Intuitive content of Loop Gravity--Rovelli's program

**Ambitwistor** · Oct23-03, 12:40 PM

Originally posted by marcus
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.

(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 · Oct23-03, 01:55 PM

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.

**Ambitwistor** · Oct23-03, 02:17 PM

Originally posted by SelfAdjoint
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.

Only a Levi-Civita connection specifies a Riemannian geometry, because only L-V connections are compatible with metrics. But starting at least with Klein, and certainly since Cartan, the notion of "geometry" has been expanded to include geometries other than Riemann's. You can think of a connections as giving a generalized kind of geometry, a special case of which are the Riemannian (metric) geometries.

**Ambitwistor** · Oct23-03, 03:33 PM

(Oops, that should be "L-C connections", i.e. "Levi-Civita".)

**Ambitwistor** · Oct23-03, 07:21 PM

Example: Yang-Mills gauge theories are geometric theories, even though they're not Riemannian geometries.

The gauge field A (e.g., the the scalar and vector electromagnetic potentials, together forming the 4-potential) is given by a connection, and the field strength tensor F (e.g., the electric and magnetic fields, together forming the Faraday tensor) is the curvature of that connection.

So, in addition to gravity, the fields of the Standard Model (electromagnetic, weak, strong) are also given by geometric theories, but it's not the Riemannian spacetime geometry of general relativity. The Ashtekar variables exploit this similarity by recasting general relativity in a form more similar to the geometry of other gauge theories. You can also go the other way, and try to recast the gauge theories in a form more similar to the geometry of conventional general relativity, in which case you get Kaluza-Klein theory.

**marcus** · Oct25-03, 10:30 AM

Originally posted by Ambitwistor
Only a Levi-Civita connection specifies a Riemannian geometry, because only L-V connections are compatible with metrics. But starting at least with Klein, and certainly since Cartan, the notion of "geometry" has been expanded to include geometries other than Riemann's. You can think of a connections as giving a generalized kind of geometry, a special case of which are the Riemannian (metric) geometries.

At the start of the thread here I was hoping to find a way of presenting an intuitive picture of loop gravity.

Now I'm recalling the explanatory job Baez did on a variety of formalisms for GR---Palatini, Ashtekar-Sen, Barbero variation---I believe it was in TWF with references to hardcopy (the book by Ashtekar, which I have not read having been spoiled by the internet). Now I am thinking that either it is impossible to do what I had in mind. Or Baez will do it and put it on his website one of these days. Or one of the others (of several talented writers in loop gravity.) Or else....the way to go is to start with what you just said "You can think of a connections as giving a generalized kind of geometry, " and (possibly by means of dervish-like handwaving) OMIT the construction of the new GR variables but just take as given that a manifold has a space of all possible connections which reflects all its possible geometries and just go from there. *Takes a deep breath*

Was delighted by one of the other poster's (Gale's) idea of a wickedly clever third grader---which you elucidated by classical anecdote--and am wondering if that approach to quantum gravity would fly with such a third grader.

**eigenguy** · Oct25-03, 11:31 AM

Originally posted by marcus
At the start of the thread here I was hoping to find a way of presenting an intuitive picture of loop gravity.

I am studying the following paper

http://arxiv.org/abs/gr-qc/0207106

Abstract: A program was recently initiated to bridge the gap between the Planck scale physics described by loop quantum gravity and the familiar low energy world. We illustrate the conceptual problems and their solutions through a toy model: quantum mechanics of a point particle. Maxwell fields will be discussed in the second paper of this series which further develops the program and provides details.

Here's an excerpt:

"We will begin with the usual Weyl algebra generated by the exponentiated position and momentum operators. The standard Schrodinger representation of this algebra will play the role of the Fock representation of low energy quantum field theories and we will construct a new, unitarily inequivalent representation called the polymer particle representation in which states are mathematically analogous to the polymer-like excitations of quantum geometry. The mathematical structure of this representation mimics various features of quantum geometry quite well; in particular there are clear analogs of holonomies of connections and fluxes of electric fields, non-existence of connection operators, fundamental discreteness, spin networks, and the spaces Cyl and Cyl*. At the basic mathematical level, the two descriptions are quite distinct and, indeed, appear to be disparate. Yet, we will show that states in the standard Schrodinger Hilbert space define elements of the analog of Cyl*. As in quantum geometry, the polymer particle Cyl* does not admit a natural inner product. Nonetheless we can extract the relevant physics from elements of Cyl* by examining their shadows, which belong to the polymer particle Hilbert space HPoly. This physics is indistinguishable from that contained in Schrodinger quantum mechanics in its domain of applicability.

These results will show that, in principle, one could adopt the viewpoint that the polymer particle representation is the `fundamental one'|it incorporates the underlying discreteness of spatial geometry|and the standard Schrodinger representation corresponds only to the 'coarse-grained' sector of the fundamental theory in the continuum approximation. Indeed, this viewpoint is viable from a purely mathematical physics perspective, i.e., if the only limitation of Schrodinger quantum mechanics were its failure to take into account the discrete nature of the Riemannian geometry. In the real world, however, the corrections to non-relativistic quantum mechanics due to special relativity and quantum eld theoretic effects largely overwhelm the quantum geometry e ects, whence the above viewpoint is not physically tenable. Nonetheless, the results for this toy model illustrate why an analogous viewpoint can be viable in the full theory: Although the standard, low energy quantum field theory seems disparate from quantum geometry, it can arise, in a systematic way, as a suitable semi-classical sector of loop quantum gravity."

**meteor** · Oct26-03, 01:00 PM

I'm trying to learn what the different spaces of LQG are useful for,for example I more or less know the utility of the Hilbert space, the configuration space and the phase space. But, what's the utility of the state space?
My resumee:
In LQG the two basic variables are a connection and a densitized triad field(sometimes called electric field). The connections are functions defined in the configuration space of the theory, and each connection represents a quantum state of spacetime.This configuration space is a vector space of functions
The connection and the densitized triad field form a canonical pair in the phase space of LQG, that is a infinite dimensional space
The Hilbert space of the theory is constructed of the connections defined in the configuration space. Spin network states (previously were used loop states) form the basis of this Hilbert space.
Now, is this Hilbert space the unique Hilbert space of the theory? I've read that there's something called "kinematical Hilbert space", and othe thing called "diffeomorphism invariant Hilbert space". They both refer to the same thing?
Would be good if you could clarify this: It's true that actually the complex SU(2) connection of Ashtekar is not used in LQG, but is used the real SO(3) connection introduced by Barbero?

**Ambitwistor** · Oct26-03, 01:40 PM

Originally posted by meteor

In LQG the two basic variables are a connection and a densitized triad field(sometimes called electric field). The connections are functions defined in the configuration space of the theory, and each connection represents a quantum state of spacetime.

Each connection (modulo an SU(2) or SO(3) gauge transformation) represents a classical state of space, not a quantum state of spacetime. (Well, not even that: it only represents space once you impose the constraints.) We haven't quantized yet.

Spin network states (previously were used loop states) form the basis of this Hilbert space. Now, is this Hilbert space the unique Hilbert space of the theory? I've read that there's something called "kinematical Hilbert space", and othe thing called "diffeomorphism invariant Hilbert space". They both refer to the same thing?

No. The kinematical Hilbert space is L^2(A/G), i.e., the (complex) Lebesgue square-integrable functions over the space of connections modulo gauge transformations. It's like saying that the configuration space of a particle is R^3 (all of space), and then saying that the space of quantum states (wavefunctions) is L^2(R^3), the space of (square-integrable) complex functions over R^3.

However, then we have to start imposing constraints. e.g., for the free particle in QM we could construct the space of states L^2(R^3), but now suppose that we really only want to quantize a particle that's constrained to move on the surface of a sphere in R^3, or something. Then we have to start chopping down the kinematical Hilbert space to get the physical Hilbert space, the wavefunctions of particles that are constrained to move on the surface of a sphere.

In loop quantum gravity, we start with the kinematical Hilbert space, which has the spin networks as a basis. It is the quantum space of states of connections (modulo gauge transformations). However, not ANY connection corresponds to a solution of Einstein's equation! Only connections which obey the Gauss, diffeomorphism, and Hamiltonian constraints are "physical", connections that represent a gravitational field. So just like we discard connections in the classical configuration space A/G that don't obey the constraints of general relativity, we have to discard states in the kinematical Hilbert space L^2(A/G) that don't obey the quantized versions of those constraints.

So, the diffeomorphism-invariant Hilbert space is what you get when you apply the diffeomorphism constraint to the kinematical Hilbert space. If you also apply the Hamiltonian constraint, you get the physical Hilbert space.

(Note: we applied the Gauss constraint before quantizing by modding out by gauge transformations to consider the space A/G, because it's easy to do that. Then we applied the other constraints after quantizing.)

See also:

http://www.lns.cornell.edu/spr/1999-05/msg0016153.html
http://www.lns.cornell.edu/spr/1999-05/msg0016258.html

Would be good if you could clarify this: It's true that actually the complex SU(2) connection of Ashtekar is not used in LQG, but is used the real SU(3) connection introduced by Barbero?

Well, there are a lot of connections floating around, actually. Some people like Ashtekar's connection. Many use Barbero's nowadays, because you don't have to deal with the reality conditions. Barbero's connection is not SU(3), it is SO(3); you can use an SU(2) connection too, but it's not the same as Ashtekar's connection.

(SU(2) and SO(3) are pretty interchangeable as far as connections are concerned, because they have the same Lie algebra. It can make a difference when global effects are concerned, but loop quantum gravity physicists are usually sloppy about such things.)

**marcus** · Oct27-03, 06:53 PM

I was hoping to arrive at some posts expressing intuitive content of loop gravity. Some of us have been reading Livine's thesis and/or work co-authored with Alexandrov or with Freidel.
I find the work admirable but difficult to assimilate. It seems to me that i am gradually having to confront a more completely lorentzian fourdimensional theory----they are extending the group to the whole lorentz group and raising the dimension. How to picture this. Maybe someone else---selfAdjoint, ambitwistor, ... has ideas about how to describe this. Or is it just plain a lot more difficult and tough to describe?

I am used to having 3D connections corresponding to a 3D spatial manifold. Quantum states of 3D geometry. Operators, which presumably can evolve a bit like the Heisenberg picture but without an absolute preferred time, only one operator you choose arbitrarily to serve as clock for the other processes. This is not too bad.

but now Livine etc make us consider 4D connections corresponding to all possible geometries on some 4D manifold. The wave functions are not just functions defined on the connections but on a pair consisting of a 4D connection and a vectorfield χ

any concerns or comments about this new material

**Ambitwistor** · Oct27-03, 07:01 PM

Originally posted by marcus
It seems to me that i am gradually having to confront a more completely lorentzian fourdimensional theory----they are extending the group to the whole lorentz group and raising the dimension.

Well, there are many approaches floating around. The 4D approaches are more related to spin foams the usual loop quantum gravity in the canonical approach. It's probably best to start by thoroughly understanding one model, such as canonical LQG with the Ashtekar-Barbero connection, or the Barrett-Crane spin foam model, rather than trying to simultaneously learn about all the different cutting-edge approaches.

**marcus** · Oct27-03, 07:21 PM

whether or not it is wise, I would like to understand the role played by this vectorfield chi, let's see how to write it

c

χ

the quantum state or wave function is defined on a pair
consisting of a connection and a vectorfield

Ψ(A, c)

as you say, Ambitwistor, the connection to spinfoam is close, but also there is a connection to the SU(2) loop gravity of the people you mentioned.

I would like to understand how this vectorfield seems to serve as a bridge between the SU(2) and the covariant (i.e. SL(2,C) or lorentzian) approaces

**eigenguy** · Oct27-03, 07:41 PM

Originally posted by Ambitwistor
Well, there are many approaches floating around. The 4D approaches are more related to spin foams the usual loop quantum gravity in the canonical approach. It's probably best to start by thoroughly understanding one model, such as canonical LQG with the Ashtekar-Barbero connection, or the Barrett-Crane spin foam model, rather than trying to simultaneously learn about all the different cutting-edge approaches.

I agree, this is excellent advice ambitwistor.

Marcus, if you want your understanding of LQG to advance beyond the impressionistic level it's on now, you really need to commit to just one or two papers on a specific topic and really go over them with a fine tooth comb, proving every intermediate result you can (if you can).

I was advised, quite wisely as it turns out, to look just at the issue of relating polymer and fock states beginning with the pedagogically effective paper I referred you to. You should listen to ambitwistor and jeff. (I must say I'm having an increasingly hard time understanding how you managed that physics expert award thing. Maybe your true calling is politics?[6)])

**marcus** · Oct27-03, 07:52 PM

the role of c[

see page 98 of the thesis
a bridge player is discussing the taking of a particular trick

right after equation (8.30) he says
"With the help of a gauge transformation, one notices that it's always possible to rotate a given c(x) to be the same fixed one eg. (1,0,0,0). So an invariant function is completely determined by its section at c=c₀"

and he defines a restricted wavefunction that now depends only on the connection

instead of f(A,c) we are now looking at
f_{χ = χ₀}(A), which I will just call f(A) for the moment

"let us remark that f(A) has a residual SU(2) invariance.
Thus we are in the process of studying functions of a lorentz connection, effectively not invariant under SL(2,C) but simply
under the compact group SU(2)!"

Livine's italics and exclamation point. so this is one of the things this vectorfield chi does.

**marcus** · Oct27-03, 08:38 PM

Originally posted by marcus
the role of c[

see page 98 of the thesis...

he calls it the "time normal" and makes it one of the configuration variables along with the connection.

he gives some more idea of how he thinks of it right there on page 98, before the part I quoted, before (8.30)

it's a vectorfield with values in the quotient SL(2,C)/SU(2)
that you can think of as a normal to the hypersurface

and he gives a reference to maybe the best article on this
chi "boost" gadget, "time normal" "internal time direction"
the reference is to
http://arxiv.org/gr-qc/0207084
Projected Spin Networks for Lorentz connection: Linking spin foams and loop gravity.
it is dated 12 April, 2003 tho the number suggests earlier.

this 15 page paper (along with the Alexandrov/Livine one we were reading earlier today gr-qc/0205109) might be the best
auxilliary reading to have handy when looking over the thesis. but the thesis is fairly self-contained as such go

selfAdjoint · Oct28-03, 11:15 AM

Well in the last paper you cite he says the chi field determines the imbedding of the 3-d space Σ in the 4-d spacetime. Which I can see, a field of little vectors normal to that hypersurface and by their direction determining just which shape it takes in 4-space. Then he goes to the network and only keeps the chis at the vertices. And by this he reduces the group action on them from Poincare SO(1,3) (he calls it Lorentz) to a product of rotation groups SO(3) over the vertices. So far so good, it seems to me. If you really want to see the origin of the chis spelled out I guess you would have to go back to Holst's paper (Red Queen, Red Queen!) or the earlier papers by Livine that he cites.