Multiple classical limits in relativistic and nonrelativistic quantum mechanics

TL;DR

This paper demonstrates that in quantum mechanics, multiple classical limits exist for systems of identical particles, showing that the classical limit is not unique and depends on the approach, with implications for particle and field descriptions.

Contribution

It proves the existence of multiple classical limits in relativistic and nonrelativistic quantum mechanics, highlighting the non-uniqueness of the classical limit as  7 0.

Findings

Multiple classical limits for interacting particles and fields are established.

Local operators can break permutation symmetry under certain conditions.

The classical limit depends on the specific approach taken in the theory.

Abstract

The existence of a classical limit describing interacting particles in a second-quantized theory of identical particles with bosonic symmetry is proved. This limit exists in addition to a previously established classical limit with a classical field behavior, showing that the limit $\hbar \to 0$ of the theory is not unique. An analogous result is valid for a free massive scalar field: two distinct classical limits are proved to exist, describing a system of particles or a classical field. The introduction of local operators in order to represent kinematical properties of interest is shown to break the permutation symmetry under some localizability conditions, allowing the study of individual particle properties.