Einstein-Maxwell-Dirac equations
Encyclopedia
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 Theorem. 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 soliton
ic solutions, i.e., classical electron
s and photon
s. This is the kind of theory Einstein was hoping for. EMD is also a totally geometricized theory 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.
Deformation Theory and Symplectic Geometry by Daniel Sternheimer, John Rawnsley
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
. The current Big Bang
Big Bang
The Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...
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
Theory of everything
A theory of everything is a putative theory of theoretical physics that fully explains and links together all known physical phenomena, and predicts the outcome of any experiment that could be carried out in principle....
, EMD violates the positivity condition in the Penrose-Hawking Singularity Theorem. 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 soliton
Soliton
In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...
ic solutions, i.e., classical electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s and photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
s. This is the kind of theory Einstein was hoping for. EMD is also a totally geometricized theory 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
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
.
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.
Program for SCESM
The Super-Classical Einstein-Standard Model:- 1. Extend Asymptotic Completeness, Global Existence and the Infrared Problem for the Maxwell-Dirac Equations to SCESM (Memoirs of the American Mathematical Society), by M. Flato, Jacques C. H. Simon, Erik Taflin.
- 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 here: The Large Scale Structure of Space-Time by Stephen W. Hawking, G. F. R. Ellis
- 3. Follow three substeps
- i. Derive approximate history of the universe from SCESM - both analytically and via computer simulation.
- ii. Compare with ESM (the QSM in a curved space-time).
- iii. Compare with observation. See: Cosmology by Steven Weinberg
- 4. Show that the solution space to SCESM, F, is a reasonable infinite dimensional super-sympletic manifold. See: Supersymmetry for Mathematicians: An Introduction
- 5. The space of fields F needs to be quotiented by a big group. One hopefully gets a reasonable sympletic noncommutative geometry, which we now need to deformation quantize to obtain a mathematically rigorous definition of SQESM (quantum version of SCESM). See:
Deformation Theory and Symplectic Geometry by Daniel Sternheimer, John Rawnsley
- 6. Derive history of the universe from SQESM and compare with observation.
See also
- Quantum Field TheoriesQuantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
- QEDQuantum electrodynamicsQuantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...
- Dirac equationDirac equationThe Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...
- Maxwell equations
- Einstein equations