Fluctuations in Aharonov-Bohm Electrodynamics

Abstract: We consider the application of the Fluctuation Dissipation Theorem (FDT) to the electrodynamics of Aharonov-Bohm (ABE), which differs from Maxwell's in that it allows for local non-conservation of charge. For the case of a system of non-conserved charges at thermal equilibrium we obtain the same spectral distribution of energy of the electromagnetic field as in Maxwell electrodynamics. However, the electric field contribution to that energy doubles that in Maxwell case, while the magnetic contribution is the same as in Maxwell theory, the electric excess energy is compensated by a negative contribution arising form the Aharonov-Bohm (AB) scalar field. For a conductor with local non-conservation of charge described by the $γ$ model, we derive the spectrum of current correlation at first order in $γ$, which results in a violet noise contribution added to the classical Johnson-Nyquist white noise result for the voltage fluctuations in a conductor.

• The paper demonstrates that applying the fluctuation dissipation theorem to Aharonov-Bohm electrodynamics yields a doubled electric field spectral density, balanced by a negative scalar contribution.
• It introduces the γ-model to account for deviations from local charge conservation, predicting a violet noise component that scales with ω², sample length squared, and the non-conservation parameter γ.
• The study provides experimentally testable predictions for voltage noise corrections in conductors, offering a robust framework for analyzing mesoscopic and quantum transport phenomena.

Fluctuation Phenomena in Aharonov-Bohm Electrodynamics

Introduction

This paper develops a formal and quantitative analysis of electromagnetic field fluctuations within the framework of Aharonov-Bohm electrodynamics (ABE), where local charge conservation is not strictly enforced. It systematically applies the Fluctuation Dissipation Theorem (FDT) to compare the equilibrium statistical properties of field fluctuations in ABE with those in Maxwell electrodynamics, and examines the physical consequences in real conductors, especially regarding their noise spectral properties. The study provides both a conceptual generalization of fluctuation theory in electromagnetic systems with non-conserved charge and proposes experimentally testable predictions derived from the extended theoretical framework.

Aharonov-Bohm Electrodynamics: Formal Structure

ABE extends Maxwellian electrodynamics by relaxing the requirement for strict local charge conservation, such that the divergence of the current density plus the time derivative of charge density, $I = \partial_t \rho + \nabla \cdot \mathbf{j}$, may be nonzero. In this context, the usual gauge freedom is restricted, as the Lorentz gauge condition is no longer generally applicable, leading to an unconstrained scalar potential sector. As a result, the scalar and vector potentials become formally independent dynamical variables, and the energy density acquires additional contributions involving a scalar field $S$ coupled to the four-potential.

The local electromagnetic energy continuity equation contains an extra source term $-I \phi$, modifying the usual Poynting theorem. The total energy density is

$$u = \frac{1}{\mu_0}\left( \frac{|\mathbf{E}|^2}{2c^2} + \frac{|\mathbf{B}|^2}{2} + \frac{\phi}{c^2}\partial_t S - \mathbf{A}\cdot\nabla S - \frac{S^2}{2} \right),$$

which implies that, in general, the equilibrium partitioning between electric, magnetic, and scalar contributions differs from Maxwell theory.

Application of the Fluctuation Dissipation Theorem

The authors recast the FDT for application in ABE by identifying the correct set of independent variables, leading to a block-diagonal (decoupled) structure for the generalized susceptibility tensor linking scalar/vector potentials to their corresponding sources. Explicitly, the retarded potential formulation remains the same as in Maxwell theory, but the presence or absence of charge conservation fundamentally alters the statistical correlations.

Key theoretical results include:

• The equilibrium spectral energy distribution (total energy density) in ABE matches the Planck result of standard quantum/statistical electrodynamics, with zero-point and thermal parts.
• The electric field contribution to the spectral energy density is doubled relative to Maxwell theory, whereas the magnetic field contribution remains unchanged. This is precisely compensated by a negative scalar field contribution; thus, the total energy density is unchanged but its internal partitioning is not.
• In the Maxwellian limit (strict local conservation), the Lorentz gauge constraint enforces a precise dependency between scalar and vector potential fluctuations, restoring the standard equal electric/magnetic energy partition.

This alteration in the partitioning of field fluctuations is a direct consequence of the increased field-theoretic degrees of freedom arising in ABE.

Electromagnetic Fluctuations in Conductors: The $\gamma$-Model

In real conductors, the authors introduce the $\gamma$-model as a phenomenological constitutive relation that parametrizes deviations from perfect local charge conservation through a parameter $\gamma$. The modified continuity equation reads

$$\partial_t \rho + (1 + \gamma) \nabla \cdot \mathbf{j} = 0,$$

where $\mathbf{j}_{nl} = \gamma \mathbf{j}$ represents a non-localized current component.

Applying the FDT to this model, the paper derives the equilibrium correlation function for current fluctuations. For a homogeneous, isotropic, non-polarizable conductor with frequency-independent conductivity $\sigma$, the current spectral density receives a correction:

• The first term reproduces the classical white Johnson-Nyquist noise ($S$0, constant spectrum).
• A second, $S$1-dependent “violet” component, proportional to $S$2, appears as a colored noise correction. This term is structurally non-local and could, in principle, be isolated experimentally at high frequencies or for sufficiently large samples, since it increases as $S$3 and $S$4.

Experimental and Theoretical Implications

The most direct physical implication of this analysis is the prediction of a measurable, frequency-dependent correction to the voltage noise spectrum (Nyquist noise) in conductors with nonzero $S$5. The violet noise correction scales linearly with the local charge non-conservation parameter $S$6, sample length squared, and the square of the measurement frequency: $S$7 For typical parameters ($S$8 cm, $S$9 GHz), this term can become comparable to an ohm-scale resistance for $-I \phi$0. The most promising experimental scenario is in low-resistance, short conductors at GHz frequencies, where the $-I \phi$1 enhancement is substantial, but the condition $-I \phi$2 remains valid, ensuring the applicability of the theory in this regime.

Theoretically, this work rigorously demonstrates that Planckian equilibrium remains intact in ABE at the total energy level, but the microscopic structure of field fluctuations is sensitive to local charge conservation. This insight is significant for quantum transport in mesoscopic or molecular-scale systems where collective effects or nonlocal quantum phenomena can activate the ABE regime, as suggested in recent ab initio studies [li2008definition, walz2015local, cabra2018simulation].

The analysis also establishes a framework for exploring open questions in high-frequency noise spectra and motivates precision noise spectroscopy as a test of extended electrodynamics.

Conclusion

The paper provides an authoritative, technically detailed application of the FDT in Aharonov-Bohm electrodynamics, highlighting distinct partitioning of field fluctuation energy and specific, experimentally accessible corrections to electrical noise spectra in realistic conductors. These predictions supply a roadmap for testing the physical relevance of ABE in laboratory settings and offer a systematic baseline for further theoretical investigations into nonlocal and collective quantum effects in condensed matter and transport systems. The formal apparatus developed here can serve as a reference framework for future studies of statistical field theory under violated charge conservation, possibly relevant in advanced mesoscopic, molecular, or topological regimes.

Reference: "Fluctuations in Aharonov-Bohm Electrodynamics" (2604.15913)