Einstein–Maxwell–Dirac equations

Not to be confused with the Einstein–Maxwell equations.

The Einstein–Maxwell–Dirac equations (EMD) are a classical field theory defined in the setting of general relativity. They are interesting both as a classical PDE system (a wave equation) in mathematical relativity, and as a starting point for some work in quantum field theory.

Because the Dirac equation is involved, EMD violates the positivity condition that is imposed on the stress-energy tensor in the hypothesis of the Penrose–Hawking singularity theorems. This condition essentially says that the local energy density is positive, an important requirement in general relativity (just as it is in quantum mechanics). As a consequence, the singularity theorems do not apply, and there might be complete EMD solutions with significantly concentrated mass which do not develop any singularities, but remain smooth forever. S. T. Yau has constructed some. It is known that the Einstein–Maxwell–Dirac system admits soliton solutions, i.e., "lumped" fields that persistently hang together, thus modelling classical electrons and photons.

The Einstein–Yang–Mills–Dirac Equations are an alternative approach to the Cyclic Universe which Penrose has been advocating as of 2020. They also imply that the massive compact objects now classified as black holes are actually quark stars, possibly with event horizons, but without singularities.

The EMD equations are a classical theory, but they are also related to quantum field theory. The current Big Bang model is a quantum field theory in a curved spacetime. No quantum field theory in a curved spacetime is mathematically well-defined; in spite of this, some theoreticians extract information from this hypothetical theory. On the other hand, the super-classical limit of the not mathematically well-defined QED in a curved spacetime is the mathematically well-defined Einstein–Maxwell–Dirac system. (One could get a similar system for the Standard Model.) The fact that EMD is, or contributes to, a super theory is related to the fact that EMD violates the positivity condition, mentioned above.

Program for SCESM

One way of trying to construct a rigorous QED and beyond is to attempt to apply the deformation quantization program to MD, and more generally, EMD. This would involve the following.

The Super-Classical Einstein-Standard Model:

1. Extend Flato et al's "Asymptotic Completeness, Global Existence and the Infrared Problem for the Maxwell–Dirac Equations" to SCESM;
2. Show that the positivity condition in the Penrose–Hawking singularity theorem is violated for the SCESM. Construct smooth solutions to SCESM having Dark Stars. See Hawking and Ellis, The Large Scale Structure of Space-Time
3. Follow three substeps:
   1. Derive approximate history of the universe from SCESM – both analytically and via computer simulation.
   2. Compare with ESM (the QSM in a curved space-time).
   3. Compare with observation. See Steven Weinberg, Cosmology
4. Show that the solution space to SCESM, F, is a reasonable infinite dimensional super-symplectic manifold. See V. S. Varadarajan, Supersymmetry for Mathematicians: An Introduction
5. The space of fields F needs to be quotiented by a big group. One hopefully gets a reasonable symplectic noncommutative geometry, which we now need to deformation quantize to obtain a mathematically rigorous definition of SQESM (quantum version of SCESM). See Sternheimer and Rawnsley, Deformation Theory and Symplectic Geometry
6. Derive history of the universe from SQESM and compare with observation.

See also

• Dirac equation
• Einstein field equations
• Maxwell's equations
• Quantum electrodynamics (QED)

Further reading

• Dereli, T.; Ozdemir, N.; Sert, O. (2010). "Einstein-Cartan-Dirac Theory in (1+2)-Dimensions". The European Physical Journal C. 73: 2279. arXiv:1002.0958. Bibcode:2013EPJC...73.2279D. doi:10.1140/epjc/s10052-013-2279-z.
• Finster, Felix; Hainzl, Christian (2011). "A Spatially Homogeneous and Isotropic Einstein-Dirac Cosmology". Journal of Mathematical Physics. 52 (4): 042501. arXiv:1101.1872. Bibcode:2011JMP....52d2501F. CiteSeerX 10.1.1.744.4551. doi:10.1063/1.3567157.
• Finster, Felix; Smoller, Joel; Yau, Shing-Tung (1999). "Particle-Like Solutions of the Einstein-Dirac Equations". Physical Review D. 59 (10): 104020. arXiv:gr-qc/9801079. Bibcode:1999PhRvD..59j4020F. CiteSeerX 10.1.1.30.3313. doi:10.1103/PhysRevD.59.104020.
• Finster, Felix; Smoller, Joel; Yau, Shing-Tung (1999). "Non-Existence of Black Hole Solutions for a Spherically Symmetric, Static Einstein-Dirac-Maxwell System". Communications in Mathematical Physics. 205 (2): 249–262. arXiv:gr-qc/9810048. Bibcode:1999CMaPh.205..249F. doi:10.1007/s002200050675.
• Finster, Felix; Smoller, Joel; Yau, Shing-Tung (2002). "Absence of Static, Spherically Symmetric Black Hole Solutions for Einstein-Dirac-Yang/Mills Equations with Complete Fermion Shells". Adv. Theor. Math. Phys. 4 (6): 1231–1257. arXiv:gr-qc/0005028. Bibcode:2000gr.qc.....5028F. doi:10.4310/ATMP.2000.v4.n6.a3.
• Bernard, Yann (2006). "Non-existence of black-hole solutions for the electroweak Einstein–Dirac–Yang/Mills equations" (PDF). Classical and Quantum Gravity. 23 (13): 4433–4451. Bibcode:2006CQGra..23.4433B. doi:10.1088/0264-9381/23/13/009.
• Finster, Felix; Smoller, Joel; Yau, Shing-Tung (2000). "The Interaction of Dirac Particles with Non-Abelian Gauge Fields and Gravity - Black Holes". The Michigan Mathematical Journal. 47 (2000): 199–208. arXiv:gr-qc/9910047. doi:10.1307/mmj/1030374678.
• Finster, Felix; Smoller, Joel; Yau, Shing-Tung (1999). "Non-Existence of Black Hole Solutions for a Spherically Symmetric, Static Einstein-Dirac-Maxwell System". Communications in Mathematical Physics. 205 (2): 249–262. arXiv:gr-qc/9810048. Bibcode:1999CMaPh.205..249F. doi:10.1007/s002200050675.
• Barrett, John W. (2007). "A Lorentzian version of the non-commutative geometry of the standard model of particle physics". Journal of Mathematical Physics. 48 (12303): 012303. arXiv:hep-th/0608221. Bibcode:2007JMP....48a2303B. doi:10.1063/1.2408400.
• Connes, Alain (2006). "Noncommutative Geometry and the standard model with neutrino mixing". Journal of High Energy Physics. 2006 (11): 081. arXiv:hep-th/0608226. Bibcode:2006JHEP...11..081C. doi:10.1088/1126-6708/2006/11/081.
• Varadarajan, V. S. (2004). Supersymmetry for Mathematicians: An Introduction. Courant Lecture Notes in Mathematics 11. American Mathematical Society. ISBN 978-0-8218-3574-6. Search this book on
• Deligne, Pierre (1999). Quantum Fields and Strings: A Course for Mathematicians. 1. American Mathematical Society. ISBN 978-0-8218-2012-4. Search this book on
• Deligne, Pierre (1999). Quantum Fields and Strings: A Course for Mathematicians. 2. American Mathematical Society. ISBN 978-0-8218-2012-4. Search this book on
• van Dongen, Jeroen (2010). Einstein's Unification. Cambridge University Press. ISBN 978-0-521-88346-7. Search this book on
• Vegt J.W. (2002). The Maxwell-Schrödinger-Dirac correspondence in Auto Confined Electromagnetic Fields. Annales de ;a Fondation Louis de Broglie. Volume 27 (1-18). Research Gate
• Vegt J.W. (1996) A particle-free model of matter based on electromagnetic self-confinement (III). Annales de la Fondation Louis de Broglie 21(4):481 - 506.

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