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Einstein–Maxwell–Dirac equations (EMD) are related to quantum field theory. The current Big Bang Model is a quantum field theory in a curved spacetime. Unfortunately, no such theory 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.) As a super theory, EMD violates the positivity condition in the Penrose–Hawking singularity theorems. Thus, it is possible that there would be complete solutions without any singularities – Yau has in fact constructed some. Furthermore, it is known that the Einstein–Maxwell–Dirac system admits of solitonic solutions, i.e., 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 a 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.
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.
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