Dirac's 1962 lecture 'Particles of finite size in the gravitational field' left a precisely stated but unfinished program: model a particle as a finite surface of radius larger than its Schwarzschild radius, coupled to general relativity through an action principle (Einstein-Hilbert exterior plus a surface area term), and settle (i) the boundary conditions at the particle surface — the gravitational analog of the perfect-conductor conditions of his extensible-electron model, including the treatment of the interior region, which the original formulation did not fix; (ii) stability of the stabilized spherical equilibrium against nonspherical (l≥1l \geq 1) surface distortions, after stability had been shown only for spherically symmetric pulsations; (iii) incorporation of spin-1/2, replacing the square-root Hamiltonian structure by one involving spin matrices; and (iv) the excitation spectrum and the tentative identification of the first spherical excited state with the muon. The subsequent literature (Sharma 1967-1969; Gnadig, Kunszt, Hasenfratz and Kuti 1978; Davidson and Paz 1993; Davidson and Rubin 2009, 2011; Trzetrzelewski 2011-2016) shows that the muon identification fails quantitatively in its original radial form (computed first excitations of about 43-137 electron masses versus the observed muon at about 207), that spin has been incorporated only in the non-gravitational model, and that nonspherical stability of the gravitating particle is explicitly still deferred in the gravitational reformulations. The open core — a stable, consistently matched, spinning action-principle model of a finite-size gravitating particle — survives.

Background

In general relativity, the Schwarzschild radius rs=2GM/c2r_s = 2GM/c^2 of a body of mass MM marks the scale below which, on Dirac's stated view, a point-particle description breaks down: signals cannot pass from inside rsr_s to outside, so the interior 'belongs to a different universe', and a physically acceptable particle coupled to the Einstein gravitational field must be a finite-sized object no smaller than its Schwarzschild radius. In a companion 1962 paper Dirac had built an 'extensible model of the electron': the electron is treated classically as a charged, perfectly conducting closed surface (a bubble) with no electromagnetic field inside, the Maxwell field outside, the potentials continuous across the surface, and an action equal to the four-dimensional Maxwell action over the exterior plus a term proportional to the three-dimensional area of the bubble's world-tube. That area term acts as a surface tension — the cohesive inward force that balances the outward Coulomb self-repulsion of the surface charge. The model has a stable spherical equilibrium whose radial pulsations follow from the action, and Bohr-Sommerfeld quantization (an early semiclassical rule imposing quantization conditions on action variables) of the radial pulsation put the first excited state at about 53 electron rest energies, which Dirac suggested might be the muon (the heavier charged lepton, about 207 electron masses); the model, however, contained no electron spin. The lecture 'Particles of finite size in the gravitational field' (Proc. R. Soc. Lond. A 270, 354, 1962) transfers this construction to gravitation: the particle is a surface of radius larger than its Schwarzschild radius, the exterior spacetime is Schwarzschild, and the action is the Einstein-Hilbert action (the standard action functional of general relativity) over the exterior plus a surface term proportional to the world-tube area. A sign reversal appears: in the gravitational case the surface term produces an inward pressure that reinforces rather than resists collapse, and the simplest model is unstable under spherically symmetric pulsations; stability in this spherical sector requires a further internal-energy term in the action — Dirac's gravitational model contained such a term, and in its earliest general-relativistic descendant the modified model was shown to remain stable only 'as long as spherical symmetry is maintained' (Sharma 1967). Three items are left unsettled in the original formulation and in the subsequent literature, together with the overall goal of a complete theory. First, the boundary conditions at the particle surface — the gravitational analog of the perfect-conductor conditions of the electromagnetic model — are not fixed within the variational principle, including the treatment of the interior region. (In modern language, matching two spacetime regions across a thin surface is governed by junction conditions relating the jump of the extrinsic curvature of the surface to its stress-energy; the general framework exists, but its implementation inside Dirac's specific action principle was not carried out.) Second, stability has been demonstrated only for spherically symmetric pulsations and only after the extra action term; nothing is established about nonspherical, higher-multipole (l≥1l \geq 1) distortions of the surface. Third, the model has no spin: the Hamiltonian is a square root of the surface degrees of freedom, and it was not known how to replace it by a structure involving spin matrices, so the model cannot directly describe fermions. Finally, the identification of the first spherical excited state with the muon was only a tentative expectation. Later related constructions — thin shells in general relativity and brane-world embeddings, in which the particle surface is treated as a dynamical membrane (brane) separating two spacetime regions — treat similar matching problems but do not complete Dirac's program.

Problem Statement

Complete Dirac's program of finite-size particles in the gravitational field. Within Einstein's general relativity, derive from an action principle a theory of a particle modeled as a finite surface of radius larger than its Schwarzschild radius — the action being the Einstein-Hilbert term over the exterior plus the surface area term and whatever additional terms are needed — and settle the questions the 1962 formulation left open, which jointly define what a physically acceptable completion means: (i) Boundary and matching conditions: determine the conditions at the particle surface, including the treatment of the interior region, that play the role of the perfect-conductor conditions of the electromagnetic model, so that the action principle is well posed. (ii) Nonspherical stability: determine whether the stabilized spherical equilibrium is stable against nonspherical distortions of the surface (multipole order l≥1l \geq 1), not only against spherically symmetric pulsations. (iii) Spin: incorporate spin-1/2 degrees of freedom, replacing the square-root Hamiltonian structure by one involving spin matrices, so that the finite-size particle can describe a fermion. (iv) Excitation spectrum: quantize the surface degrees of freedom, determine the excitation spectrum, and assess Dirac's tentative identification of the first spherical excited state with the muon. The four components support one objective — a satisfactory action-principle theory of finite-size particles interacting with the gravitational field — so that the same action and matching conditions underlie the stability analysis, the spin sector, and the spectrum. Recognized descendant formulations are admissible as the setting of the answer provided the connection to the source program is stated.

References

  1. P. A. M. Dirac, 'Particles of finite size in the gravitational field', Proc. R. Soc. Lond. A 270, 354-356 (1962). DOI: 10.1098/rspa.1962.0228, https://doi.org/10.1098/rspa.1962.0228
  2. P. A. M. Dirac, 'An extensible model of the electron', Proc. R. Soc. Lond. A 268, 57-67 (1962). DOI: 10.1098/rspa.1962.0124, https://doi.org/10.1098/rspa.1962.0124
  3. N. K. Sharma, 'Axially symmetric solution for the Dirac electron model', Il Nuovo Cimento A 48, 971-974 (1967). DOI: 10.1007/BF02721620, https://doi.org/10.1007/BF02721620
  4. N. K. Sharma, 'A stable charged sphere in general relativity', Il Nuovo Cimento A 48, 975-981 (1967). DOI: 10.1007/BF02721621, https://doi.org/10.1007/BF02721621
  5. N. K. Sharma, 'Some extensions of the general-relativistic particle model', Il Nuovo Cimento B 59, 121-124 (1969). DOI: 10.1007/BF02711025, https://doi.org/10.1007/BF02711025
  6. P. Gnadig, Z. Kunszt, P. Hasenfratz, J. Kuti, 'Dirac's extended electron model', Annals of Physics 116, 380-407 (1978). DOI: 10.1016/0003-4916(78)90238-5, https://doi.org/10.1016/0003-4916(78)90238-5
  7. T. Fliessbach, 'Simplified version of Dirac's extensible model of the electron', American Journal of Physics 49, 432-434 (1981). DOI: 10.1119/1.12692, https://doi.org/10.1119/1.12692
  8. L. Consoni, Y. Nogami, A. B. Volkov, 'Nonrelativistic simulation of Dirac's bag', Lettere al Nuovo Cimento 44, 555-559 (1985). DOI: 10.1007/BF02746760, https://doi.org/10.1007/BF02746760
  9. M. Onder, R. W. Tucker, 'Semiclassical investigation of a charged relativistic membrane model', Journal of Physics A 21, 3423-3429 (1988). DOI: 10.1088/0305-4470/21/17/012, https://doi.org/10.1088/0305-4470/21/17/012
  10. A. Davidson, U. Paz, 'Threefold family of charged spin 1/2 Dirac bubbles', Physics Letters B 300, 234-240 (1993). DOI: 10.1016/0370-2693(93)90359-P, https://arxiv.org/abs/hep-th/9302081
  11. A. Davidson, S. Rubin, 'Brane gravitational extension of Dirac's extensible model of the electron', Classical and Quantum Gravity 26, 235006 (2009). DOI: 10.1088/0264-9381/26/23/235006, https://arxiv.org/abs/0907.1189
  12. A. Davidson, S. Rubin, 'Extensible gravitational Dirac models of the electron', Classical and Quantum Gravity 28, 125005 (2011). DOI: 10.1088/0264-9381/28/12/125005, https://doi.org/10.1088/0264-9381/28/12/125005
  13. M. Trzetrzelewski, 'Dirac equation for membranes', Physical Review D 84, 087701 (2011). DOI: 10.1103/PhysRevD.84.087701, https://arxiv.org/abs/1103.1964
  14. M. Trzetrzelewski, 'Dirac equation for embedded four-geometries', International Journal of Modern Physics D 22, 1350024 (2013). DOI: 10.1142/S0218271813500247, https://arxiv.org/abs/1103.2594
  15. M. Trzetrzelewski, 'Dirac equation for strings', EPL 116, 40001 (2016). DOI: 10.1209/0295-5075/116/40001, https://arxiv.org/abs/1302.5907
  16. B. Paul, 'Gauge symmetry and Virasoro algebra in quantum charged rigid membrane — a first order formalism', Physical Review D 87, 045003 (2013). DOI: 10.1103/PhysRevD.87.045003, https://arxiv.org/abs/1212.5902
  17. A. Markou, 'On a model of the electron and the other leptons', Physics Essays 25, 547-552 (2012). DOI: 10.4006/0836-1398-25.4.547, https://doi.org/10.4006/0836-1398-25.4.547