Non-relativistic limit of quantum field theory in inertial and non-inertial frames and the Principle of Equivalence

TL;DR

This paper explores the non-relativistic limit of quantum field theory across inertial, non-inertial, and gravitational contexts, clarifying how relativistic effects influence quantum phases and wave functions in these regimes.

Contribution

It provides a detailed analysis of the non-relativistic limit of quantum field theory in various frames and gravitational fields, highlighting subtleties and deriving the associated phase effects.

Findings

Extra phase in non-relativistic wave function under Galilean transformation.

Non-relativistic limit of Klein-Gordon equation in accelerated frames reduces to quantum mechanics with gravitational effects.

Detailed description of the non-relativistic limit of Feynman propagator in weak gravitational fields.

Abstract

We discuss the non-relativistic limit of quantum field theory in an inertial frame, in the Rindler frame and in the presence of a weak gravitational field, highlighting and clarifying several subtleties. We study the following topics: (a) While the action for a relativistic free particle is invariant under the Lorentz transformation, the corresponding action for a non-relativistic free particle is not invariant under the Galilean transformation, but picks up extra contributions at the end points. This leads to an extra phase in the non-relativistic wave function under a Galilean transformation, which can be related to the rest energy of the particle even in the non-relativistic limit. (b) We show how the solution to the generally covariant Klein-Gordon equation in a non-inertial frame, which has a time-dependent acceleration, reduces to the quantum mechanical wave function in the presence of an appropriate (time-dependent) gravitational field, in the non-relativistic limit. The extra phase acquired by the non-relativistic wave function in an accelerated frame actually arises from the gravitational time dilation and survives in the non-relativistic limit. (c) We provide a detailed description of the non-relativistic limit of the Feynman propagator in a weak gravitational field, and discuss related issues. [Abridged Abstract]