LOST theorem found (important result for LQG)

[marcus]

http://arxiv.org/abs/gr-qc/0504147

this has been promised for a couple of years now.
here at PF we studied the papers leading up to it

there was one paper we discussed here by Lewandowski and Okolow (LO)
and several by Sahlmann and Thiemann (ST) or by Hanno Sahlmann solo.
The four of them have gotten together to prove the most general form of the LOST uniqueness theorem.

Uniqueness of diffeomorphism invariant states on holonomy-flux algebras

Jerzy Lewandowski, Andrzej Okolow, Hanno Sahlmann, Thomas Thiemann
38 pages, one figure
AEI-2005-093, CGPG-04/5-3

"Loop quantum gravity is an approach to quantum gravity that starts from the Hamiltonian formulation in terms of a connection and its canonical conjugate. Quantization proceeds in the spirit of Dirac: First one defines an algebra of basic kinematical observables and represents it through operators on a suitable Hilbert space. In a second step, one implements the constraints. The main result of the paper concerns the representation theory of the kinematical algebra: We show that there is only one cyclic representation invariant under spatial diffeomorphisms.
While this result is particularly important for loop quantum gravity, we are rather general: The precise definition of the abstract *-algebra of the basic kinematical observables we give could be used for any theory in which the configuration variable is a connection with a compact structure group. The variables are constructed from the holonomy map and from the fluxes of the momentum conjugate to the connection. The uniqueness result is relevant for any such theory invariant under spatial diffeomorphisms or being a part of a diffeomorphism invariant theory."

 

[marcus]

HAPPY MAY DAY TO ALL!

In other news today, Sheldon Glashow posted a paper proposing the existence of a whole other class of fermions which he calls
"terafermions"

http://arxiv.org/abs/hep-ph/0504287
A Sinister Extension of the Standard Model to SU(3)XSU(2)XSU(2)XU(1)
Sheldon L. Glashow
9 pages, adapted from talk at XI Workshop on Neutrino Telescopes, Venice

"This paper describes work done in collaboration with Andy Cohen. In our model, ordinary fermions are accompanied by an equal number `terafermions.' These particles are linked to ordinary quarks and leptons by an unconventional CP' operation, whose soft breaking in the Higgs mass sector results in their acquiring large masses. The model leads to no detectable strong CP violating effects, produces small Dirac masses for neutrinos, and offers a novel alternative for dark matter as electromagnetically bound systems made of terafermions."

 

[marcus]

http://arxiv.org/abs/gr-qc/0504147
Uniqueness of diffeomorphism invariant states on holonomy-flux algebras
Jerzy Lewandowski, Andrzej Okolow, Hanno Sahlmann, Thomas Thiemann

one thing of interest here is they introduce a new category of manifolds.

the semianalytic manifolds.

It will probably get studied in Differential Geometry.

semianalytic essentially means PIECEWISE analytic

it is a bit like what Rovelli and Fairbairn were doing but they were just saying smooth rather than analytic

it is an interesting extension of the category of analytic manifolds

For a mapping R^m -> R^m to be semianalytic the exceptional set where it is not analytic has to be defined as the zero set of another analytic function h. Or more generally the exceptional set has to be of the form {h = 0} or {h < 0} or {h > 0}.

Rovelli and Fairbairn made the exceptional set be just a finite set of points. Their paper was already pretty interesting and we discussed here at PF in a long thread. But this extends that in a certain sense because given any finite set of points it seems clear you can define an analytic function h which is zero exactly on that finite set. So the Rovelli Fairbairn finite exceptional set is also an exceptional set of the semianalytic theory.

 

[marcus]

I heard a crash in the back room.

it is a good paper

 

[marcus]

the main theorem is called theorem 4.2 and it is on page 19
it says this:

There exists exactly one invariant state on the quantum holonomy flux star-algebra A


that is the thing to remember and then you go back and pick up background details like there is this semianalytic manifold $\Sigma$
and a compact group G
and a principal bundle P on $\Sigma$ which bundle is also semianalytic
and there are connections defined on $\Sigma$
(representing the possible geometries that we we can have quantum uncertainty about which geometry it is)
and one feels these connections with one's eyes shut by doing HOLONOMIES which just means to run around loops and networks and stuff feeling one's gyroscope writhing as one goes around the loop or along pathways in the network

oops I have to go out for a moment, back soon

but that only sounds complicated, morally it has a simple enough meaning. you need some paraphernalia to catalog all the possible geometric configurations of space so that you can be uncertain about what shape it is---and so then you can embody your uncertainty, your incomplete knowledge, in a hilbert space, for such is the custom of men and nations.

it was partly selfAdjoint's intuition that we should study the papers of LO and ST carefully just about 2 years ago.
it turns out that was a good idea. the notation is essentially unchanged, the concepts have been streamlined, the theorem has gelled and looks like it will be a central one in LQG

 

[selfAdjoint]

Wonderful news Marcus! I like the generality. Any theory built on a compact group over a manifold with a connection defining the field strength along with its dual flux, will obey this theorem. Mmm, What does that say about Thiemann's quantisation of the closed string?

 

[marcus]

Wonderful news Marcus! I like the generality. Any theory built on a compact group over a manifold with a connection...

yes, they make the generality explicit in the abstract but they do not mention an important detail there----the manifold is not just m-times-differentiable (C^m) it is semianalytic.

I believe this is why the LOST paper was a year and a half delayed, so that people began joking that it was really "lost". And it is why they thank Christian Fleischhack twice (in the acknowledgments and in the appendix) for personal communication "drawing our attention to the theory of semianalytic sets". And, I suspect, why the second and third references, right after [1] Ashtekar "Lectures", are

[2]

 

[marcus]

Wonderful news Marcus! I like the generality. Any theory built on a compact group over a manifold with a connection...

yes, they make the generality explicit in the abstract but they do not mention an important detail there----the manifold is not just m-times-differentiable (C^m) it is semianalytic.

I believe this is why the LOST paper was a year and a half delayed, so that people began joking that it was really "lost". And it is why they thank Christian Fleischhack twice (in the acknowledgments and in the appendix) for personal communication "drawing our attention to the theory of semianalytic sets". And, I suspect, why the second and third references, right after [1] Ashtekar "Lectures", are

[2]Lojasiewicz, S. (1964): Triangulation of semianalytic sets. Ann. Scuola. Norm. Sup. Pisa 18, 449-474
[3]Bierstone, E. and Milman, P. D. (1988): Semianalytic and Subanalytic sets. Publ. Maths. IHES 67, 5-42

Everything is pointing to semianalytic as being very important, so the first thing I am trying to do with this paper is understand why.

 

[marcus]

you remember Fairbairn Rovelli of just a year ago
http://arxiv.org/gr-qc/0403047
"Separable Hilbert Space in LQG"
there it made an enormous difference what diffeomorphisms one actually used

now with this paper, when they say manifold they mean semianalytic manifold
when they say bundle they mean semianalytic bundle
when they say diffeomorphism invariant they mean semianalytic invariant.

it is bound to make a considerable difference (I mean it already has---they say the main point of their paper is this difference) so right now i am trying gradually to understand what the differences could be.

 

[marcus]

selfAdjoint, it looks to me like the spin networks are different.

in the old LQG one had a $C^m$ or smooth manifold $\Sigma$

and one imbedded spin networks in there to feel the connection-geometry and those imbedded spinnetworks were quantum states of geometry and formed a basis for the kinematic Hilbertspace.

and then one identified spinnetworks that were equivalent by a smooth or almost smooth mapping.

but now $\Sigma$ is supposed to be a semianalytic manifold.

the imbeddings of the spinnetworks, I presume, are to be semianalytic.
this means intuitively that the Hilbert space will be SMALLER to begin with.

maybe that is a good thing. but will it be separable?

I mean after imbedded spinnetworks are identified by semianalytic maps and formed into equivalence classes (which in Rovelli's case were knots) is there a similar reduction to a separable Hilbert space?

I am looking for reassurance that the semianalytic diffeomorphisms are a good class to be using.

so far I find little research done on semianalytic sets or functions

indications are the best reference is
E. Bierstone and P.D. Milman, Semianalytic and subanalytic sets, Publ. Math. I.H.E.S. 67 (1988), 5–42.

but this is not online and I have not checked it. If I restrict to what is available online then I find very little:

wolfram mathworld has a short entry, only one reference, to a 1997 paper
http://mathworld.wolfram.com/Semianalytic.html

an arxiv search using keyword "semianalytic" came up with only two papers, one differential geometry paper by a man at oxford
http://arxiv.org/abs/math.DG/9706227

one algebraic geometry paper which was only remotely connected with the topic
http://arxiv.org/abs/math.AG/9910064

 

[selfAdjoint]

Semianalytic sets

Marcus, it is not the manifolds which are required to be semianalytic, but the diffeomorphisms. And they are indeed still diffeomorphisms, that is $$C^{\infty}$$ functions. But instead of being analytic (convergent Tayor series) everywhere, they are so on a hierarchy of subsets of successively lower dimensions. Analytic on U except on n-1 dimensional U1, and restricted to U1 analytic except on n-2 dimensional U2, etc. These are apparently the semianalytic sets they base their theory on.
Or that's how I read the paper. Notice that the class of semianalytic functions is bigger than that of analytic ones.
But the hypotheses of the theorem are still general: principal bundle over manifold, compact group, curvature of connection forming field strength as one-form. Dual flux convertable to something that can be integrated over dual one forms, i.e.faces.

 

[marcus]

... it is not the manifolds which are required to be semianalytic, but the diffeomorphisms. And they are indeed still diffeomorphisms, that is $$C^{\infty}$$ functions. But instead of being analytic (convergent Tayor series) everywhere, they are so on a hierarchy of subsets of successively lower dimensions. Analytic on U except on n-1 dimensional U1, and restricted to U1 analytic except on n-2 dimensional U2, etc. These are apparently the semianalytic sets they base their theory on.
Or that's how I read the paper...

I think you have it about right. However see the definition of a semianalytic manifold on page 34.

"...A semianalytic structure on $\Sigma$ is a maximal semianalytic atlas. A semianalytic manifold is a differential manifold endowed with a semianalytic structure."

 

[marcus]

you know how everybody these days finds it convenient to work with categories-----a lot of times when doing setup and definitions it really does save time and makes things clearer----well I see them doing that with the semianalyitic category.

on page 34 section A.2 they are giving basic definitions they need for their main theorem. the section is called "Seminanalytic manifolds and submanifolds"

first they define what a s.a. manifold is (that is def.A10)

then given two s.a. manifolds, they define what is a s.a.map between them is (definition A11)

then they define what is a s.a. submanif. of a s.a. manifold. (def.A12)

then they define a semianalytic manifold with boundary

then they prove a property of intersections of s.a. submanifolds (proposition A14) , which is crucial for their main theorem.
what they say is special about semianalytic submanifolds is that when two of them intersect the intersection is locally a finite disjoint union of connected s.a. submanifolds.

intersections between C-infinity submanifolds can be more complicated, may not be able to write as a finite union of connected pieces, they may wiggle too much so they may intersect in pathological sets. (they hint at one example on page 35)

 

[marcus]

from equations A15 and A16 on page 35, I gather that what they mean by "edge" is semianalytic edge-----this would make a difference in what a spin network is.

and a "face" they clearly say to be a semianalytic submanifold (with other properties like codimension 1, oriented)---so it would be a s.a. face

they seem to want to work in a whole S.A. CATEGORY where everything is s.a.

and then immediately after that they give the "partition of unity" thing, at the top of page 36. they use the partition of unity (equation 121) in the proof of their main theorem (theorem 4.2 on page 19)

 

[marcus]

I am going to potter around with their notation some to see how to write it in LaTex
$$\mathfrak{A} \text{ is the *-algebra}$$

$$\omega \text{ is a positive normalized linear functional on }\mathfrak{A}$$

$\mathfrak{J}$ is the ideal consisting of all the elements of omega-norm zero, that is all a for which

$$\omega (\text{a*a}) = 0$$


then we get a Hilbert space by completing

$$\mathfrak{A}/\mathfrak{J}$$

 

[selfAdjoint]

..and then immediately after that they give the "partition of unity" thing..

Why the scare quotes, Marcus? Do you have an issue with partitions of unity? They are quite a common tool in topology, especially in regards to getting refinements of coverings.

 

[marcus]

Why the scare quotes, Marcus? Do you have an issue with partitions of unity? They are quite a common tool in topology, especially in regards to getting refinements of coverings.

no problem
seeing the words recalls happy times
"partition of unity" may be new to one or two other PFrs, though, if anyone is reading this besides you and me

 

[marcus]

$$\mathfrak{A} \text{ is the *-algebra}$$
$$\omega \text{ is a positive normalized linear functional on }\mathfrak{A}$$

$\mathfrak{J}$ is the ideal consisting of all the elements of omega-norm zero, that is all a for which
$$\omega (\text{a*a}) = 0$$

then we get a Hilbert space by completing
$$\mathfrak{A}/\mathfrak{J}$$

we have an explanation problem here. why is quantum mechanics so often done with a Hilbertspace? What is this *-algebra ("holonomies and fluxes")?

Whenever you have a *-algebra and you have a positive normalized linear functional (a *-morphism to the complex number plane with some simple properties) then that gives you a Hilbertspace. The functional provides the inner product.

And it gives you a representation of the *-algebra on that Hilbert space. And if the funtional is unique then the hilbertspace and the rep are unique.

How do we talk about this so it is intuitive?

 

[kneemo]

...why is quantum mechanics so often done with a Hilbertspace? What is this *-algebra ("holonomies and fluxes")?

The *-algebra is actually too general to be of practical use in quantum mechanics. In the usual GNS construction one usually proceeds with a specific kind of *-algebra, a C*-algebra. The self-adjoint part of the C*-algebra is the algebra of observables, and is formally a JB algebra. There is a general theorem from the 70's by J.D. Wright establishing that every JB algebra is the self-adjoint part of a C*-algebra, and also that the self-adjoint part of every C*-algebra is a JB-algebra. Thus, whenever you have a C*-algebra, you automatically get a JB algebra of observables for free.

Now it is possible to formulate quantum mechanics purely in terms of observables, meaning we base the GNS-construction on the JB self-adjoint part of a C*-algebra. In this construction, the Hilbert space is built from projection elements of the JB algebra under a trace norm. In the case of a finite dimensional JB algebra, this Hilbert space can geometrically take the form of a smooth manifold with isometries corresponding to automorphisms of the JB algebra. The isometries form a group, which has the holonomy group as a subgroup. As the JB algebra is also a *-algebra, it is possible view the holonomy group as a subgroup of the automorphisms of the JB *-algebra. This is likely how the name "holonomy flux *-algebra A" came about.

Regards,

Mike

 

[marcus]

...
Now it is possible to formulate quantum mechanics purely in terms of observables, meaning we base the GNS-construction on the JB self-adjoint part of a C*-algebra. In this construction, the Hilbert space is built from projection elements of the JB algebra under a trace norm. ...
Mike

Mike, it would be great having your company while looking over this paper.

Please say what you mean by "JB"

IIRC a *-algebra is just an algebra with involution----a unary operation analogous to complex conjugation, or to taking the conjugate transpose when it's matrices.

How about writing down some definitions? Share the labor of getting the LaTex to work?

I will put the URL for the LOST theorem in my sig, if it will fit, to keep it handy.

 

[marcus]

my feeling is that *-algebra should be like mother's milk:
a vernacular intuition common to all.

the *-algebra is the C* algebra without the norm, so it is basically just an analog of the complex numbers with complex conjugation as the "star" map.

In the present context we are sometimes working with a structure that is just a *-algebra because it does not have its norm yet.

In a way, what the LOST theorem is about is the existence and uniqueness of the norm----which makes the holonomy-flux algebra a C* algebra.
This norm appears in the shape of a "normalized positive linear functional" omega.

there are 3 equivalent things (look at the paper for clarification)

1. a diffeo-and-gauge-invariant measure on the space of connections on the manifold (this measure is called the AshtekarLewandowski and we are really proving that it is unique)

2. a diffeo-and-gauge-invariant "normalized positive linear functional" on the holonomyflux algebra (a *-algebra) built on those connections

3. a representation of the holonomyflux algebra by unitary operators on a hilbert space.

 

[marcus]

I am serious about there being an explanation problem. I would like to have an exposition that, if richard the NC were reading in this thread, would interest him and he would get something from it.

Here is Wiki about *-algebra
http://en.wikipedia.org/wiki/Star-algebra

the complex numbers are the numbers we SHOULD have learned in gradeschool (but they didnt teach us), and a *-algebra is what they are the main example of.
The complex numbers is a bunch of numbers where you get to add subtract multiply and divide as usual AND do a flipflop operation called conjugate. written with a *. so if there is a thing x then its flip, or conjugate, is x*, and if you flip again you get x** = x the same thing back again.
Look at the Wiki page it is easy enough.

the general term in mathematics for that kind of flipflop is "involution"
so well here "algebra" means a bunch of things (numbers, matrices, quantummechanical operators) where you can add multiply any two things, and also multiply by ("scale things by") complex numbers. And an "algebra with involution" is any algebra with this flipflop thing defined on it, like conjugation with complex numbers, and that is called a *-algebra.

the main problem is there are too many words for the same thing

and whenever you have a *-algebra (whether it is the complexnumber plane, or some quantummechanical operators, or some matrices) then the
SELF-ADJOINT members of the algebra are the ones that FLIPPING DOESN'T CHANGE, they are the things x for which x* = x.

that is where PF member selfAdjoint gets his handle and it is all self-adjoint means and it comes from the star-algebra context and it simply means x*=x.

these are basic things and could be taught in Junior High or Middle School, and arent (at least where I went). or a concrete example could be given at least.

and in the case of the complexnumber plane then the *-operation is just flipping the plane over the real axis, and the "self-adjoint" numbers are the ones that arent changed and they are just the REALS

(obv. flipping the plane over just reflects stuff above the real axis to stuff below it but doesn't move anything on the axle itself.)

Now we have to see how a *-algebra emerges from quantizing geometry.
(this is where the leap comes----leaping over 5 or 10 years of mathematics---we will need some radical mental image to jump the gap)

configurations of geometry get expressed as connections and the holonomyflux algebra is built up from functions defined on these connections. I am trying to look ahead and visualize how to jump the gap.
But I'm stuck for now.

Oh yeah, C* algebras. We don't need that right away because at first we won't have a norm or distance function or "size" function defined on things,
but here is Wiki about C* algebra.
http://en.wikipedia.org/wiki/C*-algebra

You can see it is just a *-algebra with the "size" of x defined, for all x.
the "size" or norm is written ||x||
and for it to be a nice norm it has to obey a nice rule copied directly from the complex numbers, namely take any x and multiply by its flip-brother x* and the norm of that is the square of norm x.
||x x*|| = ||x||²
not to worry too much about that right now.

 

[kneemo]

Please say what you mean by "JB"

By JB I mean a Jordan Banach algebra.

Definition 1.
A Jordan algebra is a vector space $$A$$ equipped with the product (i.e. bilinear form) $$(a,b)\rightarrow a\circ b$$ satisfying ($$\forall a,b\in A$$):
$$(i)\hspace{.5cm}a\circ b = b\circ a$$,
$$(ii)\hspace{.5cm}a\circ (b\circ a^2) = (a\circ b)\circ a^2$$

Definition 2.
The Jordan algebra $$A$$ is a JB Algebra if $$A$$ is also a Banach algebra with respect to the product $$\circ$$, and if the norm $$||.||$$ satisfies ($$\forall a,b\in A$$):
$$(i)\hspace{.5cm}||a^2||\leq ||a^2+b^2||$$,
$$(ii)\hspace{.5cm}||a^2||=||a||^2$$.

Regards,

Mike

 

[marcus]

By JB I mean a Jordan Banach algebra...

Mike

Thanks Mike! Banach algebra is a familiar concept to this poster, but Jordan algebra not. Please add any definitions, corrections, amplifications to this thread that occur to you!

Basically what we are groping around for is a notion of an INVARIANT probability measure, or DENSITY, ON A BUNCH OF SHAPES.

everybody has the idea of picking a point at random between 0 and 1 and that is the "uniform probability measure" . "uniform probability density".

we should be able to generalize that idea to a UNIFORM PROBABILITY MEASURE ON 3D GEOMETRIES. but it isn't obvious how. do you have a notion of the "volume" of a bunch of shapes, how much is this bunch compared with that bunch?

Mathematicians are awful name-droppers and given the right circumstances they will call this "Haar measure". When the notion of "uniform" grows up it becomes the idea of INVARIANT under shifting, sliding, mooshing around. A measure or a probability distribution on geometries is invariant if the measure of a some whatever set of geometries does not change if you shift it around. that's what the idea of "evenly distributed" gets to be when it grows up.

Just like in the interval 0 to 1 the measure of a set doesn't change if you slide it back and forth. (or form the interval into a circle and rotate)

this theorem L.O.S.and T. proved is that if you take all the geometries on for instance the 3D sphere, and tag each one with the corresponding connection machine and then you want a UNIFORM MEASURE ON THE CONNECTIONS, well there is ONLY ONE.

picture a connection as a gyroscope at each point of the manifold, where each gyro knows how to twist and writhe as you take an infinitesimal step and push it a bit in any particular direction.

Well the TRANSFORMATIONS of a connection are two: you can either
RESET ALL THE GYROSCOPES, one by one, but leave them at the same points they were at, and this is called a GAUGE transformation

or you can MOOSH the points of the manifold around which STIRS THE GYROSCOPES and this is called a DIFFEO.

This LOST theorem which is a really great theorem says that there is ONE AND ONLY ONE measure on the set of connections which is "evenly distributed" or "uniform" in the sense that the measure of any subset of connections is INVARIANT as you do gauge transformations and diffeo-mooshings and shift the connections around.

you can take a big spoon and stir the set of all possible connections (i.e. the possible shapes of the 3D manifold) and the measures or probabilities of subsets of connections do not change. each subset of connections, as it flows around, stays the same measure! (this deserves some more exclamations, like !)

so for the first time we have a natural idea of what it means to "pick a 3D shape at random"

I mean a shape for the universe.

I don't mean a 3D shape like a duck or teddybear, that lives in some larger surrounding room. I mean the geometry of the universe itself. I mean the shape of all space, with all the puckers and dimples caused by stars and the various black hole pimples and the wrinkle that is the Great Attractor. I mean all the possible geometries that the universe can have or could have.
And i say there is a notion of "pick a geometry at random," and the theorem just proved by Mr. L.O.S. and T. gives us that.

now this will be useful, for one thing because it means being able to integrate sometimes. It doesn't mean that Nature choses geometries for us at random, with a uniform or invariant distribution. What Haar measure has always been is a mathematical convenience. But it is an important convenience, like hot and cold running water, and electricity, and indoor toilets. So don't knock mathematical convenience.

OK so this is a brief orientation spiel about the LOST theorem, in case anyone is interested. and the LOST paper is
http://arxiv.org/gr-qc/0504147

 

[marcus]

want to register a simple idea of John Baez here,
the *-category
http://arxiv.org/quant-ph/0404040

he mentions nCob, and Hilb, as examples of *-categories.
idea might be helpful in the context of *-algebras

 

[selfAdjoint]

By JB I mean a Jordan Banach algebra.

Definition 1.
A Jordan algebra is a vector space $$A$$ equipped with the product (i.e. bilinear form) $$(a,b)\rightarrow a\circ b$$ satisfying ($$\forall a,b\in A$$):
$$(i)\hspace{.5cm}a\circ b = b\circ a$$,
$$(ii)\hspace{.5cm}a\circ (b\circ a^2) = (a\circ b)\circ a^2$$

Definition 2.
The Jordan algebra $$A$$ is a JB Algebra if $$A$$ is also a Banach algebra with respect to the product $$\circ$$, and if the norm $$||.||$$ satisfies ($$\forall a,b\in A$$):
$$(i)\hspace{.5cm}||a^2||\leq ||a^2+b^2||$$,
$$(i)\hspace{.5cm}||a^2||=||a||^2$$.

Regards,

Mike

According to your definition 1, numbered item ii, your JB algebras are commutative. But the *-algebras of the LOST paper certainly are not since they are quantum operator algebras.

 

[kneemo]

According to your definition 1, numbered item (i), your JG algebras are commutative. But the *-algebras of the LOST paper certainly are not since they are quantum operator algebras.

Yes, a JB algebra is indeed commutative under the Jordan product $$\circ$$. But elements of the JB algebra can be noncommutative under a different product. Let us look at a particular example to see how this works.

Consider the case of the C*-algebra $$M_n(\mathbb{C})$$ of $$n\times n$$ complex matrices. $$M_n(\mathbb{C})$$ is a noncommutative algebra under the matrix product. Now in $$M_n(\mathbb{C})$$, there is the set of $$n\times n$$ hermitian complex matrices $$\mathfrak{h}_n(\mathbb{C})$$. These hermitian matrices do not commute under ordinary matrix multiplication.

It is possible to make $$\mathfrak{h}_n(\mathbb{C})$$ an algebra using the Jordan product, which for matrices reduces to:

$$A\circ B=\frac{1}{2}(AB+BA)$$

(Note: perform matrix multiplication on two arbitrary hermitian elements for $$n=2$$ to see how the Jordan product produces a hermitian element, thus closing the algebra.)

The algebra $$\mathfrak{h}_n(\mathbb{C})$$ is an algebra of observables. After a GNS construction, the hermitian matrices become quantum operators, which are noncommutative under matrix multiplication, but commutative under the Jordan product.

Returning to the holonomy flux *-algebra $$\mathfrak{A}$$ 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 $$\mathfrak{A}$$ is a Banach algebra of observables, implies that it is a JB algebra. (Even more, the authors wish to represent $$\mathfrak{A}$$ on a Hilbert space as an algebra of bounded operators, which makes it a JC algebra).

Best Regards,

Mike

 

[selfAdjoint]

Ah! Thank you for the explanation. I see it now!. So the Jordan product is precisely the symmetrizer.

 

[kneemo]

Ah! Thank you for the explanation. I see it now!. So the Jordan product is precisely the symmetrizer.

My pleasure selfAdjoint. I'm choosing to look at $$\mathfrak{A}$$ as a JB algebra so that we can tackle the GNS representation in the purely observable formalism. That is, given $$\mathfrak{A}$$ is a JB algebra, we can build a Hilbert space using only elements of $$\mathfrak{A}$$.

In the LOST paper, the GNS construction proceeds as follows:

(1) The linear space of equivalence classes is defined by $$[\mathfrak{A}]:=\mathfrak{A}/\mathfrak{J}$$ where $$\mathfrak{J}$$ is the left ideal consisting of $$a\in\mathfrak{A}$$ such that $$\omega(a^*a)=0$$.

(2) [tex]<[a],>:=\omega(a^*b)[/tex] where $$[a]\in\mathfrak{A}/\mathfrak{J}$$ is the equivalence class defined by $$a$$. This induces a norm $$||a||_{\omega}=\sqrt{<[a],[a]>}$$ in $$\mathfrak{A}$$.

(3) The Hilbert space is defined as $$\mathcal{H}_{\omega}:=\overline{[\mathfrak{A}]}$$

(4) To every $$a\in\mathfrak{A}$$ assign a linear unbounded operator $$\pi_{\omega}(a)$$ to act on elements of $$\mathfrak{A}$$ as
[tex]\pi_{\omega}(a):=[ab]\qquad\forall b\in\mathfrak{A}[/tex].

It can be seen that the Hilbert space is built purely from the observable algebra $$\mathfrak{A}$$ in the LOST paper. This is exactly what is done in the JB formulation of quantum mechanics. However, in the JB formalism, the Hilbert space construction is based on trace.

Let us follow through a GNS construction for $$\mathfrak{A}$$ in the JB formalism.

(1) As $$\mathfrak{A}$$ is a von Neumann algebra, there always exists a semi-finite, faithful, normal trace on $$\mathfrak{A}$$. Denote the elements $$a\in\mathfrak{A}$$ for which $$tr (a^*\circ a)<\infty$$ by $$\mathfrak{A}_2$$.

(2) Trace $$tr$$ induces a bilinear, symmetric, real scalar product on $$\mathfrak{A}_2$$ as $$<a,b>=tr (a\circ b)$$. The norm then takes the form $$||a||_{tr}=\sqrt{tr (a^2)}$$.

(3) Closure with respect to the trace norm yields a Hilbert space $$\mathcal{H}_J=\overline{\mathfrak{A}_2}$$.

(4) The Jordan representation is now a linear mapping $$\pi_J:\mathfrak{A}\rightarrow Hom_R(\mathfrak{A}_2,\mathfrak{A}_2)$$. For any $$a\in\mathfrak{A}$$ we have a multiplication operator $$T_a:\mathfrak{A}_2\rightarrow\mathfrak{A}_2$$ which is explicitly $$T_a b=a\circ b$$. By continuity of the operator, this can be extended to all $$\mathcal{H}_J$$.

We now ask if there is a Yang-Mills gauge invariant and diffeomorphism invariant state on $$\mathfrak{A}$$. The LOST paper states that there is, and defines the action of $$\omega_0$$ on elements of the form $$a.\hat{Y}$$, for $$Y$$ a vector field, as $$\omega_0(a.\hat{Y}):=0$$.

$$a.\hat{Y}$$ appears to be an inner derivation on $$a$$. If this is true, let's investigate the action of trace on inner derivations of $$\mathfrak{A}$$.

For a Jordan algebra, all inner derivations are given by the associator. Explicitly, a derivation of an element $$a\in\mathfrak{A}$$ takes the form $$[r,a,s]=(r\circ a)\circ s - r\circ (a\circ s)$$.

Let us take the trace of this inner derivation of $$a\in\mathfrak{A}$$. This gives us $$tr[r,a,s]=tr((r\circ a)\circ s - r\circ (a\circ s))$$. Invoking the properties $$tr(a+b)=tr a +tr b$$ and $$tr\hspace{.1cm} (a\circ b) \circ c=tr\hspace{.1cm} a\circ (b \circ c)$$,we see that $$tr[r,a,s]=0$$, in agreement with the LOST paper.

Regards,

Mike

 

[selfAdjoint]

My pleasure selfAdjoint. I'm choosing to look at $$\mathfrak{A}$$ as a JB algebra so that we can tackle the GNS representation in the purely observable formalism. That is, given $$\mathfrak{A}$$ is a JB algebra, we can build a Hilbert space using only elements of $$\mathfrak{A}$$.

In the LOST paper, the GNS construction proceeds as follows:

(1) The linear space of equivalence classes is defined by $$[\mathfrak{A}]:=\mathfrak{A}/\mathfrak{J}$$ where $$\mathfrak{J}$$ is the left ideal consisting of $$a\in\mathfrak{A}$$ such that $$\omega(a^*a)=0$$.

(2) [tex]<[a],>:=\omega(a^*b)[/tex] where $$[a]\in\mathfrak{A}/\mathfrak{J}$$ is the equivalence class defined by $$a$$. This induces a norm $$||a||_{\omega}=\sqrt{<[a],[a]>}$$ in $$\mathfrak{A}$$.

(3) The Hilbert space is defined as $$\mathcal{H}_{\omega}:=\overline{[\mathfrak{A}]}$$

(4) To every $$a\in\mathfrak{A}$$ assign a linear unbounded operator $$\pi_{\omega}(a)$$ to act on elements of $$\mathfrak{A}$$ as
[tex]\pi_{\omega}(a):=[ab]\qquad\forall b\in\mathfrak{A}[/tex].

It can be seen that the Hilbert space is built purely from the observable algebra $$\mathfrak{A}$$ in the LOST paper. This is exactly what is done in the JB formulation of quantum mechanics. However, in the JB formalism, the Hilbert space construction is based on trace.

Let us follow through a GNS construction for $$\mathfrak{A}$$ in the JB formalism.

(1) As $$\mathfrak{A}$$ is a von Neumann algebra, there always exists a semi-finite, faithful, normal trace on $$\mathfrak{A}$$. Denote the elements $$a\in\mathfrak{A}$$ for which $$tr (a^*\circ a)<\infty$$ by $$\mathfrak{A}_2$$.

(2) Trace $$tr$$ induces a bilinear, symmetric, real scalar product on $$\mathfrak{A}_2$$ as $$<a,b>=tr (a\circ b)$$. The norm then takes the form $$||a||_{tr}=\sqrt{tr (a^2)}$$.

(3) Closure with respect to the trace norm yields a Hilbert space $$\mathcal{H}_J=\overline{\mathfrak{A}_2}$$.

(4) The Jordan representation is now a linear mapping $$\pi_J:\mathfrak{A}\rightarrow Hom_R(\mathfrak{A}_2,\mathfrak{A}_2)$$. For any $$a\in\mathfrak{A}$$ we have a multiplication operator $$T_a:\mathfrak{A}_2\rightarrow\mathfrak{A}_2$$ which is explicitly $$T_a b=a\circ b$$. By continuity of the operator, this can be extended to all $$\mathcal{H}_J$$.

We now ask if there is a Yang-Mills gauge invariant and diffeomorphism invariant state on $$\mathfrak{A}$$. The LOST paper states that there is, and defines the action of $$\omega_0$$ on elements of the form $$a.\hat{Y}$$, for $$Y$$ a vector field, as $$\omega_0(a.\hat{Y}):=0$$.

$$a.\hat{Y}$$ appears to be an inner derivation on $$a$$. If this is true, let's investigate the action of trace on inner derivations of $$\mathfrak{A}$$.

For a Jordan algebra, all inner derivations are given by the associator. Explicitly, a derivation of an element $$a\in\mathfrak{A}$$ takes the form $$[r,a,s]=(r\circ a)\circ s - r\circ (a\circ s)$$.

Let us take the trace of this inner derivation of $$a\in\mathfrak{A}$$. This gives us $$tr[r,a,s]=tr((r\circ a)\circ s - r\circ (a\circ s))$$. Invoking the properties $$tr(a+b)=tr a +tr b$$ and $$tr\hspace{.1cm} (a\circ b) \circ c=tr\hspace{.1cm} a\circ (b \circ c)$$,we see that $$tr[r,a,s]=0$$, in agreement with the LOST paper.

Regards,

Mike

So THEOREM: The JB algebra trace construction gives the same Hilbert space as the LOST ideal-factoring construction. Nifty!

Mike, where have you been all my life? Or at least the last year or so? I have been making heavy weather of the GNS construction with my obviously inadequate background for that long, beginning with Thiemann's controversial LQG quantization of the closed string.

We should email your proof to Thiemann. I know he is very kind about replying and this might even tell him something he doesn't know and can use.
Do you know how we could get LaTex into MS outlook?

 

[kneemo]

So THEOREM: The JB algebra trace construction gives the same Hilbert space as the LOST ideal-factoring construction. Nifty!

Mike, where have you been all my life? Or at least the last year or so? I have been making heavy weather of the GNS construction with my obviously inadequate background for that long, beginning with Thiemann's controversial LQG quantization of the closed string.

We should email your proof to Thiemann. I know he is very kind about replying and this might even tell him something he doesn't know and can use.
Do you know how we could get LaTex into MS outlook?

Hi selfAdjoint

I'm glad you like the dual JB construction. In the last year or so I've been working on such things for my thesis, which is why I was so excited to see LQG in a *-algebra formalism. Before the LOST paper, my thesis was primarily focused on re-writing M(atrix) theory in the JB formalism.

The LOST paper reveals to me that LQG, as well as M(atrix) theory, can be re-interpreted in the JB formalism. Thus, it should be possible to eventually express both simultaneously, formally uniting nonperturbative string theory with LQG.

It's possible for me to type up a pdf version of the trace construction using LaTeX, which can be sent to Thiemann. Then the pdf can be sent as an attachment in outlook. Would this be ok?

Regards,

Mike

 

[selfAdjoint]

It's possible for me to type up a pdf version of the trace construction using LaTeX, which can be sent to Thiemann. Then the pdf can be sent as an attachment in outlook. Would this be ok?

That would be splendid! Does your thesis have a title yet?

 

[kneemo]

That would be splendid! Does your thesis have a title yet?

Ok, I'll start TeXing something up today. After I finish, do I send it over to you?

As for my thesis, I have not settled on a final title. After the LOST paper, I'll surely have to include a section on the flux holonomy *-algebra, which will ultimately affect any final working title.

Regards,

Mike

 

[selfAdjoint]

Okay. I'll PM my email address to you.

 

[marcus]

Hi Mike, going back to your post #29 in this thread, you give an alternative line of reasoning to the LOST paper. But in step (1) of your alternative argument 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.

Perhaps you could add some explanation, at step (1)-alt,
as to 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 )

I have used asterisks *** to mark the place in your step (1)-alt where I have a question.

Any clarification would be appreciated:

...In the LOST paper, the GNS construction proceeds as follows:

(1) The linear space of equivalence classes is defined by $$[\mathfrak{A}]:=\mathfrak{A}/\mathfrak{J}$$ where $$\mathfrak{J}$$ is the left ideal consisting of $$a\in\mathfrak{A}$$ such that $$\omega(a^*a)=0$$.

(2) [tex]<[a],>:=\omega(a^*b)[/tex] where $$[a]\in\mathfrak{A}/\mathfrak{J}$$ is the equivalence class defined by $$a$$. This induces a norm $$||a||_{\omega}=\sqrt{<[a],[a]>}$$ in $$\mathfrak{A}$$.

(3) The Hilbert space is defined as $$\mathcal{H}_{\omega}:=\overline{[\mathfrak{A}]}$$

(4) To every $$a\in\mathfrak{A}$$ assign a linear unbounded operator $$\pi_{\omega}(a)$$ to act on elements of $$\mathfrak{A}$$ as
[tex]\pi_{\omega}(a):=[ab]\qquad\forall b\in\mathfrak{A}[/tex].

It can be seen that the Hilbert space is built purely from the observable algebra $$\mathfrak{A}$$ in the LOST paper. This is exactly what is done in the JB formulation of quantum mechanics. However, in the JB formalism, the Hilbert space construction is based on trace.

Let us follow through a GNS construction for $$\mathfrak{A}$$ in the JB formalism.

(1) As $$\mathfrak{A}$$ is *** a von Neumann algebra, there always exists a semi-finite, faithful, normal trace*** on $$\mathfrak{A}$$. ...

in case someone in earshot wants the relevant definition, here are two pages from Wiki and from mathworld:
http://en.wikipedia.org/wiki/W-star-algebra
http://mathworld.wolfram.com/vonNeumannAlgebra.html