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.

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. And 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 Einstein was hoping for. In fact, in 1929 Weyl wrote 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 didn't 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 pi.

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.

Besides the classical stuff, the EMD equations are 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.

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