Local states of free bose fields

TL;DR

This paper explores the concept of local states in free Bose fields, proving that finite particle states cannot be perfectly localized and clarifying misconceptions related to the Newton-Wigner position operator.

Contribution

It generalizes Knight's theorem on localization in quantum harmonic systems and clarifies the nature of localization and causality in free Bose fields.

Findings

Finite particle states cannot be perfectly localized.

Knight's theorem is extended to a broader class of states.

Difficulties with the Newton-Wigner position operator are explained without invoking relativity.

Abstract

These notes contain an extended version of lectures given at the ``Summer School on Large Coulomb Systems'' in Nordfjordeid, Norway, in august 2003. They furnish a short introduction to the theory of quantum harmonic systems, or free bose fields. The main issue addressed is the one of local states. I will adopt the definition of Knight of ``strictly local excitation of the vacuum'' and will then state and prove a generalization of Knight's Theorem which asserts that finite particle states cannot be perfectly localized. It will furthermore be explained how Knight's a priori counterintuitive result can be readily understood if one remembers the analogy between finite and infinite dimensional harmonic systems alluded to above. I will also discuss the link between the above result and the so-called Newton-Wigner position operator thereby illuminating, I believe, the difficulties associated with the latter. I will in particular argue that those difficulties do not find their origin in special relativity or in any form of causality violation, as is usually claimed.