Fresh start at a (Loop) Quantum Gravity thread

[marcus]

getting started in good order is difficult---reminds me of trying to furl sail in a high wind: there is a lot of subject-matter and it keeps flapping and getting out of hand----need to tie things down bit by bit

The two Ashtekar-Lewandowski 1994 papers seem to be basic
in fact Sahlmann refers to them and treats them that way. I put off printing out the longer of the two A-L papers until yesterday because it just seemed so long (68 pages) but I finally gave in and printed it. Somehow it seems like we can understand and describe the core of the subject and what is going on if we can
understand the main content in general terms of these two papers plus the recent (2003) ones:

A-L 1994 papers:
"Projective Techniques..."
http://arxiv.org/gr-qc/9411046
"Differential Geometry..."
http://arxiv.org/gr-qc/9412073

I will add links to recent papers by Sahlmann and others later.
I want to be able to say in general terms what is happening in these papers and nail down the analogy with the Stone-vonNeumann theorem and the historical development of quantum mechanics. They are quantizing General Relativity and it should reveal something about what is essential in quantizing classical theories.

 

[marcus]

The next place where there is something to nail down is
the 1998 paper by Ashtekar-Corichi-Zapata (ACZ)
"Quantum Theory of Geometry III..."
http://arxiv.org/gr-qc/9806041

and the next, which depends on this one is
the 2003 paper by Okolow-Landowski (O-L)
"Diffeomorphism covariant representations of the holonomy-flux *-algebra"
http://arxiv.org/gr-qc/0302059

I discussed the ACZ paper some in the "recent LQG developments" thread including this key idea on page 13.
Here is what ACZ say:

"Let us summarize. For simple finite dimensional systems, there are two equivalent routes to quantization, one starting from the Poisson algebra of configuration and momentum functions on the phase space and the other from functions and vector fields on the configuration space. It is the second that carries over directly to the present approach to quantum gravity..."

We are used to thinking of conjugate pairs of variables like (Q,P) position-momentum, and in quantum gravity what you hear most about is (A,E) the connection and the triad, as analogs of position and momentum.
What ACZ 1998 seems to say is that you don't have to have an algebra of configuration and momentum functions.

You can have an algebra comprising functions defined on the configuration space and VECTOR FIELDS also defined there. the vector field sort of takes the place of the momentum function.

They are able to get a Lie algebra this way and it will get promoted to a *-algebra of operators on a hilbertspace. But instead of looking ahead to that, I want to mull over the beginnings.

It starts with a space A of connections (on some 3D manifold) and with "holonomies" h_e:A ---> G
that means just run the connection on an edge (analytic curve) and get a group element
and then the holonomies get a bit more complicated and become
"cylindrical functions" C:A ---> C, the complex numbers,
where associated with a cyl function big C there is a graph with N edges and a group-eating function little c:G^N---> C,
and the recipe is just run the connection on the N edges and get an N-tuple of group elements and feed it to little c and get a complex number.

The cylindrical functions are called Cyl and the ones that have a smooth little c are called Cyl^oo. And Cyl is about the simplest set of functions you can define on the connections that is big enough and versatile enough to be any use.

The next thing they do is define some "derivations" X:Cyl^oo--->Cyl^oo.
This is where it gets very interesting, so I will post what I have and then see what I can say about these "derivations". They are vector fields on the infinite dimensional manifold of the connections---vector fields defined on A

 

[marcus]

The connection on a manifold tells how tangent vectors turn as you move them from point to point and it tells about the shape of the manifold. In quantum geometry or quantum relativity/quantum gravity (whatever) the underlying 3D manifold HAS NO PARTICULAR SHAPE and so the space of all possibilities is terribly important.

We are putting effort now into getting to know
A, the space of all possible connections.
In these papers we're reading now they call the group G and the Lie algebra of G is called G', and G is unspecified but probably SU(2)

Tangent vectors at any point in a manifold are DERIVATIONS and in fact the tangentspace is often defined that way. Even though A (being all possible connections encoding all possible shapes) is a very big space we can get to know it somewhat by defining things on it

1. there is Cyl the set of cylindrical functions defined by graphs in the original 3D manifold
a member of Cyl is C:A --> C, , the complex numbers. All it takes to define big C is a graph and a group-eater function little c: G^N --> C.

2. there is a unique and handy measure μ_AL defined on A that let's you integrate cylindrical functions. Ashtekar and Lewandowski finessed it from the invariant Haar measure on the group. You can also think of it
as a linear functional defined on Cyl where the linear functional simply corresponds to integrating functions by means of the measure.

3. there are vectorfields on A corresponding to DERIVATIONS of smooth Cyl functions. The notation for a vectorfield is X:Cyl^oo--->Cyl^oo.

The secret of these derivations is to make them depend both on the GRAPH underlying a cylindrical function big C and also depend on the group-eater little c. The way you make them sensitive to the graph is you specify a SURFACE in the basic 3D manifold, which is where the graphs are, and call that surface S, and then
that surface S picks out from any graph the edges that puncture it, beginning or ending at points in S.

Then if the k-th edge of the graph punctures S, you proceed to fiddle with the k-th holonomy argument of little c, which is smooth so you can differentiate it at any input component.

For the time being that is about enough about
the 1998 paper by Ashtekar-Corichi-Zapata (ACZ)
"Quantum Theory of Geometry III..."
http://arxiv.org/gr-qc/9806041

The next thing to do is see what
the 2003 paper by Okolow-Landowski (O-L) makes of this.
It is called
"Diffeomorphism covariant representations of the holonomy-flux *-algebra"
http://arxiv.org/gr-qc/0302059

In the O-L paper they credit a Penn State postdoc named Hanno Sahlmann with a lot of the ideas. Sahlmann is most recently from Berlin (Uni-Potsdamm AEI and MPI-for-Gravityphysics) where he did his PhD dissertation under Thiemann) But the fact is that the O-L paper seems, at least to me, to be slower and clearer and more careful than the original Sahlmann papers. Maybe it is because Lewandowski is older or maybe it is because he is from Warsaw. For whatever reason, what I want to understand is what Lewandowski says Sahlmann says.

First notice that if you build an algebra which is the direct sum of some functions and some derivations (vectorfields) then there is a natural well-nigh unavoidable LIE BRACKET.
Lets say it is Cyl + X
so we have elements which are ordered pairs (C,X) and (C', X')
and the bracket is going to be a new ordered pair
(X'C - XC', [X,X'])

we know how to take bracket of two vectorfields X and X', and the first component is gotten by using X and X' as derivations to differentiate the two functions.

This gives a good Lie algebra and satisfies the Jacobi identity and all that. Then there is the shifty thing that physicists do when they quantize which is to put h-bar and i onto the Lie bracket.

After a little fussing around like this (Lo and Behold, as one says) we have an algebra which Okolow and Lewandowski write with an uppercase gothic letter, just as if it were 1932 and Banach was in a Warsaw cafe inventing Banach spaces. They write Gothic_A and call it the Sahlmann algebra.

NOW of course everyone starts looking around for the Hilbert space! Once you have an algebra then where, in the name of Almighty Heaven, is the Hilbertspace of square-integrable functions!

And it will turn out, though I must post this and return to it another time, that the AL measure on A
will do just fine. You can construct L² the square integrable functions on that and then the ACTION of a pair
(C, X) in the algebra (consisting of a cylindrical function and a derviation) on some L² function F is going to be
to multiply the two functions together to get another L² function CF, and what to do with the derivation? Well
I will stop here and check the O-L paper

 

[selfAdjoint]

Marcus, this is a wonderful, well organized presentation, and I am sure it's going to be much linked-to. I have only one small remark; you describe the little c out of which a cylinder function is built as a "group-eater". My understanding is that it eats a connection and an edge and spits out a group element, as indeed you describe.

Isn't this marvelous theory? The way it all automatically fits together with the ingenuity being how to see that fitting before anybody else.

 

[marcus]

Originally posted by selfAdjoint
Marcus, this is a wonderful, well organized presentation, and I am sure it's going to be much linked-to...

That would be great! we can improve it by editing too. but
I am not sure people link to stuff at PF math forum, had not thought about this...

just a bit silly to call little c a "group eater" but I will explain why I did:

little c: G^N --> C, the complexnumbers

so it consumes an N-tuple of group elements and gives back a number

big C is a recipe which says "run A on the edges of the graph to get N group elements and feed that N-tuple to little c"

 

[marcus]

another inconsequential difference of notation
when O-L talk about cylinder functions on page 4
instead of big C and little c
they use big Ψ and little ψ

Their equation (2.4), a definition, says

Ψ(A) = ψ(A(e1), ..., A(eN) )

"run the connection A on the set of edges {e1,...,eN}
and give the resulting group elements to little ψ"

But Sahlmann uses C and c, which after all stands for "cylinder"
so I got accustomed to that.
now I am in a mixed quantum state about which notation to follow, like Schroedinger's cat, sometimes saying cee and sometimes psi

 

[marcus]

On the agenda is to extend the representation of Cyl to a representation of the whole Sahlmann algebra SAHL
Our sources write the Sahlmann algebra with an uppercase Gothic_A, which we don't have. In general I'm following the notation fairly closely in Sahlmann's 2002 paper "Some Comments on the Representation Theory of the Algebra Underlying Loop Quantum Gravity" (gr-qc/0207111) with some
attention to making it legible in PF. The electric field convention is taken from Okolow-Lewandowski 2003 paper. Here's a list of some notations we've been using or will use. Had to use a non-standard tensor product sign (x) a couple of times

Σ, the basic 3D manifold
T(Σ) and T*(Σ), tangent and cotangent bundles
G, G', and G'* the group, its Lie algebra, and the latter's dual
S, a 2D surface embedded in Σ
ƒ: S -->G', a test function defined on the surface S.
A, a connection (G' valued 1-form) in the tensor product G'(x)T*
tildE, "electric field" (G'*valued density) in G'*(x)T
A, the space of connections
E, the space of electric fields

γ = {e1,...,eN} a graph with N edges
c: G^N --> C, the complexnumbers. An N-fold group-eater
C: A --> C. A cylinder function defined using some γ and c.
E(S, ƒ): E --> C. The "flux" of an electric field thru a given surface S with a given testfunction ƒ. Defined by an integral E(S,ƒ)[tildE] on page 6 assigning to every tildE a complex number.

Cyl, the space of all cylinder functions, equipped with the supremum norm to form a C* algebra.
Cyl^oo, the infinitely differentiable (smooth) ones, meaning that the associated little c function is smooth

X, a certain class of derivations defined on the functions in Cyl^oo

X: Cyl^oo --> Cyl^oo, a derivation belonging to X, which can be seen as a vectorfield on A

SAHL, the Sahlmann algebra, direct sum Cyl + X

H, a hilbert space--essentially a vector space where the inner product (.,.) of any two vectors is defined and where any convergent sequence has a limit

spectrum(Cyl), the maximal ideal space, or Gelfand spectrum, of the commutative C* algebra Cyl. By Gelfand's theorem, Cyl is isomorphic to the C* algebra of continuous complex-valued functions on the spectrum, a compact Hausdorff space. Sahlmann and the others write the spectrum(Cyl) by "A-bar", which I can't type: A with a bar over it.
To follow Sahlmann's notation as closely as I can, I shall call the spectrum A-bar.


As the notation suggests, it can be seen naturally as a "closure" or completion of A, the space of connections. In fact Sahlmann and the others call it the space of "generalized connections." This is reasonable since the cylinder functions were originally defined on A. The idea is you take the space of connections and throw in a few more points to make it a compact space and extend the cylinder functions to be defined on this slightly larger, now compactified, space. What else to call the points in that space but "generalized connections"?



μ_AL, the measure that Ashtekar and Lewandowski defined on A-bar. Over and over again this measure comes up and turns out to be the right one. They jacked it up from the invariant Haar measure on G^N

L²(A-bar, dμ_AL), this is THE hilbert space---the square integrable functions (which includes the cylinder functions) defined on "A-bar" using the AL measure.
This is the space of the representation of the Sahlmann algebra.
The representation [pi] maps the algebra into the linear operators on this hilbert space.

 

[marcus]

SAHL, the Sahlmann algebra, direct sum Cyl + X

In an earlier (08-04) post I mentioned that Cyl + X has a Lie bracket satisfying the Jacobi identity

modulo some tweaking by a factor of i and h-bar the bracket is
defined in a straightforward way on pairs (C,X) consisting of a cylinder function C and a vectorfield X. I'll just adapt a snippet from the earlier post:

"Say we have elements of the direct sum which are ordered pairs (C,X) and (C', X'), the bracket is going to be a new ordered pair
(X'C - XC', [X,X'])

we know how to take bracket of two vectorfields X and X', and the first component is gotten by using X and X' as derivations to differentiate the two functions."

There is a March 2003 paper by Sahlmann and Thiemann (gr-qc/0303074) where they go a step farther and exponentiate these vectorfields X to get bounded operators and a variant of the algebra analogous to the Weyl algebra. Still trying to assimilate that.

 

[marcus]

The Okolow-Lewandowski paper continues to fascinate me. In this connection have been reading what Baez and other have to say about "densitized" tensors, line bundles, bundle of densities. The electric field tildE is often called a "densitized inverse triad". (A, tildE) are the Ashtekar variables. The first thing to understand about a "density" is the way it transforms under diffeomorphisms.
Judging by top of page 18 I think that if a differential form is pulled back by a diffeo, that a density is pushed forward.

Baez had a lengthy jovial argument with someone who insisted that "densitizing" a 1-form to get a 2-form was the same as the "Hodge star" because the Hodge operation will in fact convert a 1-form into a 2-form, on a 3D manifold. But the Hodge requires a METRIC which we do not have here.

Page 18 says if φ is a diffeomorphism Σ --> Σ
and tildE is a G'* valued 1-form then
φ acts on tildE to give φ^-1_*tildE

It also says φ acts on a connection (G' valued 1-form) to give
φ*A
Is not this upper asterisk saying pull back the 1-form and
the lower asterisk saying push forward the density?

This looks like it is born out by equations (4.6) and (4.7). They show, among other things, how the flux function E(S, ƒ) is affected by the diffeomorphism

φE(S,ƒ) = E( φ(S), ƒ o φ^-1)


φE(S,ƒ)[tildE] = E(S,ƒ)[φ^-1_*tildE]

I must post this to see how it looks.

The ƒ testfunction has values in G' because the "densitized" 1-form has values in G'*. Since it has values in the dual, and we want the flux to turn out a number, we have to feed in some elements of G' for the G'* to be defined on.

Will post and see how it looks and edit later

Σ, the basic 3D manifold
T(Σ) and T*(Σ), tangent and cotangent bundles
G, G', and G'* the group, its Lie algebra, and the latter's dual
S, a 2D surface embedded in Σ
ƒ: S -->G', a test function defined on the surface S.
A, a connection (G' valued 1-form) in the tensor product G'(x)T*
tildE, "electric field" (G'*valued density) in G'*(x)T
A, the space of connections
E, the space of electric fields

γ = {e1,...,eN} a graph with N edges
c: G^N --> C, the complexnumbers. An N-fold group-eater
C: A --> C. A cylinder function defined using some γ and c.
E(S, ƒ): E --> C. The "flux" of an electric field thru a given surface S with a given testfunction ƒ. Defined by an integral E(S,ƒ)[tildE] on page 6 assigning to every tildE a complex number.

Cyl, the space of all cylinder functions, equipped with the supremum norm to form a C* algebra.
Cyl^oo, the infinitely differentiable (smooth) ones, meaning that the associated little c function is smooth

X, a certain class of derivations defined on the functions in Cyl^oo

X: Cyl^oo --> Cyl^oo, a derivation belonging to X, which can be seen as a vectorfield on A

SAHL, the Sahlmann algebra, direct sum Cyl + X

H, a hilbert space--essentially a vector space where the inner product (.,.) of any two vectors is defined and where any convergent sequence has a limit

A-bar, the Gelfand spectrum or maximal ideal space of Cyl, which is isomorphic to the C* algebra of continuous complex-valued functions on the spectrum, a compact Hausdorff space. A-bar is called the space of generalized connections.

μ_AL, the measure that Ashtekar and Lewandowski defined on A-bar.

L²(A-bar, dμ_AL), the hilbert space of square integrable complex valued functions on A-bar, using the AL measure. This hilbert space, which contains the cylinder functions, hosts the representation of the Sahlmann algebra. The representation [pi] maps the algebra into the linear operators on this hilbert space.

 

[jeff]

Originally posted by marcus
The Okolow-Landowski paper continues to fascinate me.

Why?

 

[jeff]

Originally posted by marcus
I think the ƒ testfunction must be there because it IS a density.

Huh? I think you need to look up what tensor densities are and why testfunctions are needed in the definition of field operators.

 

[marcus]

this has a good list of notation

 

[marcus]

we really should have a bibliography with
some categories or annotations----of articles
written since 2000

those that give good intuition about the basic components
those dealing with matter, coupled to gravity
those exploring forms of the hamiltonian
and the low energy or semiclassical limit etc.
a number of articles pre-2000 have done these things
but recent work is of special interest

eg.
Hanno Sahlmann 2002 "Coupling matter to Loop Quantum Gravity"

Sahlmann/Thiemann 2002 "Towards the QFT on curved Spacetime
Limit of QGR"

abstr. "In this article...we address the question of how one might obtain the semiclassical limite of ordinary matter quantum fields (QFT) propagating on curved spacetimes from full-fledged Quantum General Relativity..."

When Thiemann says "Quantum General Relativity" he means LQG because he considers this to be the only going attempt to quantize General Relativity (the dynamic geometry approach to gravity)

So if you modify the Hoch-Deutsch word order the title means
towards LQG's semiclassical matter-field limit on curved spacetime manifolds.