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Global Gauge Symmetry Breaking in the Abelian Higgs Mechanism

Silvester Borsboom, Sebastian De Haro

TL;DR

This work resolves a tension in the gauge-invariant understanding of the Abelian Higgs mechanism by showing that global, not local, gauge symmetry breaking drives mass generation under boundary conditions. Using constrained Hamiltonian formalism, it identifies global gauge transformations as physical and Coulomb gauge as the appropriate setting for a gauge-invariant account, then harmonizes Struyve’s approach with the dressing-field method via a field-space, field-dependent DFM. In the quantum regime, Morchio–Strocchi results in the $C^*$-algebraic framework show that spontaneous breaking of global $U(1)$ yields massive photons and current screening, while unbroken symmetry yields massless photons and a Noether relation between charge and current. The paper thus reframes the Higgs mechanism as spontaneous global gauge-symmetry breaking, clarifying the ontology of gauge symmetries and offering a coherent bridge between classical and quantum treatments, with implications for non-Abelian generalizations and edge-mode considerations.

Abstract

This paper aims to resolve the incompatibility between two extant gauge-invariant accounts of the Abelian Higgs mechanism: the first account uses global gauge symmetry breaking, and the second eliminates spontaneous symmetry breaking entirely. We resolve this incompatibility by using the constrained Hamiltonian formalism in symplectic geometry. First we argue that, unlike their local counterparts, global gauge symmetries are physical in the presence of boundary conditions. The symmetry that is spontaneously broken by the Higgs mechanism is this global one. Second, we explain how the Coulomb gauge is the preferred gauge for a gauge-invariant account of the Abelian Higgs mechanism. Based on the existence of the physical global gauge symmetry, we resolve the incompatibility between the two accounts by arguing that the correct way to carry out the second method is to eliminate only the redundant gauge symmetries, i.e. those local gauge symmetries which are not global. We extend our analysis to quantum field theory, where we show that the Abelian Higgs mechanism can be understood as spontaneous global $U(1)$ symmetry breaking in the $C^*$-algebraic sense.

Global Gauge Symmetry Breaking in the Abelian Higgs Mechanism

TL;DR

This work resolves a tension in the gauge-invariant understanding of the Abelian Higgs mechanism by showing that global, not local, gauge symmetry breaking drives mass generation under boundary conditions. Using constrained Hamiltonian formalism, it identifies global gauge transformations as physical and Coulomb gauge as the appropriate setting for a gauge-invariant account, then harmonizes Struyve’s approach with the dressing-field method via a field-space, field-dependent DFM. In the quantum regime, Morchio–Strocchi results in the -algebraic framework show that spontaneous breaking of global yields massive photons and current screening, while unbroken symmetry yields massless photons and a Noether relation between charge and current. The paper thus reframes the Higgs mechanism as spontaneous global gauge-symmetry breaking, clarifying the ontology of gauge symmetries and offering a coherent bridge between classical and quantum treatments, with implications for non-Abelian generalizations and edge-mode considerations.

Abstract

This paper aims to resolve the incompatibility between two extant gauge-invariant accounts of the Abelian Higgs mechanism: the first account uses global gauge symmetry breaking, and the second eliminates spontaneous symmetry breaking entirely. We resolve this incompatibility by using the constrained Hamiltonian formalism in symplectic geometry. First we argue that, unlike their local counterparts, global gauge symmetries are physical in the presence of boundary conditions. The symmetry that is spontaneously broken by the Higgs mechanism is this global one. Second, we explain how the Coulomb gauge is the preferred gauge for a gauge-invariant account of the Abelian Higgs mechanism. Based on the existence of the physical global gauge symmetry, we resolve the incompatibility between the two accounts by arguing that the correct way to carry out the second method is to eliminate only the redundant gauge symmetries, i.e. those local gauge symmetries which are not global. We extend our analysis to quantum field theory, where we show that the Abelian Higgs mechanism can be understood as spontaneous global symmetry breaking in the -algebraic sense.

Paper Structure

This paper contains 17 sections, 7 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Global Gauge Symmetries as the Physical Gauge Group
  3. Gauge fields, gauge transformations and redundancies
  4. The group of physical symmetries
  5. The Gauss Law Constraint in Electromagnetism
  6. Redundant gauge symmetries are generated by constraints
  7. Electromagnetism
  8. Deriving Coulomb gauge
  9. Resolving the Incompatibility
  10. Struyve's gauge-invariant account of the Higgs mechanism
  11. The correct gauge-invariant DFM
  12. Abelian Higgs Mechanism in QED
  13. SSB in algebraic quantum theory
  14. The Higgs mechanism as quantum global gauge symmetry breaking
  15. The Noether relation and locality
  16. ...and 2 more sections

Key Result

Theorem 2.1

For the Maxwell theory as coupled to a Klein-Gordon scalar field in a simply-connected universe: given the physical content of two regions, for matter vanishing at the boundary but not in the bulk of the regions, the universal state is undetermined, resulting in a residual variety parametrized by an

Theorems & Definitions (10)

  • Theorem 2.1: Rigid variety for $U(1)$
  • Definition 4.1
  • Proposition 4.2
  • Definition 4.3
  • Definition 5.1
  • Proposition 5.2
  • Proposition 5.3
  • Proposition 5.4
  • Theorem 5.5
  • Theorem 5.6