Self-Consistent Maxwell–Pauli Framework

• Self-Consistent Maxwell–Pauli Framework is a coupled matter–field model where Pauli spinors generate electromagnetic sources via minimal coupling, incorporating spin, Darwin, and relativistic corrections.
• The framework employs variational methods and various gauge choices to derive canonical coupled equations, supporting travelling-wave and solitary structure solutions.
• It extends to ab initio implementations and many-body dynamics, linking semi-relativistic expansions with full minimal coupling for advanced light–matter simulations.

The self-consistent Maxwell–Pauli framework denotes a family of coupled matter–field models in which a two-component Pauli spinor, or Pauli–Kohn–Sham spinors, evolves under minimal electromagnetic coupling while the electromagnetic potentials satisfy Maxwell or Poisson-type equations sourced by charge and current densities constructed from the same spinorial state. Across the literature, the framework appears in one-body travelling-wave problems, semi-relativistic $1/c$ and $1/c^2$ expansions of Dirac–Maxwell theory, many-body classical-field models, and ab initio real-space light–matter simulations beyond the dipole approximation (Petersen et al., 2014, Dixit et al., 2014, Kieffer, 2020, Bonafé et al., 2024). This suggests that the expression identifies a common self-consistency principle—mutual matter–field back-reaction—rather than a single unique equation set.

1. Canonical coupled equations and gauge structure

In the one-body Coulomb-gauge formulation, the matter variable is a Pauli spinor $$\psi=(\psi_1,\psi_2)^T:\mathbb R^3\times\mathbb R\to\mathbb C^2$$, with charge $Q$ and mass $m$, coupled to electromagnetic potentials $\phi:\mathbb R^3\times\mathbb R\to\mathbb R$ and $A:\mathbb R^3\times\mathbb R\to\mathbb R^3$. The fields are

$$E=-\nabla\phi-\frac1c\partial_tA,\qquad B=\nabla\times A,$$

and Coulomb gauge imposes

$\nabla\cdot A=0,\qquad -\Delta\phi=4\pi Q|\psi|^2.$

The Pauli equation is

$i\hbar\,\partial_t\psi=H_P(A,\phi)\psi,$

with

$1/c^2$0

The corresponding current density is

$1/c^2$1

and the transverse vector potential satisfies Maxwell’s equation for the radiation field in Coulomb gauge (Petersen et al., 2014).

A distinct but closely related second-order weakly relativistic formulation keeps $1/c^2$2 and $1/c^2$3 as variational variables and starts from the Lagrangian density

$1/c^2$4

where $1/c^2$5 is the Schrödinger–Pauli Hamiltonian through $1/c^2$6. Varying the action with respect to $1/c^2$7 and $1/c^2$8 yields

$1/c^2$9

together with the homogeneous Maxwell equations. In that formulation, the closed theory is stated in terms of the six functions $$\psi=(\psi_1,\psi_2)^T:\mathbb R^3\times\mathbb R\to\mathbb C^2$$0, with gauge freedom removing one degree of freedom in $$\psi=(\psi_1,\psi_2)^T:\mathbb R^3\times\mathbb R\to\mathbb C^2$$1 and the continuity equation removing one in $$\psi=(\psi_1,\psi_2)^T:\mathbb R^3\times\mathbb R\to\mathbb C^2$$2, leaving exactly six independent field components (Cho, 2024).

Gauge fixing is not unique across the literature. Coulomb gauge is central in the travelling-wave existence theory and in full minimal-coupling Maxwell–TDDFT (Petersen et al., 2014, Bonafé et al., 2024), whereas Lorentz gauge is used in one semi-relativistic Lagrangian derivation and then expanded consistently to $$\psi=(\psi_1,\psi_2)^T:\mathbb R^3\times\mathbb R\to\mathbb C^2$$3 (Dixit et al., 2014). In the first-order semi-relativistic Pauli–Poisswell model, Coulomb gauge is again standard, although Lorenz gauge may also be adopted before dropping $$\psi=(\psi_1,\psi_2)^T:\mathbb R^3\times\mathbb R\to\mathbb C^2$$4 terms (Yang et al., 2023).

2. Source structure: charge, current, polarization, and magnetization

A defining feature of the self-consistent Maxwell–Pauli framework is that the sources are not inserted externally but derived from the same matter fields that evolve under the electromagnetic potentials. At second order in the weakly relativistic expansion, one finds

$$\psi=(\psi_1,\psi_2)^T:\mathbb R^3\times\mathbb R\to\mathbb C^2$$5

$$\psi=(\psi_1,\psi_2)^T:\mathbb R^3\times\mathbb R\to\mathbb C^2$$6

with

$$\psi=(\psi_1,\psi_2)^T:\mathbb R^3\times\mathbb R\to\mathbb C^2$$7

$$\psi=(\psi_1,\psi_2)^T:\mathbb R^3\times\mathbb R\to\mathbb C^2$$8

Defining the total electric polarization by $$\psi=(\psi_1,\psi_2)^T:\mathbb R^3\times\mathbb R\to\mathbb C^2$$9 and the total magnetization $Q$0 so that

$Q$1

one obtains the microscopic constitutive relations

$Q$2

These relations are presented as microscopic analogues of the usual macroscopic constitutive relations (Cho, 2024).

An equivalent free-plus-bound decomposition appears in the semi-relativistic Lagrangian treatment,

$Q$3

where

$Q$4

$Q$5

$Q$6

In this presentation, the Darwin correction is interpreted as smearing due to Zitterbewegung, while the spin-polarization term is interpreted as the Lorentz boost of the rest-frame magnetization (Dixit et al., 2014).

Quantity	Expression	Role
$Q$7	$Q$8	spin–orbit polarization
$Q$9	$m$0	Darwin polarization
$m$1	$m$2	spin magnetization

A recurrent point of controversy concerns “spin current.” In the second-order variational formulation, the only sources of Maxwell’s equations are $m$3 and $m$4 as defined above. One may define a spin density $m$5 and various candidate spin-current tensors $m$6, but none arise from $m$7 or $m$8, and none enter Maxwell’s equations. The local spin-continuity relation has the form

$m$9

with a spin-torque term, and is therefore not a true conservation law in the same sense as charge continuity (Cho, 2024).

3. Variational travelling waves and solitary structures

A mathematically sharp realization of the self-consistent Maxwell–Pauli framework is given by the travelling-wave problem for the one-body Maxwell–Pauli system. The monochromatic travelling-wave ansatz is

$\phi:\mathbb R^3\times\mathbb R\to\mathbb R$0

with constant velocity $\phi:\mathbb R^3\times\mathbb R\to\mathbb R$1 and unknown frequency $\phi:\mathbb R^3\times\mathbb R\to\mathbb R$2. After setting $\phi:\mathbb R^3\times\mathbb R\to\mathbb R$3, the coupled PDE reduces to a stationary system on $\phi:\mathbb R^3\times\mathbb R\to\mathbb R$4: $\phi:\mathbb R^3\times\mathbb R\to\mathbb R$5

$\phi:\mathbb R^3\times\mathbb R\to\mathbb R$6

where $\phi:\mathbb R^3\times\mathbb R\to\mathbb R$7 is the Helmholtz projector onto divergence-free fields, and

$\phi:\mathbb R^3\times\mathbb R\to\mathbb R$8

The electromagnetic energy entering this reduction is

$\phi:\mathbb R^3\times\mathbb R\to\mathbb R$9

The stationary problem is recast variationally by fixing the $A:\mathbb R^3\times\mathbb R\to\mathbb R^3$0-mass $A:\mathbb R^3\times\mathbb R\to\mathbb R^3$1 and considering

$A:\mathbb R^3\times\mathbb R\to\mathbb R^3$2

For the Pauli case, the functional is

$A:\mathbb R^3\times\mathbb R\to\mathbb R^3$3

Minimizers under the mass constraint solve the reduced travelling-wave system with a Lagrange multiplier $A:\mathbb R^3\times\mathbb R\to\mathbb R^3$4 (Petersen et al., 2014).

The existence theorem is formulated in terms of the critical speeds

$A:\mathbb R^3\times\mathbb R\to\mathbb R^3$5

where $A:\mathbb R^3\times\mathbb R\to\mathbb R^3$6 is the sharp Sobolev constant in $A:\mathbb R^3\times\mathbb R\to\mathbb R^3$7. For every $A:\mathbb R^3\times\mathbb R\to\mathbb R^3$8 with $A:\mathbb R^3\times\mathbb R\to\mathbb R^3$9, there exists $$E=-\nabla\phi-\frac1c\partial_tA,\qquad B=\nabla\times A,$$0 with $$E=-\nabla\phi-\frac1c\partial_tA,\qquad B=\nabla\times A,$$1 such that

$$E=-\nabla\phi-\frac1c\partial_tA,\qquad B=\nabla\times A,$$2

solves the Maxwell–Pauli system in Coulomb gauge, with $$E=-\nabla\phi-\frac1c\partial_tA,\qquad B=\nabla\times A,$$3. The proof proceeds by showing that $$E=-\nabla\phi-\frac1c\partial_tA,\qquad B=\nabla\times A,$$4 is bounded below on $$E=-\nabla\phi-\frac1c\partial_tA,\qquad B=\nabla\times A,$$5 below the critical speed, constructing a minimizing sequence, using concentration–compactness to prevent loss of mass at infinity, translating the sequence so that $$E=-\nabla\phi-\frac1c\partial_tA,\qquad B=\nabla\times A,$$6 is tight in $$E=-\nabla\phi-\frac1c\partial_tA,\qquad B=\nabla\times A,$$7, extracting a weakly convergent subsequence in $$E=-\nabla\phi-\frac1c\partial_tA,\qquad B=\nabla\times A,$$8, and then using lower semicontinuity to recover a minimizer (Petersen et al., 2014).

For small velocity, the energy satisfies

$$E=-\nabla\phi-\frac1c\partial_tA,\qquad B=\nabla\times A,$$9

so the leading-order term is exactly the classical kinetic energy of a particle of mass $\nabla\cdot A=0,\qquad -\Delta\phi=4\pi Q|\psi|^2.$0. The same analysis applies to the Maxwell–Schrödinger case by dropping the $\nabla\cdot A=0,\qquad -\Delta\phi=4\pi Q|\psi|^2.$1 term. No uniqueness of the minimizer is proved, and stability of the travelling waves is not addressed (Petersen et al., 2014).

4. Semi-relativistic expansions and coherent light-induced couplings

Several self-consistent Maxwell–Pauli models are derived as controlled approximations to the Dirac–Maxwell system. In one $\nabla\cdot A=0,\qquad -\Delta\phi=4\pi Q|\psi|^2.$2 derivation, the matter Lagrangian contains the Zeeman, Darwin, and spin–orbit terms, and variation with respect to $\nabla\cdot A=0,\qquad -\Delta\phi=4\pi Q|\psi|^2.$3 yields the extended Pauli equation

$\nabla\cdot A=0,\qquad -\Delta\phi=4\pi Q|\psi|^2.$4

Maxwell’s equations are expanded consistently to the same order. Writing $\nabla\cdot A=0,\qquad -\Delta\phi=4\pi Q|\psi|^2.$5 and $\nabla\cdot A=0,\qquad -\Delta\phi=4\pi Q|\psi|^2.$6, one finds

$\nabla\cdot A=0,\qquad -\Delta\phi=4\pi Q|\psi|^2.$7

so that there is no magnetic field at leading order, corresponding to the electric limit (Dixit et al., 2014).

A related semi-relativistic program starts from the Dirac equation with external and internal fields,

$\nabla\cdot A=0,\qquad -\Delta\phi=4\pi Q|\psi|^2.$8

performs a Foldy–Wouthuysen expansion to $\nabla\cdot A=0,\qquad -\Delta\phi=4\pi Q|\psi|^2.$9, and obtains a two-component Pauli Hamiltonian containing minimal coupling, Zeeman, Darwin, and spin–orbit terms built from $i\hbar\,\partial_t\psi=H_P(A,\phi)\psi,$0 and $i\hbar\,\partial_t\psi=H_P(A,\phi)\psi,$1. The charge and current are expanded as

$i\hbar\,\partial_t\psi=H_P(A,\phi)\psi,$2

with $i\hbar\,\partial_t\psi=H_P(A,\phi)\psi,$3 split into orbital, spin, and field-induced pieces, while $i\hbar\,\partial_t\psi=H_P(A,\phi)\psi,$4 is generated by $i\hbar\,\partial_t\psi=H_P(A,\phi)\psi,$5. In this formulation, the internal potentials are

$i\hbar\,\partial_t\psi=H_P(A,\phi)\psi,$6

The resulting Hamiltonian is decomposed into external-field Foldy–Wouthuysen terms, internal Breit–Pauli terms, and a coherent light-induced mean field $i\hbar\,\partial_t\psi=H_P(A,\phi)\psi,$7 (Hinschberger et al., 2016).

Within $i\hbar\,\partial_t\psi=H_P(A,\phi)\psi,$8, four coherent spin–light–induced mechanisms are identified:

1. Zeeman-like mechanism (A1): $i\hbar\,\partial_t\psi=H_P(A,\phi)\psi,$9, corresponding to $1/c^2$00 an effective light-induced magnetic field.
2. Spin–orbit-like mechanism (A2): $1/c^2$01, in which the external vector potential modifies orbital motion in the internal Coulomb field.
3. “$1/c^2$02” spin–other–orbit mechanism (B1): $1/c^2$03, where the internal spin current produces a vector potential that couples parametrically to $1/c^2$04.
4. Darwin-type spin–orbit mechanism (B2): $1/c^2$05, with

$1/c^2$06

so that a light pulse and local spin density generate an induced second-order Hartree potential (Hinschberger et al., 2016).

Taken together, these semi-relativistic derivations show that self-consistency extends beyond the bare Pauli Zeeman term to include Darwin corrections, spin–orbit structures, internal Breit–Pauli mean fields, and light-induced internal sources (Dixit et al., 2014, Hinschberger et al., 2016).

5. Many-body dynamics, energetic stability, and asymptotic limits

The many-body Maxwell–Pauli equations model $1/c^2$07 non-relativistic electrons with spin interacting with their self-generated classical electromagnetic field and $1/c^2$08 static nuclei. In the atomic units of Kieffer’s thesis, and in Coulomb gauge $1/c^2$09, the $1/c^2$10-component antisymmetric spinor $1/c^2$11 satisfies

$1/c^2$12

with the usual electron–electron, electron–nucleus, and nucleus–nucleus Coulomb terms in $1/c^2$13. Maxwell’s equations are

$1/c^2$14

$1/c^2$15

with

$1/c^2$16

and Coulomb gauge gives

$1/c^2$17

The conserved total energy is

$1/c^2$18

No further renormalization is needed (Kieffer, 2020).

The existence theory is tied to energetic stability. The absolute ground-state energy $1/c^2$19 is bounded below for general $1/c^2$20 provided

$1/c^2$21

yielding energetic stability of second kind,

$1/c^2$22

For $1/c^2$23, there is a critical nuclear charge $1/c^2$24 ($1/c^2$25) such that

$1/c^2$26

Assuming $1/c^2$27 and $1/c^2$28 are small enough so that $1/c^2$29, any initial data

$1/c^2$30

with $1/c^2$31 and divergence-free initial vector potential and time derivative, generate a global-in-time, finite-energy, weak solution

$1/c^2$32

which conserves both $1/c^2$33 and the total energy (Kieffer, 2020).

A lower-order semi-relativistic approximation is the Pauli–Poisswell system, described as the first order in $1/c^2$34 semi-relativistic approximation of the Dirac–Maxwell equation for 4-spinors coupled to self-consistent electromagnetic fields. It consists of

$1/c^2$35

coupled to the magnetostatic Poisson equations

$1/c^2$36

with

$1/c^2$37

The Hamiltonian

$1/c^2$38

is conserved. WKB analysis yields a mathematically rigorous semiclassical limit $1/c^2$39 from the Pauli–Poisswell equation to the magnetic Euler–Poisswell equation, with local wellposedness for Euler–Poisswell that is global unless a finite time blow-up occurs (Yang et al., 2023).

6. Ab initio full minimal coupling and beyond-dipole simulation

A contemporary computational realization of the self-consistent Maxwell–Pauli idea is the full minimal-coupling Maxwell–TDDFT framework implemented in Octopus. It is described as the first ab initio, non-relativistic QED method that couples light and matter self-consistently beyond the electric dipole approximation and without multipolar truncations. The matter subsystem uses a Pauli–Kohn–Sham Hamiltonian

$1/c^2$40

with transverse $1/c^2$41 in Coulomb gauge. The Maxwell subsystem evolves the transverse field according to

$1/c^2$42

or, equivalently, in the Riemann–Silberstein representation. The quantum current is split into

$1/c^2$43

with paramagnetic, diamagnetic, and magnetization-current parts, and $1/c^2$44 (Bonafé et al., 2024).

The multi-system cycle couples matter and Maxwell solvers on different real-space grids. Per global step $1/c^2$45, the code regrids $1/c^2$46 from the Maxwell grid to the matter grid, propagates the matter subsystem with a high-order exponential midpoint or Taylor propagator (4th order), evaluates $1/c^2$47 and the total current $1/c^2$48, regrids $1/c^2$49 onto the Maxwell grid, propagates the Maxwell subsystem for $1/c^2$50 steps of $1/c^2$51 with $1/c^2$52, solves the Poisson equation for $1/c^2$53 and gauge-corrects to maintain $1/c^2$54, then regrids the updated $1/c^2$55 back to the matter grid. Matter grids typically use $1/c^2$56 spacing, while the Maxwell grid is coarser, $1/c^2$57. Perfectly matched layers absorb outgoing radiation, and Dirichlet conditions can introduce plane waves (Bonafé et al., 2024).

System	Setup	Reported result
Cherenkov radiation of an electronic wavepacket	$1/c^2$58 in a refractive medium with $1/c^2$59	back-reaction changes spatially resolved density by $1/c^2$60 versus $1/c^2$61 in dipole-only back-coupling
Non-chiral benzene under XUV	$1/c^2$62, polarization and $1/c^2$63 both in-plane	magneto-optical spectrum has a distinct resonance around $1/c^2$64, absent in pure dipole spectra
$1/c^2$65 dimer	gap $1/c^2$66; full Maxwell–matter back-coupling	plasmon resonance red-shifts by $1/c^2$67; hot-spot phase shifts by $1/c^2$68

The framework is presented as origin-independent, with all multipole orders included non-perturbatively and self-consistently together with retardation and radiation reaction. It also retains exact treatment of paramagnetic and diamagnetic currents and spin–orbit (Pauli) coupling. The stated limitations are higher computational cost due to 3D Maxwell propagation, gauge issues with non-local pseudopotentials, and discretization constraints associated with Courant-type stability (Bonafé et al., 2024).

7. Alternative formulations, interpretive variants, and open directions

A distinct static-field usage of Maxwell–Pauli language appears in the Schrödinger–Pauli theory reformulated by the Quantal Newtonian first law,

$1/c^2$69

Here the external field is the sum of electrostatic and Lorentz fields, while the internal field is the sum of electron–interaction, differential-density, kinetic, and internal magnetic fields, each defined from expectation values of Hermitian operators. Because the total field is a known functional of the wave function, the binding potential is written as

$1/c^2$70

leading to the generalized self-consistent Schrödinger–Pauli equation

$1/c^2$71

This formulation further supports a local effective-potential mapping within quantal density-functional theory (Sahni, 2019).

Another interpretive variant introduces an additional vector field $1/c^2$72 into a unified action for $1/c^2$73, the electromagnetic field, and a Klein–Gordon-type $1/c^2$74-field. Variation produces a Maxwell–Pauli–$1/c^2$75 system in which the Pauli equation contains the nonlinear term $1/c^2$76, and $1/c^2$77 satisfies

$1/c^2$78

Within that framework, spontaneous emission is said to emerge from the atom’s own dipole radiation field, and spin density $1/c^2$79 together with magnetic moment density $1/c^2$80 are given a classical field interpretation, with $1/c^2$81 and $1/c^2$82 (Rashkovskiy, 2022).

A further construction treats $1/c^2$83 electrons and $1/c^2$84 photons through a single configuration-space wave function $1/c^2$85. In that approach, the expectation values of photon-field operators satisfy the inhomogeneous Maxwell equations with source terms given by the electron charge and current densities, and the algebraic structure of bosonic creation and annihilation operators is recovered in first-quantized guise without second quantization of the classical Maxwell field (Kiessling, 2020).

The framework also carries explicit unresolved issues. In the travelling-wave existence theory, no uniqueness of minimizers is proved and stability is not addressed (Petersen et al., 2014). In the many-body Maxwell–Pauli equations, open problems include local well-posedness without smallness assumptions, blow-up beyond the stability threshold, inclusion of full Coulomb self-interaction, semiclassical and non-relativistic limits, and extension to quantized fields (Kieffer, 2020). Possible extensions identified in the travelling-wave and ab initio literatures include many-body mean-field limits, Maxwell–Dirac, pseudo-relativistic solitary waves, twisted light and orbital angular momentum beams, strong-field physics, inelastic light scattering, and cavity QED or polaritonic chemistry (Petersen et al., 2014, Bonafé et al., 2024). At the level of source variables, the second-order variational theory explicitly rejects an additional “spin current” as an independent electromagnetic variable, retaining $1/c^2$86 as the minimal set (Cho, 2024).

Taken together, these developments define the self-consistent Maxwell–Pauli framework as a technically diverse but conceptually unified program: Pauli spinors or spinor Kohn–Sham orbitals generate electromagnetic sources, those sources determine fields through Maxwell or Poisson equations, and the resulting fields feed back into the matter Hamiltonian through minimal coupling, Zeeman, Darwin, spin–orbit, and, in semi-relativistic settings, higher-order internal-field terms.