Equivalence of two definitions of the effective mass of a polaron

TL;DR

This paper proves that two common definitions of the effective mass of a polaron are mathematically equivalent in models similar to the Froehlich polaron, clarifying the conceptual understanding of polaron mass.

Contribution

It demonstrates the equivalence of the low-momentum energy-based and external potential-based definitions of polaron effective mass.

Findings

The two definitions of effective mass are shown to be equal in Froehlich-like models.

Provides a rigorous mathematical proof of the equivalence.

Clarifies the conceptual framework for polaron effective mass.

Abstract

Two definitions of the effective mass of a particle interacting with a quantum field, such as a polaron, are considered and shown to be equal in models similar to the Froehlich polaron model. These are: 1. the mass defined by the low momentum energy $E(P) \approx E(0)+P^2/2M$ of the translation invariant system constrained to have momentum $P$ and 2. the mass $M$ of a simple particle in an arbitrary slowly varying external potential, $V$, described by the nonrelativistic Schroedinger equation, whose ground state energy equals that of the combined particle/field system in a bound state in the same $V$.