A modification of Einstein–Schrödinger theory that contains Einstein–Maxwell–Yang–Mills theory

Abstract

The Lambda-renormalized Einstein-Schrödinger theory is a modification of the original Einstein-Schrödinger theory in which a cosmological constant term is added to the Lagrangian, and it has been shown to closely approximate Einstein-Maxwell theory. Here we generalize this theory to nonAbelian fields by letting the fields be composed of d × d Hermitian matrices. The resulting theory incorporates the U(1) and SU(d) gauge terms of Einstein-Maxwell-Yang-Mills theory, and is invariant under U(1) and SU(d) gauge transformations. The special case where symmetric fields are multiples of the identity matrix closely approximates Einstein-Maxwell-Yang-Mills theory in that the extra terms in the field equations are < 10 −13 of the usual terms for worst-case fields accessible to measurement. The theory contains a symmetric metric and Hermitian vector potential, and is easily coupled to the additional fields of Weinberg-Salam theory or flipped SU(5) GUT theory. We also consider the case where symmetric fields have small traceless parts, and show how this suggests a possible dark matter candidate.

Unification of Gravity and Electromagnetism

“…The electromagnetic field strength tensor (F vσ ) is given by And their lowered index counterpart. The first term of Electrodynamics Lagrangian for the electromagnetic field is given by (8) Canonical energy momentum tensor for electromagnetic field Lagrangian is (9) Using the identity we find (10) Equation ( 10) is not antisymmetric due to the asymmetric Tensor (-F μv F μ σ ) [7], for this let's suppose that the asymmetric tensor is the sum of symmetric and antisymmetric tensors as follow (11) The divergence tensor is arbitrary antisymmetric tensor in their first two indices (χ σvμ = -χ vσμ ), it is constructed from electromagnetic field strength tensor( F vσ ) and electromagnetic vector potential (A μ ). Equation (11) in terms of this definition can be rewritten as ( 12) Employing the Maxwell equation, we obtain (13)…”