The Casimir spectrum revisited

TL;DR

This paper analyzes the spectral density sigma(w) related to the Casimir effect, clarifying its mathematical properties, physical interpretation, and limitations in revealing force sign information.

Contribution

It demonstrates that sigma(w) is a regulator-independent spectral distribution, interprets it as a difference between spectral densities, and clarifies its physical significance and limitations.

Findings

sigma(w) is regulator independent

sigma(w) is a spectral distribution, not a function

The sign of the Casimir force cannot be directly inferred from sigma(w)

Abstract

We examine the mathematical and physical significance of the spectral density sigma(w) introduced by Ford in Phys. Rev. D38, 528 (1988), defining the contribution of each frequency to the renormalised energy density of a quantum field. Firstly, by considering a simple example, we argue that sigma(w) is well defined, in the sense of being regulator independent, despite an apparently regulator dependent definition. We then suggest that sigma(w) is a spectral distribution, rather than a function, which only produces physically meaningful results when integrated over a sufficiently large range of frequencies and with a high energy smooth enough regulator. Moreover, sigma(w) is seen to be simply the difference between the bare spectral density and the spectral density of the reference background. This interpretation yields a simple `rule of thumb' to writing down a (formal) expression for sigma(w) as shown in an explicit example. Finally, by considering an example in which the sign of the Casimir force varies, we show that the spectrum carries no manifest information about this sign; it can only be inferred by integrating sigma(w).