What is the significance of the Sahlmann algebra in loop quantum gravity?

[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"?

 

[marcus]

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 can't 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]

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]

Couldn't resist. I just came across this ! We should use it for derivations?

 

[selfAdjoint]

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 E_S,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 E_S,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.

 

[marcus]

Originally posted by selfAdjoint

But did you notice that in the integral for E_S,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 don't remember appearing in the later, more detailed, exposition, where there are proper respectable subscripts and superscripts that give a firmer idea of what's going on. Or let's hope.

I will let you know if it dawns on me what that asterisk before the E is doing

 

[marcus]

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 μ_γ
defined on subspaces A_γ.

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]

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 don't remember appearing in the later, more detailed, exposition, where there are proper respectable subscripts and superscripts that give a firmer idea of what's going on. Or let's 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.

 

[marcus]

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 doesn't 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]

Comparing Salhman's definition of E_S,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!

 

[marcus]

Originally posted by selfAdjoint
Comparing Salhman's definition of E_S,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]

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]:T_gG -> T_eG by
[the]: X -> (L_g^-1)_*X = (L_g)_*^-1X

 

[marcus]

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]:T_gG -> T_eG by
[the]: X -> (L_g^-1)_*X = (L_g)_*^-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*θ[/size]_MC, ...)

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[/size] 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]

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?

 

[marcus]

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 anyone 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 = tildE_i^α 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 haven't 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]

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 Ý'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.

 

[marcus]

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]

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_E^a_iτ^*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]

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>

 

[marcus]

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.

 

[marcus]

diffeo covariance of the representation/page 18

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 X_S,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 = X_{φS, ƒ(φ^-1)} φC

I had better post this before a computer glitch loses it.

 

[marcus]

Equation 4.9 on page 18 of O-L

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
Mult_C Cyl^oo --> Cyl^oo , which just sends C' into CC'.

What equation 4.9, the first part, says is that
φ Mult_C φ^-1 = Mult_φC

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 (Mult_C (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 Mult_φC 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 Cyl^oo --> Cyl^oo 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: Cyl^oo --> Cyl^oo

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 Cyl^oo 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]

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 X_S,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 = X_{φS, 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.

 

[marcus]

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 don't 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]

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.

 

[marcus]

Originally posted by selfAdjoint
...I think these current papers are giving us a better school than a textbook right now.

I agree!

 

[marcus]

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. don't know if they are known to you or not

 

[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_E^a_iτ^*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?

 

[marcus]

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_E^a_iτ^*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 g^ab = tildE^a_itildE^b_i

a,b are spatial indices and i,j are "internal" referring to a basis chosen for the Lie algebra.

approach #2, E^a_i is introduced as a (distributional) functional derivative with respect to Aⁱ_a which must be integrated against test functions ("suitably smeared") in order to be well-defined

E^a_i(x) = - i hbar G δ/(δAⁱ_a(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 E^a with the Levi-Civita density gives the dual of the triad, which is a 2-form
E = ε_abc E^adx^b dx^c.
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 I've 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 don't 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

 

[marcus]

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 = Aⁱ_aτ_i(X)dx^a

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 don't explain that to a large extent it is the compelling naturalness of the construction