# Higher Derivative Scalar Quantum Field Theory in Curved Spacetime

## TL;DR

This paper investigates higher derivative scalar quantum fields in curved spacetime, deriving energy-momentum tensors, exploring quantisation issues, and establishing non-existence of static solutions, with applications to black holes and cosmology.

## Contribution

It provides a detailed analysis of higher derivative scalar fields in curved backgrounds, including energy-momentum tensors, quantisation currents, and no-hair theorems, with novel insights into stability in cosmological settings.

## Key findings

- Derived energy-momentum tensors for higher derivative scalar fields.
- Established non-existence of static scalar solutions around black holes.
- Showed Hubble friction stabilizes Pais-Uhlenbeck fields in de Sitter space.

## Abstract

We study a free scalar field $\phi$ in a fixed curved background spacetime subject to a higher derivative field equation of the form $F(\Box)\phi =0$, where $F$ is a polynomial of the form $F(\Box)= \prod_i (\Box-m_i^2)$ and all masses $m_i$ are distinct and real. Using an auxiliary field method to simplify the calculations, we obtain expressions for the Belinfante-Rosenfeld symmetric energy-momentum tensor and compare it with the canonical energy-momentum tensor when the background is Minkowski spacetime. We also obtain the conserved symplectic current necessary for quantisation and briefly discuss the issue of negative energy versus negative norm and its relation to Reflection Positivity in Euclidean treatments. We study, without assuming spherical symmetry, the possible existence of finite energy static solutions of the scalar equations, in static or stationary background geometries. Subject to various assumptions on the potential, we establish non-existence results including a no-scalar-hair theorem for static black holes. We consider Pais-Uhlenbeck field theories in a cosmological de Sitter background, and show how the Hubble friction may eliminate what would otherwise be unstable behaviour when interactions are included.