A general worldline quantum inequality

TL;DR

This paper establishes a general worldline quantum inequality for the energy density of a scalar quantum field along arbitrary observer trajectories in any globally hyperbolic spacetime, extending previous static cases.

Contribution

It generalizes existing static-worldline inequalities to arbitrary smooth timelike trajectories and states, using microlocal analysis techniques.

Findings

Valid for arbitrary smooth timelike trajectories

Applicable to any Hadamard quantum state

Includes a compact form for stationary trajectories

Abstract

Worldline quantum inequalities provide lower bounds on weighted averages of the renormalised energy density of a quantum field along the worldline of an observer. In the context of real, linear scalar field theory on an arbitrary globally hyperbolic spacetime, we establish a worldline quantum inequality on the normal ordered energy density, valid for arbitrary smooth timelike trajectories of the observer, arbitrary smooth compactly supported weight functions and arbitrary Hadamard quantum states. Normal ordering is performed relative to an arbitrary choice of Hadamard reference state. The inequality obtained generalises a previous result derived for static trajectories in a static spacetime. The underlying argument is straightforward and is made rigorous using the techniques of microlocal analysis. In particular, an important role is played by the characterisation of Hadamard states in terms of the microlocal spectral condition. We also give a compact form of our result for stationary trajectories in a stationary spacetime.