QG and diffeomorphism group cocycles

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Thomas Larsson

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Mar 16, 2004, 3:48:30 AM3/16/04
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"Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote in message news:<404f064e$1...@news.sentex.net>...

"I believe that LQG-like quantization of gravity could make
sense for some V=complicated correction. Maybe.

But how should we find this V? And is anyone looking for it?

[...]

Ok, so it seems that I am proposing a new LQG program:

Find a V=something quantum correction to the diffeomorphism
group such that a sensible semiclassical limit is obtained. "

>
> I'll admit or deny authorship after having seen the reaction of the
> experts... ;-)
>

I will come back to this point, because I want you to ponder the logical
consequences of your suggestion.

I think you mean that LQG should treat the diffeo constraint in a
similar way as string theory handle its conformal worldsheet symmetry.
In particular, the diffeo group should acquire some kind of quantum
correction, at least if you restrict yourself to the fields. There
should perhaps also be a negative contribution from ghosts, making the
total anomaly zero, but the field contribution should be non-zero.

If this is not what you mean, then you should please correct me.
However, if I have indeed understood you correctly, one question
immediately arises: why just LQG? The main complaint about Thomas
Thiemann's string quantization is that he gets a different conformal
anomaly than string theory does. If there is essentially a unique
way to quantize the string, as I believe, then there should be a
single correct value for the central charge (for the fields alone).
If so, shouldn't there also be a single correct value for a diffeo
anomaly, to be found in any correct quantization of gravity?

In short: if the diffeo group should be anomalous in LQG, shouldn't it
be anomalous in any correct formulation of QG, based on strings, fields,
loops, or whatever?

Urs Schreiber

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Mar 16, 2004, 4:08:42 AM3/16/04
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"Thomas Larsson" <thomas_l...@hotmail.com> schrieb im Newsbeitrag
news:24a23f36.04031...@posting.google.com...

> I think you mean that LQG should treat the diffeo constraint in a
> similar way as string theory handle its conformal worldsheet symmetry.

The Polyakov action is nothing but 1+1 dimensional gravity coupled to scalar
fields. It would seem that by looking at it from this point of view one can
learn something about the quantization of 3+1d gravity. (Of course some
things in 1+1 d are unique to these numbers of dimensions, like the
classical conformal invariance.)

> In particular, the diffeo group should acquire some kind of quantum
> correction, at least if you restrict yourself to the fields. There
> should perhaps also be a negative contribution from ghosts, making the
> total anomaly zero, but the field contribution should be non-zero.
>
> If this is not what you mean, then you should please correct me.

As far as my current understanding goes this is what I mean. Often when I
say something outright wrong Aaron Bergman is so kind to point out the
mistake. So let's see if he does! :-)

> However, if I have indeed understood you correctly, one question
> immediately arises: why just LQG? The main complaint about Thomas
> Thiemann's string quantization is that he gets a different conformal
> anomaly than string theory does.

I wouldn't put it this way. The main complaint is rather that the procedure
by which he arrives at that result is not what is commonly called 'canonical
quantization'. It does for instance neither follow from path integral
quantization, BRST quantization or Dirac/Gupta-Bleuler (OCQ) quantization.

> If there is essentially a unique
> way to quantize the string, as I believe, then there should be a
> single correct value for the central charge (for the fields alone).

How about (D/12)(m^3 - m) ? :-)

> If so, shouldn't there also be a single correct value for a diffeo
> anomaly, to be found in any correct quantization of gravity?

Yes, maybe, that's at least what I currently expect based on my current
understanding.

> In short: if the diffeo group should be anomalous in LQG, shouldn't it
> be anomalous in any correct formulation of QG, based on strings, fields,
> loops, or whatever?

Well, this is indeed the question which I posed a few days ago in
http://groups.google.de/groups?selm=4051f3fc%241%40news.sentex.net . I have
also asked Ashtekar about this, but he didn't know. I should talk to a
string expert.

In http://golem.ph.utexas.edu/string/archives/000330.html#c000786 I
speculate about what would happen if LQG tried to handle the BRST extended
(including ghost sector) version of the EH action.

But it is not clear to me yet that extrapolations from quantization of 1+1d
gravity to the background equations motion found in string theory is
possible. I am not even sure if it makes sense to quantize the effective
action of striung backgrounds.

I have once talked about this with Thomas Mohaupt and his group, who are
working on string cosmology questions. Thomas Mohaupt himself told me that
he wasn't sure if it makes sense to quantize the string background effective
action, or if it must rather be regarded already as a quantized theory, the
quantum corrections being all the higher-order terms in alpha'. So perhaps
the answer is that in string theory the gravitational field is on a
completely different footing than is assumed in other approaches of QG, like
LQG or minisuperspace quantum cosmologies. I don't know. Comments are
appreciated.

Urs Schreiber

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Mar 16, 2004, 11:16:18 AM3/16/04
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"Thomas Larsson" <thomas_l...@hotmail.com> schrieb im Newsbeitrag
news:24a23f36.04031...@posting.google.com...

> I think you mean that LQG should treat the diffeo constraint in a


> similar way as string theory handle its conformal worldsheet symmetry.
> In particular, the diffeo group should acquire some kind of quantum
> correction, at least if you restrict yourself to the fields.

I have now talked with Hermann Nicolai about this point. He tells me that
this is more or less what he would expect. He says that he does not like the
space+time splitting of LQG and that he would rather see LQG handle the
constraints of 3+1d gravity in precisely the same (or at least analogous)
way as is done for 1+1d gravity in standard quantization of the Polyakov
action.

(My own opinion about that is, in case anyone cares to know, that the
space-time splitting itself and as such is not the problem. When doing
canonical quantization of the string in the Schroedinger picture instead of
in the Heisenberg picture, which is usually used in field theory, there is
also a space/time split on the worldsheet. But since this is just a question
of quantum pictures this does not affect in any way the resulting formulas
and results, as can easily be checked explicitly.)

I told H. Nicolai that you told me that all possible anomalies for the
diffeo algebra in arbitrary dimensions have been classified and understood.
(Hope that's about right.) He told me that he instead had been told that
there are infinitely many possibilities and that people don't know what to
do with them.

Unfortunately I couldn't say more regarding this point than what I knew
second-hand. Since I didn't have any references with me you should perhaps
contact Nicolai and point him to the respective papers. I'll be visiting the
AEI in April and will maybe have time to further clarify this then.

I also asked him if and how he thinks the equivalent problem is dealt with
in string theory. I.e. I asked if he thinks that it even makes sense to try
to canonically quantize the effective background action.

He said that he wouldn't try that, because nobody knows how to do it.
Rather, one should try to understand closed string field theory. Of course
there are proposals for how this could be done (using limits of open string
field theory or of Matrix Models), but it has not been done yet.

He didn't say so explicitly, but my summary of what he said is that if we
were able to do closed string field theory we would probably see a spacetime
diffeomorphism anomaly with respect to the physical fields which vanishes
when the ghosts are taken into account.

> There
> should perhaps also be a negative contribution from ghosts, making the
> total anomaly zero, but the field contribution should be non-zero.

Yes, that's the impression I got about what Hermann Nicolai is thinking, and
it sounds pretty plausible to me. (But please note very well that everything
I report here is only my possibly incomplete and maybe unintentionally
distorted summary of our conversation.)

> In short: if the diffeo group should be anomalous in LQG, shouldn't it
> be anomalous in any correct formulation of QG, based on strings, fields,
> loops, or whatever?

Ok, I hope I have successfully reported a possible answer to that! :-)

Thomas Larsson

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Mar 17, 2004, 9:09:20 AM3/17/04
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"Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote in message news:<40572852$1...@news.sentex.net>...


>
> I told H. Nicolai that you told me that all possible anomalies for the
> diffeo algebra in arbitrary dimensions have been classified and understood.
> (Hope that's about right.) He told me that he instead had been told that
> there are infinitely many possibilities and that people don't know what to
> do with them.
>

We probably compare apples with oranges. What has been classified are
abstract extensions of the diffeo algebra; more precisely, the algebra
of polynomial vector fields in N dimensions by modules of tensor
fields. There are many concrete nonlinear realizations of each abstract
cocycle using a given set of fields.

However, the anomaly by itself is nothing. What singles out the Virasoro
algebra in any dimension is that it has nice representations, which
essentially come from canonical quantization. One starts from a classical
oscillator realization, introduce a Fock vacuum, and normal order w.r.t.
this vacuum. However, when N > 1 one must start from a particular type of
nonlinear realization, which makes the extension non-central.

It is instructive to look at the algebra of gauge transformations instead.
It has an abstract cocycle of the form

[J(X), J(Y)] = J([X,Y]) + K^{abij}(d_i X^a d_j Y^b),

where X and Y are g-valued functions, d_i = d/dx^i and the extension
K^{abij} transforms as the indices indicate: as adjoint x adjoint under
g and as a two-tensor under diffeos. This abstract cocycles can be
concretely realized in 3D as

K^{abij}(F_{abij}) = e^ijk d^abc \int d^3x F_{abij}(x) A_ck(x)

for any smearing function F_{abij}. Here e^ijk is the epsilon tensor
(hence this only works in 3D), d^abc are the totally symmetric
structure constants associated to the third Casimir, and A_ck is an
adjoint-valued vector. In higher dimensions, the same abstract cocycle
can be realized in other ways, see http://www.arxiv.org/abs/hep-th/9401027 .

It turns out that Jacobi identities still hold if A_ck is a connection
rather than an adjoint vector. The algebra is then known as the
Mickelsson-Faddeev algebra, which pertains to the Hamiltonian formulation
of the chiral anomaly. Some kind of no-go theorem has been proven for
this algebra, which essentially states that it has no good representation,
see

D. Pickrell, On the Mickelsson-Faddeev extensions and unitary representations,
Comm. Math. Phys. 123 (1989) 617.

There is also a compelling physical reason to avoid this kind of anomaly:
the fermion content of the standard model is chosen to make d^abc = 0.

The gauge algebra also has a different kind of anomaly, which generalizes
affine Kac-Moody algebra to higher dimensions:

[J(X), J(Y)] = J([X,Y]) + delta^ab S^i(X^a d_i Y^b)

S^i(d_i F) = 0.

delta^ab is the Killing metric proportional to the second Casimir, which
does not vanish in the standard model, so the usual demand for anomaly
cancellation does evidently not apply here. S^i not quite a tensor field,
due to the last condition, but rather dual to a closed one-form. This
is the kind of extension that arises from normal ordering and has a
nice representation theory.

Let me also explain how to build representations. The critical idea is
to expand all fields in a Taylor series, written in 1D for simplicity:

f(x) = sum_k f_k (x-q)^k.

Classically at least, we can express everything in terms of Taylor data

q, f_0, f_1, f_2, ...

instead of field data f(x). This may be inconvenient and there may be
problems with convergence, but I see no conceptual problems here. Diffeos
evidently act on the Taylor data as well. One may ask what such a trivial
reformulation is good for. The answer is that the cocycle S^i depends on
the expansion point q. There is no way to express the cocycle in terms of
the field f(x) which is indendent of q.

This is in agreement with the well-known fact that the only anomalies
in 4D is a pure gauge anomaly proportional to the third Casimir and a
mixed gravity-U(1) anomaly proportional to the first Casimir, see e.g.
Weinberg II, chapter 22. To express the Kac-Moody anomaly proportional
to the second Casimir, or the pure gravitational Virasoro anomaly, you
need to use Taylor data, which evades Weinberg's axioms.

The situation is slightly more complicated. Instead of expanding around
a point q we must expand around a closed 1D line q^i(t); the Taylor
coefficients are then replaced by functions f_k(t). We must also
truncate the Taylor series at some finite order p. This can be viewed as
a kind of regularization, with the very peculiar property that diffeos
act in a natural manner. We now have a realization on finitely many
functions of a single variable t, which is exactly the situation where
normal ordering works.

The cocycle takes the explicit form

S^i(F_i) = \int dt dq^i(t)/dt F_i(q(t)).

The closedness condition follows from

S^i(dF) = \int dt dF(q(t))/dt = 0,

because there are no endpoints. The cocycle is central in 1D because
there is a single circle along which to integrate.

>
> Ok, I hope I have successfully reported a possible answer to that! :-)

I am very grateful for your efforts.

Thomas Larsson

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Mar 22, 2004, 10:51:25 AM3/22/04
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Urs Schreiber <Urs.Sc...@uni-essen.de> skrev i
diskussionsgruppsmeddelandet:4056c41a$1...@news.sentex.net...

> I have once talked about this with Thomas Mohaupt and his group, who are
> working on string cosmology questions. Thomas Mohaupt himself told me that
> he wasn't sure if it makes sense to quantize the string background effective
> action, or if it must rather be regarded already as a quantized theory, the
> quantum corrections being all the higher-order terms in alpha'. So perhaps
> the answer is that in string theory the gravitational field is on a
> completely different footing than is assumed in other approaches of QG, like
> LQG or minisuperspace quantum cosmologies. I don't know. Comments are
> appreciated.

If gravity is on a different footing from the other fields, you would
lose your case against LQG, wouldn't you? LQG quantization of the
string
may be wrong (perhaps), but the real issue is whether LQG quantization
of gravity is right. If gravity is different (and 4D gravity is very
different from 2D gravity), LQG still may be ok. Or maybe gravity
shouldn't be quantized at all, as suggested in other threads on this
newsgroup.

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