Probability Current - Definition (relativistic 4-current)

Definition (relativistic 4-current)

Above, the components of the probability 3-current are:

The fact that the density is positive definite and convected according to a continuity equation;

implies that one may integrate the density over a certain domain and set the total to 1, and this condition will be maintained by the conservation law. A proper relativistic theory with a probability density current must also share this feature. To maintain the notion of a convected density, we must generalize the Schrödinger expression of the density and current so that the space and time derivatives again enter symmetrically in relation to the scalar wave function. We are allowed to keep the Schrödinger expression for the current, but must replace by probability density by the symmetrically formed expression (in SI units):

which now becomes the zeroth component of a space-time vector, and the entire 4-current density has the relativistically covariant expression

where (translating the usual Cartesian-subscript notation into vector indices):

This is compatible with relativity, though the expression for the density is no longer positive definite – the initial values of both ψ and ∂ψ/∂t may be freely chosen, and the density may thus become negative.