Einstein–Maxwell–Dirac 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.

Not to be confused with the Einstein–Maxwell equations.

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. Indeed, S. T. Yau has constructed some. Furthermore, 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.

This is the kind of theory Albert Einstein was hoping for. In fact, in 1929 Weyl wrote to Einstein that any unified theory would need to include the metric tensor, a gauge field, and a matter field. Einstein considered the Einstein–Maxwell–Dirac system by 1930. He probably did not develop it because he was unable to geometricize it. It can now be geometricized as a non-commutative geometry; here, the charge e and the mass m of the electron are geometric invariants of the non-commutative geometry analogous to π.

The Einstein–Yang–Mills–Dirac Equations provide an alternative approach to the Cyclic Universe which Penrose has recently been advocating. 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. Unfortunately, no quantum field theory in a curved spacetime is mathematically well-defined; in spite of this, theoreticians claim to 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"[1] 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[2]
  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[3]
  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[4]
  6. Derive history of the universe from SQESM and compare with observation.

See also

References
  1. Flato, Moshé; Simon, Jacques Charles Henri; Taflin, Erik (1997). "Asymptotic Completeness, Global Existence and the Infrared Problem for the Maxwell-Dirac Equations". Memoirs of the American Mathematical Society. American Mathematical Society. ISBN 978-0198526827.
  2. Weinberg, Steven (2008). Cosmology. Oxford University Press. ISBN 978-0198526827.
  3. V. S. Varadarajan (2004). Supersymmetry for Mathematicians: An Introduction. Courant Lecture Notes in Mathematics. 11. American Mathematical Society. ISBN 978-0821835746.
  4. Sternheimer, Daniel; Rawnsley, John; Gutt, Simone, eds. (1997). "Deformation Theory and Symplectic Geometry". Mathematical Physics Studies. 20. Kluwer Academic Publishers. ISBN 978-0792345251. Cite journal requires |journal= (help)
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