Obtaining the Non-relativistic Quantum Mechanics from Quantum Field Theory: Issues, Folklores and Facts

TL;DR

This paper explores the conceptual and technical challenges in deriving non-relativistic quantum mechanics from quantum field theory, emphasizing the fundamental differences in their descriptions and the role of antiparticles and causality.

Contribution

It clarifies the issues preventing a straightforward limit from QFT to NRQM and advocates for a language closer to QFT to achieve a seamless transition.

Findings

Standard procedures fail to produce NRQM from QFT directly.

Antiparticles and causality are key factors in the conceptual gap.

A closer language to QFT is needed for a proper NRQM limit.

Abstract

Given the classical dynamics of a non-relativistic particle in terms of a Hamiltonian or an action, it is relatively straightforward to obtain the non-relativistic quantum mechanics (NRQM) of the system. These standard procedures, based on either the Hamiltonian or the path integral, however, do not work in the case of a relativistic particle. As a result we do not have a single particle description of relativistic quantum mechanics (RQM). Instead, the correct approach requires a transmutation of dynamical variables from the position and momentum of a single particle to a field and its canonical momentum. Particles, along with antiparticles, reappear in a very non-trivial manner as the excitations of the field. The fact that one needs to adopt completely different languages to describe relativistic and non-relativistic free particle implies that obtaining the NRQM limit of QFT is conceptually non-trivial. I examine this limit in several approaches (like, for e.g., Hamiltonian dynamics, Lagrangian and Hamiltonian path integrals, field theoretic description etc.) and identify the precise issues which arise when one attempts to obtain the NRQM from QFT in each of these approaches. The dichotomy of description between NRQM and QFT does not originate just from the square root in the Hamiltonian or from the demand of Lorentz invariance, as it is sometimes claimed. The real difficulty has its origin in the necessary existence of antiparticles to ensure a particular notion of relativistic causality. Because of these conceptual issues, it turns out that one cannot, in fact, obtain some of the popular descriptions of NRQM by any sensible limiting procedure applied to QFT. To obtain NRQM from QFT in a seamless manner, it is necessary to work with NRQM expressed in a language closer to that of QFT. (Abridged)