From Quantum Field Theory to Quantum Mechanics

TL;DR

This paper explicitly constructs the quantum mechanical operator algebra for identical particles from quantum field operators in free field theory, clarifying the relation between quantum field theory and quantum mechanics.

Contribution

It provides an explicit relation between quantum field operators and quantum mechanical position and momentum operators for free fields, highlighting the properties of these operators.

Findings

Position and momentum operators obey canonical commutation relations.

The Newton-Wigner position operator is identified for relativistic particles.

Position operators do not transform as tensors under Lorentz transformations.

Abstract

We construct the algebra of operators acting on the Hilbert spaces of Quantum Mechanics for systems of $N$ identical particles from the field operators acting in the Fock space of Quantum Field Theory by providing the explicit relation between the position and momentum operators acting in the former spaces and the field operators acting on the latter. This is done in the context of the non-interacting Klein-Gordon field. It may not be possible to extend the procedure to interacting field theories since it relies crucially on particle number conservation. We find it nevertheless important that such an explicit relation can be found at least for free fields. It also comes out that whatever statistics the field operators obey (either commuting or anticommuting), the position and momentum operators obey commutation relations. The construction of position operators raises the issue of localizability of particles in Relativistic Quantum Mechanics, as the position operator for a single particle turns out to be the Newton-Wigner position operator. We make some clarifications on the interpretation of Newton-Wigner localized states and we consider the transformation properties of position operators under Lorentz transformations, showing that they do not transform as tensors, rather in a manner that preserves the canonical commutation relations.