New constraints for canonical general relativity

TL;DR

This paper investigates the invariance properties of Ashtekar's canonical formulation of general relativity, introduces a new fully 4-diffeomorphism invariant theory with stronger constraints, and discusses implications for quantum loop gravity.

Contribution

It derives a new canonical formulation of GR from Plebanski's action that is fully 4-diffeomorphism invariant and explores its classical and quantum implications.

Findings

Ashtekar's theory is invariant under infinitesimal 4-diffeomorphisms but not under some finite ones when the dreibein is degenerate.

A new invariant theory has stronger constraints, allowing flux loop births and deaths.

Implications for quantum GR include finite amplitudes for loop creation and annihilation, affecting the loop representation.

Abstract

Ashtekar's canonical theory of classical complex Euclidean GR (no Lorentzian reality conditions) is found to be invariant under the full algebra of infinitesimal 4-diffeomorphisms, but non-invariant under some finite proper 4-diffeos when the densitized dreibein, $\tilE^a_i$, is degenerate. The breakdown of 4-diffeo invariance appears to be due to the inability of the Ashtekar Hamiltonian to generate births and deaths of $\tilE$ flux loops (leaving open the possibility that a new `causality condition' forbidding the birth of flux loops might justify the non-invariance of the theory). A fully 4-diffeo invariant canonical theory in Ashtekar's variables, derived from Plebanski's action, is found to have constraints that are stronger than Ashtekar's for $rank\tilE < 2$. The corresponding Hamiltonian generates births and deaths of $\tilE$ flux loops. It is argued that this implies a finite amplitude for births and deaths of loops in the physical states of quantum GR in the loop representation, thus modifying this (partly defined) theory substantially. Some of the new constraints are second class, leading to difficulties in quantization in the connection representation. This problem might be overcome in a very nice way by transforming to the classical loop variables, or the `Faraday line' variables of Newman and Rovelli, and then solving the offending constraints. Note that, though motivated by quantum considerations, the present paper is classical in substance.