Maxwell–Like Relations in Unified Theories

• Maxwell–like relations are mathematical structures that generalize classical Maxwell equations to diverse contexts such as gravity, thermodynamics, and quantum field theories.
• They enforce symmetry and integrability through generalized Bianchi identities and mixed derivative equalities that underpin conservation laws in various physical systems.
• These relations enable unified modeling across gauge theories, gravitational frameworks, and experimental protocols, offering insights into nonlinear, higher-dimensional, and topological dynamics.

Maxwell–like relations are mathematical structures, equations, and identification schemes that generalize or abstract the forms, symmetries, and operational principles of classical Maxwell equations into a broad range of physical contexts. These include relativistic gravity, gauge and field theories with or without torsion, thermodynamics of black holes and quantum critical systems, correlation-based quantum thermodynamics, and nonassociative or higher-geometrical frameworks. Maxwell–like relations also encompass generalized constitutive laws, integrability constraints, and mixed partial symmetry principles applied far beyond electromagnetism.

1. Foundations and Prototypical Maxwell–Like Equations

The classical Maxwell equations for electromagnetism provide the archetype: a set of antisymmetric tensorial relations, derived via local gauge invariance, that unite field strengths, sourced and homogeneous equations, and their energetic content. In gauge theory language, they follow from commutators of covariant derivatives, Bianchi identities, and local conservation laws. Many modern theories generalize these ideas by:

• Defining field strengths as commutators of appropriate derivatives.
• Enforcing homogeneous relations via generalized Bianchi identities.
• Establishing inhomogeneous equations, with “currents” dictated by matter or geometric sources.
• Introducing gauge compensating mechanisms that restore local invariance under generalized transformation groups.

Generalized Maxwell–like systems often exchange the physical content of charge, current, and electromagnetic fields for gravitational, thermodynamic, or more abstract geometric analogues, preserving the structural form of the equations (Resconi et al., 2017, Arbab, 2013).

2. Maxwell–Like Structures in Gravitation and Geometry

Gravitoelectromagnetism and Maxwell–Like Gravity

Various approaches have recast aspects of General Relativity in Maxwell–like form:

• Geroch (3+1) decomposition: For spacetimes admitting a Killing field (e.g., Schwarzschild), the metric can be decomposed such that one constructs a gravitational four-potential $V^\mu$ and antisymmetric field tensor $F^{\mu\nu}$ defined as $$F_{\mu\nu} = \partial_\mu V_\nu - \partial_\nu V_\mu$$. This enables a reformulation where Einstein's equations reduce to Maxwell–like field equations with four-current sources $J^\mu = \partial_\nu F^{\mu\nu}$; their structure ensures the equivalence to the standard Einstein tensor formulation when contracted with suitable source terms (Ogonowski et al., 2013).
• Gravity with torsion in generalized gauge approach: Using fully affine covariant derivatives, both curvature and torsion tensors arise as "field strengths," and Jacobi-type identities yield the homogeneous (Bianchi) equations. Inhomogeneous equations take the Maxwell-like form $$[D_\gamma, R_{\alpha\beta}] = \chi J_{\gamma\alpha\beta}$$, with currents $J_{\gamma\alpha\beta}$ generically including non-conservation terms tied to matter-energy non-conservation, providing mechanisms for novel couplings between matter, energy, gravity, and torsion (Resconi et al., 2017).
• Post-Newtonian/linearized gravity: The Damour–Soffel–Xu formalism expresses the first post-Newtonian metric and dynamics using gravitoelectric and gravitomagnetic fields $(\mathbf{E}_g, \mathbf{B}_g)$ that obey Maxwell-like equations, directly connecting with the momentum flow, stress tensors, and energetic flux of General Relativity (0808.2510).
• Maxwell–like equations for gravity from continuity equations: Even absent GR, starting from mass conservation and Gauss’s law for gravity, one obtains a complete set of Maxwell–like equations for gravitoelectric and gravitomagnetic fields (including energy density, stress, and analogous radiation) (Lopez, 2017).

Geometric and Algebraic Extensions

• Multi-time Hamilton–Jet frameworks: Maxwell–like equations are derived for electromagnetic-type fields on dual 1-jet bundles $J^{1*}(T, M)$ using polymomentum Hamiltonians, yielding Bianchi-type homogeneous and inhomogeneous (current-encoded) equations that generalize Maxwell’s system to multi-time/multimomentum geometry (Oana et al., 2012).
• Quaternionic Dirac and split-octonion formulations: Maxwell–like systems appear as symmetry-closed consequences of quaternionic or octonionic representations for Dirac fields, with field strengths and equations following from algebraic analyticity/Fueter-type conditions. These yield both standard and symmetrized (monopole-including) Maxwell-type equations, energy–momentum flux laws, and dualities (“triality” in the octonionic case) (Arbab, 2013, Gogberashvili et al., 2020).

3. Maxwell–Like Relations in Thermodynamics and Quantum Systems

Thermodynamic Maxwell Relations

Maxwell relations in thermodynamics arise from the equality of mixed second derivatives of thermodynamic potentials, enforced by the exactness of differentials. These have extensive applications:

• Kerr Black Hole Thermodynamics: The fundamental thermodynamic relation $U(S, J)$ admits Maxwell-type relations between derivatives of entropy, angular momentum, temperature, and angular velocity. The four Maxwell identities are most transparently visualized in “thermodynamic squares,” directly analogous to those for fluids or magnetic materials. These relations allow expressing response functions (heat capacities, susceptibilities, expansion coefficients) in terms of one another and connect black hole mechanics with conventional condensed-matter systems (Escamilla et al., 2015).

Generalized Relations in Quantum Many-Body Systems

• Correlation-based Maxwell relations: In quantum simulators and quantum statistical mechanics, generalized Maxwell–like relations permit the extraction of difficult-to-measure thermodynamic properties from more directly accessible local (e.g., two-point) correlation functions. If $$\hat H(\{\lambda_i\}) = \hat H_0 + \sum_i \lambda_i \hat C_i$$, then one has

$$\frac{\partial \langle \hat C_j \rangle}{\partial \lambda_i} = \frac{\partial \langle \hat C_i \rangle}{\partial \lambda_j}$$

for all pairs of conjugate variables. This provides “universal” access to pressure, chemical potential, magnetization, heat capacity, etc., via local observables, a powerful tool for quantum simulation of complex thermodynamics (Rist et al., 24 Jun 2025).

• Experimental protocols for topological states: In charge controlled quantum devices, thermodynamic entropy changes (e.g., across a quantum point contact in a topological phase) can be extracted via Maxwell relations connecting the derivative of entropy with respect to chemical potential to the derivative of charge with respect to temperature, enabling the experimental determination of topological entanglement entropy in fractional quantum Hall systems (Sankar et al., 2022).

4. Nonlinear, Higher-Component and Extended Maxwell–Like Systems

Four-Boson Maxwell Extensions

• Four–four Maxwell equations: The $U_q(1)$ invariant theory introduces a quadruplet of vector bosons—$A_\mu$ (photon), $U_\mu$ (neutral massive photon), and $V^\pm_\mu$ (charged vector bosons)—and constructs a coupled set of Maxwell–like equations, separating spin-1 (transverse) and spin-0 (longitudinal) sectors. Antisymmetric, symmetric (“granular”), and “collective” field strengths are formulated, leading to a richer phenomenology including induced polarization, magnetization, monopole sources, and photon self-coupling. This framework interpolates between ordinary Maxwell theory and nonlinear, photonics, spintronics, and electroweak-like regimes, each characterized by a different subset of the boson quartet and couplings (Doria et al., 2022).

Higher-Order and Multi-Derivative Frameworks

• Abstract geometric Maxwell–like equations: In advanced jet-space and multi-time Hamiltonian frameworks, Maxwell–like equations govern horizontal-vertical 2-forms built from deflection tensors and induced by quadratic polymomentum Hamiltonians, extending the classical Maxwell structure to multi-time, multi-momentum and higher-order tensorial domains (Oana et al., 2012).

5. Common Mathematical Structures and Physical Consequences

Table: Maxwell–Like Structures Across Contexts

Context	Field Strength/Equation	Physical/Mathematical Analogues
General Relativity (GEM, Geroch)	$F^{\mu\nu}, J^\mu$	Curvature, Einstein tensor, energy-stress
Non-conservative Gauge Gravity	$[D_\gamma, R_{\alpha\beta}]$	Torsion, non-conservation, matter coupling
Thermodynamics (Kerr, QMC)	Mixed derivatives of thermodynamic potentials	Response functions, entropy-charge relations
Quantum Simulation	$\partial_\lambda \langle \hat C \rangle$	Pair correlations, thermodynamical observables
Extended EM (Four–four Maxwell)	$F^I_{\mu\nu}$, $S^I_{\mu\nu}$	Multiboson fields, polarization, magnetization, monopoles

Maxwell–like relations universally impose:

• Symmetry and integrability constraints (commutative mixed partials, generalized Bianchi identities).
• Relations between response, susceptibility, and cross-correlation functions in thermodynamics.
• Conservation laws and local compensative mechanisms underpinning interactions of fields and matter.
• Generalized field-strength definitions for complex or higher-dimensional objects.

6. Applications, Regimes, and Experimental Protocols

Maxwell–like relations underpin:

• The computation of gravitational radiation, energy, and momentum transfer in both weak-field/post-Newtonian and exact frameworks.
• The derivation and measurement of thermodynamic response functions in black holes and quantum many-body systems.
• Universal characterization of strongly correlated and topological systems via experimentally accessible correlations.
• Unified and nonlinear electromagnetic scenarios, “spintronic,” photonic, and electroweak-like models, via the extension to multiple gauge bosons and nonlinear constitutive relations.
• The theoretical formulation and identification of topologically protected entropy and emergent phenomena in condensed matter.

Experimental protocols leveraging Maxwell–like thermodynamic relations, such as entropy-charge relations in quantum dots coupled to quantum point contacts, enable the extraction of quantities (e.g., topological entanglement entropy) previously considered inaccessible (Sankar et al., 2022).

7. Outlook and Integration with Broader Frameworks

The persistence of Maxwell–like structures in gravitational, thermodynamic, quantum, and geometric frameworks reflects the underlying mathematical symmetry and universality of gauge- and commutator-based field equations. Whether in the context of the Einstein equations, modern gauge extensions, topological quantum matter, or quantum simulation, these relations serve as organizing principles connecting conjugate variables, conservation laws, and collective response phenomena. Future research aims to exploit these structures for deeper insight into emergent physical law, efficient experimental measurement schemes, and theoretical unification across physical domains (Ogonowski et al., 2013, Resconi et al., 2017, Rist et al., 24 Jun 2025, Doria et al., 2022).