[Paper Review] The gauge non-invariance of Classical Electromagnetism

Germain Rousseaux|ArXiv.org|Jun 28, 2005

Quantum and Classical Electrodynamics11 references3 citations

TL;DR

This paper challenges the conventional view of gauge invariance in classical electromagnetism by arguing that gauge transformations introduce physical paradoxes—such as time-dependent potentials in stationary systems—rendering them unphysical. It concludes that potentials are not gauge-invariant, and instead, the potentials are only undetermined up to a constant, with gauge conditions better interpreted as physical constraints rather than mathematical conveniences.

ABSTRACT

"Physical theories of fundamental significance tend to be gauge theories. These are theories in which the physical system being dealt with is described by more variables than there are physically independent degree of freedom. The physically meaningful degrees of freedom then reemerge as being those invariant under a transformation connecting the variables (gauge transformation). Thus, one introduces extra variables to make the description more transparent and brings in at the same time a gauge symmetry to extract the physically relevant content. It is a remarkable occurrence that the road to progress has invariably been towards enlarging the number of variables and introducing a more powerful symmetry rather than conversely aiming at reducing the number of variables and eliminating the symmetry" [1]. We claim that the potentials of Classical Electromagnetism are not indetermined with respect to the so-called gauge transformations. Indeed, these transformations raise paradoxes that imply their rejection. Nevertheless, the potentials are still indetermined up to a constant.

Motivation & Objective

• To challenge the widely accepted notion that gauge transformations are physically meaningful symmetries in classical electromagnetism.
• To demonstrate that gauge transformations lead to paradoxes in stationary systems, such as time-dependent vector potentials in static fields.
• To argue that the indeterminacy of potentials is not due to gauge freedom but only up to a constant, with gauge conditions being physical constraints rather than mathematical choices.
• To reframe the interpretation of gauge conditions as physical continuity equations rather than arbitrary conventions.
• To advocate for the rejection of gauge transformations as physically invalid, preserving only a weaker form of invariance under constant shifts.

Proposed method

• Analyzes a one-dimensional stationary electric field with static potentials $ V = -Ex $, $ \mathbf{A} = 0 $, showing that a time-dependent gauge function $ f = -Ext $ leads to a time-varying vector potential $ \mathbf{A}' = -Et $.
• Applies the Stokes-Helmholtz-Hodge decomposition to separate the vector potential into longitudinal and transverse components, highlighting the role of gradients in gauge transformations.
• Examines the physical inconsistency of associating a time-varying vector potential with a static electric field, as it implies time-varying sources contrary to experimental observation.
• Uses the hydrodynamic analogy and the Riemann-Lorenz theory to interpret the vector potential as a physical electromagnetic impulse, analogous to momentum differences.
• Reinterprets the Coulomb and Lorenz gauge conditions as physical constraints (electromagnetic continuity equations), not mathematical fixes.
• Argues that gauge invariance should be replaced by invariance under constant shifts of the potentials, with the constant set to zero at infinity or via reference point.

Experimental results

Research questions

• RQ1 Do gauge transformations in classical electromagnetism lead to physically inconsistent descriptions, such as time-dependent potentials in stationary systems?
• RQ2 Is the indeterminacy of electromagnetic potentials truly due to gauge freedom, or is it only up to a constant?
• RQ3 Are gauge conditions merely mathematical conveniences, or do they represent physical constraints related to source continuity?
• RQ4 Can the vector potential be considered a physical quantity if it is not gauge-invariant, and what is its physical interpretation?
• RQ5 Is the standard view of gauge invariance in classical electromagnetism fundamentally flawed due to paradoxes arising from gauge transformations?

Key findings

• Gauge transformations lead to a paradox in stationary systems: a time-dependent vector potential arises from a static electric field, implying unphysical time-varying sources.
• The vector potential in a static capacitor setup cannot be time-dependent if the electric field is static, contradicting the predictions of gauge transformations.
• The potentials are not gauge-invariant; instead, they are only undetermined up to a constant, which is fixed by boundary conditions (e.g., vanishing at infinity).
• The Coulomb and Lorenz gauge conditions are not equivalent; they represent distinct physical constraints—specifically, continuity equations for charge and current.
• Gauge conditions should be renamed 'constraints' to reflect their physical role, not their mathematical convenience.
• Gauge invariance is preserved only in a weaker sense: the physical content is invariant under constant shifts of the potentials, not under arbitrary gauge transformations.