# Weinberg's gauge-fixed quantum gravity

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Summary:
Anyone completed the derivation of Einstein's Equation?
In this 1965 paper by Weinberg, https://journals.aps.org/pr/abstract/10.1103/PhysRev.138.B988, he describes a quantum field theory of the graviton in a Coulomb-like fixed gauge, where the free graviton has only space-space components and is traceless. This of course makes the field dynamics non-covariant; he then shows that to get back covariance, you need to add a nonlocal "Newtonian" term to the Hamiltonian and also have the graviton couple to a conserved tensor. After a long calculation he gets back the linear form of Einstein's equations, and argues that the tensor on the right-hand side will include a gravitational energy term that is equivalent to the nonlinear parts of the left-hand side in Einstein's equations. But he does not prove this. He also does not prove that certain noncovariant "gradient terms" in his graviton propagator will not contribute to physical amplitudes; he conjectures that this requirement will in fact fix the form of the gravitational energy term.

Has this approach been taken up by others? Have these conjectures ever been proven?

ohwilleke, atyy, Demystifier and 1 other person

Hey, good to see you're still around! I've been away from PF for a while, but when I come back I get my first response from an old friend!

DeWitt's work is certainly very central and powerful, but I'm specifically interested in the "Coulomb-gauge" approach developed by Weinberg in that paper. I like it because it gives the field operators a fully explicit interpretation, in terms of creating and annihilating (on-shell) gravitons. OTOH the explicitly nonlocal Hamiltonian is a bit of a steep price... though maybe not for a Bohmian like you!

vanhees71 and Demystifier
Demystifier
Gold Member
Not an expert, but perhaps the gravitational Coulomb gauge is studied more in classical gravitational wave literature.

vanhees71
Not an expert, but perhaps the gravitational Coulomb gauge is studied more in classical gravitational wave literature.
It is actually quite similar to the standard Transverse Traceless (TT) gauge. It might even be identical; I'm not sure about that. Kind of ironic that Weinberg calls his gauge "too ugly to deserve a name"! Was TT gauge in use for gravitational waves, back in 1965?

Anyhow, none of the classical GR literature will address the the issue of the gradient terms in the propagator. They also are unlikely to have used Weinberg's Hamiltonian much, because of the nonlocality. But it would be interesting if someone did a detailed comparison between this Hamiltonian and the ADM version.

As for the gravitational energy pseudotensor, it's obvious that moving the nonlinear terms in ##G_{\mu\nu}## to the RHS of the Einstein Equation does give a noncovariantly-conserved (and symmetric) form for the total SEM pseudotensor - assuming Einstein's Equation holds. The interesting question is finding some set of assumptions that make this form unique, beyond the also-obvious point that it serves as the source for the linear part of ##G_{\mu\nu}##. (Linear here means first-order in ##h_{\mu\nu} = g_{\mu\nu}-\eta_{\mu\nu}##)

vanhees71