Ashtekar Constraint Surface as Projection of Hilbert-Palatini One

TL;DR

This paper investigates the relationship between the Ashtekar and Hilbert-Palatini formulations of general relativity, revealing that their quantum theories differ due to the specific form of constraints relating selfdual and antiselfdual variables.

Contribution

It demonstrates that the Ashtekar constraint surface can be viewed as a projection of the Hilbert-Palatini constraint surface, clarifying the non-equivalence of their quantum theories.

Findings

Ashtekar and Hilbert-Palatini quantum theories are non-equivalent.

Constraints relating selfdual and antiselfdual variables are of a 'loose relation' type.

Quantization procedures do not commute with the exclusion of antiselfdual variables.

Abstract

The Hilbert-Palatini (HP) Lagrangian of general relativity being written in terms of selfdual and antiselfdual variables contains Ashtekar Lagrangian (which governs the dynamics of the selfdual sector of the theory on condition that the dynamics of antiselfdual sector is not fixed). We show that nonequivalence of the Ashtekar and HP quantum theories is due to the specific form (of the "loose relation" type) of constraints which relate self- and antiselfdual variables so that the procedure of (canonical) quantisation of such the theory is noncommutative with the procedure of excluding antiselfdual variables.