The Theory of a Quantum Noncanonical Field in Curved Spacetimes

TL;DR

This paper investigates quantum field theories with noncanonical commutation relations in curved spacetimes, examining whether their DSR-like symmetries can be gauged or imply Lorentz symmetry breaking, and explores their implications for quantum gravity models.

Contribution

It extends the analysis of noncanonical quantum fields with DSR-like symmetry to curved spacetimes, clarifying their gauge properties and relation to Lorentz invariance.

Findings

DSR-like symmetry cannot be gauged in curved spacetime.

An auxiliary spacetime can describe the theory as standard QFT.

Breaking of boost invariance leads to foliation of spacetime for gravity coupling.

Abstract

Much attention has been recently devoted to the possibility that quantum gravity effects could lead to departures from Special Relativity in the form of a deformed Poincar\`e algebra. These proposals go generically under the name of Doubly or Deformed Special Relativity (DSR). In this article we further explore a recently proposed class of quantum field theories, involving noncanonically commuting complex scalar fields, which have been shown to entail a DSR-like symmetry. An open issue for such theories is whether the DSR-like symmetry has to be taken as a physically relevant symmetry, or if in fact the "true" symmetries of the theory are just rotations and translations while boost invariance has to be considered broken. We analyze here this issue by extending the known results to curved spacetime under both of the previous assumptions. We show that if the symmetry of the free theory is taken to be a DSR-like realization of the Poincar\'e symmetry, then it is not possible to render such a symmetry a gauge symmetry of the curved physical spacetime. However, it is possible to introduce an auxiliary spacetime which allows to describe the theory as a standard quantum field theory in curved spacetime. Alternatively, taking the point of view that the noncanonical commutation of the fields actually implies a breakdown of boost invariance, the physical spacetime manifold has to be foliated in surfaces of simultaneity and the field theory can be coupled to gravity by making use of the ADM prescription.