A New Non-Perturbative Approach to Quantum Theory in Curved Spacetime Using the Wigner Function

TL;DR

This paper introduces an exact, non-perturbative Wigner function-based approach to quantum theory in curved spacetime, enabling systematic quantum effect calculations and applications to quantum gravity.

Contribution

It proposes a novel, exact Wigner equation for curved spacetime that differs from previous models and allows systematic approximation schemes.

Findings

Derived the trace of the renormalised energy-momentum tensor.

Linked conformal anomaly to non-conservation of currents in 2D and 4D.

Extended the method to arbitrary spins.

Abstract

A new non-perturbative approach to quantum theory in curved spacetime and to quantum gravity, based on a generalisation of the Wigner equation, is proposed. Our definition for a Wigner equation differs from what have otherwise been proposed, and does not imply any approximations. It is a completely exact equation, fully equivalent to the Heisenberg equations of motion. The approach makes different approximation schemes possible, e.g. it is possible to perform a systematic calculation of the quantum effects order by order. An iterative scheme for this is also proposed. The method is illustrated with some simple examples and applications. A calculation of the trace of the renormalised energy-momentum tensor is done, and the conformal anomaly is thereby related to non-conservation of a current in d=2 dimensions and a relationship between a vector and an axial-vector current in d=4 dimensions. The corresponding ``hydrodynamic equations'' governing the evolution of macroscopic quantities are derived by taking appropriate moments. The emphasis is put on the spin-1/2 case, but it is shown how to extend to arbitrary spins. Gravity is treated first in the Palatini formalism, which is not very tractable, and then more successfully in the Ashtekar formalism, where the constraints lead to infinite order differential equations for the Wigner functions.