Gr-qc/921001 - the constraint algebra of general relativity

[shoehorn]

Hi. I'm trying to work my way through Chris Isham's "Canonical Quantum Gravity and the Problem of Time", gr-qc/921001. However, I've gotten a bit stumped by the constraint algebra of general relativity. By "stumped" I don't mean that I can't understand the reasoning behind the constraint algebra, but rather that I can't actually figure out the calculations behind it.

To be more precise, if we take $g_{ij}$ and $\pi^{ij}$ as the fundamental quantities in the Hamiltonian description of general relativity, we know that we can write down two constraints: the Hamiltonian constraint $H$ and the momentum constraint $J_i$, where

$$H = g^{-1/2}(g_{ik}g_{jl} - \frac{1}{2}g_{ij}g_{kl})\pi^{ij}\pi^{kl} - g^{1/2}R$$

$$J_i = -2D_j\pi_i^{\phantom{i}j}$$

Here I've used $R$ to denote the scalar curvature of a spatial slice in the spacetime. On page 32 (equations 3.3.30-3.3.32) Isham presents the "constraint algebra" of general relativity. (I won't type them out here because the expressions are quite long.) The "algebra" is composed of four elements, namely the Hamiltonian constraint and the three components of the vector momentum constraint, and the algebraic operation on this set is the Poisson bracket. I've seen precisely this algebra in other papers so I'm assuming that it's correct.

My problem, however, is in actually deriving these results. I think that part of my confusion stems from the fact that Isham is using a definition of the Dirac delta distribution that seems strange, at least to me. He calls it a Dirac bidensity. What he seems to mean by this is that the Dirac function $\delta(x,x')$ is a scalar density of weight zero in its first argument $(x)$ and a scalar density of weight one in its second argument $(x')$. (He actually defines this quantity on page 22 but that doesn't seem to shed any light on the situation for me.)

So, I guess my questions are as follows:

(1) Can anyone here ever recall actually working through the calculations required to derive the constraint algebra?

(2) If so, are there any hints you could give me about things to watch out for when doing the calculations? For example, does the fact that the Dirac distribution he uses is a "bidensity" throw up any nasty little surprises or subtleties that an amateur like me wouldn't necessarily spot?

(3) I guess that even a hint about what the correct form of the functional derivatives

$$\frac{\delta J_k(x)}{\delta g_{ij}(y)}$$

and

$$\frac{\delta J_l(x)}{\delta\pi^{ij}(y)}$$

would be of great help to me.

Thanks in advance for any responses!(I should also probably point out that I've got a copy of John Baez's book on this topic and I can indeed repeat the calculations concerning the constraint algebra that is found there. The difference, however, is that Baez's version of the constraint algebra involves *smeared* constraints (these are roughly the equivalent of equations 3.3.34-3.3.36 in Isham's paper). This is what leads me to think that I'm missing something important about the properties of the bidensity Dirac distribution.)

 

[shoehorn]

Anyone?

....

 

[Chris Hillman]

I was planning to suggest, if no-one knowledgebable replied after a few days, that you repost this in sci.physics.research. But it seems you have already done that.

 

[coalquay404]

First off, congratulations. Isham's paper is beautiful and is a wonderful place from which to learn about canonical gravity. In case you haven't seen it already, I can also heartily recommend Thiemann's review paper gr-qc/0110034 - it's well worth a look if you're interested in this sort of thing.

Now to the main point. You're correct in suspecting that the bidensity Dirac distribution is at the heart of all of the calculational difficulties in the constraint algebra. Unfortunately, when DeWitt first introduced this sort of thing in a famous Phys. Rev. D paper in the sixties, he didn't give any details of how to perform the calculation: he simply presented the result, completely forgetting that mere mortals would have difficulty with it.

I won't give you a full run down of the calculations required to reproduce the constraint algebra. What I will do, however, is to point out one crucial identity involving the bidensity Dirac distribution; given this, you should be able to figure out the calculations yourself. The situation is as follows:

Any time you see $\delta(x,x')$ in a paper on canonical gravity, treat it as a bidensity, i.e., $\delta(x,x')$ is a density of weight zero in the first argument and a density of weight one in the second. Thus, for some scalar function $F(x)$ we have

$$F(x) = \int d^3x F(x')\delta(x,x')$$

The important identity which you need to remember when trying to compute the constraint algebra is the following (I'm using commas to denote partial differentiation):

$$F(x')\delta_{,i}(x,x') = F(x)\delta_{,i}(x,x') + F_{,i}(x)\delta(x,x')$$

With this result you should now be able to figure out the elements of the algebra. If you're really having trouble with it, post back here and I'll try to come up with something a bit more detailed.