We seek a relativistic covariant equation of the form

which is first order in the time derivative and will have positive definite probability density. Assuming such an equation is also linear in the space derivatives, we can write

(5.2) |

Using operator notation we have

For invariance under spatial rotation, the can not be numbers and can not be a scalar. In analogy with the spin wave function of nonrelativistic quantum mechanics, we choose to be a vector, and and to be matrices. Explicitly,

(5.4) |

We thus have coupled first-order equations.

These equations must

- have free-particle solutions that satisfy ,
- yield a continuity equation and probability interpretation of , and
- be Lorentz covariant.

For the first condition to be satisfied, each component of must satisfy the Klein-Gordon equation. Applying the operator (5.1) twice gives

(5.5) |

To obtain the Klein-Gordon equation the following must be satisfied

where represents an unit matrix. We will not write the unit matrix explicitely in the wave equation unless it is required for clarity. This should not create any confusion since matrices can only equal matrices.

No terms in the Hamiltonian can have any space or time coordinates. Such terms would have the property of space-time dependent energies and would give rise to forces. Space and time derivatives can only appear in and , but not in and , since the equation is to be linear in these derivatives. Thus and are independent of and hence commute with them.

Since the Hamiltonian must be hermitian, and must be hermitian matrices. Since the matrices are hermitian they must be square.

Since and anticommute according to equation 5.7, they are traceless. This can be seen as follows:

(5.9) |

where the last line follows from the cyclic property of the trace. The choice of and is not unique. All matrices related to these by any unitary matrix (which preserves the anti-commutation relations) are allowed: and .

Since , the eigenvalues of and are . Since the trace is the sum of eigenvalues and , must be of even dimensions. For , only three anti-commuting matrices exist (the Pauli matrices). Thus the smallest dimension allowed is .

If one matrix is diagonal, the others can not be diagonal or they would commute with the diagonal matrix. We can write a representation that is hermitian, traceless, and has eigenvalues of :

where are the Pauli matrices and is the unit matrix.

2004-03-18