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.
An understanding of the classical properties of light was developed by 19th century scientists on the basis of Thomas Young??s suggestion in 1817 that light waves consist of transverse vibrations such as occur in an elastic solid. For a comprehensive history, see Whittaker . There have also been recent attempts to revive this model of the aether [Dmitriyev 1992, Hatch 1992, Karlsen 1998]. According to this model light consists of transverse waves whose evolution is described by a second order differential equation. Actually, the appropriate wave equation for ideal elastic waves has not been conclusively established. Wave equations have been derived for an ideal elastic solid from analysis of stress and strain, but there is some confusion as to how the theory should accommodate rotations. For instance, in Kleinert (1989) additional elastic constants were introduced ad hoc for this purpose.
The wave nature of matter was first proposed by de Broglie  and subsequently confirmed in experiments by Davisson and Germer , and independently by Thomson and Reid . However the equations developed to describe these ??matter waves?? are first order equations rather than second-order wave equations. Although these waves are commonly interpreted as probability waves, the quantum mechanical Dirac equation is also a deterministic equation for the evolution of angular momentum density and other physical observables. As such, it should correspond to classical wave theory. Others have reformulated the Dirac theory in terms of deterministic relations between local physical observables [Takabayashi 1957, Hestenes 1973]. However these investigators did not construct a corresponding classical wave theory.
Several attempts have been made to describe elementary particles as soliton (or particle-like) wave solutions of a nonlinear Dirac equation. See Ra??ada  for a short review and Gu  for a more recent discussion of this approach to understanding matter. The soliton solutions found to date do not appear to correspond to matter, although some similarities have been claimed. It is not clear how Dirac solitons relate to ordinary classical wave solitons.
In this paper we focus on analyzing the solutions to simple wave equations. We find that the bispinors which are associated with matter in quantum mechanics are in fact the general solutions of ordinary scalar or vector wave equations. However, the mathematical structure of classical wave bispinors differs significantly from that of Dirac??s bispinors. Unlike the Dirac algebra, classical wave bispinors can be factored with independent rotations of velocity and polarization. We subsequently discuss how these solutions may be applicable to the equations of waves in an ideal elastic solid, and how they relate to the electron equation, electromagnetic potentials, and the parity transformation. We also describe briefly how classical wave theory can explanation various physical phenomena such as special relativity and gravity.