An Algebraic Bootstrap for Dimensionally Reduced Quantum Gravity

TL;DR

This paper introduces an algebraic bootstrap approach to quantize cylindrical gravitational waves in Einstein gravity, using recursive functional equations derived from a quadratic algebra, avoiding renormalization issues and revealing symmetry breaking.

Contribution

It presents a novel algebraic bootstrap method for quantizing the Ernst system in 4D gravity, linking spectral-transformed variables to conserved charges.

Findings

Exact solutions for matrix elements of the metric operator.

Avoidance of renormalization problems in quantum gravity.

Identification of spontaneous SL(2,R) symmetry breaking.

Abstract

Cylindrical gravitational waves of Einstein gravity are described by an integrable system (Ernst system) whose quantization is a long standing problem. We propose to bootstrap the quantum theory along the following lines: The quantum theory is described in terms of matrix elements e.g. of the metric operator between spectral-transformed multi-vielbein configurations. These matrix elements are computed exactly as solutions of a recursive system of functional equations, which in turn is derived from an underlying quadratic algebra. The Poisson algebra emerging in its classical limit links the spectral-transformed vielbein and the non-local conserved charges and can be derived from first principles within the Ernst system. Among the noteworthy features of the quantum theory are: (i) The issue of (non-)renormalizability is sidestepped and (ii) there is an apparently unavoidable ``spontaneous'' breakdown of the SL(2,R) symmetry that is a remnant of the 4D diffeomorphism invariance in the compactified dimensions.