LOST theorem found (important result for LQG)

[marcus]

Hi Marcus

Certainly one can refer informally to an algebra as an algebra of basic, quantum observables. However, the words are not the problem.

Self-adjoint elements are required because of their real spectrum...

yup, I know about real spectrum of s.a. operators, natch.

but you are still not getting it.

\frak{A} does not consist of stuff that is all s.a., or hermitian, or whatever you want to call a*=a.

It has some elements that DO satisfy a*=a
but in general the elements of \frak{A} do not.

My question to you is why do you Mike imagine that the author's want to have \frak{A} consist only of such elements?

Can you point to a particular line of mathematics on a particular page?

I see no indication that Lewandowski et al ever dreamed that anyone might suppose that \frak{A} consists only of such (a* = a) things.

Indeed the Cylinder functions are obviously not such, and they are effectively a large subset of \frak{A}.

 

[marcus]

Mike I do not understand why you haven't caught on to this yet, it seems very simple.

It occurred to me how you might grasp it---try this as an experiment.
Take your copy of the Lewandowski et al paper and turn to page 6, beginning of section 2, and
STRIKE OUT THE WORDS "of basic, quantum observables".

consider those words to be a non-essential verbal slip having nothing to do with the mathematics, or the rest of the paper.

now re-read the paper and see if you understand it.

Believe me, you have not found a mistake in the proof of theorem 4.2

those words might as well have been some other casual phrase like
"of basic, quantum variables".

If you continue presuming that they mean \frak{A} consists solely of a* = a type stuff you will never get to first base with the paper.
Cause the first thing you will find out when you see how \frak{A} is constructed is that it is based on stuff which is a* NOT = a.

 

[kneemo]

Mike I do not understand why you haven't caught on to this yet, it seems very simple.
...
Believe me, you have not found a mistake in the proof of theorem 4.2

Hi Marcus

Re-read page 5 and consider the motivation behind the LOST GNS construction. Here are some important points:

A simple formulation of these properties can be given by asking for a state (i.e. a positive, normalized, linear functional) on A that it is invariant under the classical symmetry automorphisms of A. Given a state on A one can define a representation via the GNS construction.

Finally, if the state is invariant under some automorphism of A, its action
is automatically unitarily implemented in the representation.

Finding a state that is invariant under automorphisms of the holonomy-flux *-algebra is the real issue. In the Jordan GNS construction, state is given by a hermitian form, trace. Trace vanishes for any infinitesimal action of the automorphism group G, once the holonomy-flux *-algebra is take to be a Jordan algebra. Therefore there exists a state (trace) in the Jordan GNS construction that is invariant under classical symmetries of the holonomy-flux *-algebra. This is an interesting result that I'm sure Thiemann would appreciate.

Regards,

Mike

 

[marcus]

Hi Marcus

Re-read page 5 and consider the motivation behind the LOST GNS construction. Here are some important points:
...

Hello Mike, it seems to me that whatever points you have to make are apt to be premised on your belief that elements of \frak {A} must satisfy the condition a* = a. The points would therefore be invalid.

1. If you don't mind, I would like a clear explanation for why you thought that. Quote some specific part of the LOST paper. Please be explicit. Maybe you were misinterpreting something out of context.

2. I want to hear definite news from you that you have changed your view on that point and are no longer assuming that the authors want a* = a.

If necessary one of us can recommend that in the first sentence of section 2 on page 6 of the LOST paper the non-essential words "of basic quantum observables" could be deleted.

they don't add anything, nothing else depends on them, and the words may conceivably have confused other people besides yourself.

 

[marcus]

Mike, maybe this will help

https://www.physicsforums.com/showthread.php?p=561200#post561200

I copied out a key passage from one of Jerzy Lewandowski's papers that exemplifies how some of the terminology is being used.

 

[marcus]

I have reliable confirmation that (even though the term "basic kinematical observables" was used) it was not intended to suggest that the elements of the *-algebra should be thought of as self-adjoint.

More precisely, one should not assume that a* = a, a general element of the star-algebra is not a fixed point of the involution.

Unfortunately, as I expected, it seems that Mike Rio's paper
http://arxiv.org/abs/gr-qc/0505038
is more or less empty. The paper is alleged to be about the case where the holonomy-flux algebra is a Jordan algebra, but since elements of the hol-flux algebra usually don't have a* = a, that case would not seem to come up.

More specifically, I do not believe that Mike, or anybody, could exhibit a case of a manifold \Sigma with its accompanying space of connections \mathcal {A} \text{ } and a holonomy-flux algebra \mathfrak {A} \text{. . .} defined in the usual way on the connections, where the *-algebra is also a Jordan algebra.

Unless Mike can show us a hol-flux *-algebra which is Jordan, we have to conclude that the case Mike's paper purports to be about simply does not arise. There ain't no such animal.

 

[marcus]

I'd like to encourage Mike to research and write something which is actually accurate and relevant to Loop Quantum Gravity, since it is an area of growing interest and activity.

IMHO it does no good, though, to pretend that the present paper hits the mark.