A Dirac Equation
Robert A. Close
The time evolution of electrons and other fermions is described by the first-order Dirac equation.
Although typically interpreted probabilistically, the Dirac equation is fundamentally a deterministic
equation for the evolution of physical observables such as angular momentum density. The Dirac
equation can be considered as a second-order wave equation if the wave function is a representation of
the first derivatives. The conventional Dirac formalism has two serious flaws. First, the conventional
derivation of the parity operator is incorrect. Conventional theory holds that the wave function of a
Dirac particle is its own mirror image but certain mirrored interactions do not occur. Such mirror
particles have never been observed. Experimental evidence, such as beta decay, supports the
alternative hypothesis that the mirror image of matter is antimatter. This problem is solved by
identifying a flaw in the conventional derivation of the parity operator, then deriving a new parity
operator based on the algebraic properties of vectors. Second, the conjugate momenta (pi) in the freeparticle
Hamiltonian (H) do not have the proper relation ( pi =δH δqi ) to the time derivatives of
coordinates ( qi ). This problem is solved by replacing the mass term with convection and rotation
terms. We then show that the resultant bispinor equation of evolution is equivalent to a classical
second-order wave equation for angular momentum density in an elastic solid. The co-existence of
forward- and backward-propagating waves along a single axis is the basis of half-integer spin. Wave
interference produces both the Lorenz force and the Pauli Exclusion Principle. Mass is associated with
radially inward acceleration of the wave such as occurs in a soliton. Angular correlations between spin
states are equal to the quantum correlations. Bell?s Theorem is not applicable to classical bispinors.
Matter and anti-matter are related by spatial inversion, consistent with experimental observations. The
classical wave formulation therefore provides a conceptually clear interpretation of fermion dynamics.
