Quantum backreaction (Casimir) effect I. What are admissible idealizations?

TL;DR

This paper analyzes the quantum backreaction (Casimir) effect, emphasizing the importance of algebraic structures and idealizations, and clarifies conditions under which energy densities are well-defined in quantum field models.

Contribution

It introduces a framework based on algebraic quantum theory to identify acceptable idealizations and clarifies the nature of energy densities in the Casimir effect.

Findings

Normal ordered energy density is a well-defined distribution.

Incompatibility of observable algebras causes infinities in traditional models.

Zero point expressions are inappropriate but can appear due to manipulations.

Abstract

Casimir effect, in a broad interpretation which we adopt here, consists in a backreaction of a quantum system to adiabatically changing external conditions. Although the system is usually taken to be a quantum field, we show that this restriction rather blurs than helps to clarify the statement of the problem. We discuss the problem from the point of view of algebraic structure of quantum theory, which is most appropriate in this context. The system in question may be any quantum system, among others both finite as infinite dimensional canonical systems are allowed. A simple finite-dimensional model is discussed. We identify precisely the source of difficulties and infinities in most of traditional treatments of the problem for infinite dimensional systems (such as quantum fields), which is incompatibility of algebras of observables or their representations. We formulate conditions on model idealizations which are acceptable for the discussion of the adiabatic backreaction problem. In the case of quantum field models in that class we find that the normal ordered energy density is a well defined distribution, yielding global energy in the limit of a unit test function. Although we see the "zero point" expressions as inappropriate, we show how they can arise in the quantum field theory context as a result of uncontrollable manipulations.