Quantum Diffeomorphisms and Conformal Symmetry

TL;DR

This paper explores how quantum conformal theories in four dimensions behave under coordinate transformations, revealing a unique quantum modification of the Hamiltonian constraint on the Einstein universe and proposing a weaker physical state condition.

Contribution

It demonstrates a specific quantum modification of the Hamiltonian constraint for conformal theories on $R imes S^3$ and introduces a new perspective on physical state conditions.

Findings

Quantum constraints simplify on $R imes S^3$

The conformal symmetry algebra determines a finite shift in the Hamiltonian

A weaker condition $raket{ abla T^{00}}=0$ is proposed for physical states

Abstract

We analyze the constraints of general coordinate invariance for quantum theories possessing conformal symmetry in four dimensions. The character of these constraints simplifies enormously on the Einstein universe $R \times S^3$. The $SO(4,2)$ global conformal symmetry algebra of this space determines uniquely a finite shift in the Hamiltonian constraint from its classical value. In other words, the global Wheeler-De Witt equation is {\it modified} at the quantum level in a well-defined way in this case. We argue that the higher moments of $T^{00}$ should not be imposed on the physical states {\it a priori} either, but only the weaker condition $\langle \dot T^{00} \rangle = 0$. We present an explicit example of the quantization and diffeomorphism constraints on $R \times S^3$ for a free conformal scalar field.