Abstract
We address a long-standing debate over whether classical magnetic forces can do work, ultimately answering the question in the affirmative. In detail, we couple a classical particle with intrinsic spin and elementary dipole moments to the electromagnetic field, derive the appropriate generalization of the Lorentz force law, show that the particle’s dipole moments must be collinear with its spin axis, and argue that the magnetic field does mechanical work on the particle’s elementary magnetic dipole moment. As consistency checks, we calculate the overall system’s energy-momentum and angular momentum, and show that their local conservation equations lead to the same force law and therefore the same conclusions about magnetic forces and work. We also compute the system’s Belinfante–Rosenfeld energy–momentum tensor.
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Notes
For a more extensive treatment of the results in this paper, see [2].
This group-theoretic definition of the particle’s phase space is the classical counterpart to Wigner’s classification [17] of quantum particle-types based on irreducible Hilbert-space representations of the Poincaré group.
We thank David Griffiths for pointing out that a defining feature of Maxwell’s original theory is the inclusion of magnetic dipoles solely of the Ampère type, without magnetic monopoles, dyons, Gilbert dipoles, or elementary magnetic dipoles.
For maximum generality and to avoid introducing any unnecessary constraints into the particle’s Lagrangian formulation, it is convenient to wait until after deriving the particle’s equations of motion before imposing the simplifying condition that \(\lambda\) is the particle’s proper time \(\tau\).
The quantum-mechanical analogue of this classical collinearity condition follows from the Wigner-Eckart theorem.
Explicitly, we have \(\dot{\theta }^{\mu \nu }=(i/2)\mathrm {Tr}[\sigma ^{\mu \nu }\dot{\Lambda }\Lambda ^{-1}]\), where \([\sigma ^{\mu \nu }]^{\alpha }{}_{\beta }=-i\eta ^{\mu \alpha }\delta _{\beta }^{\nu }+i\delta _{\beta }^{\mu }\eta ^{\nu \alpha }\) are the generators of the Lorentz group.
We thank Sebastiano Covone for suggesting the relevance of these results to the Bohr-van Leeuwen theorem [6, 13]. The Bohr-van Leeuwen theorem assumes the original Lorentz force law without contributions from elementary dipole moments, and asserts that a non-rotating system of particles, when treated classically, always has a vanishing average magnetization at thermal equilibrium. The theorem’s implication is that phenomena like diamagnetism can only be understood in terms of quantum effects, a view challenged by our results, at least in principle.
The authors of [10] decompose the overall energy-momentum tensor by including the interaction terms with the energy-momentum tensor for the particle, an approach that obscures the work being done by the electromagnetic field on the particle.
For a review, see [8].
This formula differs from the corresponding result in [7], whose energy-momentum tensor yields the correct equations of motion for the particle only after an unjustified four-dimensional integration by parts.
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Acknowledgements
J. A. B. has benefited from personal communications with Gary Feldman, Howard Georgi, Andrew Strominger, Bill Phillips, David Griffiths, David Kagan, David Morin, Logan McCarty, Monica Pate, Alex Lupsasca, and Sebastiano Covone.
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Barandes, J.A. On Magnetic Forces and Work. Found Phys 51, 79 (2021). https://doi.org/10.1007/s10701-021-00483-4
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DOI: https://doi.org/10.1007/s10701-021-00483-4