LQG Dirac Quantization in Parametrized Field Theory: Madhavan Varadarajan

[f-h]

This is something I thought I should do if nobody else would by the time I had learned enough to do it:

gr-qc/0607068:
Title: Dirac Quantization of Parametrized Field Theory
Authors: Madhavan Varadarajan
Comments: 33 pages

Parametrized field theory (PFT) is free field theory on flat spacetime in a diffeomorphism invariant disguise. It describes field evolution on arbitrary foliations of the flat spacetime instead of only the usual flat ones, by treating the `embedding variables' which describe the foliation as dynamical variables to be varied in the action in addition to the scalar field. A formal Dirac quantization turns the constraints of PFT into functional Schrodinger equations which describe evolution of quantum states from an arbitrary Cauchy slice to an infinitesimally nearby one.This formal Schrodinger picture- based quantization is unitarily equivalent to the standard Heisenberg picture based Fock quantization of the free scalar field if scalar field evolution along arbitrary foliations is unitarily implemented on the Fock space. Torre and Varadarajan (TV) showed that for generic foliations emanating from a flat initial slice in spacetimes of dimension greater than 2, evolution is not unitarily implemented, thus implying an obstruction to Dirac quantization.
We construct a Dirac quantization of PFT,unitarily equivalent to the standard Fock quantization, using techniques from Loop Quantum Gravity (LQG) which are powerful enough to super-cede the no- go implications of the TV results. The key features of our quantization include an LQG type representation for the embedding variables, embedding dependent Fock spaces for the scalar field, an anomaly free representation of (a generalization of) the finite transformations generated by the constraints and group averaging techniques. The difference between 2 and higher dimensions is that in the latter, only finite gauge transformations are defined in the quantum theory, not the infinitesimal ones.


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This is of great importance, there have been repeated doubts whether LQG quantization can reproduce standard results. This implies that it can, and indeed that in a diffeomorphism invariant context the techniques employed are necessary (as Lee Smolin and Thomas Thiemann have repeatedly argued).

I'll be giving this a careful reading over the next couple of days, if anybody want's to discuss this I'd be very happy to do so...

 

[Careful]

***
This is of great importance, there have been repeated doubts whether LQG quantization can reproduce standard results. This implies that it can, and indeed that in a diffeomorphism invariant context the techniques employed are necessary (as Lee Smolin and Thomas Thiemann have repeatedly argued). ***

I am reading it (page 14) - and I shall reserve some comments for later. Let me say already that this is not LQG, it is an old idea of Kuchar (dynamical embeddings) in which the LQG quantization of finite diffeomorphisms can be extended to the entire spacetime diff-group.

Careful

 

[f-h]

Careful, yes, definately.

I meant LQG in the widest sense, as in a collection of techniques to construct quantum theories. This is an important test of these techniques.

There is of course, by the failure of Stone von Neumann, the possibility of different physically inequivalent quantizations of field theories. In the case of the flat space time field theory we know which one is the physically correct one.

If you write flat spacetime+field as a diffeo invariant theory and use the quantization techniques developed for diffeo invariant theories it reproduces this physically correct quantization. That was in doubt for a long time, especially by people who understood the new quantization method badly (like me) and misapplied it, or drew wrong conclussions about how it compared to other quantization methods.
This doesn't clinch it, but it's encouraging! (If the result holds up to scrutiny of course)

It increasingly looks like LQG methods give a highly non-trivial generalization of Quantum Field Theory. That in itself is pretty exciting.

 

[Careful]

Wait a moment f-h, here comes the more serious stuff (I read the first 20 pages now)

***
This is of great importance, there have been repeated doubts whether LQG quantization can reproduce standard results. This implies that it can, and indeed that in a diffeomorphism invariant context the techniques employed are necessary (as Lee Smolin and Thomas Thiemann have repeatedly argued). ***

Let me say already that this is not LQG, it is an old idea of Kuchar (dynamical embeddings) in which the LQG quantization of finite diffeomorphisms can be extended to the entire spacetime diff-group. I quickly read the first twenty pages (so I would be *glad* to see if my comments are off track):
(a) the paper starts by treating scalar field theory on the pages 9,10,11 - the author assumes the knowledge of the Bogoliubov transformations associated to a pair of hypersurfaces (a,b) in spacetime. More accurately, one should say parametrized surfaces, since indeed nowhere does the intrinsic geometry enter in the definition of the creation/annihilation operators, instead a global coordinate system is used (4.14). Also, it is nowhere treated in any detail how to make sense of Bogoliubov transformations attached to nonsmooth/exotic parametrizations (these parametrizations could just be bijections and they could be spacelike, null or timelike). They are simply assumed to exist and all resulting expressions are assumed to make sense, e.g (5.30). Actually, for the purposes of the paper, it does not even matter what they are, as long as the properties C(a,b)C(b,c) = C(a,c), C(a,a) = 1 as well as matching to the Minkowski C's are satisfied !
(b) the fact that the construction goes through and does not even depend upon detailed properties of the ``gauge'' group (as said : even foldings etcetera are allowed and the latter just has to be transitive) is simple given point (a) : the dynamical meaning of the diffeomorphism invariance is put in by hand (5.6 and 5.7) through the knowledge of the b operators (which are only known if the C's are).
(c) Restriction to diffeomorphisms which map smooth spacelike embeddings to smooth spacelike embeddings = restriction to the Poincare group. The author does not say it that bluntly but it is clear from his comments on pages 14 and 15 that he doesn't think otherwise. In this case, albeit all remarks in (a) are cancelled, the result is trivial by construction.
(d) What does all of this change when we take for example an interacting scalar field (for which we have no Hilbertspace representation yet) ? The Bogoliubov transformations are not going to be matrices, but include higher order tensors in which normal ordering problems etc. will occur. One will have to know all the C's again in order to *define* the action of the diff-group, how to do this in practice, and what guarantuees me that this representation even has a well defined meaning (see (a))?

Anyways, this is a starter. Let me add that I do appreciate that the physical result itself does not depend upon kinematical Hilbert space construction involved as long as all intermediate steps are well defined. Some things which do worry me more are

(1) What about non-linear theories ??
(2) What does this do for me concerning quantum covariance in LQG : in the old formalism the spacelike diffeomorphism constraints are treated very differently from the Hamiltonian constraint and clearly the technique in this paper does not apply.
Here you *start* from a well known quantization and show that one can implement covariance (without making the metric a dynamical variable) by making use of known results. The goal of LQG is actually the reverse :-), to provide a quantization where standard techniques fail.


Careful

 

[f-h]

I'm not getting deep reading done, but just some comments:

5.6 and 5.7 seem to be the precise way in which we usually construct a nonseperable HIlbertspace carrying a unoitary representation of the diffeomorphism group from cyclical functions in LQG. There is nothing "put in by hand" here as far as I can see.

"Restriction to diffeomorphisms which map smooth spacelike embeddings to smooth spacelike embeddings = restriction to the Poincare group."

No. This would be for flat plane to flat plane, but not for smooth surface to smooth surface. The diffeo invariance is not broken down to poincare invariance, it's avaraged out in the LQG way.

"The goal of LQG is actually the reverse :-), to provide a quantization where standard techniques fail. "
Yes, but that the new techniques seem to coincide with the old in a natural way when the old apply is an important check for the physical viability of this way of constructing quantum theories.

If we want to build a new more powerfull and general way to build quantum theories which can't even construct the ones we know to work then one can very reasonably question whether this is a physically viable route at all.
So what this paper does is ail a powerfull a priori doubt about the physical viability of these techniques.
It happens to be a doubt that had been on my mind quite a bit recently, which is why I was very keen to dig deeper into this paper...

 

[Careful]

***
5.6 and 5.7 seem to be the precise way in which we usually construct a nonseperable HIlbertspace carrying a unoitary representation of the diffeomorphism group from cyclical functions in LQG. There is nothing "put in by hand" here as far as I can see. ***

What is ``put in by hand'' is the action of the diffeo's on the b operators of the matter sector - in standard quantization such action for the generators is derived (and might contain anomalous terms which do not enter here because your theory is free of course).

**
"Restriction to diffeomorphisms which map smooth spacelike embeddings to smooth spacelike embeddings = restriction to the Poincare group."

No. This would be for flat plane to flat plane, but not for smooth surface to smooth surface. The diffeo invariance is not broken down to poincare invariance, it's avaraged out in the LQG way.***

You seem to misunderstand what I wrote here. I said that the maximal group which maps *any* *spacelike* hypersurface to a *spacelike* hypersurface (they do not need to be flat planes at all) is the Poincare group (added with a dilatation and four accelerations - at least this result holds for Minkowski in 4 dimensions). Even the author says that on page 14 (I guess) where he wonders whether $$G(\eta)$$ would be bigger than the group of conformal diffeomorphisms (which is only 15 dimensional).

***
"The goal of LQG is actually the reverse :-), to provide a quantization where standard techniques fail. "
Yes, but that the new techniques seem to coincide with the old in a natural way when the old apply is an important check for the physical viability of this way of constructing quantum theories. ***

The construction of the kinematical representation is full of ambiguities (what group to choose, how to pick out the relevant Bogoliubov transformations and how to even define them in some cases). Also, the way the paper treats diff invariance is not how you do it LQG, so what does this paper teach us apart from the fact that I can setup an extravagant (and rather obvious) procedure starting from a known result to reproduce the latter again ?

***
So what this paper does is ail a powerfull a priori doubt about the physical viability of these techniques. **

Well, I don't know about anomalies yet, but someone else who is better in this might comment on that? These techniques are only useful once you can employ them where standard techniques fail, otherwise it is an expensive way to run in circles.

Careful

 

[f-h]

Hu? Poincare maps spacelike to spacelike, yes, but to go from *any* spacelike to *any* spacelike, like they say, you need diffeos. And I don't see why they say they would need (conformal) isometries of the metric, after all all spatial diffeomorphisms that act on the embedding without moving it, surely satisfy this property as well. But I'll have to give this some more thought...

 

[Careful]

Hu? Poincare maps spacelike to spacelike, yes, but to go from *any* spacelike to *any* spacelike, like they say, you need diffeos. And I don't see why they say they would need (conformal) isometries of the metric, after all all spatial diffeomorphisms that act on the embedding without moving it, surely satisfy this property as well. But I'll have to give this some more thought...

You are confusing this construction with what you do in LQG where one works with a *fixed* embedding. Imagine the space S of all spacelike embeddings and the natural action of the spacetime diffeo's from the left. Then, the only group whose action remains within S is the group of conformal diffeomorphisms.

Careful

 

[f-h]

No I saw that difference, but thought the following:

Let's see, the condition that we have a spacelike embedding is that X^A (t,x) is a spacelike surface in our usual minkowski metric on the X^A. The condition they write for this is that $$q_{ab}(x) = \eta_{AB}\frac{\partial X^A}{\partial x^a}\frac{\partial X^B}{\partial x^b}$$ is a valid metric of positive signature. But equivalently I can write the condition:
$$q_{00}(x) = \eta_{AB}\frac{\partial X^A}{\partial t}\frac{\partial X^B}{\partial t} < 0$$

That is, the t direction should be timelike.
If we now take a diffeomorphism that acts only on the x coordinates they clearly can't take an embedding that satisfies this condition into one that doesn't, since it only changes the labeling of the points in this condition but the condition has to hold at all points anyways.

Am I making a stupid mistake or drastically misreading something?

 

[Careful]

***No I saw that difference, but thought the following:

Let's see, the condition that we have a spacelike embedding is that X^A (t,x) is a spacelike surface in our usual minkowski metric on the X^A. The condition they write for this is that $$q_{ab}(x) = \eta_{AB}\frac{\partial X^A}{\partial x^a}\frac{\partial X^B}{\partial x^b}$$ is a valid metric of positive signature. But equivalently I can write the condition:
$$q_{00}(x) = \eta_{AB}\frac{\partial X^A}{\partial t}\frac{\partial X^B}{\partial t} < 0$$

That is, the t direction should be timelike. ***

No, that is incorrect. Take as counterexample $$(t,x) -> (t,x+2t)$$ then the above expression becomes - 1 + 4 = 3 > 0; therefore $$\partial_t$$ and $$\partial_x$$ are both spacelike.

***
If we now take a diffeomorphism that acts only on the x coordinates they clearly can't take an embedding that satisfies this condition into one that doesn't, since it only changes the labeling of the points in this condition but the condition has to hold at all points anyways. ***

The diffeomorphism applies to the left, in other words it acts on the (X,T) embedding coordinates (which are the dynamical variables), not on the (x,t) ones (see around equation 2.6).

Careful

 

[f-h]

With acting on the x I meant a map from x->x (acting on X) instead of a map from (x,t) -> (x,t).

I have to say I find the paper highly confusing especially in the details. Maybe it's just the language, but it seems to be rather unclean conceptually.

I think what they (attempt to) do is important but it doesn't look like it's quite there yet. Maybe I'll get back to it later.

 

[f-h]

Oh and thanks for the discussion careful. Really helpful

 

[Careful]

With acting on the x I meant a map from x->x (acting on X) instead of a map from (x,t) -> (x,t).

I have to say I find the paper highly confusing especially in the details. Maybe it's just the language, but it seems to be rather unclean conceptually.

I think what they (attempt to) do is important but it doesn't look like it's quite there yet. Maybe I'll get back to it later.

Hi,

The underlying idea is more or less the same than in LQG but it is important to realize the details of the latter. There, you start out with a coordinate system (x,t) and an embedding $$T : (x,t) -> Manif$$ where indeed it is *required* that $$\{T(x,t) | x \in R^n \}$$ are spacelike hypersurfaces and $$\partial_t$$ is timelike with norm $$- N^2$$ where N is the lapse (that is just how we decompose the spacetime metric). Here, the dynamical variables are the spacelike metric and it's conjugate momentum, added with four lagrange multipliers (the lapse and shift) which correspond to the diffeomorphism freedom. Again, the diffeomorphisms act from the left on the metric, but it's action can be canonically pulled back to (x,t) by means of the fixed embedding T which allows you to write $$\psi (x,t)$$. This is not the case anymore in the work of Kuchar (Varadarajan) since here the embedding is the dynamical object at hand and there is no way to pull back the action of diffeo's (there is no reference embedding so to speak).

The details of the paper (apart from some technicalities which I mentioned before) seem to be all right.

Careful

 

[f-h]

What do you mean by "act from the left on the metric"?

The metric transforms like any 2-form under diffeos. Now in LQG the foliation we use breaks 4-diffeo invariance and reduces it to 3-diffeos + lapse.

Here, too, the 4-diffeo invariance is broken (right above 2.4 where a time coordinate is singled out (even though as you pointed out, it's not actually required to be timelike)), without giving a full foliation. But if I specify the diffeomoprhisms acting on the dynamical fields using this information they should respect it, too.

If I want to treat PFT as a proper Background independent theory then the X are simply fields, it's misleading to define the diffeomorphisms on the manifold by "vector fields on the X" by abusing the notation and using X as coordinates.

 

[Careful]

**
The metric transforms like any 2-form under diffeos. Now in LQG the foliation we use breaks 4-diffeo invariance and reduces it to 3-diffeos + lapse. **

Huh ?! The foliation *by itself* in LQG does not break 4-diffeo invariance at all ! Classically, you have the four constraints which define an (anti) representation of the 4-diff Lie algebra. With ``by the left'' I mean that the diffeo's are defined in terms of F(x,t) (where F is the foliation) and not (x,t). You must distinguish a diffeomorphism from a coordinate transformation - I have seen many people make that mistake (while it is actually elementary differential geometry). I am free to choose my coordinates (foliation) without ``touching'' diff-invariance (actually the coordinates on M in this paper are the Minkowski inertial coordinates and not the (x,t) !).

**
Here, too, the 4-diffeo invariance is broken (right above 2.4 where a time coordinate is singled out (even though as you pointed out, it's not actually required to be timelike)), without giving a full foliation. But if I specify the diffeomoprhisms acting on the dynamical fields using this information they should respect it, too. **

No, this theory is 4-diff invariant ! Diffeo invariance has a dynamical meaning in the canonical formalism, not a kinematical one : it is defined in terms of functional derivatives of the dynamical fields (the metric in this paper is not a dynamical field).

**
If I want to treat PFT as a proper Background independent theory then the X are simply fields, it's misleading to define the diffeomorphisms on the manifold by "vector fields on the X" by abusing the notation and using X as coordinates. ***

This is not an abuse of notation, this was the point Kuchar and Isham tried to make The author does not use the X as coordinates at all, X : R^{n+1} -> M where R^{n+1} is considered as a different manifold. The X^A(x,t) on the other hand are the Minkowski coordinates of X(x,t) (the inertial coordinate system itself is left untouched by the diffeomorphism).

Cheers,

Careful

 

[f-h]

I'm well aware of the difference between diffeomorphisms and coordinate transformations.
I can give a foliation of a manifold without specifying coordinates, and this will usually not be taken to itself, or into the same class of foliations by by a diffeomorphism.

And of course only explicit invariance is broken, the canonical analysis singles out a time coordinate (the evolution in which is a gauge transformation of course). If the canonical analysis is done correctly the theories are of course equivalent, even if it's no longer explicitly diffeomorphism invariant (and we get the hypersurface deformation algebra). I think this shows up in the far to drastic restriction on the vectorfields they get.

But I don't properly understand the point of view taken wrt the diffeomorphisms here, maybe I need to go back and read the Kuchar+Isham papers.

 

[Careful]

**I'm well aware of the difference between diffeomorphisms and coordinate transformations.
I can give a foliation of a manifold without specifying coordinates, and this will usually not be taken to itself, or into the same class of foliations by by a diffeomorphism. **

But this autor does give foliations without specifying coordinates (his X), in LQG the foliation *is* a choice of coordinates.

***
And of course only explicit invariance is broken, the canonical analysis singles out a time coordinate (the evolution in which is a gauge transformation of course). If the canonical analysis is done correctly the theories are of course equivalent, even if it's no longer explicitly diffeomorphism invariant (and we get the hypersurface deformation algebra). I think this shows up in the far to drastic restriction on the vectorfields they get. ***

Huh ? Their vectorfields in this paper are as general as they can be given that the author would specify Bogoliubov transformations between spacelike and timelike surfaces - say.

Careful

 

[f-h]

Well as you pointed out they restrict it to conformal isometries of the metric. That restriction would kill quantization because averaging over these transformations surely is not enough to kill the gauge degrees of freedom. That's why they have to enlarge their group of transformations in a completely ad hoc manner.

 

[Careful]

Well as you pointed out they restrict it to conformal isometries of the metric. That restriction would kill quantization because averaging over these transformations surely is not enough to kill the gauge degrees of freedom. That's why they have to enlarge their group of transformations in a completely ad hoc manner.

Right, the conformal isometries are too small for the killing purposes (no transitive property on the space of spacelike hypersurfaces) The enlargement is rather free (transitive property needs to be satisfied) - which is ``unnatural'', anyway go to sleep now :zzz: :zzz: