Dirac-electron hidden momentum beyond order $1/c^2$: higher-order spin corrections and force balance under nonzero average acceleration

Van Vleck and Huang (Physics Letters A 28, 768 (1969)) analyzed the hidden momentum of the Dirac electron — the spin-dependent linear momentum that an electron in an electric field carries with no corresponding motion of its centre of energy — via a four-component Dirac-spinor expansion carried to order $1/c^2$. They obtained the spin-induced displacement of the position expectation value, the velocity expectation value, and the spin-dependent momentum change, matching Shockley's earlier four-component result and reconciling with the two-component (Pauli-based) treatment of Coleman and Van Vleck once the condition of vanishing average acceleration is imposed. Two restrictions are evident from the analysis: all expansions are truncated at order $1/c^2$, and systems with nonzero average acceleration were not settled. The open problem posed here is the completion of that analysis: compute the relativistic corrections of the next nonzero order beyond $1/c^2$ to the position, velocity and spin-momentum observables (equivalently, the corrections to the electron hidden momentum $\mathbf{m}\times\mathbf{E}/c^2$), and establish the two- versus four-component force balance for accelerated systems without the no-acceleration condition.

Background

A magnetic dipole at rest in a static electric field carries a purely mechanical linear momentum $\mathbf{m}\times\mathbf{E}/c^2$ (with $\mathbf{m}$ the magnetic-dipole moment, $\mathbf{E}$ the electric field and $c$ the speed of light) even though nothing in the system is visibly moving. Shockley and James discovered this hidden momentum in 1967 in the charge–magnet system: the momentum is hidden in the internal relativistic motion of the current carriers (an electric field does different amounts of work on carriers moving at different speeds around the loop) and is exactly cancelled by the momentum of the electromagnetic field, so that the centre of energy — the relativistic analogue of the centre of mass, mass-weighting replaced by energy-weighting — remains at rest. Hidden momentum is now a standard ingredient in resolving momentum-balance and torque paradoxes of classical electromagnetism; the review by Babson, Reynolds, Bjorkquist and Griffiths (2009) surveys the classical theory.

For the electron the same question arises in relativistic quantum mechanics, and it proved subtle. Shockley (1968) computed from the Dirac equation the spin-dependent change in the linear momentum of a Dirac-electron wave packet in an external electric field, a hidden linear momentum associated with the $\boldsymbol{\alpha}\cdot\mathbf{E}$ term of the Dirac Hamiltonian, where $\boldsymbol{\alpha}$, together with $\beta$, are the 4×4 Dirac matrices. Coleman and Van Vleck (1968) derived the corresponding back-force on a magnet from a two-component treatment — a Lagrangian formulation built on the two-component Pauli spinor and a nonrelativistic Hamiltonian with spin–orbit coupling, whose velocity-dependent terms encode Thomas precession, the kinematic relativistic rotation of the rest frame of a spinning, accelerating particle. Their derivation required an extra condition: the magnet must suffer no average acceleration. This left an apparent discrepancy, of a factor of two, between the two-component (Pauli-based) and four-component (Dirac-based) predictions of the spin-dependent momentum change unless the no-acceleration condition is imposed.

Van Vleck and Huang (1969) re-examined the question with the full four-component Dirac formalism. Writing the small components $\chi$ of the Dirac spinor in terms of the large components $\varphi$ (an expansion $\chi=\Gamma\varphi$ in powers of $1/c$) and normalizing the large-component spinor correctly to order $1/c^2$, they obtained three results to order $1/c^2$: (i) the position expectation value $r_4$ evaluated with the four-component spinor equals the two-component position expectation $r_2$ plus a spin-dependent displacement

$$\Delta r=\frac{\hbar}{4m^2c^2}(\mathbf{p}\times\boldsymbol{\sigma}),$$

where $m$ is the electron mass, $\mathbf{p}$ the canonical momentum and $\boldsymbol{\sigma}$ the vector of Pauli matrices — the displacement by which spin shifts the electron's centre of energy; (ii) the velocity expectation

$$c\langle\boldsymbol{\alpha}\rangle=\left\langle\frac{\mathbf{p}}{m}-\frac{\mathbf{p}^3}{2m^3c^2}+\frac{e\hbar}{2m^2c^2}\,\mathbf{E}\times\boldsymbol{\sigma}\right\rangle,$$

with $e$ the electron charge; and (iii) the spin-dependent change of linear momentum in an electric field, $\Delta\mathbf{p}_{spin}=-e\hbar(\mathbf{E}\times\boldsymbol{\sigma})/2mc^2$, which matches Shockley's result and is the electron-spin hidden momentum $\mathbf{m}\times\mathbf{E}/c^2$. They noted that the four-component formalism contains the centre-of-energy and Thomas effects automatically, and that the averaged back-force agrees with the two-component treatment precisely when the no-average-acceleration condition holds. The analysis, however, is carried only to order $1/c^2$ — terms of order $1/c^3$ and higher in the expansions are omitted — and the behaviour of accelerated systems is not settled.

The modern framework for such comparisons is the Foldy–Wouthuysen transformation, a unitary transformation that block-diagonalizes the Dirac Hamiltonian and so connects the four-component Dirac description with the two-component Pauli description order by order in $1/c$. Within it one distinguishes the position of the electron's charge (simple in the Dirac representation) from its mean position (simple in the Foldy–Wouthuysen representation); the two differ by the spin-dependent shift above, which is the leading term of an exact free-electron expression. The Dirac position operator executes the Zitterbewegung, a rapid oscillation of amplitude comparable to the Compton wavelength that underlies the usual microscopic picture of the electron's spin and magnetic moment. Using this framework, Filho and Saldanha (2015) established hidden momentum in a hydrogen atom perturbatively, and Cameron and Cotter (2018a,b) re-derived the electron and nuclear hidden momentum $\mathbf{m}\times\mathbf{E}/c^2$, its cancellation by field momentum in atoms and molecules, and the general molecular momentum bookkeeping — but, like Van Vleck and Huang, only to order $1/c^2$, with the $O(1/c^3)$ remainders left unevaluated, and without revisiting the accelerated-regime comparison of the two-component and four-component formalisms.

Problem Statement

Complete the Van Vleck–Huang (1969) analysis of the spin-dependent position, velocity and momentum of the Dirac electron by removing its two standing restrictions. The two requested results are independent completions of those restrictions in, respectively, relativistic order and dynamical regime; each is to be established in its own stated conventions and at its own stated relativistic order, within both the four-component Dirac description and the two-component Pauli/Foldy–Wouthuysen description.

(1) Higher-order relativistic corrections. Compute the corrections of the next nonzero order beyond $1/c^2$ (the order at which the next nonzero corrections appear in $1/c$ is itself part of the result and must be determined, not assumed) to the three observables of the Van Vleck–Huang analysis: (i) the difference $\Delta r$ between the four-component and two-component position expectation values, whose leading term is $\hbar(\mathbf{p}\times\boldsymbol{\sigma})/4m^2c^2$; (ii) the velocity expectation $c\langle\boldsymbol{\alpha}\rangle$, whose leading terms are $\mathbf{p}/m-\mathbf{p}^3/2m^3c^2+(e\hbar/2m^2c^2)\mathbf{E}\times\boldsymbol{\sigma}$; and (iii) the spin-dependent change of the electron's linear momentum induced by an external electric field, whose leading term is $-e\hbar(\mathbf{E}\times\boldsymbol{\sigma})/2mc^2$ — equivalently, the corrections beyond leading order to the electron hidden momentum $\mathbf{m}\times\mathbf{E}/c^2$. The conventions (position operator used, normalization of the two-component spinor, treatment of the scalar and vector potentials, and the weak-field, low-momentum domain of the asymptotic expansion) must be stated, and the results must reduce to the known order-$1/c^2$ expressions and, for vanishing external fields, to the exact free-electron expressions.

(2) Nonzero average acceleration. Establish the force balance and hidden-momentum accounting for a system in which the magnet (or electron) undergoes nonzero average acceleration, without imposing the Coleman–Van Vleck condition of vanishing average acceleration: either show that, with all terms of the working order retained consistently, the two-component and four-component treatments still yield the same averaged back-force (identifying the terms that replace the condition, including the time derivative of the hidden momentum), or exhibit a concrete system in which the two treatments give genuinely different physical predictions and interpret the difference.

Each of the two results is to be established at its own explicitly stated working order and conventions (part (2) at the leading relativistic order at which the Coleman–Van Vleck comparison is made), so that the spin-dependent position, velocity and momentum of the Dirac electron in external electromagnetic fields are known beyond order $1/c^2$, and the back-force is accounted for in the accelerated regime, with the two- and four-component descriptions shown to agree or shown to differ with a physical account of the difference.

References

1. J. H. Van Vleck and N. L. Huang, 'Note on the Dirac electron and hidden momentum forces', Physics Letters A 28, 768–769 (1969). DOI: 10.1016/0375-9601(69)90612-4, https://doi.org/10.1016/0375-9601(69)90612-4
2. W. Shockley and R. P. James, ''Try Simplest Cases' Discovery of 'Hidden Momentum' Forces on 'Magnetic Currents'', Physical Review Letters 18, 876–879 (1967). DOI: 10.1103/PhysRevLett.18.876, https://doi.org/10.1103/PhysRevLett.18.876
3. S. Coleman and J. H. Van Vleck, 'Origin of "Hidden Momentum Forces" on Magnets', Physical Review 171, 1370–1375 (1968). DOI: 10.1103/PhysRev.171.1370, https://doi.org/10.1103/PhysRev.171.1370
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