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

David M. A. Stuart

TL;DR

This work develops a Hamiltonian framework for quantizing solitons in the φ^4_{1+1} theory, focusing on the semiclassical limit where the coupling g is small. It constructs two CCR representations in the soliton sector and derives a precise description of the soliton as a nonrelativistic quantum particle with mass $ rac{4m^3}{3g^2}$, while also accounting for transverse bosonic modes in the soliton background. A Dashen–Hasslacher–Neveu semiclassical mass shift is obtained via unitary equivalence between representations, and the paper then extends modulation theory to quantum field theory by coupling the soliton to an external electromagnetic field, proving a Born–Oppenheimer–type approximation. The main result is an explicit effective dynamics for the soliton and its radiation, including an emergent quadratic term in the soliton position that captures dispersion, established on time scales $t\, ext{up to}\, au_{loc}/\, oot 2 ext{ extancy}{g}$. These findings connect classical soliton modulation with quantum soliton dynamics and provide a rigorous framework for soliton behavior in quantum field theories with external fields.

Abstract

We first carry out the soliton sector quantization of the spatially cut-off $φ^4_{1+1}$ theory with double well potential in the semiclassical limit, deriving the nonrelativistic Schrödinger 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 $φ^4_{1+1}$ 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.

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

TL;DR

This work develops a Hamiltonian framework for quantizing solitons in the φ^4_{1+1} theory, focusing on the semiclassical limit where the coupling g is small. It constructs two CCR representations in the soliton sector and derives a precise description of the soliton as a nonrelativistic quantum particle with mass , while also accounting for transverse bosonic modes in the soliton background. A Dashen–Hasslacher–Neveu semiclassical mass shift is obtained via unitary equivalence between representations, and the paper then extends modulation theory to quantum field theory by coupling the soliton to an external electromagnetic field, proving a Born–Oppenheimer–type approximation. The main result is an explicit effective dynamics for the soliton and its radiation, including an emergent quadratic term in the soliton position that captures dispersion, established on time scales . These findings connect classical soliton modulation with quantum soliton dynamics and provide a rigorous framework for soliton behavior in quantum field theories with external fields.

Abstract

We first carry out the soliton sector quantization of the spatially cut-off theory with double well potential in the semiclassical limit, deriving the nonrelativistic Schrödinger 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 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.

Paper Structure

This paper contains 5 sections, 2 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Classical Theory.
  3. The soliton as a nonrelativistic quantum particle.
  4. Bosons in the soliton background.
  5. Modulation Theory in the presence of an external electric field

Key Result

Theorem 1.1

In the limit $\kappa\to+\infty$ the operators $\setstackgap{S}{1.25pt}. . \pmb{H}^{sol}_{g,{\pmb{b}},\kappa}\setstackgap{S}{1.25pt}. . $ determine a self-adjoint operator $\setstackgap{S}{1.25pt}. . \pmb{H}^{sol}_{g,{\pmb{b}}}\setstackgap{S}{1.25pt}. . $ on ${L^2(\mu_0)}$ which is bounded below an Here $\setstackgap{S}{.75pt}. . . H^{sol}_0\setstackgap{S}{.75pt}. . . $ is defined in simh while

Theorems & Definitions (7)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Example 1.4
  • Remark 1.5
  • Theorem 1.6
  • proof