View Full Version : recent developments in LQG
I was just reading a paper co-authored by Jerzy Lewandowski
which is dated June 24 2003----but also dated February 2003, I dont know which applies.
http://arxiv.org/gr-qc/0302059
It begins "the quantum holonomy operators and the quantum flux operators are the basic elements of quantum geometry..."
"...Recently, Sahlmann [1] proposed a new, more algebraic point of view. It opens the door to a representation theory of quantum geometry. The main idea is to spell out a definition of a *-algebra constructed from the holonomies and fluxes, that underlies all the loop quantum gravity framework, and to study its representations..."
Lewandowski calls a certain kind of holonomy-flux *-algebra a "Sahlmann algebra". In his papers Hanno Sahlmann is more modest and does not call the algebra that. (naming things after oneself is not customary in mathematics). Hanno is pretty young to have something named after him. I am trying to understand this a little better.
A good many of the key papers (1994--present) in loop quantum gravity (quantum-general-relativity in effect) have been by Ashtekar and Lewandowski---if there are central figures in LQG then those two would be my pick of the lot. Hanno Sahlmann just appeared---his thesis date is, I think, 2002.
In November 2002 Ashtekar, Lewandowski, and Sahlmann posted
http://arxiv.org/gr-qc/0211012
"Polymer and Fock representations for a Scalar field"
here is the abstract:
"In loop quantum gravity, matter fields can have support only on the polymer-like excitations of quantum geometry, and their algebras of observables and Hilbert spaces of states can not refer to a classical, background geometry. Therefore, to adequately handle the matter sector, one has to address two issues already at the kinematic level. First, one has to construct the appropriate background independent operator algebras and Hilberts spaces. Second, to make contact with low energy physics, one has to relate this polymer description of matter fields to the standard Fock description in Minkowski space. While this task has been completed for gauge fields, imporatant gaps remained in the treatment of scalar fields. The purpose of this letter is to fill these gaps."
edit: forgot earlier to include link to the first paper mentioned
selfAdjoint
Jul31-03, 01:32 PM
Is the first paper you mentioned, the one coauthored by Lewandowski, online? If so, could we have a link to it? If this is the Ashtekar school's approach to putting quantized matter into their quantized spacetime, it is very important indeed.
Originally posted by selfAdjoint
Is the first paper you mentioned, the one coauthored by Lewandowski, online? If so, could we have a link to it? If this is the Ashtekar school's approach to putting quantized matter into their quantized spacetime, it is very important indeed.
selfAdjoint thanks for flagging my omission. I intended to include a link to that paper but forgot as I was typing. It is:
http://arxiv.org/gr-qc/0302059
At almost the same time (Feb 2003) that paper by Okolow and Lewandowski appeared, there also came out one by Sahlmann and Thiemann which is doing essentially the same thing:
http://arxiv.org/gr-qc/0302090
"On the Superselection Theory of the Weyl Algebra for Diffeomorphism Invariant Quantum Gauge Theories"
Here is a quote from the beginning to give an idea:
<<Abstract
Much of the work in loop quantum gravity and quantum geometry rests on a mathematically rigorous integration theory on spaces of distributional connections. Most notably, a diffeomorphism invariant representation of the algebra of basic observables of the theory, the Ashtekar-Lewandowski representation, has been constructed. This representation is singled out by its mathematical elegance, and up to now, no other diffeomorphism invariant representation has been constructed. This raises the question whether it is unique in a precise sense.
In the present article we take steps towards answering this question. Our main result is that upon imposing relatively mild additional assumptions, the AL-representation is indeed unique. As an important tool which is also interesting in its own right, we introduce a C*-algebra which is very similar to the Weyl
algebra used in the canonical quantization of free quantum field theories.
1. Introduction
Canonical, background independent quantum field theories of connections [1] play a fundamental role in the program of canonical quantization of general relativity (including all types of matter), sometimes called loop quantum gravity or quantum general relativity. For a review geared to mathematical physicists see [2], for a general overview [3]).
The classical canonical theory can be formulated in terms of smooth connections A on principal G-bundles over a D-dimensional spatial manifold Σ for a compact gauge group G and smooth sections of an associated (under the adjoint representation) vector bundle of Lie(G)-valued vector densities E of weight one. The pair (A, E) coordinatizes an infinite dimensional symplectic manifold (M , s) whose (strong) symplectic structure s is such that A and E are canonically conjugate.
In order to quantize (M , s), it is necessary to smear the fields A, E. This has to be done in such a way, that the smearing interacts well with two fundamental automorphisms of the principal G-bundle, namely the vertical automorphisms formed by G-gauge transformations and the horizontal automorphisms formed
by Diff(Σ) diffeomorphisms. These requirements naturally lead to holonomies and electric fluxes, ...>>
selfAdjoint
Jul31-03, 08:29 PM
Funny thing is, I had found the Sahlmann-Thiemann paper just browsing the archive back when it came out. All that G-bundle theory you quoted brought it back to me. How are you on that stuff? I can do part of it but the vertical homomorphisms are out of my orbit. I have another paper where a very similar construction is used to define the BRST transformation in terms of topology.
Originally posted by selfAdjoint
Funny thing is, I had found the Sahlmann-Thiemann paper just browsing the archive back when it came out. All that G-bundle theory you quoted brought it back to me. How are you on that stuff? I can do part of it but the vertical homomorphisms are out of my orbit. I have another paper where a very similar construction is used to define the BRST transformation in terms of topology.
I'm willing to try to connect with Sahlmann's work, by going thru the easiest most pedagogical of his recent papers. I've looked over everything from early 2002 onwards (his PhD thesis is 2002 and called "Coupling Matter to Loop Quantum Gravity") and the most accessible, I think, is
http://arxiv.org/gr-qc/0207112
"When Do Measures on the Space of Connections Support the Triad Operators of Loop Quantum Gravity"
Have a look at it, selfAdjoint, and see if you'd be interested in reading some of it with me
selfAdjoint
Aug1-03, 01:08 PM
Yes, I printed off and I would very much like to go through it with you and work on questions either of us might have along they way. I am currently ready to start on section 2, Measures on thespace of generalized Connections.
Do you want to do this here on the boards or by email?
Originally posted by selfAdjoint
Yes, I printed off and I would very much like to go through it with you and work on questions either of us might have along they way. I am currently ready to start on section 2, Measures on thespace of generalized Connections.
Do you want to do this here on the boards or by email?
I have it printed out too, and section 2 seems like a good place to start (right after the introduction). I see no harm in proceeding with it in this thread (maybe Hurkyl, who has a longterm interest in LQG, or Lethe will join in) and we can always retire into a corner
(either with PF's personal messaging, or email) if there are too many interruptions.
The "projective limit" mechanism excites me. It is a beautiful device for constructing (not just measures but other) utilities on infinite dimensional spaces. The projective limit is how A-L originally defined the measure μAL on the space of
connections in the landmark 1994 paper "Projective techniques and functional integration for gauge theories"
http://arxiv.org/gr-qc/9411046
What we must understand is two things----invariant Haar measure, which is just the analog of the uniform measure on the circle, and which every compact group has-----and how A-L promoted Haar measure (by projective limit) up to a measure on the space of connections.
It is a neat business. to take the projective limit of a family of measures, they need a DIRECTED SET (basically a partial ordering with something inclusive of each pair) and the GRAPHS in the manifold are a directed set! Given any two you can always merge them to make a larger one containing both.
And associated with each graph there is a bunch of numerical valued "cylindrical functions" c[A] of the connection. To specify a cyl function you give a graph----a set of N edges embedded in the manifold---and you give a function c(h1, h2, ...., hN) defined on GN the cartesian product of N copies of the group. (The group is probably just SU(2) and has an invariant Haar measure.)
What could be simpler than one of these cylindrical functions?!
To evaluate it on a connection A you just run holonomy on the edges of the graph and get an N-tuple of group elements, to which you apply the function c(h1, h2, ...., hN) and get a number c[A]. (By abuse of notation, i'm using the letter c in two related senses.)
To define the measure on the whole of A we just need to know how to integrate cyl functions. A measure can be identified with a linear functional on a function space, corresponding to integrating those functions with that measure. So get ready to do the projective limit----for each graph we need a measure or linear functional defined on the cyl functions with that graph----but for that the old N-fold Haar measure on GN will work!
In their section 2, Sahl/Thiemann, snuck a density function ƒ into the picture for a bit more generality but then in the last paragraph they remark that the really classical application of it is with the density identically equal to 1, in which case you get the A-L measure. But you get John Baez measures and other variants by letting ƒ vary.
selfAdjoint
Aug1-03, 05:38 PM
I am right with you on most of your exposition, and section 2, but I can't quite get my mind around the statement on page 5 where they have taken their projective limit of spaces of connections (bar_A) and say that the closure of the cylindrical funtions on bar_A under the sup norm is a C* algebra. OK so far, then they say the spectrum of the algebra can be identified with bar_A thus endowing it with a Hausdorf topology.
I have been vaguely aware that this is a pretty standard move in function theory, to get the space back from some algebraic operation on the functions defined on it. But I don't have the foggiest how this is achieved. Could you give me a for dummies quicky? Also the statement of the cited Riesz-Markov theorem?
As you say this graph projection approach is obviously powerful and very neat. Seems to be worth while to work a bit to get my head around it.
I do thank you for pointing to the paper and your excellent presentation.
Originally posted by selfAdjoint
I am right with you on most of your exposition, and section 2, but I can't quite get my mind around the statement on page 5 where they have taken their projective limit of spaces of connections (bar_A) and say that the closure of the cylindrical funtions on bar_A under the sup norm is a C* algebra. OK so far, then they say the spectrum of the algebra can be identified with bar_A thus endowing it with a Hausdorf topology.
I have been vaguely aware that this is a pretty standard move in function theory, to get the space back from some algebraic operation on the functions defined on it...
Let's look at page 11 of the A-L paper they are drawing from:
http://arxiv.org/gr-qc/9411046
<<2.2 The Gel'fand spectrum of bar_Cyl...
A basic result in the Gel'fand-Naimark representation theory assures us that every Abelian C*-algebra ...with identity is realized as the C*-algebra of continuous functions on a compact Hausdorff space, called the spectrum...Furthermore, the spectrum can be constructed purely algebraically...[as a space of maximal ideals]...>>
We have to understand the space of maximal ideals---I, as you, have seen this over and over as a "pretty standard move" as you say. There must be something good about this move that we should understand. I will try to talk about it some---to get and share intuition.
As far as that Riesz Markov theorem goes, I wont forget and will get around to it, but I want to focus on this maximal ideal space business for a while.
As for thanks, mine to you likewise. This is topnotch math and its fun to have someone to talk to about it.
selfAdjoint
Aug1-03, 07:07 PM
Oh yeah, those maximal ideals. I know I have run into that somewhere recently - not as real exposition, but some kind of historical note in Mathematical Intelligencer. I'll try to find it.
selfAdjoint
Aug1-03, 07:49 PM
Via google I found this description of the Gelfond, Naimark, Segal construction (http://www.mth.kcl.ac.uk/~iwilde/notes/calg/s8.ps), whic establishes the theorem. It's in terms of Hilbert spaces and states rather than compact sets and points, but should carry through.
Warning, the file is in postscript.
this is something which Hurkyl and Lethe may have seen before and could join us on----it is a very cool trick (not being an expert I only halfway recall or understand it, I have to rediscover it, but I dimly know it to be cool)
Some gypsies give you a ring with commutative multiplication and they conceal from you the fact that this ring is simply the continuous complex valued functions on the interval [0,1]. Those crafty bastards! They give it to you in abstract form as a bag of different colored M and M candy which you can add and multiply to get other M and Ms of different color. They dont tell you that the thing originally arose in all its algebraic splendor simply as a space of functions on [0,1]. The next morning the gypsies are nowhere in sight and the chickens are gone too.
But each point x in the interval [0,1] corresponds to an IDEAL in the space of functions on [0,1] consisting of the set of functions which are zero at x
Ix = {f such that f(x) = 0} is an ideal because it has the sticky quality that if you take any g at all and multiply g by some f in the ideal you wind up in the ideal.
for any g in the ring, and any f already in the ideal, gf is in the ideal. Like a black hole or the tarbaby, you touch it, and you're in.
And that Ix is maximal in that it is not contained in any larger ideal except the whole ring.
So there is a one-one correspondence between points x in the interval [0,1] and the set of maximal ideals.
so the gypsies gave you an abstract ring---a mere bag of M&Ms---but you suspect that it arose as the ring of continuous functions on some space X. And you are able to RECOVER the space X as the set of maximal ideals of the ring.
so far its only heuristic. there is more: a way to put a topology on the space of maximal ideals that makes it compact. maybe I will post something, or someone else will, about that later.
there's even a hint of a technique for "compactification" here. Start with a space that isnt compact, make a ring of functions that are well behaved on it in some fashion, take the maximal ideals, with some topology and it may turn out to be a compact space "including" the original. I seem to recall the "almost periodic" functions on the real line being handled this way, as functions on a compact ideal space. maybe someone has heard of this. Ashtekar used that method of compactification in a cosmology paper recently calling it "the Bohr compactification of the real line". I couldn't remember having heard of the Bohr compactification before but I harbor a deep suspicion that maximal ideals were being used there too.
And an algebra? What that? just a ring with scalar multiplication. Lethe! Is that right? And a *-algebra is just an algebra with something analogous to complex conjugation.
It if is the ring of continuous complex-valued functions on something, then it automatically has complex conjugate----f* is just what you think it should be. And if there is a norm too then it is a C*-algebra. so we are just talking rings----function rings, rings with norms, conjugation, addition & multiplication of functions. It does not get any more natural. You can define these things in your sleep without having ever seen the definition.
OK and Hanno Sahlmann has taken the holonomy+flux operators of LQG and made a C*algebra of them (by taming the flux operators to make them bounded) and proved that there is essentially only one irreducible representation of that algebra as operators on a hilbertspace. And it does not even seem to be all that gnarly! Blows me away. Must get down and learn this stuff. "Get my mind around it" to use selfAdjoint's expression.
selfAdjoint, I just saw your post with the link to
discussion of GNS, for which thanks! I personally
have not the skill to read PS, and read everything in PDF,
but others very likely can profit from that file.
I'm inclined to move ahead at catch up on GNS details
later. but also if anyone is especially interested in
this point or has questions about it we could dwell
on it for a while.
BTW they are going to take the projective limit of
QUOTIENT spaces to get a measure on something
they call A/G
(notation gets awkward because of lack of symbols)
which is the connections with the diffeomorphisms modded out
I am referring a lot to the original A-L 1994 paper because
it is less condensed than the one we are reading in its
coverage of these things.
selfAdjoint
Aug2-03, 10:48 AM
Marcus, I have just gone through all the projective development in the 1994 paper, and surprize! I understood it. Ashtekar is a super expositer, and the only difficulty with jis relaxed approach in comparison to the lemma, lemma, theorem, corollary, corollary approach of my youth is that the onus of rigor falls on the reader. It's so easy to skim and say "Sure, sure, yes that's right". Well I worked through about half of their basic proofs and sorted of fuzzed the other half, but I feel that I am well grounded now and we can discuss the application of this projective technique to Sahlmann's theorem.
BTW I finally twigged to the brand of Category thoeory that would apply here. Isham's toposes (or topoi if you want to be clever) include a partially ordered "index" component. They would be naturals for projective reasoning.
Originally posted by selfAdjoint
Marcus, I have just gone through all the projective development in the 1994 paper, and surprize! I understood it. Ashtekar is a super expositer,...
You are ahead now and I need to catch up
I agree with you about Ashtekar's writing style.
In case anyone wants to join us the paper is
http://arxiv.org/gr-qc/9411046
"Projective techniques and functional integration for gauge theories" by Ashtekar and Lewandowski
maybe this is a good clear place to start reading
and this years papers by Sahlmann and others attach
on to this as an extension
Anyway I should make myself a strong cup of coffee and
do my homework: give a more careful reading to the 1994
paper you mention
selfAdjoint
Aug3-03, 11:48 AM
I have now reviewed Sahlmann's paper down through the section of diffeomorphism-invariant measures. I need to re-review a lot of the stuff, starting in section 3 where he suddenly jumps from the principle bundle with group G to spacetime with a group SU(2). I think his graphs also suddenly morph into those simplex edges carrying spinor reps of SU(2), taken I suppose directly from his reference 13, the 1997 paper by A&L where they quantize the area operator. I don't really want to go back and review that paper too - it strarts to feel Like I'm Achilles and Sahlmann is the tortoise!
What I need is some help unpacking his notation - what are the X's from and to, and in what sense are the f's "co-vectors"? Mappings from something to the complex numbers, but from what? G (aka SU(2)? Any help would be appreciated.
this is great
having some definite questions from you will help
me get some traction
so I wont keep slipping and sliding all over the place
You are referring, I see, to
http://arxiv.org/gr-qc/0207112
"When do measures..."
I have this in hand and will try to respond. I was proposing this
paper earlier as a good place to start and it may turn out to
be.
Originally posted by selfAdjoint
I have now reviewed Sahlmann's paper down through the section of diffeomorphism-invariant measures. I need to re-review a lot of the stuff, starting in section 3 where he suddenly jumps from the principle bundle with group G to spacetime with a group SU(2). I think his graphs also suddenly morph into those simplex edges carrying spinor reps of SU(2), taken I suppose directly from his reference 13, the 1997 paper by A&L where they quantize the area operator. I don't really want to go back and review that paper too - it strarts to feel Like I'm Achilles and Sahlmann is the tortoise!
What I need is some help unpacking his notation - what are the X's from and to, and in what sense are the f's "co-vectors"? Mappings from something to the complex numbers, but from what? G (aka SU(2)? Any help would be appreciated.
WOAH! I forgot to respond earlier to your question about the X's.
The clearest explanation is by Lewandowski on page 8 of
http://arxiv.org/gr-qc/0302059
This is the Okolow-Landowski paper which we talked a bit about earlier. It presents itself as a parallel presentation of Sahlmann's ideas with maybe a slight correction. Landowski is a senior person and it was this paper that put me onto Sahlmann's papers. Because it is more expository and takes more time with the definitions---slow and careful---it is easier in some ways to read than the original. On page 8 it says what those X's are.
Tomorrow I will try to give an intuitive reading of page 8. This is the key step of exponentiating the flux operator thru a given surface S in order to "tame" it, and then including it in the holonomy-flux algebra. This may be the gnarliest place---hope
once thru it we have smooth sailing.
Im backing up and getting into low gear with this paper "When do measures..."
To review what he is trying to do, he starts on page 2 with the standard Ashtekar or "new" variables of GR, namely the connection A and the triad field E and he writes two equations (1) and (2) which introduce holonomies and fluxes:
he[A] is the holonomy along edge e using connection A
ES,f[E] is the flux of E through surface S integrated with the help of a covector f that I think of as collapsing the triad so he can get a number (each choice of f gives a different value for the integral so he puts f into the subscript along with the surface S.
"All this makes it worthwhile to study the representation theory of the observables (1) and (2) in somewhat general terms."
He notes that representation theory of the algebra of HOLONOMIES is already well studied. In fact the cyclic representations are in 1 to 1 correspondence with measures
on the space A-bar. What is missing is to also include
the FLUXES in the algebra and then to characterize the representations of the larger algebra----if possible as before by
finding they are in 1-1 correspondence with measures.
The general theory of putting reps into correspondence with measures is an extremely efficient (powerful) math tool which I can describe. I should do this. It leads to uniqueness theorems
and a good control of the reps. Stuff you can actually calculate with!
But the section you asked about is where he begins to TINKER with the fluxes so that he can trick them into coming into the algebra and joining the holonomies in one big happy algebra.
I just got a telephone call and must do something else for an hour or so but will be back to this soon
BTW a cyclic representation is even more general than an irreducible one.
Any irreducible representation is cyclic
So this great GelfandNaimark theorem that makes the
cyclic reps correspond to measures on a certain maximumidealspace applies very generally
A cyclic rep of some algebraic "thing" (a normed ring, an algebra...) on a hilbertspace is one where there is at least one ONE point in the hilbertspace that goes everywhere under the action of the "thing"
say it is a ring R and the rep is denoted π:R --> GL(H) the operators on H
Now π(R) is a whole bunch of operators on H, the collective image of the whole ring
and cyclic just means there is at least one x in H that gets moved around so much by the bunch of operators that it forms a dense set of points in the hilbertspace
that is, the set of points π(R)[x] is dense in H
x is called a cyclic vector because it "cycles thru" the whole space as you apply successive ring elements to it
I just pulled a book by Naimark himself off a bookshelf and it fell open at a page stating the theorem we need and it isnt even very hard to prove. just one of those good ideas that somebody has and then get used so much the become classics
Naimark's book is called "Normed Rings", the first american edition 1964 with a brief introduction by Naimark himself written in Russian and on page 245 he states the theorem
"Every cyclic representation of a complete completely regular commutative ring R with identity is equivalent to its representation, defined by means of the formula
Ax ξ(m) = x(m)ξ(m)
in some space L2(ƒ) where ƒ is an integral in the ring C(M) of all continuous functions on the space M of maximal ideals of the ring R."
What he means by an integral defined on C(M) is what we call a MEASURE defined on M-----it is a means of integrating functions on M. So we normally call such a thing a measure μ and write the integral as ∫ dμ But it is just a semantic difference.
So his hilbertspace L2(ƒ) we would write as
L2(M, μ) the square-integrable functions defined on the maximalidealspace M using the measure μ on M.
And the representation action is simple MULTIPLICATION.
A point in the hilbertspace is just a function ξ(m) defined on M
and you want to know how a ring element x in R is going to act
on ξ to give another function on M, another point in the hilbertspace.
And he refers to the main theorem on page 230 which says that
a the ring R is isomorphic to the continuous functions on its maximal ideals. So there is a natural correspondence between x in the ring and x(m) a function on the maximal ideal space.
So this at-first-mysterious equation
Ax ξ(m) = x(m)ξ(m)
finally becomes clear. A point x in the ring corresponds to
a function x(m) defined on M
and the generic representation of the ring goes like this:
x acts on ξ in the hilbertspace simply by multiplying
the two functions x(m) and ξ(m) together to get yet another
function defined on M.
It looks to me as if what Naimark was calling normed rings in 1963 are essentially what are now called C* algebras or
*-algebras. And what he calls the maximalidealspace is also called the GelfandNaimark "spectrum" of the ring or algebra.
Terrible how alternative terminology builds up like treebark
wish we could slough it off once and for all and have the ideas
in clean pristine beauty but it never happens.
selfAdjoint
Aug3-03, 06:21 PM
Thank you for both posts.
I am still puzzled by the f's, they are co-vectors in the case of SU(2) but simple functions in the case of U(1). Is this because of the densities defined on the triads? That's the only place the group might come into the E's that I can see.
Your second post had a lot of stuff I didn't know, and will have to ponder. So this is the "trading reps for measures" you wrote of in your first post?
Originally posted by selfAdjoint
Thank you for both posts.
I am still puzzled by the f's, they are co-vectors in the case of SU(2) but simple functions in the case of U(1). Is this because of the densities defined on the triads? That's the only place the group might come into the E's that I can see.
Your second post had a lot of stuff I didn't know, and will have to ponder. So this is the "trading reps for measures" you wrote of in your first post?
Hi, I just edited an earlier post to respond to your question
about those X's.
I found page 8 of the Okolow-Landowski paper helpful. It goes over Sahlmann's definition of the X's and the exponentiated E's
in a more pedagogical way. I put a link to it in the earlier post.
I will try to give a reading of that page 8 material tomorrow.
selfAdjoint
Aug4-03, 08:59 AM
HI, I followed your advice and looked up the Okolow-Lewandaowski paper, and you're right. Where Sahlmann just skims the bases he needs for rigor, they go into details and try to motivate what's going on. In the spirit of trying to show you where my difficulties are I am going to make some quotes from the paper with my understanding of them. Bold type is the paper.
Tilde_E is a vector dessity of weight 1 which takes values in the space G'* dual to the Lie algebra G'.
A vector density is an integrand, when you change coordinates it changes like a vector but with an extra factor of the Jacobian of the change, raised to the power of the weight (here 1). The dual space of the Lie algebra is the set of homomorphisms from the Lie algebra to the coefficient module. In the case of SU(2) or U(1) this is the complex numbers. More light on this comes from this quote.
...the fields A and tilde_E are expressed by their components with rspect to a basis ([tau]i[/i]) i= 1,...,n of the Lie algebra G', and the dual basis ([tau]'[sub]i) of G'*.
The dual basis is the values the homomorphisms take on the Lie algera basis elements. Now for SU(2) there are three of those, but for U(1) only 1, so tilde_E has vector components for SU(2) but only a scalar component for U(1). The we saee the f's as shown by the following quote.
To define the flux of E fix (i) a finite d-1 submanifold S, (ii) an orientation in the normal bundle to S, and (iii) a function S -> G'.
So the value of f in the expressions will be a vector based on the ([tau]) basis og G'.
So far so good, but I am having trouble in seeing how all of this represents a flux.
Hi selfAdjoint, I've been reading more in the O-L paper this morning even before turning on the computer (!) and feel growing confidence that this is the right paper to study (plus dipping into the O-L references as necessary). I'm glad your judgement confirms this.
I found the passages in O-L which you quoted, from page 4 and 6, and made some typographical changes for legibility (my own benefit) and do very much appreciate your accurate paraphrasing. Restating something definitely helps and I will try to do some of that with O-L.
Originally posted by selfAdjoint
HI, I followed your advice and looked up the Okolow-Lewandaowski paper, and you're right. Where Sahlmann just skims the bases he needs for rigor, they go into details and try to motivate what's going on. In the spirit of trying to show you where my difficulties are I am going to make some quotes from the paper with my understanding of them. Bold type is the paper.
Tilde_E is a vector density of weight 1 which takes values in the space G'* dual to the Lie algebra G'.
A vector density is an integrand, when you change coordinates it changes like a vector but with an extra factor of the Jacobian of the change, raised to the power of the weight (here 1). The dual space of the Lie algebra is the set of homomorphisms from the Lie algebra to the coefficient module. In the case of SU(2) or U(1) this is the complex numbers. More light on this comes from this quote.
...the fields A and tilde_E are expressed by their components with respect to a basis ([tau]i) i= 1,...,n of the Lie algebra G', and the dual basis ([tau]'i) of G'*.
The dual basis is the values the homomorphisms take on the Lie algera basis elements. Now for SU(2) there are three of those, but for U(1) only 1, so tilde_E has vector components for SU(2) but only a scalar component for U(1). The we see the f's as shown by the following quote.
To define the flux of E fix (i) a finite d-1 submanifold S, (ii) an orientation in the normal bundle to S, and (iii) a function S -> G'.
So the value of f in the expressions will be a vector based on the ([tau]) basis of G'.
So far so good, but I am having trouble in seeing how all of this represents a flux.
selfAdjoint
Aug4-03, 10:43 AM
Hi, I did go to the Okolow-Lewandowski paper and as you say it is much clearer. In order to give you a picture of where I am now, and where you can help me, I will quote some text from O-L (bold) with my own interpretations (normal text).
Tilde_E is a vector density of weight 1 which has values in the space G'* dual to the Lie algebra G'.
...the fields A and tilde_E are expressed by their components with respect to a basis of the Lie algebra G', and the dual basis of G'*.
This makes sense for A, which is a sort of connection. A connection in a bundle takes values in the Lie algebra of the group of the bundle. I motivate this by thinking of the old Ricci way of developing the connection in GR by parallel transport. As you moved the vector it encountered fresh geometry which changed the definition of "parallel", so you go a covariant derivative and a Christoffel symbol or connection component.
In the bundle, G acts on the manifold and to represent that action in practice you go to the Lie algebra, which is the differential actions of the group from the identity. To get the action over a space you would integrate these differentials over the space, and in the range of integration you would encounter at each point different actions, so the differantial actions form a field on the manifold and the local values of the field are the images of the Lie algebra under the connection. Thus A.
Now tilde_E. At each point of the manifold you have the image of the Lie algebra, alias the local differentials of the group action. If the group is more complicated than U(1) these differentials will be a vector, with as many components as the Lie algebra has basis elemenets. For SU(2) the number is 3, for the 3 Pauli matrices which generate its Lie algebra. This is a vector of complex numbers. Now tilde_E is another vector of complex numbers, representing not differential group actions but linear functions of those actions. Which function would depend on the point of the manifold so this vector varies as a field too. I am tentitavely interpeting this as a "wave". Each differential group motion in the different basis directions is done with a certain "wave" represented as a complex number with magnitude and phase. Please correct me if you understand this differently.
To define the flux fix (i) a finite d-1 submanifold S..., (ii) an orientation in the normal bundle to S, and (iii) a function f: S -> G'.
So f is Lie Algebra valued and has vector components like A for SU(2) but only a scalar component for U(1). That solve that puzzle.
This is where I am today. I am also very interested in your teaser about trading representations for measures and getting theorems out of that.
WOAH! I did not see your 8:43 post while I was writing this. So this is not a reply to the immediately preceding, but here it is anyway:
------------
selfAdjoint, have been reading more in the O-L paper expecially with your question about the X's in mind and I think it may gradually be dawning on me what the X's are about
Note that on page 6 they begin section 2.2 "The Ashtekar-Corichi-Zapata Poisson Algebra" and they refer to a 1998 paper by ACZ and on page 8 they define the ACZ algebra of classical variables.
Then they say on page 9, at the beginning of section 2.3 "The Sahlmann holonomy-flux *-algebra", that the quantization simply amounts to assigning every elementary classical variable (in the ACZ Lie algebra) to some operator in the Sahlmann algebra.
I still need to study this to get a satisfactory grasp and am mentioning it here only because it shows their general direction.
They SEEM to be saying, in a not unkind way, "hey this guy Sahlmann didn't invent all this, he just redid the last step, a quantization step, in what seems to be a better way, and got a result. But the basic setup was already there in the 1998 ACZ paper." Verbally they give Sahlmann a lot of credit and well-deserved congratulations. But they also redo everything he did very carefully and link it clearly and explicitly to the earlier work.
I am going to fetch some quotes from that 1998 paper because they motivate the invention of the X's.
I think the secret to understanding them is to picture the configuration space A as a manifold. Only for a minute (because we wont have to actually work with it) think of the set of all connections (on the original 3D manifold) as a manifold itself. And then the Cylindrical functions are numerical-valued functions defined on
A and the question is ......what are VECTOR FIELDS defined on A ?
Well vector fields on any manifold are the "derivations"----they can be identified with the operation of taking directional derivatives of functions defined on the manifold.
And ACZ (that is, Ashtekar and co-workers) found out how to use the triads, together with any surface S and test-function or smearing function f, to actually take a kind of derivative of the cylinder functions defined on the connections.
That is what these XS,f things are----they are in a sense vector fields on A because they are "derivations" defined on Cyl---the cylindrical functions.
This is still a vague notion for me, so I will post this and go hunt some quotes from the ACZ paper that hopefully will clarify it.
selfAdjoint
Aug4-03, 12:11 PM
Originally posted by marcus
I think the secret to understanding them is to picture the configuration space A as a manifold. Only for a minute (because we wont have to actually work with it) think of the set of all connections (on the original 3D manifold) as a manifold itself. And then the Cylindrical functions are numerical-valued functions defined on
A and the question is ......what are VECTOR FIELDS defined on A ?
Well vector fields on any manifold are the "derivations"----they can be identified with the operation of taking directional derivatives of functions defined on the manifold.
And ACZ (that is, Ashtekar and co-workers) found out how to use the triads, together with any surface S and test-function or smearing function f, to actually take a kind of derivative of the cylinder functions defined on the connections.
That is what these XS,f things are----they are in a sense vector fields on A because they are "derivations" defined on Cyl---the cylindrical functions.
This is still a vague notion for me, so I will post this and go hunt some quotes from the ACZ paper that hopefully will clarify it.
This sounds right to me. I saw where they defined the X's as derivations, but was unsure as to how to interpret that.
Posting will be spotty during the day today, as I have a bunch of chores to do, but I should be able to concentrate on this tonight.
The 1998 ACZ paper
http://arxiv.org/gr-qc/9806041
is very much a working paper in the sense that it spend the first half (pages 1-10 out of a total 20 pages) exploring things that dont work and saying why they dont.
it is sending out signals to co-workers "dont try this one, we did and it doesnt work"
so this is not a pedagogical paper and we only need to dip into it at a few points to get what we need for the O-L paper
As usual there is a 3D manifold Σ but now very much center-stage is A the set of all SU(2) connections on that original Σ
And along with A comes Cyl, the ring of cylindrical functions defined on A.
A cylindrical function Cγ,c is defined using holonomy on a graph γ and a group-gobbling function c(g1,...,gN) where N is the number of gamma's edges. I'm using ACZ notation and they distinguish between little c : GN --> complex numbers and big C which is a complex valued function of the connection A obtained by running the connection on the graph to get an N-tuple of group elements which you then plug into c(g1,...,gN) to get a complex number C[A].
now on page 12 ACZ say:
"XS,f is a derivation on the ring of cylindrical functions that is
XS,f : Cyl ---> Cyl
such that the map is linear and satisfies the Leibnitz rule:
XS,f (C + λC') = XS,f C + λXS,f C [[linearity]]
XS,f (CC') = CXS,f C' + (XS,f C)C' [[product rule]]
for all cylindrical functions C and C' and all complex numbers λ"
And they give a definition of how XS,f works on a sample cylindrical function C------the definition is equation (3.3) on page 12 and it uses a more primitive X term defined earlier on page 8. I am going to discuss this definition (3.3) but first I will motivate it in vague terms.
The neat thing about cylindrical functions is that the guts of one is simply this little c(g1,.....,gN) function defined on N copies of the group G. It is completely loose from the original manifold and the graph and the connection and diffeomorphisms and all that complicated stuff. It is just a complexvalued function on GN
This incidentally was what made it so easy to define a measure and integrate these things because you say "forget the graph, I will just use Haar measure on GN!" It is amazing that something that simple works.
And in this case we can define a "derivation" of big C just by infinitesimally scootching so to speak in the k-th component of the N-tuple of group elements that we are going to feed little c. It is ridiculously simple-minded but apparently it works. You scootch in the k-th component if the k-th edge of the graph hits the surface.
Before we do anything at all we have first specified a surface S and a testfunction f. And we now look at a Cyl element C, which gives us a graph. And we ONLY MESS with little c at endpoints of edges that are actually in the surface (!)
Now there are only two cases to consider---either the beginning of the edge is in the surface or the endpoint is in the surface---well maybe four cases because the surface is oriented so we can say the edge is "upwards" out of the surface or downwards into it etc etc---it is moderately messy but not hideous.
And if the k-th edge of the graph has an endpoint in the surface we are going to tweak little c (.......gk....) in its k-th argument according as the edge of the graph is in or out relative to the surface. And the triads----the flux idea----gets in at this point.
Now this by any measure of sanity would seem totally loony but here is what Ashtekar et al say about it on page 13:
"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..."
That is what the X's are, they are vector fields (derivations) on the space of connections.
Now back to Okolow-Lewandowski on page 8 they say
"The Ashtekar-Corichi-Zapata algebra can be written as the direct sum
Gothic_A sub ACZ = Cyl + X,
where X are the derivations obtained by taking all the X operators given by all the submanifolds S and [test]functions ƒ and all the [linear combinations of] commutators."
Inching closer to understanding this. And Sahlmann somehow just put in a factor of i at one point and a minus sign somewhere and got lucky and for some reason hit the jackpot with a Stone vonNeumann theorem and various uniqueness results. Or so it seems, the story is not at all clear to me. I will post this and look at the O-L paper again.
selfAdjoint
Aug4-03, 07:08 PM
Here is where I am now. We are converging I think.
A cylindrical function is built out of integrating A (i.e. values of connections A, differential group actions) along an edge e, obtaining, ta-da, group actions, i.e. elements of G, and O&L show that this integration respects the group multiplication. Integrate along the concatenation of two edges and you get the product of the two group elements from the individual edges. And reverse the direction of integration and get the inverse of the group element.
So a cylindrical function is just a function of some n-ntuple of thes Ae intgrated group elements, or as they say, it maps into the n-fold product of G.
But the domain of a cylindrical function is A, the space of connections, crossed with Gamma the set of relevant edges.
And the X's are literally derivations on the space of cylinder functions CYL; in the O&L Poisson bracket you can see the X's defined by d/ds.
BTW, I didn't understand before that the Lie algebra valued function f was a test or smearing function. I'm going to have to think about that.
Originally posted by selfAdjoint
BTW, I didn't understand before that the Lie algebra valued function f was a test or smearing function. I'm going to have to think about that.
I am not sure how to think of the function f, although I have called it a testfunction several times, so I would be glad to
know your thinking about its purpose in the scheme of things
if you get some insight about that
another thing it seems to be there for is to boil triads down to numbers and letting it vary (and letting the ES,f and the XS,f depend on it seem more flexible relaxed than making some choice of basis once and for all (as conceivably you might have to? if you wanted to get rid of the f's?) Well I am not sure about this so for now I just accept it.
BTW I have gone back to having a look at Sahlmann's
"Some comments on the Representation Theory..."
http://arxiv.org/gr-qc/0207111
which as you say is condensed, just touching the bases
he needs for rigor, and I find that it is more understandable
and I have more confidence in what he says now, after
looking at O-L and AGZ papers. So things are, as you say,
converging
The definition of the representation "pi" is interesting
we already have it on the Cyl functions (just multiply an L2 function by a cylinder function) but we dont yet have it
on the "derivations" X, what should [pi](X) do to an L2
function? There is a nice construction for this im beginning
to understand. Great stuff!
Hi selfAdjoint,
I noticed an inconsequential difference between O-L
and Sahlmann's "Some comments...." that just effects
what you call things.
on page 4 of O-L they introduce two classical canonical conjugate fields
A and E-tilde
A has values in the Lie algebra G'
E-tilde has values in the dual of that G'*
so the "test function" ƒ, when it makes its appearance later,
is vector valued, not covector
(is that right?, these things are pure formalities I would think)
but Sahlmann has ƒ be covector
and instead of E-tilde he starts out with a plain E
so I guess for him A and E are a classical canonical conjugate
pair both having values in the Lie algebra G'
Mox Nix, but it's one reason math gets irritating at times
is this bird-like hopping from one notation to another
I'm going to look at the representation of the S. algebra today
and see how the derivation component acts on the hilbert space
selfAdjoint
Aug5-03, 09:25 AM
No, the direct and dual are not just notation, they are different things. I know you think worrying about duals and twisted duals in some theories is just for the math's sake, but the whole point of mathematical physics is that the quiddities of the math have physical consequences.
I thought I understood why tilde_E was dual and now I am going to have to go back to Sahlmann and rethink. What does come out of this is a tentative guess; Whatever E is, f is the opposite. Could this be the mathematical way of saying "smearing"?
Originally posted by selfAdjoint
No, the direct and dual are not just notation, they are different things. I know you think worrying about duals and twisted duals in some theories is just for the math's sake, but the whole point of mathematical physics is that the quiddities of the math have physical consequences.
I thought I understood why tilde_E was dual and now I am going to have to go back to Sahlmann and rethink. What does come out of this is a tentative guess; Whatever E is, f is the opposite. Could this be the mathematical way of saying "smearing"?
You are right of course, part of understanding is keeping track of whether something is in the dual or not. So I should say "minor" difference rather than "inconsequential". I think we both like O-L style of exposition---slower and more pedagogical---but I am inclined to favor Sahlmann's notation where they differ slightly!
I cannot write E-tilde, and Sahlmann uses a plain E and point out the the ƒ is a covector (what you say is right, they have to be opposite because they have to react together to give a number).
Another thing, merely about vagaries of notation: for the cylinder functions (which you have to write all the time being so basic) I like Sahlmann's choice of C and c
for purely typographical reasons (!)
In the standard PF font I cant see a lot of difference between
the upper and lowercase psi
do you ever envy the Russians who have three alphabets to choose notation from instead of just two? Four if you count Gothic.
Here I am reduced to the penury of C and c, which already mean the complex numbers and the speed of light
must stop complaining and get to the piece de resistance
selfAdjoint
Aug5-03, 11:20 AM
I liked your comment on the gothic_A. I suppose there is some involved HTML sequence to generate that, but I think our notation here is fine. I agree with you about C and c. We should settle on that for our posting here. I'll try to go over the relevant part of Sahlmann today and review my thinking. I may have been too reificational, if that's a word.
selfAdjoint
Aug5-03, 11:28 AM
Couldn't resist. I just came across this (http://www.sci.fi/~alboin/tengwartutorial.htm) ! We should use it for derivations? [6)]
selfAdjoint
Aug5-03, 09:05 PM
OK, after rereading Sahlmann's "When do Measures..." I see I was wrong about c functions. It is holonomies that eat a connection and an edge and spit out a group element. Then the c functions depend on those group elements, and for the edges in a graph, the c function depends on them separately so it's a function on Gn as you and Sahlmann both said. So my reification still works for the A variables.
But did you notice that in the integral for ES,f (Sahlmann's display (2)) that E enters the integrand as *E ? So that's a dual, at least according to the notations I learned. And f isn't multplied by *E, rather it is evaluated at *E. So let's see. f takes vectors into the reals. E is a nontrivially densitized triad. So *E would take that triad into the reals, producing 3 real numbers. If the triad has beins 1,2, and 3, associated with densities r1, r2, and r3, and *E takes the beins into reals a1, a2, and a3, then the result of *E I would think would be the triple (a1r1,a2r2,a3r3) and of course f as a covector could map that into some real number. Then you integrate those numbers over S and get ES,f.
Also when he writes (p.5) "Now to each graph one can define a certain equivalence class of connections" does he mean two connections are equvalent if their holonomies on the edges of the graph are equal? I think that would produce the behaviors he cites.
Sorry I haven't got any further after your excellent explanations. I plead craziness in real life.
Originally posted by selfAdjoint
But did you notice that in the integral for ES,f (Sahlmann's display (2)) that E enters the integrand as *E ? So that's a dual, at least according to the notations I learned.
I know the place you mean. That asterisk, coming before the symbol and about belt height, may indicate the "Hodge star"
operation by which a 1-form can be converted to a 2-form (in a 3D context). Differential geometers are a shifty lot and you have to watch them like a hawk.
Eric Weisstein would say something about "Hodge star".
But that is only one equation anyway and that particular asterisk I dont remember appearing in the later, more detailed, exposition, where there are proper respectable subscripts and superscripts that give a firmer idea of whats going on. Or lets hope.
I will let you know if it dawns on me what that asterisk before the E is doing
Originally posted by selfAdjoint
Also when he writes (p.5) "Now to each graph one can define a certain equivalence class of connections" does he mean two connections are equvalent if their holonomies on the edges of the graph are equal? I think that would produce the behaviors he cites.
I think your reading is absolutely right. He wants to take a projective limit of measures μγ
defined on subspaces Aγ.
You and I have been over this ground already I think. He is just
reviewing what A-L did to get their measure on A
The key thing is that the graphs form a directed set. Given any two graphs there is always a larger one that includes both.
Using the graphs as an index set for spaces and measures sets
the scene for passing to a projective limit.
selfAdjoint
Aug5-03, 10:38 PM
Originally posted by marcus
I know the place you mean. That asterisk, coming before the symbol and about belt height, may indicate the "Hodge star"
operation by which a 1-form can be converted to a 2-form (in a 3D context). Differential geometers are a shifty lot and you have to watch them like a hawk.
Eric Weisstein would say something about "Hodge star".
But that is only one equation anyway and that particular asterisk I dont remember appearing in the later, more detailed, exposition, where there are proper respectable subscripts and superscripts that give a firmer idea of whats going on. Or lets hope.
I will let you know if it dawns on me what that asterisk before the E is doing
Looked up Hodge star. Found this (for 3-space). *(u/\v/\w) = u.v x w. But E is a triad, and f is really fi with 3 components. I am having trouble parsing this.
Originally posted by selfAdjoint
Looked up Hodge star. Found this (for 3-space). *(u/\v/\w) = u.v x w. But E is a triad, and f is really fi with 3 components. I am having trouble parsing this.
for me the trouble stems from minor differences in conventions between papers
the two representative papers I'm comparing are
O-L (gr-qc/0302059) and S-T(gr-qc/0303074)
Okolow-Lewandowski use careful consistent conventions that I think I can understand and there is nothing looking like a Hodge star----nothing with an asterisk coming before the E.
On the other hand Sahlmann-Thiemann have a nice result and there are seeming advantages to their approach to proving it. But I too have trouble parsing some of their notation.
There is this probably trivial difference in conventions that comes to the same result either way----the E and the smearing function ƒ have to be of opposite type and O-L happens to
make ƒ G'-valued and E-tilde G'* valued
while S-T happens to make ƒ covector (G'* valued) and doesnt have a tilde on the E, and E is G' valued.
I am leaning toward adopting O-L because of the clarity.
For example look at their page 4. It says EXACTLY what A and E-tilde are with no ambiguity at all.
A = a certain tensor product, E = a certain other tensor product.
I am going to have to teach myself to read Sahlmann papers and translate the conventions as I go along, so I can paraphrase them in O-L notation. There is BARELY any difference, but just enough to be bothersome now and then. One must simply pick one or the other and right now it seems to me that Lewandowski is a slightly safer choice.
selfAdjoint
Aug6-03, 10:25 AM
Comparing Salhman's definition of ES,f with O-L, i find they are within a scoche of the same. With a little more thought maybe they are the same.
Look at O-L page 6. We have 1/(n-1)! times an integral over S of the components of tilde_E in its dual G' basis and local chart basis times local G'-valued components of f times epsilon function that are +1 if the subscript is an even permutation, -1 if an odd permutation, and 0 otherwise times the differential (d-1)-form on S.
The bolded items are the definition of the Hodge star of tilde_E!
Originally posted by selfAdjoint
Comparing Salhman's definition of ES,f with O-L, i find they are within a scoche of the same. With a little more thought maybe they are the same.
Look at O-L page 6. We have 1/(n-1)! times an integral over S of the components of tilde_E in its dual G' basis and local chart basis times local G'-valued components of f times epsilon function that are +1 if the subscript is an even permutation, -1 if an odd permutation, and 0 otherwise times the differential (d-1)-form on S.
The bolded items are the definition of the Hodge star of tilde_E!
Bingo! That is a nice observation.
I may have come across something that explicates another line of O-L
On page 4 they give a "more formal" description of the gauge transformation associated with a map a:Σ --> G
which involves two technical things: (1) the adjoint action of G on its Lie algebra G' and (2) the Maurer-Cartan G'-valued 1- form Θ on G. I just happened across a "spr" post by John Baez,
dated sometime June 2003:
-------quote from sci.physics.research-------
>More and more often I encounter the so-called maurer-cartan forms and
>their structural equations. In my attempt to look for a basic
>explanation of what these objects are and where they are needed for in
>high-energy physics (why do we want to use them?) I had little success.
>Does anybody know an easy explanation? (Some terminology about Lie
>algebras may be used ...)
The Maurer-Cartan form is a 1-form on a Lie group taking
values in its Lie algebra. In other words, it's a beast
that eats a tangent vector anywhere on the Lie group and
spits out an element of the Lie algebra, in a linear way.
How do we define this beast? At the identity element of
our Lie group, it's easy. The tangent space at the
identity element of a Lie group *is* its Lie algebra,
by definition. So, the Maurer-Cartan form just eats
a tangent vector at the identity and spits out the
very same thing - but calling it a Lie algebra element!
It's also not hard to define the Maurer-Cartan form
at any other point of a Lie group. The reason is that
we can map any point of our Lie group to the identity
by left multiplication by a suitable element. This
in turn gives a way to map tangent vectors at any point
to tangent vectors at the identity. So, to get the
Maurer-Cartan form we just do that and then say
"Hey, but now it's a Lie algebra element"!
In short, the Maurer-Cartan form is a completely
tautologous thing. For that reason it must either
be completely boring or very, very interesting.
In fact it's very, very interesting, mainly because
its exterior derivative is not zero. There's a very
pretty formula for its exterior derivative, calledthe Maurer-Cartan formula.
(Nota bene: I'm thinking of the Maurer-Cartan form as
a single 1-form taking values in the Lie algebra. People
fond of indices will instead pick a basis for the
Lie algebra and get a list of "Maurer-Cartan forms",
which are ordinary 1-forms, one for each basis vector.
It's just a slightly less elegant way of doing the same
thing, so don't sweat it.)
---------------------
selfAdjoint
Aug6-03, 12:06 PM
Wow! define trivially at the identity, and define anywhere else by multplying by the inverse - which gets you to the identity - and then doing trivial again.
Only a genius would think up something so vacuous and so useful!
Here it is in Nakahara:
We define a Lie-algebra-valued one-form [the]:TgG -> TeG by
[the]: X -> (Lg-1)*X = (Lg)*-1X
Originally posted by selfAdjoint
Wow! define trivially at the identity, and define anywhere else by multplying by the inverse - which gets you to the identity - and then doing trivial again.
Only a genius would think up something so vacuous and so useful!
Here it is in Nakahara:
We define a Lie-algebra-valued one-form [the]:TgG -> TeG by
[the]: X -> (Lg-1)*X = (Lg)*-1X
O-L refer to the Maurer-Cartan form on page 4 (undoubtably you've seen the passage) where they give a "more formal"
description of a gauge transformation using a map a:Σ-->G
instead of (A, tildE) --> (a-1Aa + a-1da, a-1tildE a)
they say (ad(a-1)A + a*θMC, ...)
where since a:Σ-->G is just a map of manifolds I guess the notation a* means the lift of a to a map of tangent spaces
so that unless I am mistaken
a*: T(Σ) --> T(G)
so that it could indeed pull θ back to a form on Σ
I suspect youve been over that part already and one of youir books like Nakahara probably as a parallel development of gauge transformations! I like the expression "parsing" for this. Very necessary activity.
PS (added by edit): have been looking through Rovelli's new book "Quantum Gravity" and finding it interesting
selfAdjoint
Aug7-03, 10:25 AM
Hi marcus, I'v been through this passage again with your analysis in mind and I do agree with your representation of their syntax. I have only one question. I notice they don't spend any explanation on the tensor product of tilde_E and partial wrt x, but as the images of both of these are just numbers maybe they don't need to?
Originally posted by selfAdjoint
Hi marcus, I'v been through this passage again with your analysis in mind and I do agree with your representation of their syntax. I have only one question. I notice they don't spend any explanation on the tensor product of tilde_E and partial wrt x, but as the images of both of these are just numbers maybe they don't need to?
we are mulling over stuff on page 4 of O-L (in case any one wanders in and asks)
and there is this IMO quite interesting equation halfway down
the page which I need a tensorproduct symbol (x) to write--this old browser is not very font-smart.
tildE = tildEiα τ*i (x) ∂α.
this is what you were asking about? or this applied to something else?
in case anyone asks (ICAA?) the taus are a basis of G', the Lie algebra which selfAdjoint you surely know by heart but I always have to look up or figure out each time, and the ∂'s are just a basis of the tangent space derived from whatever coord chart is being used when it comes down to numbers.
and the tau*----- in this default font it is τ*----are the basis of G'*, the dual of the Lie algebra.
I still have to check over that list of notation in the (08-05) post in the "Fresh LQG start" thread, to make sure havent absentmindedly dropped a stitch. Here is what I said:
-------
A, a connection (G' valued 1-form) in the tensor product G'(x)T*
tildE, "electric field" (G'*valued density) in G'*(x)T
-------
So A, the space of connections, consists of maps
A:σ --> G'(x)T*σ
or sections of a bundle G'(x)T*(Σ)
we both know what I'm trying to say, if you think of an efficient way to put it, tell me
selfAdjoint
Aug7-03, 11:22 AM
Hi again,
in case anyone asks (ICAA?) the taus are a basis of G', the Lie algebra which selfAdjoint you surely know by heart but I always have to look up or figure out each time, and the 's are just a basis of the tangent space derived from whatever coord chart is being used when it comes down to numbers.
The taus are part of the dual basis of G' and the dels are dual to the tangent space. Both of them spit out things that map into the base field - previously specified to be R in this case.
But now take that map a:&Sigma -> G. How does it act on the tensor product to give their result a-1 tilde_E a?
BTW, do you know how to make the greek letter dingus work? I have tried every combination of & and [] and nothing works.
Originally posted by selfAdjoint
Hi again,
The taus are part of the dual basis of G' and the dels are dual to the tangent space. Both of them spit out things that map into the base field - previously specified to be R in this case.
But now take that map a:&Sigma -> G. How does it act on the tensor product to give their result a-1 tilde_E a?
BTW, do you know how to make the greek letter dingus work? I have tried every combination of & and [] and nothing works.
You need a semicolon after spelling out the greek letter
so without the spaces it would be
& Sigma ;
what you type is
&whatever;
I was wondering the same thing as you! about
the action of the a on the tilde_E
since adjoint action works on G'
maybe it works one flight up on G'* by
stirring the argument before you feed it to the linear functional
selfAdjoint
Aug7-03, 02:13 PM
Let's follow the authors. tilde_E takes values in the dual space of the Lie algebra. Now the dual space is the set of linear maps from the vector space G' to its base field, I suppose C. This set of functions is itself a vector space, and its basis elements are the τ*i. So tilde_Eaiτ*i is a vector in this dual vector space, that is, it is one particular linear function G' -> C.
Now a: Σ -> G induces through the adjoint: G -> G' a linear vector space function a:Σ -> G', and this induces a pullback in the reverse direction on the dual spaces. a*: G'* -> Σ*. This will map tilde_E to a linear function from Σ to R, aka a one-form on Σ. So the only "E-thing" that a can consistently act on is this one_form on Σ. And a will take it back to some element of G where using the MC form it becomes an element a-1tilde_E a in G'.
Unless of course I have dropped the ball somewhere.
selfAdjoint
Aug9-03, 01:39 PM
I am going to continue studying the O-L paper, using Sahlman only for a reference. O-L includes Sahlman's result and generalizes it, and I think this is the big breakthrough in establishing a purely algebraic way to study LQG>
Originally posted by selfAdjoint
I am going to continue studying the O-L paper, using Sahlman only for a reference. O-L includes Sahlman's result and generalizes it, and I think this is the big breakthrough in establishing a purely algebraic way to study LQG>
I agree and shall continue with the O-L paper likewise.
They do things the nice way----elegant, efficient...
I am interested in Rovelli's book not for the development
of LQG which he has in the second half (p 160 ff) but
for a perspective he gives on classical (non-quantum) general
relativity. I will try not to let my reading in Rovelli interfere
with continuing to make progress with O-L.
I need to study the decomposition of the hilbert space into
cyclic subspaces.
The hard part (which you need the decomposition for) is defining the action of the X's (the derivations or vectorfields).
The action of the cylinderfunctions is just ordinary multiplication of one function by another so there is nothing especially novel about it. But the action of the X's needs some study. Hope to get to it today. My wife just returned from a trip---I have been batching and have let the house become slightly disorderly.
selfAdjoint,
there is a nice easy parsing job to be done on page 18---just interpreting how a diffeomorphism φ:Σ --> Σ
acts on cylinder functions! and on the vectorfields X.
Remember that cylinder functions C:A --> Complexneumbers
(I'm making an exception and writing Salmann's notation C instead of the uppercase Psi that O-L use)
So cylinder finctions are not even defined on the basic 3D manifold! How (naive question) are they to be acted on by a diffeomorphism? You could work this out just by following the customs, without being told, but O-L says how anyway.
C is constructed from an edge set {e1,...,eN} and an N-fold group-eater c. So we just let the diffeo act on the edges!!!
The new edge-set is {φ(e1),...,φ(eN)}
the new cylinderfunction φC is what you get using that, and the same N-fold group-eater c as before.
O-L writes it out:
(φC)[A] = c[A(φ(e1)),...,A(φ(eN))]
What is left to do? The Sahlmann algebra is made of two things, cylinder functions and XS,ƒ operators each of which is constructed using a 2D surface S and a testfunction ƒ:S --> G'
The X does some differential fiddling with an edge where it punctures the surface. So if we are going to move the edge set (using the diffeomorphism) we have to move the whole kit and kaboodle along with it----the surface S --> φS
HERE I BELIEVE THEY HAVE A TYPO see what you think. You see what they have on page 18 defining the new testfunction ƒ-tilde
What they should have, I think (correct me if I'm mistaken) is
ƒ-tilde: φS --> G' defined by ƒ(φ-1).
It is just a composition of the two functions---you are on φS and you first go by φ-1 which takes you to S and then you go by ƒ and get to G'.
Anyway instead of writing S-tilde and ƒ-tilde the way they do I will just write (unless you find an error) φS and ƒ(φ-1)
Then φ applied to E(S,ƒ) = E(φS, ƒ(φ-1)
Now to finish saying how φ acts on the Sahlmann we have to say what φX is for any of those derivations X. That means take a cylinder function C and say what the new derivation φX does to it. Well move C over to φC and work on it:
(φX) C =
(φX) C = XφS, ƒ(φ-1) φC
I had better post this before a computer glitch loses it.
Now we know how a diffeo φ acts on a cylinder function C
to give a new cylinder function φC
The cylinder functions act on themselves by multiplication. For any C we have
MultC Cyloo --> Cyloo , which just sends C' into CC'.
What equation 4.9, the first part, says is that
φ MultC φ-1 = MultφC
Let's check that by applying both sides to a cylinder function C'.
Imagine that the diffeo φ maps you from Ohio to Kentucky and that C' is defined using a graph somewhere in Kentucky. The left hands side is
φ (MultC (φ-1 C'))
which means "take C' back to Ohio and multiply it by C and ship it out to Kentucky again"
The right hand side is MultφC C',
which means "ship C out to Kentucky and multiply C' by it on site".
So both have the same effect. That is how diffeomorphism covariance is supposed to be.
The other part of equation 4.9 is about derivations X
and says if you take a cylinder function C and bring it back to Ohio and do the (S,ƒ) derivation to it and then return it to Kentucky this has the same effect as taking the whole operation over to Kentucky and doing the derivation on site using (φS, ƒ(φ-1)
So recall that the Sahlmann algebra SAHL is an algebra of linear operators Cyloo --> Cyloo that is generated by the Mults and the X's. What we now have is a DIFFEOMORPHISM ACTION on SAHL which we have looked at working on the generators in a bit of detail.
φ started out as Σ --> Σ and we jacked it up to a linear map φ: Cyloo --> Cyloo
And now we can conjugate elements a of SAHL by it
φ a φ-1 is going to be a new element of SAHL which means it is a new linear map on Cyloo which we have looked at in the cases of the two types of generators.
Now. what does it mean for a REPRESENTATION of the Sahlmann algebra to be covariant?
It is going to turn out that the representation based on the Ashtekar-Lewandowski measure is covariant and also conversely that any representation of SAHL if it meets certain criteria including diffeo-covariance will be equivalent to one based on that measure. So we should get an idea of what a covariant rep of the algebra is.
selfAdjoint
Aug10-03, 09:42 AM
Originally posted by marcus
selfAdjoint,
there is a nice easy parsing job to be done on page 18---just interpreting how a diffeomorphism φ:Σ --> Σ
acts on cylinder functions! and on the vectorfields X.
....
O-L writes it out:
(φC)[A] = c[A(φ(e1)),...,A(φ(eN))]
What is left to do? The Sahlmann algebra is made of two things, cylinder functions and XS,ƒ operators each of which is constructed using a 2D surface S and a testfunction ƒ:S --> G'
The X does some differential fiddling with an edge where it punctures the surface. So if we are going to move the edge set (using the diffeomorphism) we have to move the whole kit and kaboodle along with it----the surface S --> φS
HERE I BELIEVE THEY HAVE A TYPO see what you think. You see what they have on page 18 defining the new testfunction ƒ-tilde
What they should have, I think (correct me if I'm mistaken) is
ƒ-tilde: φS --> G' defined by ƒ(φ-1).
It is just a composition of the two functions---you are on φS and you first go by φ-1 which takes you to S and then you go by ƒ and get to G'.
Anyway instead of writing S-tilde and ƒ-tilde the way they do I will just write (unless you find an error) φS and ƒ(φ-1)
Then φ applied to E(S,ƒ) = E(φS, ƒ(φ-1)
Now to finish saying how φ acts on the Sahlmann we have to say what φX is for any of those derivations X. That means take a cylinder function C and say what the new derivation φX does to it. Well move C over to φC and work on it:
(φX) C =
(φX) C = XφS, ƒ(φ-1) φC
I had better post this before a computer glitch loses it.
I went around and around on this, and finally came down on your side. Your explanation is so clear, natural, and simple, and I couldn't make any dual space pullback work (that's what their notation would suggest). I am still troubled because it isn't what you would expect from a typo, and I can only suppose that in an early draft of this paper they were working from a dual perspective, a la Sahlmann, and then decided to convert to direct vector spaces, and this was overlooked in the conversion.
Originally posted by selfAdjoint
I went around and around on this, and finally came down on your side. Your explanation is so clear, natural, and simple, and I couldn't make any dual space pullback work (that's what their notation would suggest). I am still troubled because it isn't what you would expect from a typo, and I can only suppose that in an early draft of this paper they were working from a dual perspective, a la Sahlmann, and then decided to convert to direct vector spaces, and this was overlooked in the conversion.
I took another look at (the third line of) equation (4.7) on page 18
it looks to me as if what was meant was ƒ o φ-1, that is, ƒ( φ-1)----the only thing that is defined on the right domain---but what got written down was
(φ-1|S)* ƒ
BTW I believe that might parse better if one simply added a tilde to the S----tilde S was their notation for φ(S) the image of the surface as mapped by the diffeomorphism.
then one would have
(φ-1|φ(S))*
and φ-1 would be restricted to the new
surface φ(S). That seems now to make good sense and
I'm surprised I didnt think of it earlier.
It would have the same effect if one took a couple of the symbols in a different order, first restricting φ to the domain S and then taking inverse map
((φ|S)-1)*
Then one could appeal to a possible meaning of the much over-used * and say that with any diffeomorphism ζ the operation
ζ* applied to a function ƒ MEANS to take ƒ(ζ).
In fact that is just how they used the (much overused!) asterisk
five lines further down the page in a slightly different context where they say "...diffeomorphisms act naturally on connections considered as functions defined on edges by
(φ* A) (e) = (A o φ) (e) = A(φ(e))..."
or words to that effect
it is one or two lines down from equation (4.8)
I feel all right about that. If you feel confident of the interpretation then it is just an omitted tilde, or a couple of transposed symbols, and we dont have to make an issue with Okolow.[;)]
My esteem for O-L continues unabated, while my enthusiasm for rovelli's draft textbook is no longer quite up to what it was earlier
selfAdjoint
Aug15-03, 03:08 PM
Let me push on with O-L. I have been totally distracted by home events the last few days, but I had printed off your comments and was studying the paper with their aid. I have to resume that.
After making the old college try on Rovelli, I have concluded that I can postpone it. If I need more depth on the Ashtekar variables I can go to Ashtekar himself, he discusses them in several papers. But I think these current papers are giving us a better school than a textbook right now.
Originally posted by selfAdjoint
...I think these current papers are giving us a better school than a textbook right now.
I agree!
Hello again S.A.,
just to say I'm still around and reading in that group
of recent preprints. Despite the fact that I much prefer
Lewandowski's style (or more exactly that of O-L)
They never get to the point of exponentiating the
X's.
Today I was looking in arxiv for a followup paper---they reference an "in progress" four-way collaboration O+L+S+T but that has not appeared, as far as I could tell.
In a couple of the Sahlmann-Thiemann papers they exponentiate
the X operators and get an analog of the Weyl algebra. If I get
too impatient for the next Lewandowski paper I will have
another look at one or the other of the S-T papers. I am wondering if you looked at either and got much of an impression.
I find Thiemann's writing style a bit hard to take for some reason.
Was also looking today at some papers by Daniele Oriti (cambr.) and Etera Livine (marseilles)---both postdoctorate fellows. dont know if they are known to you or not
selfAdjoint
Aug19-03, 05:10 PM
I've spent half a day meditating on this sentence.
"..tilde_E is a vector density of weight 1 which takes values in the space G*' dual to the Lie Algebra G'."
I know what all the words mean, but I can't wrap my head around the concept of a vector density which takes values in a dual L.A.
The coordinate representation, which we went over earlier is not much clearer.
tilde_E = tilde_Eaiτ*i(X)∂a
Thus the space in which tilde_E is defined is spanned by the n x d tensor products τ*i(X)∂a.
The τ*i, as he says span the dual L.A. and the ∂a span the covectors in Σ.
But what image does this call up in your mind?
Originally posted by selfAdjoint
I've spent half a day meditating on this sentence.
"..tilde_E is a vector density of weight 1 which takes values in the space G*' dual to the Lie Algebra G'."
I know what all the words mean, but I can't wrap my head around the concept of a vector density which takes values in a dual L.A.
The coordinate representation, which we went over earlier is not much clearer.
tilde_E = tilde_Eaiτ*i(X)∂a
Thus the space in which tilde_E is defined is spanned by the n x d tensor products τ*i(X)∂a.
The τ*i, as he says span the dual L.A. and the ∂a span the covectors in Σ.
But what image does this call up in your mind?
This will be wordy and not, I fear, well directed to you as reader, selfAdjoint. I will probably say things you've already figured out or read elsewhere.
the "ashekar new variables" are a pair (A,E)
what does the E represent intuitively and geometrically?
in the literature the E is called "electric field" and also
"inverse densitized triad"
the E is approached from 2 separate directions
approach #1, assume a classical situation with a metric and say with a foliation into constant time 3D manifolds---or, it does not matter how, we somehow have a spatial (3D) manifold with the metric restricted to it.
the inverse triad is like the square root of the metric
here's a typical equation, g is the determinant of the metric:
g gab = tildEaitildEbi
a,b are spatial indices and i,j are "internal" referring to a basis chosen for the Lie algebra.
approach #2, Eai is introduced as a (distributional) functional derivative with respect to Aia which must be integrated against test functions ("suitably smeared") in order to be well-defined
Eai(x) = - i hbar G δ/(δAia(x))
"The functional derivative with respect to the connection 1-form A(x) is a vector density of weight one, or equivalently, a 2-form. Contracting the vector density Ea with the Levi-Civita density gives the dual of the triad, which is a 2-form
E = εabc Eadxb dxc.
Hence they may be identified. Since 2-forms are naturally integrated against 2-surfaces..."
Im quoting page 20 of a 1999 paper by Gaul/Rovelli but Ive seen the same thing over and over. I will post this without completing the idea, which goes something like this:
the question is mental pictures surrounding the E's
the essence of the gravitational field can be seen in various things----the metric, and the E's are the squareroot of the metric---and in the "locally inertial frame" of which the triad is a kind of 3D proxy----and in the connection, with respect to which the E turns out to be a functional derivative.
and because of nature is such a tease we dont get to have the E's until they are smeared out as FLUXES thru sample 2D surfaces. We have to use 2D surfaces to catch the E-flux, which is a pain in the neck. But it is the only way we get to know them---by integrating them
a tensor density is just something that tranforms with an extra factor of the jacobian when you change coordinates (as you know well) and a vector density of "weight one" takes the jacobian to the power one---so it is really begging to be made into a 2-form
now some people just identify things that are the same information in a slapdash fashion but O-L dot their eyes and cross their tees, our job is to figure out why they have the values be in the Lie algebra dual!!!
I am ashamed to say so far my post is mere preamble!
When a basis is chosen, as it has been with the L.A., then a vectorspace and its dual look virtually the same to most people. People like Rovelli---practical types---identify them without even noticing that they are doing so. Others may say "this is not really the same but by *abuse of notation* we will use the same symbol...". And still others will painstakingly use distinct symbols.
I must stop typing and have a look
Gradually by observing the obvious it gets clearer.
Relativists are accustomed to having a metric, so they can raise and lower indices with complete innocence whenever they feel the urge.
But if you have a frame of vectors in V the functions which correspond to expressing something IN THAT BASIS live in V*
and the bigger the vees are the smaller, in some sense, are the vee-star functions that tell you the coefficients to use in expressing something in that basis.
So when they talk about the "inverse triad" it has to be in the dual of whatever the triad is in.
Also since I get to explain some stuff that a random passerby might wonder about (even tho you know it), one thing
the tensor product (X) does is give you a way to say "with values in".
If you have some real-valued linear functions W and you want to make them have values in the plane R2 then you say
R2 (X) W-----which is a new vectorspace consisting of the same linear functions except now they have values in the plane. So some of what we were writing before,
quoting from O-L, is for instance just a compact way of writing A explicitly as a 1-form with values in the Lie algebra.
A = Aiaτi(X)dxa
Then there is the bundlebusiness, which turns out to be easier than it seemed at first. It only enters early on in some optional streamlining of definitions
You can always construct a manifold as a principal bundle with some group and then make various associated bundles where the fiber can be any homogeneous space of the group---anything the group acts vigorously enough on. This means that arranging for the LIE ALGEBRA to be the fiber is a natural. Because the group acts on it by adjoint!
there is an irrestistible mathematical urge to do this
once you have a manifold, think of a good structure group for it, and make the (P,G,M) the principal bundle----a frames bundle will do----and once you have that you can hardly refuse to call up the "associated" bundle which has G' as fiber
so as novices we may well walk in and be amazed to see all these G'--valued functions and G'* valued functions and no one will think to explain. Or, if you ask, they will just say oh that is just how it is, we have all these Lie algebra valued things. It must be connections---infinitesimal rotations as you take an infinitesimal step in some direction---oh yes! it must be that, or some other physical reason. But they dont explain that to a large extent it is the compelling naturalness of the construction
I've been looking at papers by the current postdocs, often they will be by two or more postdocs, or several plus a senior person.
People age so fast in mathematical physics research already
Ashtekar, Rovelli, even Baez?, Lewandowski, Thiemann are old
and they have to bring up a new generation.
Daniele Oriti (a guy, Italians spell Daniel that way)
Hanno Sahlmann
Martin Bojowald
Etera Richard Livine (dont know if he or she)
Robert Oeckl
Florian Conrady (new name, Heidelberg and Berlin)
Andrzej Okolow (only the one paper that i know of)
Sergei Alexandrov
Funny thing, when you see an online photo of someone like
Rovelli he looks really young. Baez too.
I do not think the list is complete or representative but I notice some things: like there seem to be a lot of people from
Cambridge, Potsdam (MPI and U. Berlin), Marseilles, Lyon, Warsaw. People from Madrid and Mexico City also come up, though as it happens I havent listed them.
May be a thought here, maybe not. The fact that Rovelli seems to have moved back from Pittsburgh etc to Marseilles. Maybe the weather is nicer.
In a previous post you asked what mental image comes to mind regarding, I think, the "electric field" or "inverse densitized triad" E.
It is a good kind of question to consider. As you say "how to get one's mind around" the Ashtekar new variables (A, E)
It is interesting that A gets integrated along curves and E gets integrated over 2D surfaces so that what one ends up dealing with are not (A,E) but (holonomies, fluxes).
Notations like he[A] for integrate the connection A along the edge e.
And like ES, ƒ[E] for flux of E thru surface S sampled by function ƒ.
So in the end it is cylinder functions (holonomies) and derivations (made using fluxes)
This generates an urge to turn around and go back to the roots.
What is the physical meaning of A,E?
Connections are pretty conceptual and graphic---not much trouble visualizing a tangent vector veering as it is moved about.
But how to imagine E?
What kind of feel do you get about the manifold from knowing E, what kind of grip on it do you get.
the fact that E is essentially an inverse triad seems important.
something that will take a vector and write it out for you in a special basis or frame
and that it remembers the metric
well the events of the day are tugging, must go
from Ashtekar's not-too-technical 2002 paper "Q. Geom. in Action"
http://arxiv.org/math-ph/0202008
page 5
"Let me now turn to specifics. It is perhaps simplest to begin with a Hamiltonian or symplectic description of general relativity. The phase space is the cotangent bundle. The configuration variable is a connection, A on a fixed 3-manifold Σ representing 'space' and (as in gauge theories) the momenta are the 'electric field' 2-forms E, both of which take values in the Lie-algebra of SU(2)."
Notice he has them be 2-forms valued in G' from the start (others might begin with vector densities valued in G'* and then take the dual and get 2-forms to integrate over surfaces, but he is just a little more direct) Then he refers to the "orthonormal triad" interpretation, to give a little geometric intuition about them.
"In the present gravitational context, the momenta acquire a geometrical significance: their Hodge-duals *E can be naturally interpreted as orthonormal triads (with density weight 1) and determine the dynamical, Riemannian geometry of Σ. Thus, (in contrast to Wheeler's geometrodynamics)
the Riemannian structures on Σ are now built from momentum variables.
The basic kinematic objects are holonomies of A, which dictate how spinors are parallel transported along curves, and the 2-forms E, which determine the Riemannian metric of Σ.
(Matter couplings to gravity have also been studied extensively [2, 1].)..."
Continuing the quote from Ashtekar on page 5 of the same paper, there is another bit of intution about E.
"...In the quantum theory, the fundamental excitations of geometry are most conveniently expressed in terms of holonomies [3, 4]. They are thus one-dimensional, polymer-like and, in analogy with gauge theories, can be thought of as 'flux lines of the electric field'. More precisely, they turn out to be flux lines of areas: an elementary flux line deposits a quantum of area on any 2-surface S it intersects. Thus, if quantum geometry were to be excited along just a few flux lines, most surfaces would have zero area and the quantum state would not at all resemble a classical geometry.
Semi-classical geometries can result only if a huge number of these elementary excitations are superposed in suitably dense configurations [13, 14]. The state of quantum geometry around you, for example, must have so many elementary excitations that 1068 of them intersect the sheet of paper you are reading, to endow it an area of 100 cm2.
Even in such states, the geometry is still distributional, concentrated on the underlying elementary flux lines; but if suitably coarse-grained, it can be approximated by a smooth metric. Thus, the continuum picture is only an approximation that arises from coarse graining of semi-classical states..."
selfAdjoint
Aug20-03, 04:34 PM
Thanks Marcus, for all these explanations. I am going to carefully go over the Ashtekar 2002 paper you linked to. 2-forms with values in G' at first sound more reasonable than vector densities with values in the dual space of G', but in fact - what is a 2-form? It's something times dx/\dy (being cheesy about the coordinates). Is he saying that for every such dx/\dy we have an associated map into G' (linear? preserving product?)
Or consider that tensor product from O-L, τ* {X} ∂ (suppressing indexes). Can we look on that as a (dual) 2-form?
BTW I think the relation of a vector density to a 2-form is via that completely antisymmetric Levi-Civitta density. A vector density of weight one changes sign on reflection (because the determinant of the Jacobian of a reflection is -1).
Originally posted by selfAdjoint
BTW I think the relation of a vector density to a 2-form is via that completely antisymmetric Levi-Civitta density. A vector density of weight one changes sign on reflection (because the determinant of the Jacobian of a reflection is -1).
I agree, Levi-Civita usually written with epsilon
I just happened to have the Sahlmann/Thiemann paper (gr-qc/0303074) open to page 3 where it says:
ES,ƒ = ∫S Eai ƒiεabcdxbdxc
selfAdjoint, your question "what mental image" related to E or tilde-E has gotten me focussed on getting some intuition. After lunch I was thinking one thing space is doing all the time is expanding----the metric, distances between pairs of stationary objects like galaxies, is increasing all the time..........so does this show up in E?
What happens to the electric field or the "densitized inverse triads" or the fluxes thru sample surfaces?
All this stuff at some level is fairly straightforward and I think there is a straightforward answer----E gets bigger.
The area and volume operators---selfadjoint operators on the hilbert space, observables---they are calculated from the E's
There is a derivation, we did not go thru it here yet but it is
in several papers.
So as the E's get bigger all the areas and volumes of things will get bigger. because E is at the root of those observables.
E is something, on the tongue-tip, it is an idea---cant quite think of the word. But it is understandable, with some intuitive content. Maybe something will occur to you
Something has happened to Thomas Thiemann's expository
style. His October 2002 "Lectures on Loop Quantum Gravity"
is loaded with intuitive perceptions about the basic
elements of the theory with even some almost cartoon-like sketches and is considerably more readable than his
October 2001 "Introduction to Modern Canonical Quantum General Relativity" (written for LivingReviews).
One might try to dispose of the difference by saying he's targeting
a different audience but I doubt that explains even half
of it
Here's the link to "Lectures" in case anyone wants to check it out
http://arxiv.org/gr-qc/0210094
selfAdjoint
Aug22-03, 09:24 PM
It's not just the audience but the occasion. The lectures are, well, _lectures_. To what looks lkike a bunch of bright grad students. The living reviews piece is a high-falutin' non-working, get it right or rue it megillah.
BTW I found a quick and dirty intro to these matters in Week 7 of Baez's This week's finds.
Originally posted by selfAdjoint
It's not just the audience but the occasion. The lectures are, well, _lectures_. To what looks like a bunch of bright grad students. The living reviews piece is a high-falutin' non-working, get it right or rue it megillah.
BTW I found a quick and dirty intro to these matters in Week 7 of Baez's This week's finds.
Glad you mentioned Week 7 of Baez finds. I had looked at it some time ago but got more out of it when I went back to it yesterday
I will prefer any day Thiemann hobnobbing with the grad students to Thiemann making pronouncements ex cathedra like he is the pope (in an ecumenical spirit I have broadened the ethnic scope----stuffed shirts are universal---but must concede the metaphor of the megillah is hard to beat)
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