Dirac Equation - Mathematical Formulation

Mathematical Formulation

The Dirac equation in the form originally proposed by Dirac is:

where

• ψ = ψ(x, t) is a complex four-component field ψ that Dirac thought of as the wave function for the electron,
• x and t are the space and time coordinates,
• m is the rest mass of the electron,
• p is the momentum, understood to be the momentum operator in the Schrödinger theory,
• c is the speed of light, and ħ = h/2π is the reduced Planck constant.

Modern textbooks write this equivalently as:

Dirac's purpose in casting this equation was to explain the behavior of the relativistically moving electron, and so to allow the atom to be treated in a manner consistent with relativity. His rather modest hope was that the corrections introduced this way might have bearing on the problem of atomic spectra. Up until that time, attempts to make the old quantum theory of the atom compatible with the theory of relativity, attempts based on discretizing the angular momentum stored in the electron's possibly non-circular orbit of the atomic nucleus, had failed - and the new quantum mechanics of Heisenberg, Pauli, Jordan, Schrödinger, and Dirac himself had not developed sufficiently to treat this problem. Although Dirac's original intentions were satisfied, his equation had far deeper implications for the structure of matter, and introduced new mathematical classes of objects that are now essential elements of fundamental physics.

The new elements in this equation are the 4 × 4 matrices α_k and β, and the four-component wave function ψ. The matrices are all Hermitian and have squares equal to the identity matrix:

and they all mutually anticommute:

when i and j are distinct. The single symbolic equation thus unravels into four coupled linear first-order partial differential equations for the four quantities that make up the wave function. These matrices, and the form of the wave function, have a deep mathematical significance. The algebraic structure represented by the Dirac matrices had been created some 50 years earlier by the English mathematician W. K. Clifford. In turn, Clifford's ideas had emerged from the mid-19th century work of the German mathematician Hermann Grassmann in his "Lineale Ausdehnungslehre" (Theory of Linear Extensions). The latter had been regarded as well-nigh incomprehensible by most of his contemporaries. The appearance of something so seemingly abstract, at such a late date, and in such a direct physical manner, is one of the most remarkable chapters in the history of physics.