#### marcus

Science Advisor

Gold Member

Dearly Missed

- 24,713

- 783

There has been a lot of internet discussion of a paper by Ashtekar, Fairhurst, and Willis called "Quantum gravity, shadow states, and quantum mechanics" gr-qc/0207106, and one of the authors, Josh Willis, replied on SPR today (in the thread "LQG and diffeomorphism group cocycles")

Here is a representative exerpt of the longer post:

--------exerpt from Willis post------

Well, since Urs keeps referring to this paper I co-authored, I feel I

ought to respond with what I see.

So I will start by replying to the remarks about gr-qc/0207106, and

then work backwards towards some more general remarks about the

relationship between group averaging and anomalies, as I see it. This means I take the quoted article in reverse order. My apologies if this is confusing, but I think in the long run it will be clearer.

So, let me start with:

Urs Schreiber wrote:

> Sometimes LQG papers like gr-qc/0207106 are mentioned which supposedly show

> that the LQG-like 'form of quantization' reproduces ordinary quantization in

> some limit. But having a closer look at this paper shows that this _only_

> works when the usual quantum corrections are copied to the LQG-formalism

> (lower half of p.14). This is however not the case in the 'LQG-string' or

> the LQG treatment of the spatial diffeo constraints of 3+1d gravity.

I do not understand your assertion that the "quantum

corrections" are copied into the model system we study, and that this

is somehow very different from what is done with, for instance, the

diffeomorphism constraint in LQG.

So, for people who haven't read page 14 of gr-qc/0207106, let me

summarize quickly what is going on.

And let me start by saying that page 14 of that paper is the wrong

place to be looking to begin with: there we are finding candidate

semiclassical states, not considering commutation relations. Instead,

look at pages 7 and 8, to start.

We are looking at the one-dimensional point particle in quantum

mechanics. We have not yet considered any particular Hamiltonian, but

are instead at this point in the paper looking only at the canonical

commutation relations.

One normally learns these as:

{q,p} = 1

and then seeks in the quantum theory for self-adjoint operators

\hat{q} and \hat{p} that satisfy the corresponding relations:

[\hat{q},\hat{p}] = i\hbar

But these commutation relations among the p's and q's of course imply

commutation relations among the exponentiated operators, which---if

\hat{q} and \hat{p} are self-adjoint---will be unitary. This is the

relation:

(*) U(l)V(m) = exp{-ilm}V(m)U(l)

where:

U(l) = exp{il\hat{q}}

V(m) = exp{im\hat{p}}

The relation (*) is that of the Weyl-Heisenberg algebra, and what we

do in our paper is look at a non-standard unitary representations of

this algebra, rather than looking at a self-adjoint representation of

the CCR. The particular representation we look at is motivated by

analogy with LQG, and the main point of the paper is to look at the

low-energy limit of this theory, using techniques that it is hoped

will be relevant for LQG. However, as that does not seem to be what

the questions are about at the moment, I will not say more about that

here.

And this is the key point: the relation (*)---including the factor of

exp{-ilm}, which is what in other places on the net Urs seems to think

is put in by hand, i.e. by "knowing" what the quantum theory should

be---is in fact dictated *classically* by the Poisson algebra of the

basic observables. Any representation by unitary operators which

purports to be a canonical quantization *must* have this relationship,

and in particular the representation we consider does (as of course

also does the standard Schroedinger representation)...

--------end quote-------

Here is a representative exerpt of the longer post:

--------exerpt from Willis post------

Well, since Urs keeps referring to this paper I co-authored, I feel I

ought to respond with what I see.

So I will start by replying to the remarks about gr-qc/0207106, and

then work backwards towards some more general remarks about the

relationship between group averaging and anomalies, as I see it. This means I take the quoted article in reverse order. My apologies if this is confusing, but I think in the long run it will be clearer.

So, let me start with:

Urs Schreiber wrote:

> Sometimes LQG papers like gr-qc/0207106 are mentioned which supposedly show

> that the LQG-like 'form of quantization' reproduces ordinary quantization in

> some limit. But having a closer look at this paper shows that this _only_

> works when the usual quantum corrections are copied to the LQG-formalism

> (lower half of p.14). This is however not the case in the 'LQG-string' or

> the LQG treatment of the spatial diffeo constraints of 3+1d gravity.

I do not understand your assertion that the "quantum

corrections" are copied into the model system we study, and that this

is somehow very different from what is done with, for instance, the

diffeomorphism constraint in LQG.

So, for people who haven't read page 14 of gr-qc/0207106, let me

summarize quickly what is going on.

And let me start by saying that page 14 of that paper is the wrong

place to be looking to begin with: there we are finding candidate

semiclassical states, not considering commutation relations. Instead,

look at pages 7 and 8, to start.

We are looking at the one-dimensional point particle in quantum

mechanics. We have not yet considered any particular Hamiltonian, but

are instead at this point in the paper looking only at the canonical

commutation relations.

One normally learns these as:

{q,p} = 1

and then seeks in the quantum theory for self-adjoint operators

\hat{q} and \hat{p} that satisfy the corresponding relations:

[\hat{q},\hat{p}] = i\hbar

But these commutation relations among the p's and q's of course imply

commutation relations among the exponentiated operators, which---if

\hat{q} and \hat{p} are self-adjoint---will be unitary. This is the

relation:

(*) U(l)V(m) = exp{-ilm}V(m)U(l)

where:

U(l) = exp{il\hat{q}}

V(m) = exp{im\hat{p}}

The relation (*) is that of the Weyl-Heisenberg algebra, and what we

do in our paper is look at a non-standard unitary representations of

this algebra, rather than looking at a self-adjoint representation of

the CCR. The particular representation we look at is motivated by

analogy with LQG, and the main point of the paper is to look at the

low-energy limit of this theory, using techniques that it is hoped

will be relevant for LQG. However, as that does not seem to be what

the questions are about at the moment, I will not say more about that

here.

And this is the key point: the relation (*)---including the factor of

exp{-ilm}, which is what in other places on the net Urs seems to think

is put in by hand, i.e. by "knowing" what the quantum theory should

be---is in fact dictated *classically* by the Poisson algebra of the

basic observables. Any representation by unitary operators which

purports to be a canonical quantization *must* have this relationship,

and in particular the representation we consider does (as of course

also does the standard Schroedinger representation)...

--------end quote-------

Last edited: