The classical relativistic connexion between the energy
p
t
of a free article and its momentum
p
x
, p
y
, p
z
, namely,
p
t
2
—
p
x
2
—
p
y
2
—
p
z
2
—
m
2
= 0, (1) leads in the quantum theory to the wave equation {
p
t
2
—
p
x
2
—
p
y
2
—
p
z
2
—
m
2
} ψ = 0, (2) where the
p
's are understood as the operators
iħ
∂/∂
t
, —
iħ
∂/∂
x
. . The general theory of the physical interpretation of quantum mechanics requires a wave equation of the form {
p
t
— H} ψ = 0, (3) where H is a Hermitian operator not containing
p
t
, and is called the Hamiltonian. The obvious equation of the form (3) which one gets from (2), namely, {
p
t
— (
p
x
2
+
p
y
2
+
p
z
2
+
m
2
)
½
} ψ = 0, is unsatisfactory on account of the square root, which makes the application of Lorentz transformations very complicated. By allowing our particle to have a spin, we can get wave equations of the form (3) which are consistent with (2) and do not involve square roots. An example, applying to the case of a spin of half a quantum, namely, the equation {
p
t
+
α
x
p
x
+
α
y
p
y
+
α
z
p
z
+
α
m
m
} ψ = 0, (4) where the four
α
's are anti-commuting matrices whose squares are unity, is well known, and has been found to give a satisfactory description of the electron and positron. The present paper will be concerned with other examples, applying to spins greater than a half. The elementary particles known to present-day physics, the electron, positron, neutron, and proton, each have a spin of a half, and thus the work of the present paper will have no immediate physical application. All the same, it is desirable to have the equation ready for a possible future discovery of an elementary particle with a spin greater than a half, or for approximate application to composite particles. Further, the underlying theory is of considerable mathematical interest.
Exact analytic solutions for nonlinear waves in cold plasmas