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shoehorn
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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.)
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.)
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