[Paper Review] A Note on Gauge Principle and Spontaneous Symmetry Breaking in Classical Particle Mechanics

Naohisa Ogawa|ArXiv.org|Jan 17, 1998

Relativity and Gravitational Theory4 references3 citations

TL;DR

This paper proposes a classical formulation of U(1) gauge theory with spontaneous symmetry breaking, deriving the Higgs mechanism and London equation without relying on quantum field theory. By introducing gauge symmetry from the outset and using a hydrodynamical interpretation of classical fields, it establishes a classical analog of superconductivity, linking gauge principles directly to macroscopic electromagnetic behavior.

ABSTRACT

The U(1) gauge field is usually induced from the gauge principle, that is, the extension of global U(1) phase transformation for matter field. However the phase itself is realized only for quantum theory. In this paper we introduce the U(1) gauge field and gauge coupling from the gauge principle classically. The gauge symmetry is spontaneously broken from the out set. The Higgs mechanism occurs and we obtain the London equation. The Hydrodynamical interpretation of classical field we utilized is given, and the relation to super conductivity is discussed.

Motivation & Objective

• To establish a classical framework for U(1) gauge theory that incorporates spontaneous symmetry breaking.
• To derive the Higgs mechanism and the London equation in a purely classical setting.
• To provide a hydrodynamical interpretation of classical fields that parallels quantum field theory mechanisms.
• To clarify the connection between classical gauge theories and superconductivity through symmetry breaking.
• To show that the gauge principle and spontaneous symmetry breaking can be consistently applied in classical particle mechanics without invoking quantum mechanics.

Proposed method

• Introduce the U(1) gauge field and coupling via the gauge principle at the classical level, treating phase transformations as fundamental symmetries.
• Implement spontaneous symmetry breaking from the outset by assuming a non-zero vacuum expectation value for the scalar field.
• Use a hydrodynamical analogy for the classical field, interpreting the phase and amplitude as fluid-like variables.
• Derive the effective dynamics of the gauge field after symmetry breaking, leading to the London equation.
• Apply the resulting equations to model the Meissner effect and magnetic screening in a classical superconducting context.
• Use the Lagrangian formalism to derive equations of motion that include both the gauge field and matter fields with explicit symmetry breaking.

Experimental results

Research questions

• RQ1 Can the Higgs mechanism and the London equation be derived in a classical field theory without relying on quantum field theory?
• RQ2 How can the gauge principle be consistently applied in classical particle mechanics to generate a massive gauge field?
• RQ3 What is the role of spontaneous symmetry breaking in generating mass for the gauge field in a classical setting?
• RQ4 How does the hydrodynamical interpretation of classical fields relate to the phenomenology of superconductivity?
• RQ5 Can a classical analog of the Higgs mechanism be constructed that reproduces key features of superconducting behavior?

Key findings

• The paper successfully derives the London equation from a classical gauge theory with spontaneous symmetry breaking.
• The Higgs mechanism is realized classically, resulting in a massive gauge field due to the Higgs mechanism without requiring quantum field theory.
• The hydrodynamical interpretation of the classical field allows for a fluid-like description of the phase and amplitude, enabling a classical analog of the Higgs mechanism.
• Spontaneous symmetry breaking is imposed from the beginning, leading to a non-zero vacuum expectation value that generates mass for the gauge field.
• The resulting theory reproduces key features of superconductivity, such as the Meissner effect, in a classical framework.
• The gauge field acquires mass through the Higgs mechanism, and the dynamics are consistent with the London penetration depth in superconductors.