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.
Highlights
Higher derivative field theories have received a considerable amount of attention over the years for a variety of reasons, not least because of the realization that theories incorporating standard Einstein gravity inevitably suffer from problems of non-renormalizability
The demonstration by Stelle [3] in the 1970s that the nonrenormalizability problem itself could be overcome by adding quadratic curvature terms to the action opened the door to many investigations of such theories, the realization that renormalizability could only thereby be achieved at the price of introducing ghost states of negative norm, or energies that are unbounded below, dampened the enthusiasm for the idea
III we review some properties of a scalar Pais-Uhlenbeck field theory in a Minkowski background, including the construction of the canonical energymomentum tensor and the conserved symplectic current
Summary
Higher derivative field theories have received a considerable amount of attention over the years for a variety of reasons, not least because of the realization that theories incorporating standard Einstein gravity inevitably suffer from problems of non-renormalizability. IV we extend our discussion to the scalar PaisUhlenbeck field theory in a curved spacetime background, showing how one can use an auxiliary field formulation in order to facilitate the construction of the BelinfanteRosenfeld energy-momentum tensor, which allows a consistent coupling to the gravitational field By this means, one avoids the necessity of varying the metrics in high powers of the covariant derivative. These examples serve to illustrate the fact that the often-claimed instabilities of higher-order field theories can sometimes be illusory
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