Extension of Theory of Gravitomagnetism and Spinor Quantum Mechanics with Dynamics of Free Electromagnetic Field

The theory of Gravitomagnetism and spinor quantum mechanics describing the interaction between the Dirac spinor field, the electromagnetic field, and a weak gravitational field is extended by including the Lagrangian density of the free electromagnetic field. It is shown that the newly added term in the Lagrangian density is necessary to restore a symmetric energy-momentum tensor in the interaction term of the Lagrangian density within the theory. We prove that when the electromagnetic energy term is missing, the tensor in the interaction Lagrangian density does not represent the energy-momentum tensor of the Dirac field minimally coupled to the electromagnetic field on flat space-time. Moreover, it follows that the Lagrangian density remains unchanged regardless of whether the theory is developed from the principle of general covariance or is defined as a flat space-time theory with the gravitational field treated as a regular tensor field, in analogy to the electromagnetic field. This contradicts previous findings.