# Hamiltonian quantization of solitons in the $\phi^4_{1+1}$ quantum field theory

## TL;DR

This paper quantizes solitons in the 1+1 dimensional $^4$ quantum field theory, deriving a Schrödinger equation for soliton dynamics and proving a mass shift formula, extending classical soliton modulation to quantum regimes.

## Contribution

It introduces a Hamiltonian quantization approach for solitons in $^4_{1+1}$ theory and extends soliton modulation techniques to quantum field theory with electromagnetic coupling.

## Key findings

- Derived the nonrelativistic Schrödinger equation for soliton dynamics.
- Proved the semiclassical mass shift formula of Dashen, Hasslacher, and Neveu.
- Extended soliton modulation theory to quantum field theory with external fields.

## Abstract

We first carry out the soliton sector quantization of the spatially cut-off $\phi^4_{1+1}$ theory with double well potential in the semiclassical limit, deriving the nonrelativistic Schr\"odinger equation as an equation describing the limiting soliton dynamics. In the process we prove the semiclassical mass shift formula of Dashen, Hasslacher and Neveu, which is interpreted in terms of a unitary equivalence between normal ordered semiclassical quadratic Hamiltonians in two different representations of the Heisenberg relations. Secondly, we consider the $\phi^4_{1+1}$ theory coupled topologically to an external electromagnetic field and prove the main result, which is an approximation theorem reminiscent of the Born-Oppenheimer method, which describes the nonrelativistic dynamics of the soliton coupled to infinitely many transverse bosonic degrees of freedom, extending the techniques of soliton modulation theory from classical to quantum field theory.