Gao, Shan (2026) The Gauge-Invariant Higgs Mechanism Is Incomplete: From Angular Momentum to Flux Quantization. [Preprint]

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Abstract

Gauge-invariant reformulations of the Higgs mechanism, such as Struyve's treatment of the Abelian Higgs model, eliminate gauge-dependent quantities in favor of local gauge-invariant variables: the Higgs magnitude \(\rho = |\phi|\) and the composite vector field \(B_\mu = A_\mu + \frac{1}{e}\partial_\mu\theta\). While mathematically equivalent to the standard theory for local calculations, these formulations suffer from a fundamental incompleteness that exactly parallels the well-known incompleteness of the Madelung hydrodynamic formulation of quantum mechanics. Just as the Madelung equations require an externally imposed quantization condition \(m\oint \mathbf{v}\cdot d\mathbf{r} + e\Phi = 2\pi n\) to enforce wave function single-valuedness, Struyve's formulation requires an analogous condition \(\oint_C B_\mu dx^\mu = \Phi + \frac{2\pi n}{e}\). This condition involves the global circulation of \(B_\mu\) and the magnetic flux \(\Phi\); it cannot be derived from the local equations of motion for \(\rho\) and \(B_\mu\). We demonstrate that this incompleteness already manifests for a single Higgs boson in a state with orbital angular momentum, where the circulation of \(B_\mu\) around the axis must be quantized even in the absence of vortices. Moreover, the problem persists and deepens for exotic topological defects such as Nielsen-Olesen vortices, where flux quantization requires the same externally imposed condition. Our analysis suggests that the gauge potential \(A_\mu\) and the phase \(\theta\) remain indispensable for a complete, self-contained description of the Higgs mechanism, carrying essential global and topological information that cannot be fully encoded in local gauge-invariant composites.

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Item Type:	Preprint
Creators:	Creators Email ORCID Gao, Shan sgao7319@uni.sydney.edu.au
Subjects:	Specific Sciences > Physics > Cosmology Specific Sciences > Physics > Quantum Field Theory Specific Sciences > Physics > Symmetries/Invariances
Depositing User:	Prof. Shan Gao
Date Deposited:	08 Mar 2026 01:31
Last Modified:	08 Mar 2026 01:31
Item ID:	28474
Subjects:	Specific Sciences > Physics > Cosmology Specific Sciences > Physics > Quantum Field Theory Specific Sciences > Physics > Symmetries/Invariances
Date:	7 March 2026
URI:	https://philsci-archive.pitt.edu/id/eprint/28474

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