ABSTRACT
In the search for a relativistic wave equation for a free particle of rest mass m one must demand that the energy(E)-momentum (p) [dispersion] relation
E2 = m2c4 + p2c2 (25.1)
is satisfied. By the substitutions
E → i~ ∂ ∂t , p→ −i~∇ (25.2)
in Eq. (25.1) one obtains an operator, which when acting on a scalar wave function, gives us the Klein-Gordon wave equation for a spinless particle. The Klein-Gordon equation is of second order in the space (r) and time (t) coordinates. If one seeks a covariant first-order differential equation, r and t must play symmetric roles. A natural starting point therefore would be a linear dispersion relation of the form
E = βmc2 + cα · p, (25.3)
where β and α = (α1, α2, α3) are real and dimensionless. Since Eq. (25.3) must be compatible with Eq. (25.1), squaring of Eq. (25.3) leads to the following relations [already given in Eqs. (10.77)-(10.79)]:
β2 = 1, (25.4)
αiβ + βαi = 0, (25.5)
αiαj + αjαi = 2δij , (25.6)
where i, j = 1, 2, 3 (x, y, z). It is clear from Eqs. (25.4)-(25.6) that β and α cannot be numbers, but one can find matrices which satisfy these equations. Since the Hamiltonian appearing upon a substitution of the connections in (25.2) into Eq. (25.3) must be Hermitian, the αi and β matrices also have to be Hermitian, i.e.,
α†i = αi, i = 1− 3, (25.7) β† = β. (25.8)
One can show that the matrices must have even rank and that the rank must be at least 4, see, e.g., [88]. The Dirac equation, which describes the quantum mechanics of a spin1/2 elementary particle, e.g., an electron, relates to a four-dimensional realization of Eqs. (25.4)-(25.8). Representations of a given rank are not unique.
