The quantum electrodynamical law which applies to spin-1/2 particles and is the relativistic generalization of the Schrödinger equation.
In dimensions (three space dimensions
and one time dimension), it is given by
(1)
(Bjorken and Drell 1964, p. 6), where is h-bar, is the speed of light, is the wavefunction, is the mass of the particle, and are the Dirac matrices
(with
being called
by Bjorken and Drell 1964, p. 8; Berestetskii et al. 1982, p. 78).
The Dirac equation can also be written in the concise form
Alvarez, A.; Pen-Yu, K.; and Vazquez, L. "The Numerical Study of a Nonlinear One-Dimensional Dirac Equation." Appl. Math. Comput.18,
1-15, 1983.Berestetskii, V. B.; Lifshitz, E. M.; and Pitaevskii,
L. P. Quantum
Electrodynamics, 2nd ed. Oxford, England: Pergamon Press, 1982.Bethe,
H. A. and Salpeter, E. Quantum
Mechanics of One- and Two-Electron Atoms. New York: Plenum, p. 37, 1977.Bjorken,
J. D. and Drell, S. D. "The Dirac Equation." §1.3 in Relativistic
Quantum Mechanics. New York: McGraw-Hill, pp. 6-9, 1964.Dirac,
P. A. M. "The Quantum Theory of the Electron." Proc. Roy.
Soc. LondonA117, 610-624, 1928.Dirac, P. A. M.
"The Quantum Theory of the Electron, Part II." Proc. Roy. Soc. LondonA118,
351-361, 1928.Dirac, P. A. M. Principles
of Quantum Mechanics, 4th ed. Oxford, England: Oxford University Press, 1982.Griffiths,
D. J. "The Dirac Equation" and "Solutions to the Dirac Equation."
§7.1-7.2 in Introduction
to Elementary Particles. New York: Wiley, pp. 213-222, 1987.Zwillinger,
D. Handbook
of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 137,
1997.
dimensions (three space dimensions
and one time dimension), it is given by
is h-bar, 
is the speed of light, 
is the wavefunction, 
is the mass of the particle, and 
are the Dirac matrices
(with 
being called 
by Bjorken and Drell 1964, p. 8; Berestetskii et al. 1982, p. 78).
are Dirac
matrices in the "Dirac basis" (Griffiths 1987, p. 216), and Einstein summation has been used to sum over 
, 1, 2, 3.
dimensions, a generalization of the Dirac equation is given by the system of partial
differential equations
corresponds to the quantum electrodynamical equation.
