In his 1952 Nature letter 'Is there an Æther?' — a reply to Infeld — Dirac conceded that the field equations of his 'new electrodynamics' can be rewritten so as not to require an aether, but argued that this is insufficient for a complete dynamical theory: an action principle and a Hamiltonian formulation suitable for quantization are needed, and 'for this the æther velocity is required'. The problem is Dirac's necessity claim for this constrained theory — the fixed-norm timelike structure tied to the electromagnetic four-potential by AμAμ=k2A_{\mu}A^{\mu}=k^{2}: whether the theory admits an action principle and a Hamiltonian formulation suitable for quantization, and whether the æther velocity is required for that construction, as Dirac insisted against Infeld. Later literature resolves only part of the broad titular question: Dirac constructed classical action principles and Hamiltonian formulations for the aether theory but never quantized it; QED is quantized without any aether; and modern preferred-frame (bumblebee/Einstein-aether, Lorentz-violation) theories and experiments leave the necessity question (quantization of the constrained theory) open; the wider existence question (Lorentz-symmetry violation in the photon and electron sectors) is modern successor framing, background to the problem rather than part of it.

Background

The nineteenth-century electromagnetic aether — an all-pervading medium assumed to carry light and electric forces — was abandoned after 1905 because a preferred rest frame of such a medium appeared incompatible with the principle of relativity, which requires all uniformly moving laboratories (inertial frames) to be equivalent for every law of physics. In a 1951 Nature letter Dirac reopened the question, concluding (in words quoted by Wright 2018) that 'we are rather forced to have an æther'. His grounds were a 'new classical theory of electrons' developed in a series of papers in the Proceedings of the Royal Society. In standard electrodynamics the electromagnetic four-potential AμA_{\mu} (the relativistic combination of scalar and vector potential) is physically redundant: it can be shifted by a gauge transformation without changing the observable fields. Dirac proposed to destroy this gauge freedom by imposing the pointwise constraint AμAμ=k2A_{\mu}A^{\mu}=k^{2}, with fixed constant kk and indices raised by the Minkowski metric of special relativity, so that the redundant variables acquire physical meaning as a stream of electric charge; the normalized vector Aμ/kA_{\mu}/k is a unit timelike vector field (at each spacetime point it defines a possible charge velocity, hence a local rest frame), which Dirac interpreted as the velocity a small test charge would have if introduced — the velocity of a revived aether. The field equations also contain a Lagrange multiplier λ\lambda (an auxiliary scalar enforcing the constraint), which Dirac read as signaling the presence of charge.

Leopold Infeld objected in April 1952 that eliminating λ\lambda from the equations leaves a system in which the constraint, and with it the aether, is no longer manifest. Dirac's reply — the source of the present problem — conceded the algebra but denied the conclusion: the letter's abstract states that Infeld 'has shown how the field equations of my new electrodynamics can be written so as not to require an æther', but that 'this is not sufficient to make a complete dynamical theory. It is necessary to set up an action principle and to get a Hamiltonian formulation of the equations suitable for quantization purposes, and for this the æther velocity is required.' An action principle is a variational formulation from which the field equations follow as Euler–Lagrange equations; a Hamiltonian formulation rewrites the theory in canonical coordinates and conjugate momenta, the structure from which canonical quantization proceeds by promoting them to operators. Dirac also expected that the classical theory would need to be modified by introducing spin variables (degrees of freedom describing the electron's intrinsic angular momentum) before a satisfactory quantum theory of electrons could be obtained. Dirac's own series continued to 1954, constructing classical action principles and Hamiltonian forms for the aether theory, but the quantization — the avowed goal since the first paper — never appeared.

Two later developments frame the modern question. First, quantum electrodynamics (QED), the quantized theory of electrons and photons, had already been constructed from an action principle with no aether or preferred frame, and its Lorentz-invariant form became established physics; so no aether is required for a quantum theory of the electromagnetic field and electrons as we possess it. Dirac nevertheless regarded renormalized QED — QED treated by renormalization, the standard procedure that absorbs the divergences of quantum-field calculations into redefined masses and charges — as unsatisfactory, and his question was whether a more fundamental quantized theory would force the aether back in. Second, the aether idea returned in a new guise: modern field theories built on a fixed-norm timelike vector field (a vector constrained exactly as Dirac constrained the potential) describing a dynamical preferred frame — the 'bumblebee' models of spontaneous Lorentz-symmetry breaking (in which formally symmetric equations are destabilized into a state singling out one frame) and Einstein-aether gravity (a theory of gravitation in which spacetime carries, besides the metric, a dynamical unit timelike vector field of just this kind), whose authors explicitly identify Dirac's theory as the non-gravitational sector. These theories, together with the experimental program searching for preferred-frame effects — Lorentz-symmetry violation, systematized in the Standard-Model Extension, a cataloguing framework for all such effects — keep Dirac's titular question alive as a modern successor framing.

Problem Statement

The problem is Dirac's necessity claim in the primary formulation of the 1952 Dirac–Infeld exchange: Infeld showed that the field equations of Dirac's new electrodynamics — the electromagnetic four-potential subject to the pointwise constraint AμAμ=k2A_{\mu}A^{\mu}=k^{2}, with Aμ/kA_{\mu}/k the unit-timelike 'aether velocity' stream — can be written so as not to require an æther, but Dirac held that this is not sufficient to make a complete dynamical theory: it is necessary to set up an action principle and to get a Hamiltonian formulation of the equations suitable for quantization purposes, and 'for this the æther velocity is required'. The open problem is to decide this necessity claim for the constrained theory itself: does Dirac's aether electrodynamics admit an action principle and a Hamiltonian formulation suitable for quantization, and is the aether velocity indispensable to that construction — as Dirac claimed against Infeld — or does a quantizable action/Hamiltonian formulation of the same constrained theory exist that dispenses with it?

Two features of the target are fixed by the source exchange:

  • The constrained theory is the one in which the fixed-norm condition is a physical constraint, not a mere gauge choice: Dirac destroyed the gauge freedom of the four-potential so that the redundant variables describe a stream of electric charge, and the unit timelike vector plays the dual role of gauge-fixed vector potential and flow vector of the charge stream (the identification later made explicit for the non-gravitational sector of Einstein-aether theory). A rewriting of the field equations alone — the point Dirac conceded to Infeld and declared insufficient for a complete dynamical theory — including one that treats AμAμ=k2A_{\mu}A^{\mu}=k^{2} merely as a gauge-fixing condition on ordinary Maxwell theory, in which the aether velocity is pure gauge and no physical stream survives, does not engage the claim.

  • The criterion is the source's own: an action principle and a Hamiltonian formulation 'suitable for quantization purposes' — that is, a formulation from which the canonical quantization of the constrained theory proceeds. Whether the aether velocity is 'required' is a question about such formulations of this constrained theory. For this task, a formulation dispenses with the aether velocity when it supplies a quantizable action principle and Hamiltonian for the constrained theory — the four-potential AμA_{\mu} held to AμAμ=k2A_{\mu}A^{\mu}=k^{2} as a physical constraint — without introducing the aether-velocity field Aμ/kA_{\mu}/k (or an equivalent velocity field) as an ingredient of the construction; exhibiting one such aether-free formulation would refute Dirac's necessity claim, which is confirmed only if no such formulation exists. The question concerns only the quantization of the constrained theory; the spin extension and the experimental preferred-frame (Standard-Model Extension) program are background, not part of the question.

References

  1. P. A. M. Dirac, 'Is there an Æther?', Nature 168, 906–907 (1951), DOI 10.1038/168906a0, https://doi.org/10.1038/168906a0
  2. P. A. M. Dirac, 'A new classical theory of electrons', Proceedings of the Royal Society of London A 209, 291–296 (1951), DOI 10.1098/rspa.1951.0204, https://doi.org/10.1098/rspa.1951.0204
  3. P. A. M. Dirac, 'Is there an Æther?', Nature 169, 146 (1952), DOI 10.1038/169146b0, https://doi.org/10.1038/169146b0
  4. K. J. Le Couteur, 'Dirac's New Electrodynamics', Nature 169, 146–147 (1952), DOI 10.1038/169146c0, https://doi.org/10.1038/169146c0
  5. E. Schrödinger, 'Dirac's New Electrodynamics', Nature 169, 538 (1952), DOI 10.1038/169538a0, https://doi.org/10.1038/169538a0
  6. L. Infeld, 'Is there an Æther?', Nature 169, 702 (1952), DOI 10.1038/169702a0, https://doi.org/10.1038/169702a0
  7. P. A. M. Dirac, 'Is there an Æther?', Nature 169, 702 (1952), DOI 10.1038/169702b0, https://doi.org/10.1038/169702b0
  8. P. A. M. Dirac, 'A new classical theory of electrons. II', Proceedings of the Royal Society of London A 212, 330–339 (1952), DOI 10.1098/rspa.1952.0085, https://doi.org/10.1098/rspa.1952.0085
  9. Banesh Hoffmann, 'Dirac's New Classical Theory of Electrons', Physical Review 87, 703–705 (1952), DOI 10.1103/physrev.87.703, https://doi.org/10.1103/physrev.87.703
  10. R. W. Iskraut, 'Angular Momentum in Dirac's New Electrodynamics', Nature 170, 1125–1126 (1952), DOI 10.1038/1701125a0, https://doi.org/10.1038/1701125a0
  11. P. A. M. Dirac, 'A new classical theory of electrons. III', Proceedings of the Royal Society of London A 223, 438–445 (1954), DOI 10.1098/rspa.1954.0129, https://doi.org/10.1098/rspa.1954.0129
  12. V. Alan Kostelecký and Stuart Samuel, 'Gravitational phenomenology in higher-dimensional theories and strings', Physical Review D 40, 1886–1903 (1989), DOI 10.1103/PhysRevD.40.1886, https://doi.org/10.1103/PhysRevD.40.1886
  13. S. Caser, 'Electrodynamics in Dirac's Gauge: A Geometrical Equivalence', Foundations of Physics Letters 14, 263–274 (2001), DOI 10.1023/A:1012286111139, https://doi.org/10.1023/A:1012286111139
  14. Ted Jacobson and David Mattingly, 'Gravity with a dynamical preferred frame', Physical Review D 64, 024028 (2001), DOI 10.1103/PhysRevD.64.024028, arXiv:gr-qc/0007031, https://arxiv.org/abs/gr-qc/0007031
  15. Marcelo Carvalho and Alexandre L. Oliveira, 'A New Version of the Dirac's Æther and Its Cosmological Applications', Foundations of Physics Letters 16, 255–263 (2003), DOI 10.1023/A:1025919310001, https://doi.org/10.1023/A:1025919310001
  16. Marcelo Carvalho, Alexandre L. Oliveira, and Carlos R. Rabaça, 'Dirac's æther in curved spacetime-II: the geometric amplification of the cosmic magnetic induction', Anais da Academia Brasileira de Ciências 76, 9–17 (2004), DOI 10.1590/S0001-37652004000100002, https://doi.org/10.1590/S0001-37652004000100002
  17. M. Gomes, J. R. Nascimento, A. Yu. Petrov, and A. J. da Silva, 'Aetherlike Lorentz-breaking actions', Physical Review D 81, 045018 (2010), DOI 10.1103/physrevd.81.045018, https://doi.org/10.1103/physrevd.81.045018
  18. A. F. Santos and Faqir C. Khanna, 'Aether field in extra dimensions: Stefan-Boltzmann law and Casimir effect at finite temperature', Physical Review D 95, 025021 (2017), DOI 10.1103/physrevd.95.025021, https://doi.org/10.1103/physrevd.95.025021
  19. Alexander B. Balakin and Anna O. Efremova, 'Interaction of the axionic dark matter, dynamic aether, spinor and gravity fields as an origin of oscillations of the fermion effective mass', European Physical Journal C 81, 674 (2021), DOI 10.1140/epjc/s10052-021-09341-z, https://doi.org/10.1140/epjc/s10052-021-09341-z
  20. Jakub Rembieliński and Jacek Ciborowski, 'Electrodynamics with a preferred frame', Physical Review A 98, 042107 (2018), DOI 10.1103/physreva.98.042107, https://doi.org/10.1103/physreva.98.042107
  21. Jacek Ciborowski and Jakub Rembieliński, 'Search for a preferred frame of the photon', Physical Review A 100, 032103 (2019), DOI 10.1103/physreva.100.032103, https://doi.org/10.1103/physreva.100.032103
  22. J. Rembieliński and K. A. Smolinski, 'Quantum Preferred Frame: Does It Really Exist?', EPL 88, 10005 (2009), DOI 10.1209/0295-5075/88/10005, arXiv:0910.5324, https://arxiv.org/abs/0910.5324
  23. Iver Brevik, 'Quantum electrodynamics and the ether', arXiv:2108.04671 (2021), https://arxiv.org/abs/2108.04671
  24. Richard D. Bateson, 'From Dirac's Aether to the Dirac Equation', Journal of High Energy Physics, Gravitation and Cosmology 10, 1450–1466 (2024), DOI 10.4236/jhepgc.2024.104081, https://doi.org/10.4236/jhepgc.2024.104081
  25. Aaron Sidney Wright, 'An Ether by Any Other Name?', in Jaume Navarro (ed.), Ether and Modernity, Oxford University Press (2018), DOI 10.1093/oso/9780198797258.003.0012, https://doi.org/10.1093/oso/9780198797258.003.0012
  26. David Mattingly, 'Modern Tests of Lorentz Invariance', Living Reviews in Relativity 8, 5 (2005), DOI 10.12942/lrr-2005-5, arXiv:gr-qc/0502097, https://arxiv.org/abs/gr-qc/0502097