Is the Holonomy-Flux Algebra Generally Hermitian in LQG?

[marcus]

Back in 2003 we here at PF studied papers by Sahlmann and by Lewandowski/Okolow about a basic algebra of LQG called the
"holonomy-flux algebra".

It is a star-algebra and anyone familiar with its construction will realize that it is NOT generally the case that a = a* for some element a in $$\frak{A}$$.

Pretending that a = a* for all stuff in the holonomy-flux algebra is kind of like supposing that all complex numbers are real----that is, saying that each z is equal to its own conjugate.

If you assume something that wrong then you can prove or disprove pretty much anything you want as a logical consequence. Like, if 1 = 2 then Moonbear is the Queen of France, or whatever. QED.

But although we studied $$\frak{A}$$ quite a bit back in 2003, people forget and it is probably time for a refresher (I am sure I would benefit from a review). And it is especially appropriate because a long-awaited paper by Lewandowski-et-al just came out.

 

[marcus]

The most recent paper by Lewandowski et al about the hol-flux algebra is
http://arxiv.org/gr-qc/0504147

but that paper is relatively condensed because much of it is going over what should be familiar from earlier papers so they can put the finishing touches on.

So here are some earlier Lewandowski papers about $$\frak{A}$$ which will give you a good background for gr-qc/0504147 and an easier more gradual introduction.

This is the one we read at PF in 2003:

http://arxiv.org/abs/gr-qc/0302059
Diffeomorphism covariant representations of the holonomy-flux star-algebra.
Andrzej Okolow, Jerzy Lewandowski
37 pages
Class.Quant.Grav. 20 (2003) 3543-3568

"Recently, Sahlmann proposed a new, algebraic point of view on the loop quantization. He brought up the issue of a star-algebra underlying that framework, studied the algebra consisting of the fluxes and holonomies and characterized its representations. We define the diffeomorphism covariance of a representation of the Sahlmann algebra and study the diffeomorphism covariant representations. We prove they are all given by Sahlmann's decomposition into the cyclic representations of the sub-algebra of the holonomies by using a single state only. The state corresponds to the natural measure defined on the space of the generalized connections. This result is a generalization of Sahlmann's result concerning the U(1) case."

This is a follow-up that proves the same thing under somewhat more general conditions:

http://arxiv.org/abs/gr-qc/0405119
Automorphism covariant representations of the holonomy-flux *-algebra
Andrzej Okolow, Jerzy Lewandowski
34 pages, 1 figure
Class.Quant.Grav. 22 (2005) 657-680

"We continue an analysis of representations of cylindrical functions and fluxes which are commonly used as elementary variables of Loop Quantum Gravity. We consider an arbitrary principal bundle of a compact connected structure group and following Sahlmann's ideas define a holonomy-flux *-algebra whose elements correspond to the elementary variables. There exists a natural action of automorphisms of the bundle on the algebra; the action generalizes the action of analytic diffeomorphisms and gauge transformations on the algebra considered in earlier works. We define the automorphism covariance of a *-representation of the algebra on a Hilbert space and prove that the only Hilbert space admitting such a representation is a direct sum of spaces L^2 given by a unique measure on the space of generalized connections. This result is a generalization of our previous work (Class. Quantum. Grav. 20 (2003) 3543-3567, gr-qc/0302059) where we assumed that the principal bundle is trivial, and its base manifold is R^d."

 

[marcus]

There is a simple reason that you don't generally have a*=a in the holo-flux algebra which is that it is built up starting with complex-valued functions called cylinder functions.

and the star is defined on these simply by taking the complex conjugate!

Like f*(x) is defined to be the complex conjugate of f(x)

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

I shouldn't have to explain this. In fact the title of this thread ("Holo-flux alg aint all a=a*") really should be a joke. It should be obvious to anyone who knows the basics. But as it happens we currently have a stubborn misconception at PF to the effect that the holo-flux algebra consists of "self-adjoint" elements satisfying a* = a.

I am not sure that "self-adjoint" is a good term to describe a complex number which happens to be real, or a complex-valued function that happens to be real-valued. But when I hear the condition a* = a interpreted that way (i.e. that the element a of the star-algebra is "self-adjoint") then at least I get the idea. And I can assure anyone interested that a general element of $$\frak{A}$$ simply aint "self-adjoint" in that sense.

Well, there is some gentle introductory material in the two papers I just mentioned. I guess it is time to get some exerpts and page references.

 

[marcus]

BTW we should have a bit of info about the authors of the LOST paper just to get a sense of who they are:

Here's a list of Jerzy Lewandowski's papers since 1993.
http://arxiv.org/find/grp_physics/1/au:+Lewandowski_Jerzy/0/1/0/all/0/1

There are some 39 papers listed, I notice that the first one was co-authored with Ashtekar as have been many subsequently.
Lewandowsi is Associate Professor of Physics at Warsaw University
http://www.fuw.edu.pl/~lewand/
and concurrently also
Visiting Associate Professor, Center for Gravitational Physics and Geometry, Penn State, 2001 - present.

Here's a list of Thomas Thiemann's papers, 58 since 1993:
http://arxiv.org/find/grp_physics/1/au:+Thiemann/0/1/0/all/0/1
Thiemann is at AEI-Potsdam and also concurrently at Perimeter Institute in Waterloo, Canada.
This AEI page has a link to Thiemann's homepage
http://www.aei.mpg.de/english/contemporaryIssues/members/db_members/index.php

 

[marcus]

Here is a key quote from a Lewandowski paper illustrating the terminology "elementary variable". this is what the cylinder functions are in LQG. This passage gives a clue as to what is meant in the LOST paper by "basic quantum observable". (not necessarily s.a. or hermitian).
the terminology can be confusing, so one has to focus on the actual mathematics.

-----gr-qc/0405119, page 2-----

A crucial step of the canonical quantization procedure is the choice of elementary variables which are (complex) functions on the phase space of the theory (see e.g. [4]). Once such a set is chosen, the quantization procedure consists in assigning to any elementary variable an operator on a Hilbert space in such a way that
(i) $$(i \hbar)^{-1}$$ times the commutator of the two operators assigned to a pair of variables corresponds to the Poisson bracket of these variables and
(ii) the conjugate of the operator assigned to a variable corresponds to the complex conjugate of this variable--if

$$\hat{f}, \hat{g}$$ are the operators assigned respectively to variables f, g then

(i) $$(i \hbar)^{-1}[\hat {f}, \hat {g}]$$ $$=\widehat{\{ f,g \} }$$

(ii) $$\hat{f}\text{*} = \hat{ \bar {f} }$$

The assignment $$f \rightarrow \hat{f}$$ is called a representation of elementary variables. The present paper concerns the theory of representations of the commonly used elementary variables of LQG which are cylindrical functions and fluxes.
---end quote---

maybe this will help clarify. $$\frak{A}$$ is a *-algebra of operators corresponding to certain elementary variables (holonomies and fluxes) which by abuse of nomenclature we also will call holonomies and fluxes.
A significant chunk of $$\frak{A}$$ consists of the operators (OBVIOUSLY NOT HERMITIAN IN ANY SENSE) which are simply multiplication by some cylinder function. The star of such a thing corresponds to the complex conjugate of the original cylinder function.

And these are not operators on a Hilbert space, we haven't gotten there yet and still have a ways to go.

So although one can call them (stressing the word "basic" in the sense of root or primitive) basic quantum observables, or elementary variables, or whatever one choses to call them, one should NOT expect them to satisfy the a* = a condition in general.

Hope this helps.