LOST theorem found (important result for LQG)

[selfAdjoint]

Marcus, from post #27 on this thread,

Returning to the holonomy flux *-algebra in gr-qc/0504147, notice the authors refer to it as a *-algebra of basic, quantum observables (page 6). They also require that it be a Banach algebra (page. 5) to avoid domain complications. That is a Banach algebra of observables, implies that it is a JB algebra. (Even more, the authors wish to represent on a Hilbert space as an algebra of bounded operators, which makes it a JC algebra).

 

[marcus]

Marcus, from post #27 on this thread,

yeah, I read that same passage and noted that they "wish to" represent.
that is my point. the representation (at this stage) is in the future and has not been shown to be unique
so at that point it seems to me (perhaps you will explain clearly otherwise) that it begs the conclusion to take the hol.flux algebra as consisting of bounded operators.

at that stage it is not AFAICS

Returning to the holonomy flux *-algebra in gr-qc/0504147, notice the authors refer to it as a *-algebra of basic, quantum observables (page 6). They also require that it be a Banach algebra (page. 5) to avoid domain complications. That is a Banach algebra of observables, implies that it is a JB algebra. (Even more, the authors wish to represent on a Hilbert space as an algebra of bounded operators, which makes it a JC algebra).

maybe we need to look at this a bit more.

See what Mike says
"notice the authors refer to it as a *-algebra of basic, quantum observables (page 6)."

that might give you the impression that $$\mathfrak {A}$$ is an operator algebra. But it is not, at that stage, and they are not saying that. Look at what they actually say on page 6.

2 The Holonomy-Flux *-algebra

The goal of this section is a definition of the *-algebra $$\mathfrak {A}$$ of basic, quantum observables. We have already mentioned the algebra in the introduction and explained its meaning...

but at that point there is no Hilbert space and no operators on a Hilbert space. The algebra $$\mathfrak {A}$$ has indeed been discussed in terms of connections and the cylindrical functions Cyl.
Just because you see the word "observables" in the quote on page 6 does not mean you have a von Neumann algebra! (there are no operators at that stage)

I think it would be wise to look more carefully at Mikes alternative step (1) which assumes a von Neumann algebra and starts off with a trace defined on it.

 

[marcus]

self-Adjoint you referred me to a passage in Mike's post #27 which was, in fact, one that had caused my misgivings. Here is another problem with it:

He refers to page 5 and says that the authors assume $$\frak {A}$$ to be a Banach algebra. but if you look on page 5 for yourself you will see that they do NOT assume that----and they talk about the fact that they don't assume that.

So that is something that can have happened easily enough to Mike thru hasty reading.

BTW Cyl does have a norm. But $$\frak {A}$$ is not the same as Cyl and $$\frak {A}$$ does not have a norm. It is NOT a Banach algebra----until you prove something.

Here is what the authors say

... Finally, if $$\frak {A}$$ is not a Banach-algebra, one has to worry about domain questions and it is somewhat natural to consider representations first that have simple properties in this respect. A simple formulation of these properties can be given by asking for a state (i.e. a positive, normalized, linear functional) on $$\frak{A}$$ that it is invariant under the classical symmetry automorphisms of $$\frak{A}$$. Given a state on $$\frak{A}$$... one can define a representation via the GNS ...

But here is what Mike says:

Returning to the holonomy flux *-algebra in gr-qc/0504147, notice the authors refer to it as a *-algebra of basic, quantum observables (page 6). They also require that it be a Banach algebra (page. 5) to avoid domain complications. That is a Banach algebra of observables, implies that it is a JB algebra. (Even more, the authors wish to represent on a Hilbert space as an algebra of bounded operators, which makes it a JC algebra).

You see Mike says that they require it to be a Banach algebra, but he has read too hastily and indeed they do NOT and in fact their MOTIVATION for considering "states" omega (positive normalized linear functionals) on $$\frak{A}$$

they do not require $$\frak{A}$$ to be a Banach algebra and the fact that $$\frak{A}$$ MIGHT NOT BE a Banach algebra provides part of the motivation for their approach which is to prove the existence and uniqueness of a certain kind of linear functional on $$\frak{A}$$.

In fact this linear functional that they prove the existence and uniqueness of is the celebrated Ashtekar-Lewandowski measure on the space of connections. Its existence and uniqueness are non-trivial. Once you have PROVEN the theorem then you do have a norm and you do have a unique invariant representation of the *-algebra by operators on a Hilbert space.

Until you prove their theorem you could in principle have many inequivalent representations. So you cannot just slap on a norm and say you have a Banach algebra or von Neumann algebra or a trace etc etc.

Anyway that's my provisional take on it. I think considerable clarification is needed of Mike's step (1) where he declares he has a von Neumann algebra and a trace (the trace he thinks he has seems to substitute for the Ashtekar-Lewandowski measure, or, if you prefer, the functional omega that they are proving exists and is unique!)

So I would suggest a bit more thought before you dash off a letter to Thiemann that you have an alternative proof.

However, of course that is up to you

 

[kneemo]

Hi Mike...you begin by assuming the hol.-flux algebra is a von Neumann algebra (bounded operators on a Hilbert space) and therefore has a trace.

However at that point $$\mathfrak{A}$$ is not an operator algebra and there is no Hilbert space. that remains to be constructed and it is not clear that the representation will be unique.

...why you can assume that $$\mathfrak{A}$$ is a von Neumann algebra and has a trace.

(Without additional argument, it seems almost like assuming the conclusion )

Hi Marcus

But you're right, we should keep everything Jordan. So let us clean up the construction by throwing out the von Neumann condition altogether and replace it with JBW algebra. A JBW algebra has all the properties of a JB algebra (as stated above), with the extra condition that the Jordan algebra be a dual Banach space. The trace still exists, and there are no Hilbert space assumptions whatsoever.

As for other matters, I've been looking more closely at the LOST GNS construction. If $$\mathfrak{A}$$ is indeed a Banach algebra of observables, then its product is the Jordan product $$\circ$$. This allows us to re-write

$$\mathfrak{J}=\{a\in\mathfrak{A}\hspace{.1cm}|\hspace{.1cm}\omega(a^*a)=0\}$$

as

$$\mathfrak{J}=\{a\in\mathfrak{A}\hspace{.1cm}|\hspace{.1cm}\omega(a^*\circ a)=\omega(a\circ a)=\omega(a^2)=0\}$$.

Thus $$\mathfrak{J}$$ is not an ideal in $$\mathfrak{A}$$, and $$\mathfrak{A}$$ does not act on the Hilbert space $$\mathfrak{A}/\mathfrak{J}$$. This means once $$\mathfrak{A}$$ is identified as a Banach algebra of observables (which needs the symmetrizer product for closure), the LOST GNS construction breaks down. It should still break down even without the Banach requirement, as the ideal argument is based on the Jordan product and even makes no reference to trace.

Regards,

Mike

 

[marcus]

Hi Marcus, as selfAdjoint pointed out, the LOST authors implicitly made the assumption that $$\mathfrak{A}$$ is a von Neumann algebra by calling it an algebra of basic, quantum observables.
...

You are referring to this quote, I take it, which you cited earlier from their page 6

2 The Holonomy-Flux *-algebra

The goal of this section is a definition of the *-algebra $$\mathfrak {A}$$ of basic, quantum observables. We have already mentioned the algebra in the introduction and explained its meaning...

But in fact in that section they do NOT in that section define it as an algebra of operators on a hilbert space.

So it seems to be a legalistic or semantic point. you seem to be suggesting that by "quantum observables" they have to mean something which in fact they do not mean.

I think you may be misled by what you take to be the conventional meaning of the words. Have to go, be back later.

 

[kneemo]

So it seems to be a legalistic or semantic point. you seem to be suggesting that by "quantum observables" they have to mean something which in fact they do not mean.

Hi Marcus

I threw out the von Neumann assumption, because all that is needed for trace is a JBW algebra.

Please read my last post to see that only the Jordan algebra assumption is relevant to break the LOST GNS construction.

Regards,

Mike

 

[marcus]

Hi Mike, I'm back.

I know that at the very beginning of section 2 (on page 6) they say that in that section they are going to define $$\frak{A}$$
as a *-algebra of "basic quantum observables"

and you could get the idea that they are talking about defining it as a Banach algebra or even better as a von Neumann algebra as you assume----operators on a Hilbert space. But if you actually read section 2 they don't do that. there is no Hilbert space in section 2. there are no operators. there is no Banach-algebra norm defined on $$\mathfrak {A}$$

YAY! I just saw your most recent post! You threw out the von Neumann algebra assumption. Great!

Now how do you get the trace?

The things in $$\mathfrak {A}$$ are so far not operators on anything nice that I can see. Where does the trace come from.

what exactly does JBW mean?

 

[marcus]

A JBW algebra has all the properties of a JB algebra (as stated above), with the extra condition that the Jordan algebra be a dual Banach space.

Now the question is how do you know that $$\frak{A}$$
is a "dual Banach space"?

maybe wd be good to define "dual Banach space"----spell it out in a little detail

the suspicion is always that any assumption that involves the existence of something like a trace may actually be assuming something tantamount to
what the the theorem is trying to prove!

(the existence and uniqueness of this diffeo-and-gauge-invariant linear functional omega)

It is a quiet rainy Mother's day here. Thanks for livening things up!

 

[kneemo]

Hi Marcus

Before I answer your JBW question, note that the holonomy-flux *-algebra, with only a *-algebra structure, already has a product implicitly defined. Stating that $$\mathfrak{A}$$ is a *-algebra, is actually more precise than saying $$\mathfrak{A}$$ is a Jordan algebra with involution *, as the Jordan algebra does not come with an involution.

I use the Jordan product because the authors want the holonomy-flux *-algebra to be an algebra of observables. What are observables? Traditionally, they are self-adjoint elements of a $$C^*$$-algebra. Pascual Jordan, one of the founders of quantum mechanics, figured out how to make an algebra out of only the self-adjoint elements, and these algebras are Jordan *-algebras.

In the LOST paper, the theorem is about the existence and uniqueness of a Yang-Mills/Diffeomorphism invariant state for the holonomy-flux *-algebra. If you look closely on page 18, you'll notice that they define instead of derive the existence of an invariant state by setting:

$$\omega_0(a\cdot \hat{Y}):=0$$.

Truly, it would be more powerful to derive such a result for all inner derivations of $$\mathfrak{A}$$. This is what happens with the trace GNS construction as all inner derivations of a Jordan algebra are given by the associator, and the trace of the associator always vanishes.

Now the question is how do you know that $$\frak{A}$$
is a "dual Banach space"?

By saying $$\mathfrak{A}$$ is a JBW algebra, we are saying it is the dual of some Banach space, i.e., it consists of linear functionals over some Banach space. This seems like a safe assumption considering that $$\mathfrak{A}$$ consists of functions of finitely many holonomies along curves in $$\Sigma$$.

Now, I've also mentioned (in my recent posts) that the LOST GNS construction breaks down by assuming $$\mathfrak{A}$$ is a Jordan algebra. That this happens for observable-only constructions led to the Jordan GNS construction in the first place. All I had to do was notice the LOST GNS construction is the traditional one used for $$C^*$$-algebras. The failing of the LOST GNS construction means that when we want the holonomy-flux *-algebra to be an algebra of observables, we must use the trace construction. This is a point Thiemann should be made aware of ASAP.

Regards,

Mike

 

[selfAdjoint]

The failing of the LOST GNS construction means that when we want the holonomy-flux *-algebra to be an algebra of observables, we must use the trace construction. This is a point Thiemann should be made aware of ASAP.

Absolutely! Both the LOST GNS failure and your fix for it! Do you want me to email him tomorrow with a link to this thread, or to wait for your essay on the JBW-algebra trace construction?

 

[kneemo]

Do you want me to email him tomorrow with a link to this thread, or to wait for your essay on the JBW-algebra trace construction?

Hi selfAdjoint

I'm nearly finished typing up the trace GNS paper for the holonomy-flux *-algebra. After I take my Mother out to dinner, I'll put the finishing touches on it and send it over to you.

Regards,

Mike

 

[marcus]

I use the Jordan product because the authors want the holonomy-flux *-algebra to be an algebra of observables. What are observables? Traditionally, they are self-adjoint elements of a $$C^*$$-algebra. Pascual Jordan, one of the founders of quantum mechanics, figured out how to make an algebra out of only the self-adjoint elements, and these algebras are Jordan *-algebras.

I think I see where you are confused, Mike. the authors do NOT want elements of $$\mathfrak{A}$$ to be self-adjoint.
they explicitly do not assume that a = a*
for example see equation (46)

You however insist that they want them to be self-adjoint. You presume that every element a of $$\mathfrak{A}$$ satisfies a = a*. This is very far from being the case! And far from the authors' intentions! From this you reason that the paper must be wrong.

Also the authors nowhere use the word "Jordan", but you argue that they assume $$\mathfrak{A}$$ is a Jordan algebra. They do not.
From this you conclude that their GNS construction does not work. That is based on your own erroneous assumption, not the author's error.

For example, here is a quote from you, Mike:

Now, I've also mentioned (in my recent posts) that the LOST GNS construction breaks down by assuming $$\mathfrak{A}$$ is a Jordan algebra.

But THEY DONT ASSUME THAT, the elements of $$\mathfrak{A}$$ are not assumed to be self-adjoint (indeed explicitly not) and they never anywhere say "Jordan" and they simply do not assume $$\mathfrak{A}$$
is a Jordan algebra.

I would suggest that, before anyone bothers Thomas Thiemann with this, Mike should explain why he thinks the authors suppose all the elements of $$\mathfrak{A}$$ are self-adjoint.

Maybe we can clear this up ourselves and not look inept in front of Thiemann.

I assume what Mike means by self-adjoint is a = a*,
since elements of Cyl are not operators, although * is defined on them by simple complex conjugate. With cylinder functions it is not the case that a = a* as a rule, and cylinder functions are the original founding members of $$\mathfrak{A}$$.

 

[kneemo]

I think I see where you are confused, Mike. the authors do NOT want elements of $$\mathfrak{A}$$ to be self-adjoint.

Hi Marcus

In quantum mechanics self-adjoint operators are called observables. By self-adjoint we mean, not only that the operators are hermitian (a=a*), but that the domains match up. Self-adjoint elements have a real spectrum, which physically corresponds to what is "observed", hence the name observables. The real spectrum is why anybody even bothers with self-adjoint elements in the first place.

The LOST authors assert that the holonomy-flux *-algebra is an algebra of observables, rather than a C*-algebra, which is the completion of Cyl (see gr-qc/0302059). Yet, the authors use the C* GNS construction as if it is applicable to algebras of observables. That the C* GNS construction breaks down for algebras of observables is not my original finding. It is an old result in the representation theory of Jordan algebras. It is a new result, however, that the trace GNS construction works for algebras of observables, and that is what I am applying to the holonomy-flux *-algebra.

Jordan algebras are more the mathematician's playground these days, so it wouldn't surprise me if Thiemann is not familiar with their structure. The case for JB algebras and JBW algebras I suspect is even worse, and I have only seen John Baez and Lee Smolin mention a potential application in quantum gravity.

I am merely following Baez and Smolin's lead by applying Jordan algebra theory explicitly to abstract loop quantum gravity. Maybe Baez and Smolin have already read Thiemann's paper and pointed out the GNS complication. Either way, Theimann should know about this subtlety for future reference.

Regards,

Mike

 

[marcus]

Hi Marcus

In quantum mechanics self-adjoint operators are called observables.

that is true. self-adjoint operators are called observables. the converse is not true. one can refer to something informally as a "basic quantum observable" without meaning that it is a self-adjoint operator on a hilbert space.

You have allowed yourself to be misled, if I understand you correctly, by the first sentence of section 2. on page 6.

It is clear from the way it is constructed that in general elements of $$\frak {A}$$
do not have the property a* = a.

See my discussion in a separate thread
https://www.physicsforums.com/showthread.php?t=74924

However, because of a single reference in the paper (on page 6)
to "basic quantum observables", which you take out of context and insist must mean self-adjoint, you are persistently misconstruing the paper.

If you will simply go through the construction of $$\frak {A}$$ you will see that the authors do not intend it to satisfy the condition a* = a in general. (Some elements do satisfy this condition but a generic element does not.)

 

[kneemo]

that is true. self-adjoint operators are called observables. the converse is not true. one can refer to something informally as a "basic quantum observable" without meaning that it is a self-adjoint operator on a hilbert space.

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.

I followed the link you provided and see there is some confusion concerning self-adjoint complex operators.

...the only way you would have a*=a for something like that is if the complex-valued cylinder function was actually always real-valued!

Take the case of a 2x2 complex hermitian matrix. Only the diagonal elements are real, while the off-diagonal elements are complex.

Regards,

Mike

 

[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.