What is the Precise Meaning of Canonical in Quantum Gravity Context

[inflector]

What is the Precise Meaning of "Canonical" in Quantum Gravity Context

I keep seeing "canonical" in the context of the expression "canonical quantum gravity" and it is clear the meaning has a precise definition. But I haven't seen it articulated anywhere and a search doesn't bring up anything useful, although it does bring up a surprising number of old PH threads. I know what the word means, in general, but nowhere do I see it described specifically in the context as used in the term "canonical quantum gravity."

What precisely does "canonical" mean as a descriptor for quantum gravity?

From context it seems to mean something like: "using a straightforward/naive/uncomplicated/standard quantization strategy," but clearly that's not the precise definition. If possible, I'd like a two types of answers. First, in laymen's terms and second in more precise mathematical/physics language.

Or is it as simple as a gravity theory that uses "canonical quantization?" It seems to be something more specific than that. But if this is all it means, then what precisely does "canonical quantization" mean in the context of the history of LQG development?

As always, any pointers to good papers or references are appreciated.

 

[tom.stoer]

"canonical" means nothing else but canonical quantization, i.e. promoting the Hamilton function, "canonically conjugate variables" [like x p; in the context of LQG A(x) and E(x)] plus their Poisson brackets to operators acting on a Hilbert space of physical states.

The first steps are not so much different from canonical quantization as applied to other field theories like QED and QCD.

http://en.wikipedia.org/wiki/Canonical_quantization

 

[marcus]

I hope some other people will answer. My simple view is that there are two well-established formats for a quantum field theory---the Hamiltonian format and the Path Integral (or sum over histories, also called "covariant" because it is 4D)

Canonical is just a codeword for the Hamiltonian approach. The basic picture involves choosing a spatial slice. I think it is just an historical accident that they used the word "canonical"----having to do with the time-honored mathematics of "canonically conjugate pairs" of variables.

Classical theories can be put either in canonical or in covariant form. If I'm not mistaken the first time GR was put into canonical (i.e. "3+1 D" rather than "4D") format was around 1970 by Arnowitt Deser Misner---the socalled ADM paper. It is on arxiv.

I could be wrong, there could have been an earlier Hamiltonian formulation of GR, before ADM.

LQG was originally given a canonical formulation, circa 1990. There was a 3D spatial slice, quantum states of geometry were defined on the set of all 3D connections* (states of geometry in effect meant states of 3D geometry. States were described by spin-networks. More to talk about, like the form and role of the Hamiltonian in that setup, but I'll stop here.)

Then in late 1990s people began to talk about an equivalent covariant (i.e. 4D) formulation using spinfoams. That being the evolution-track of a spin-network. Again "covariant" is serving as a codeword---essentially for the feature that in that approach you DON'T focus on a 3D slice.

Offhand I can't think of any references. Maybe someone else can give a more concise, better organized account, and some references.

*A connection is a manifold's parallel-transport function---how tangent vectors pitch and roll as you push them around. The possible connections on a manifold describe the geometries just like the possible metrics do. It's two alternative handles on geometry. The distance function (metric) is one and the connection is another. The set of possible connections on the manifold is one type of "configuration space" or set of possible geometric configurations.

Loops and multilooped spin-networks were originally introduced as a way to test or feel connections. You could measure what the connection did as you moved a tangent vector around the loop. You could use the loop or the network to get numbers, from a connection. The network was then a numerical valued function defined on a configuration space----analogous to the wavefunction of a particle defined on the real line---its possible positions i.e. configurations.

So spin-networks started being used as quantum states of geometry----because they were functions defined on the set of connections on the chosen 3D slice.
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EDIT: Thanks Tom! You already replied, while I was writing. Didn't see your post.

 

[inflector]

Thanks tom and marcus, that clarifies things a lot.