Intuitive content of Loop Gravity--Rovelli's program

meteor

Hi here,
With respect to the Poisson algebra, the only thing that I've discovered is that in 1987, Smolin and Rovelli introduced an infinite set of gauge invariant loop variables on the phase space of the theory (called then the Ashtekar phase space). These variables form a closed Poisson algebra
I don't know if Poisson algebras are anymore important in LQG since loops were substituted by spin networks
Marcus reading your anterior post, there are some ideas that have popped up in my mind, could you clarify, please? [t)]
Is it possible that this Ashtekar phase space is really the Hilbert space of LQG?. I mean, is possible that the loop variables introduced by Rovelli and Smolin are functions in this Hilbert space?
Is it possible that before the introduction of the loop variables, the functions in the Hilbert space were the connections?
Best wishes and keep fighting the stringers!

nonunitary

Hi there,

I am kind of new in this forum and found this tread about LQG which interests me a lot. I don´t know if I am misleading but I think that poisson algebras are still important in LQG, but they are not so much
talked about. Rovelli and Smolin indeed had a Poisson algebra that was the starting point for quantization. I think that Ashtekar and co-workers wrote a paper showing that the Rovelli-Smolin algebra was not closed and changing the poisson algebra to some other algebra I don't remenber. Recently, Sahlmann, Lewandowski and Thiemann have
taken this proposal and expanded it.
Just a few more comments. The phase space of the theory is not the same as the Hilbert space. Normaly, this is constructed out of functions of the configuration space, in this case, connections.
The loop variables are functions of the connection, but labelled by loops (or graphs in the case of spin networks).

sol1

Gravity

Well the question to me then raises what the foundation of this whole topic is built upon? Philospohically Smolin was able to unite three roads to form algebraic topology.

The basis of the all the maths including the geometries must also follow the logic? Venn logic, and geometrical Intuitional developement? Category theory and topos(Algebraic Topology) was a integration of Smolins Three Roads?

It had to begin from supersymmetry, and from this, the gravity is understood, as well as, weak field settling to boundaries and defintion in cooling and discrete forms?

But in all of this, there is a exchange between energy/matter and the mobius can see things turn over as well as the klien bottle turning inside out. Where is that? Twisting in differential rotations?

You had to be able to see it from Kaluza and Kliens perspective and how we got there. U(1)=5d......and this includes all the covers?

Kip thorne help us to visualize, and in this great distance Ligo reads and in the quantum world how much more is this energy that continuity says, listen, things seem very smooth. But we have discrete structures, and how is the gravity revealed from the perspective of tangible objects, but in the recogniton of the distances?

Intuitively it must come to the distances? We do not disregard the structure that arises from the movement of the energy into objects(crystalization)

Sol

selfAdjoint

Sol, it was a rhetorical question. Marcus is going to tell us what a connection is. I'll bet he's working on his metaphors right now.

sol1

Oh good:) You know how intuition can be sometimes trying to find the right words to explain the essence of things.

Sol

marcus

Hi meteor, welcome nonunitary,
everything you two said in your posts seems right to me, and in particular this:

This is a good thing to be clear about----the configuration space is the set of possible connections (reflecting all the possible geometries there could be on the manifold). In the process of quantizing a classical theory a hilbertspace is constructed consisting of (complex number valued) functions defined on the configuration space.

Spaces of functions are typically convenient to use because they are linear---you can add two functions just like you add two vectors and so on----spaces of functions are typically infinite dimensional vectorspaces and they are handy just the way vectorspaces are. (This is hardly news to you meteor and nonunitary but might as well be said) And a hilbertspace is a type of vectorspace that commonly comes up as a set of real or complex-valued functions defined on something---in this case the classical theory's configuration space: the set of all possible geometries, as represented by the connections associated with them.

A classical theory will have a poisson algebra of various readings off of phase space corresponding to classical deterministic measurements you can make and in the process of quantizing the theory one will want to find an algebra of OPERATORS on the hilbertspace that these things correspond to.

I agree with nonunitary about the role played the poisson algebras and the algebras of quantum observables that people seem able to discover corresponding to them. Also that Sahlmann, Thiemann, Lewandowski and others have been busy with these things recently.

In effect, you are ahead of me right now and there is no need to wait for me to catch up. Anything you know about the theory that you want to explain, you should go ahead! We have no special responsibilities to anyone and no need to follow any special order.

What iterests me right now is this: how would you explain to someone with a minimum of math a way to think about connections.

A manifold is just a set equiped with coordinates around any point so that you CAN use those coordinate patches to give it geometrical shape if you so desire. But a bare manifold is devoid of geometry----there is almost nothing interesting about it unless it has some kinky topological features.

One way to put some starch in your manifold is to define a METRIC on it (which you can do because you have the kind of minimal amenities, namely coordinate patches). Once you have said what the distance between each pair of points in the manifold is, the thing snaps to attention and takes on shape.

But you can also proceed a different way. Because you have this minimal structure of coordinates you can define the tangent "plane" at every point---actually for a 3D manifold it is a tangent 3-space not a tangent plane. And the thing still looks like a ruppled shirt or wet paper bag, it just has tangentspaces stuck to it. But NOW you can decide on the infinitesimal "roll" that happens as you go from one point to the nearest neighboring points. That is, you can pick a "connection". And by continuing along a path and integrating the tiny roll at each point you get
by the end of the path a reall substantive rotation. So a "connection" is basically a contraption that tells how tangent vectors are supposed to rotate as you move along a definite path thru the manifold.

Well that bespeaks geometry too, just like a metric does. Technically it doesn't completely determine it but intuitively it goes a long ways towards defining what shape the thing is.

Now physicists learn Lie groups and Lie algebras long before they put on long pants and start shaving and having dates, so for them it is a kneejerk response that an infinitesimal "roll" is an element of su2, the Lie algebra of SU2. It is like the things you have to learn to pass your drivers test. An element of su2 is a particularly cute kind of 2x2 matrix of complex numbers.

So what is a connection? It is a program I have on my palmpilot that if you show me a point in the manifold and show me a DIRECTION in which to set out from that point, I will tell you the infinitesimal "roll" that is I will tell you a cute 2x2 matrix of complex numbers which is a member of su2.

And in a very rough sense all the possible geometries the manifold can have are reflected usefully in the set of all possible connections that can be defined on it

Ambitwistor

The labels of spin network have always been representations; it's just that in the case of SU(2), representations can be simply labeled by numbers.

The Lie group can be anything, but in the connection variables, an SU(2) spatial connection is typically used, which leads to a kinematical Hilbert space of SU(2) spin networks.

Spin networks don't have to be trivalent. In fact, in LQG, trivalent spin networks have zero volume.

The group representations are irreducible, since the point of spin networks is to form an orthonormal basis of the space of connections (modulo gauge transformations); for that, you want networks labelled with irreps, as follows from the Peter-Weyl theorem; see http://arXiv.org/abs/gr-qc/9504036.

marcus

I checked this in Rovelli's textbook and you are right: he says a node must be at least quadrivalent to have nonzero volume. I like everything in your post and look forward to many more. welcome. there is a considerable need for loop-knowledgeable people here.

I guess I need to look again at the volume formula, for some reason until I checked just now, I thought that a trivalent vertex contributed a unit of volume. I know that any vertex can be broken down by a series of surgical steps into a collection of trivalent one, I must go back and try to understand what happens to the volume.

Ambitwistor

At this level, it might be worth emphasizing that these "complex-valued functions on configuration space" are what some people might know better as "wavefunctions": the configuration space describes the system (like the position of a particle), and the value of the wavefunction (a complex number) is the probability amplitude for finding the system in that particular configuration (like the probability of finding a particle in a particular location).

I'm not sure that I understand this side remark correctly, but if you want to speak technically, a connection does completely determine the geometry: if you have a Levi-Civita connection, then that is equivalent to having a metric. If you have some other kind of connection, then it defines a more general kind of geometry (not Riemannian) that does not arise from a metric.

marcus

absolutely right! In another thread I stressed the term "wavefunction" for this. I should always mention that as a synonym. I like to try a few alternative ways of saying things to allow for people coming to the subject from different backgrounds.

I am so glad you are on hand, ambitwistor! I will try (if the board permits it) to edit some of my posts to remove the vagueness about that in accordance with what you say.

My intuitive feel is that the connection describes the geometry, and I am puzzled that the Ashtekar variables are not simply A (the connection) but are various pairs, like [A, E] where E is the densitized triad or "electric field"(a term sometimes used depending on a possibly confusing analogy). A and E are presented as "conjugate" variables. Yet the configuration space is just the collection of all possible A's. what is the essential additional information given by E? You are probably familiar with the notation I'm using, for brevity omiting subscripts and so on.

Ambitwistor

As mentioned in Baez's Week 55, Loll showed that trivalent vertices do not contribute volume (http://arXiv.org/abs/gr-qc/9511030).

As for breaking down vertices into trivalent vertices, you're probably thinking of those tangle diagrams, where the spin network edges are decomposed into "virtual" nodes and edges, wired up according to recoupling theory. You can decompose spin networks into trivalent diagrams of that sort, but they aren't spin networks -- though they are equivalent to (not-necessarily trivalent) spin networks.

See http://relativity.livingreviews.org/...TheVirtualNode

marcus

I'm impressed. In 5 minutes Ambitwistor came up with two
precisely-on-target online references---one to original work by a remarkable woman named Renata Loll and one to the exact place in Rovelli's classic LivingReviews exposition. That means Ambitwistor is a pro. Probably his/her time is too valuable to hang around much. Came thru like a big train through a small station and that was it.

Well well

I'd stay up and see what else happens tonight, but its after midnight here and time to turn in

(today selfAdjoint plugged PF, and the loop gravity threads in particular, on SPR Usenet. this could explain unexpected visits.
nice if some of these people stayed around)

selfAdjoint

Marcus,
I have two thoughts on this. One, Ashtekar doesn't use all of his connection, but only the "anti-self-dual" part. Two, the Ashtekar variables specify not only a geometry but a kinematics. Thiemann's intro does a lot of degree-of-freedom counting, maybe that would be a reference on this issue.

marcus

I was counting on being able to edit some of this expository writing, but discovered yesterday that the PF rules have changed. there is a time limit of 30 minutes afterwhich I cannot edit a post
It makes it easier to write if you can put in placeholder stuff at some point, then go research it and fix it up if necessary.

If you have more extensive editing capability (a mentor perk?) than I do, you are welcome to fix vague points in my discussion, correct errors, improve style or whatever. editing is half of writing

I was a bit sleepy when Ambitwistor passed through---not sure but I got the impression of someone who may actually do research in quantum gravity----knows work of Baez and of Renate Loll with quick exactitude.

nonunitary

quote:
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My intuitive feel is that the connection describes the geometry, and I am puzzled that the Ashtekar variables are not simply A (the connection) but are various pairs, like [A, E] where E is the densitized triad or "electric field"(a term sometimes used depending on a possibly confusing analogy). A and E are presented as "conjugate" variables. Yet the configuration space is just the collection of all possible A's. what is the essential additional information given by E? You are probably familiar with the notation I'm using, for brevity omiting subscripts and so on.
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Hi there,

I think there is a confusion here. It is true that a metric and its (Levi Civita) connection carry almost the same information (up to constant re-scaling of the metric). In the basic variables of LQG the metric information of the manifold is in the triad E. The connection "A" that is known as the Ashtekar-Barbero connection has more infromation than just the metric. It also knoes about the ADM conjugate variable, namely the extrinsic curvature $K_{ab}$. Then the connection $A$ is given by $A_a^i=\Gamma^i_a- (const) K^i_a$, where the constant in the formula is the infamous Immirzi parameter.
Thus, even when the connection knows about the metric, it also has information about the extrinsic curvature, and that is why it serves as a conjugate variable for the $E$'s (that is, if the Immirzi parameter were zero, the variables would be all "configuration variables", and its Poisson bracket would vanish).

Another comment. Te self dual connections were the original variables introduced in 1986 by Ashtekar, but they were replaced in the 90's by the Ashtekar-Barbero connection with a "real" Immirzi parameter (instead of $i$ for the original self-dual case). The nice geometrical interpretation is however, lost.

Ambitwistor

(A,E) are a conjugate pair, i.e., a point in phase space (not configuration space). It's analogous to how in QM, position and momentum (x,p) are a conjugate pair (but you only pick one of them them to form your quantum Hilbert space). In the ADM geometrodynamic variables, the 3-metric and the extrinsic curvature form the conjugate phase space variables, and you generally form your Hilbert space over the 3-metric. In the Ashtekar variables, you pick the Ashtekar connection and the densitized 3-triad.

Classically, you use the configuration space variable to describe the kinematics, and the conjugate momentum to describe the dynamics. (e.g., position in QM or the 3-metric in geometrodynamics describes the system at a given instant of time, and momentum or the extrinsic curvature describes how that state will evolve).

You can reconstruct the (densitized) 3-geometry of a spatial slice from E, not A (since the triad is basically just the "square root" of the 3-metric, as nonunitary mentioned). A itself determines a "geometry" on space, just like any connection does, but it's not the kind of metric geometry that a Levi-Civita connection defines. A actually carries information that one can use to reconstruct the spacetime geometry -- like extrinsic curvature does, and in fact A involves the extrinsic curvature (as nonunitary also pointed out).

selfAdjoint

In Thiemann's derivation of the Ashtekar variables he first enlarges the phase space of the Palatini action, spanning this larger space with canonical variables K and E, K will go away but E will remain in the Ashtekar variables. He shows that the new (K,E) coincide with the Palatini (p,q) variables when a constraint is satisfied; this constraint is satisfied identically in the Palatini geometry. Only then is the connection A introduced, and it replaces the nonce variable K, and the new variables (A,E) are canonical and span the big phase space.

In general is it really true that a connection by itself specifies a geometry? Recall that in traditional Riemann you have first a metric - specified by a symmetric tensor, which restricts your choice of geometries, and then define the connection as a function of your metric (through the Christoffel symbols). This then gives you the curvature tensor and all the rest. But the contribution of the symmetric metric was important.