> an operator _only_ in its exponentiated form of a holonomy U[A,\gamma].
> (e.g. equation (36)).
I don't get this. The only thing that matters is that the A's
and E's generate a Heisenberg algebra, which acts on a Hilbert
space that consists of functionals of the A's alone. Then it
immediately follows that the E's are lowering operators and
that bilinears AE are already normal ordered. Smearing and
exponentiation cannot possibly change this!?
The main thing that triggered my interest was Rehren's comment
that Thiemann gives up positive energy. For somebody whose
worldview is shaped by the success of CFT in statistical
physics, this looks very wrong, since there quantization is
the same thing as building lowest- energy representations.
This seems to lead to a conclusion that I don't particularly
like. If LQG has anything to do with what I have written above
(and this is not clear to me), then it is not a genuine
quantum theory.
Urs Schreiber
news:24a23f36.04022...@posting.google.com...
> Urs Schreiber <Urs.Sc...@uni-essen.de> skrev i
> diskussionsgruppsmeddelandet:4039e031$1...@news.sentex.net...
>
> > > Bilinears of the form
> > >
> > > A(x)E(y)
> > >
> > > generate a gl(infinity), and we can embed diffeomorphism and current
> > > algebras into this gl(infinity).
> ...
> > The point is that there is _no_ quantization whatsoever of the Virasoro
> > algebra which avoids the m^3/12 term of the anomaly, no matter which
> > ordering you choose.
>
> This is true if you start with a single set of oscillators a_m
> satisfying [a_m, a_n] = m delta_m+n.
Ok. I thought that's what we are talking about, because that's what you get
by canonically analyzing the Nambu-Goto action or any other equivalent
action.
> However, I have two sets
> A_m, E_m which are canonically conjugate, [E_m, A_n] =
> delta_m+n. In this case, you can even add an extra term which
> modifies the central charge with standard order.
>
> Set
>
> L_m = sum_k (k - lambda m) A_k E_m-k.
Ok, but now you are playing by different rules because these objects are not
obtained by canonically quantizing the NG action.
> But this is maybe not related to what Thiemann does. Or maybe
> it is.
What Thomas Thiemann does in his paper is something different.
> This could mean that he requires
>
> L_m |vac> = 0 for all m, positive and negative,
>
> which is what I got above.
In Thomas Thiemann's approach the L_m are _not_ represented as operators on
his Hilbert space, hence no equation of this sort is considered by him.
> I don't get this. The only thing that matters is that the A's
> and E's generate a Heisenberg algebra, which acts on a Hilbert
> space that consists of functionals of the A's alone.
Yes, one would think so. But this is not what happens in LQG. See for
instance page 2 of gr-qc/0207106, last paragraph:
"[...] the connection A fails to be a well-defined operator(-valued
distribution); only the holonomies are well-defined operators".
So there is no Heisenberg algebra in LQG. What we have is only the Weyl
algebra. Since this is represented non-weakly continuously, it is not
equivalent to exponentiated E and A.
Thomas Larsson
"Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote in message news:<403b27d6$1...@news.sentex.net>...
> > However, I have two sets
> > A_m, E_m which are canonically conjugate, [E_m, A_n] =
> > delta_m+n. In this case, you can even add an extra term which
> > modifies the central charge with standard order.
> >
> > Set
> >
> > L_m = sum_k (k - lambda m) A_k E_m-k.
>
> Ok, but now you are playing by different rules because these objects are not
> obtained by canonically quantizing the NG action.
I am not sure about that. Define
a_m = E_m - m/2 A_m
b_m = i(E_m + m/2 A_m)
a_m and b_m form two commuting sets of oscillators, from which
one can build two commuting Virasoro algebras. The vacua are
defined by
a_-m |0> = b_-m |0> = 0, for all -m < 0
(a_m - ib_m) |vac> = 0, for all m, positive or negative.
The catch is probably that the second vacuum connects the
two Virasoro algebras.
Actually, normal ordering becomes non-trivial, but there is
still no extension (I think). Consider one of the Virasoro
algebras, e.g.
L_m = sum_n :a_n a_m-n:
= sum_n :(E_n - n/2 A_n)(E_m-n - (m-n)/2 A_m-n):
= sum_n ( a_n a_m-n + n/2 delta_m )
= sum_n a_n a_m-n
because sum_{n=-infinity}^infinity n = 0. In standard order,
the relevant sum runs from 1 to infinity instead.
Less formally, note that
L_m |vac> = sum_n n(m-n)/4 A_n A_m-n |vac>,
because normal-ordered terms with at least one E annihilates
|vac>. When we compute [L_-m, L_m] |vac>, we get terms
proportional to AA|vac> and AAAA|vac>. Since all A's commute,
no central terms arise (and we know that non-central terms
will take care of themselves). This is different from standard
order, where we have to reorder terms like aa|0>.
There might be something subtle about that already L_m|vac> is
an infinite sum, though. I find all of this very confusing.
Urs Schreiber
> a_-m |0> = b_-m |0> = 0, for all -m < 0
>
> (a_m - ib_m) |vac> = 0, for all m, positive or negative.
>
> The catch is probably that the second vacuum connects the
> two Virasoro algebras.
While that does look strange, it should not affect the result about the
anomaly. The calculation at
http://golem.ph.utexas.edu/string/archives/000299.html#c000572 assumes
nothing about the representation of the oscillators nor of the vacuum. The
only input is the commutation relation [a_n,a_m] = n delta_{n+m}.
> Actually, normal ordering becomes non-trivial, but there is
> still no extension (I think). Consider one of the Virasoro
> algebras, e.g.
>
> L_m = sum_n :a_n a_m-n:
>
> = sum_n :(E_n - n/2 A_n)(E_m-n - (m-n)/2 A_m-n):
>
> = sum_n ( a_n a_m-n + n/2 delta_m )
>
> = sum_n a_n a_m-n
>
> because sum_{n=-infinity}^infinity n = 0. In standard order,
> the relevant sum runs from 1 to infinity instead.
Ok, so the non-standard normal ordering that you are using here is
Weyl-ordering
(http://golem.ph.utexas.edu/string/archives/000299.html#c000575). But no
matter which ordering one uses the universal part of the anomaly should
remain the same.
> Less formally, note that
>
> L_m |vac> = sum_n n(m-n)/4 A_n A_m-n |vac>,
>
> because normal-ordered terms with at least one E annihilates
> |vac>. When we compute [L_-m, L_m] |vac>, we get terms
> proportional to AA|vac> and AAAA|vac>.
I haven't done this calculation, but how can it be that you get A^4 terms?
These won't be expressible in terms of first powers of L_m at all!?
> Since all A's commute,
> no central terms arise (and we know that non-central terms
> will take care of themselves). This is different from standard
> order, where we have to reorder terms like aa|0>.
Hm, I currenly don't follow this. I should perhaps sit down and try
recapitulate what you are doing here. Do you think you are using different
assumptions than [a_n,a_m] = n delta_{n+m} and L_m = quadratic in a_i? If
so, how can it be that you get somthing differing from
http://golem.ph.utexas.edu/string/archives/000299.html#c000572 ?
Maybe your concern is similar to Robert Helling's
http://golem.ph.utexas.edu/string/archives/000299.html#c000610 ?
> There might be something subtle about that already L_m|vac> is
> an infinite sum, though. I find all of this very confusing.
Agreed, the infinite sums are subtle. Mayby you should rewrite your proposed
reps of the Virasoro algebra in functional form, this might make it more
transparent if and where the problem is.
I am saying this because in functional form the mere presence of the anomaly
is much better visible than in Fourier-decomposition. The object
Y(s) := const sum_n=-infty^infty a_n e^{-ins}
is an operator-valued _distribution_ and the Virasoro generators are
formally
L_m = (1/2) int ds e^{ims} Y(s) Y(s) .
But this should already signal trouble, since it involves the product of two
distributions. The trouble becomes more manifest when the commutator is
computed
[Y(s), Y(s')] = -i delta'(s,s') Y(s) Y(s') + delta'(s,s') delta'(s,s') .
The second term is clearly ill defined. Depending on how I choose to
formally integrate over this term I can make it look like 0, infinity or 42.
Hence the mere problem is well visible in this functional form. The correct
value of the proper regularization of the (delta')^2 term is still subtle to
compute.
On the other hand, something nice is going on. Pick any approximation
f'(s-s') of delta'(s-s'). Then the anomaly is the m^3 term of
pi int e^{-im s} f'^2(s-s') .
Thomas Larsson
> > (a_m - ib_m) |vac> = 0, for all m, positive or negative.
..
> > L_m = sum_n a_n a_m-n
>
> Ok, so the non-standard normal ordering that you are using here is
> Weyl-ordering
> (http://golem.ph.utexas.edu/string/archives/000299.html#c000575). But no
> matter which ordering one uses the universal part of the anomaly should
> remain the same.
But I never have to reorder! The crux is that the definition
of the vacuum is invariant under translations in Fourier space,
m -> m + k. This is very peculiar, and somewhat similar to the
classical situation. In particular, it means that I can freely
shift the summation variables in the four terms that arise in
[L_m, L_n].
This could have something to do with what Thiemann does.
Translation invariance in momentum space means that positive
energy is abandoned and that the central charge vanishes, as
desired.
>
> > Less formally, note that
> >
> > L_m |vac> = sum_n n(m-n)/4 A_n A_m-n |vac>,
> >
> > because normal-ordered terms with at least one E annihilates
> > |vac>. When we compute [L_-m, L_m] |vac>, we get terms
> > proportional to AA|vac> and AAAA|vac>.
>
> I haven't done this calculation, but how can it be that you get A^4 terms?
> These won't be expressible in terms of first powers of L_m at all!?
The A^4 terms will be the same for L_-m L_m and L_m L_-m and
will thus cancel; I only worried about central terms. However,
I was wrong to conclude that no central terms arise. I assumed
that A transforms as a primary field, but in fact [L_m,A] ~ A + E.
Both L_-m L_m |vac> and L_m L_-m |vac> seems to give the same
central terms. However, they are proportional sum_n n^2 which
in infinite, so I subtract two equal infinite constants here,
which is a dubious operation.
Another observation: L_m |vac> is an infinite sum, which
probably means that L_m is an ill defined operator, which also
reminds us of LQG quantization.
I think it should be possible to approximately understand LQG
quantization of the string within the standard framework, even
if one has to resort to formal manipulations with ill-defined
operators. Just saying that Thiemann does something different
is not very satisfactory.
Urs Schreiber
> I think it should be possible to approximately understand LQG
> quantization of the string within the standard framework, even
> if one has to resort to formal manipulations with ill-defined
> operators. Just saying that Thiemann does something different
> is not very satisfactory.
Right. And we know exactly what it is that is different. Thomas Thiemann
constructs a Hilbert space on which he defines operators U_\pm(phi) which by
definition represent the diff x diff group without anomaly. The Virasoro
generators are not represented on his Hilbert space, only these U operators
are and they are _defined_ to produce the classical group without anomaly.
Such operators do exist, no problem, they just don't drop out of usual
quantization prescriptions.
The same is done in LQG for the spatial diffeomorphism constraints. There
Operators U(phi) are defined which represent the spatial diffeo group on the
space of spin network states. Only the Hamiltonian constraint is really
quantized itself and imposed as a Dirac constraints.
Thomas Larsson
>
> Another observation: L_m |vac> is an infinite sum, which
> probably means that L_m is an ill defined operator, which also
> reminds us of LQG quantization.
>
I do not longer think that my construction has anything to do
with LQG, but it shares the important property that the L_m
are not genuine operators. We have to be careful how we define
the Hilbert space, because it is an infinite-dimensional vector
space.
A natural basis for the Hilbert space consists of vectors
obtained from the vacuum by applying *finitely* many creation
operators, and a general vector is a *finite* linear
combination of basis vectors. Now, |vac> is evidently a vector
in the Hilbert space, but L_m|vac> is not, because it consists
of an *infinite* linear combination of basis vectors. Hence L_m
is not an operator.
I suppose that the computation of the central charge depends
critically on the assumption that the L_m's are operators. In
Hilbert spaces where this is not true, this need not be true.
In fact, it does not really make much sense to talk about the
central charge in that case.
What we see here is a breakdown of the Stone-von Neumann
theorem (which says that all quantizations are unitarily
equivalent) in infinite dimension. We start with the same
classical set of oscillators (a_m, b_m or E_m, A_m), and fix a
polarization by demanding that half of them annihilate the
vacuum. Either polarization defines a good quantization, but
the conformal generators only act as operators on the standard
Hilbert space.
Once we have two inequivalent quantizations, there might be
several. E.g., let
a(t) = sum_n exp(int) a_n
be the oscillators in real space. Then you can define a third
Hilbert space by demanding that a(t) annihilates that vacuum
for all t < 0. I think this is inequivalent to both the other
two. LQG quantization presumably gives us a fourth
possibility. In fact, I would bet that we have infinitely many
inequivalent quantizations of the string.
So which of these infinitely many quantization is the right
one? One answer is to leave the verdict to experiments. In that
case, I believe that the right answer is none of them, since
the evidence that any quantization of a classical string should
describe nature is at best very weak. However, from one
particular theoretical point of view, the standard quantization
is better:
If we have some classical system with an important symmetry,
even if it is only a gauge symmetry like conformal symmetry in
string theory, then a quantization which promotes the symmetry
generators to genuine operators is a privileged quantization.
Now apply this insight to general covariance...
Arnold Neumaier
> A natural basis for the Hilbert space consists of vectors
> obtained from the vacuum by applying *finitely* many creation
> operators, and a general vector is a *finite* linear
> combination of basis vectors. Now, |vac> is evidently a vector
> in the Hilbert space, but L_m|vac> is not, because it consists
> of an *infinite* linear combination of basis vectors. Hence L_m
> is not an operator.
The Hilbert space consists of the closure of the set of finite linear
combinations, hence of all absolutely convergent infinite such
combinations. Thus L_m is a (densely defined) linear operator.
Arnold Neumaier
Thomas Larsson
Arnold Neumaier <Arnold....@univie.ac.at> wrote in message news:<404449AC...@univie.ac.at>...
>
> The Hilbert space consists of the closure of the set of finite linear
> combinations, hence of all absolutely convergent infinite such
> combinations. Thus L_m is a (densely defined) linear operator.
>
This sum is not absolutely convergent, is it?
L_m |vac> = sum_n n(m-n)/4 A_n A_m-n |vac>.
The norm of A_n A_m-n |vac> is 1, right? So the norm of
L_m |vac> is infinite.
Anyway, all I wanted to do is to convince myself that there
are exotic quantizations of the string where L_m is not a
nice (bounded?) operator. I don't claim, and don't believe,
that such exotic quantizations are good for anything.
Arnold Neumaier
Thomas Larsson wrote:
> Arnold Neumaier <Arnold....@univie.ac.at> wrote in message news:<404449AC...@univie.ac.at>...
>
>>The Hilbert space consists of the closure of the set of finite linear
>>combinations, hence of all absolutely convergent infinite such
>>combinations. Thus L_m is a (densely defined) linear operator.
>
> This sum is not absolutely convergent, is it?
>
> L_m |vac> = sum_n n(m-n)/4 A_n A_m-n |vac>.
>
> The norm of A_n A_m-n |vac> is 1, right? So the norm of
> L_m |vac> is infinite.
Sorry, yes. I jumped from my first correct observation with too
little thinking to the second, wrong one...
Arnold Neumaier