10:30 am

Lorentz Covariance of the Maxwell Equations

Rollin S. Armour, Jr.

and Jose L. Balduz, Jr.
Department of Physics
Mercer University

We seek all linear transformations of the Maxwell variables and spacetime coordinates that leave Maxwell's equations form-invariant. Form-invariance forces coordinate transformations to leave the Minkowski interval invariant allowing five different four-dimensional Lorentz spacetimes, one real and four complex, corresponding to coordinate transformations under the (1/2,1/2), (0,0)+(0,1), (0,0)+(1,0), (1/2,0)+(1/2,0), and (0,1/2)+(0,1/2) representations of the Lorentz group. In each spacetime, Maxwell's equations remain covariant under at least two different Lorentz transformation rules for the Maxwell variables, with charge invariance, gauge invariance, and a covariant Lorentz four-force accompanying at least one of these rules. (In four-vector spacetime, the second rule is spin-1/2. See Found. Phys. 34, 815, 2004.) The Maxwell Lagrangian density is the same in every case, and primary field invariants are always formed with the Minkowski metric, yielding a common set of Maxwell invariants and conservation laws under every Lorentz transformation of the Maxwell variables in all five Lorentz spacetimes.

2:30 pm

Quantum Dynamical Manifolds

Dillon F. Scofield
Department of Physics
Oklahoma State University

& ApplSci Inc.

A quantum geometrodynamical approach is developed for the computation of unified mass-spacetimes (MST). The resulting theory of the evolution of quantum dynamical manifolds (QDMs) is shown to contain contemporary quantum mechanics. This approach overcomes a major obstacle in Yang-Mills theories that generalize Maxwell’s equations. The process used transforms the basis of abstract vectors defining the geometry of the QDM so that the equations of the geometrodynamical extension of quantum mechanics, called quantum dynamical manifold equations (QDMEs), can be put into a canonical form. This places the effects causing problems in Yang-Mills theories into the abstract vector space basis in a way that the resulting equations are integrable. The QDMEs are shown to be completely classifiable according to the Cartan-Killing theorem of Lie algebra theory. This leads to a hierarchy of equations that include, at lower energies, ones for manifolds with u(1), su(2), and su(3),…, Lie algebraic symmetry. The physical picture of the resulting unified MST is of a self-consistently curved and twisted QDM in which each effective (quasi-) particle appears free, particles can be created and destroyed and interact through the quantum dynamical manifold they collectively create. Examples of the solution of the QDMEs are given. In particular, it is shown that the Dirac-Maxwell system of equations, extended so they have manifold solutions, are QDMEs with Lie group symmetric QDMs as solutions. Some methods for computing the masses and coupling “constants” appearing as parameters in the Standard Model will be outlined.