Conservation of probability current

         By defining the adjoint spinor

where  is the conjugate transpose of , and noticing that

,

we obtain, by taking the Hermitian conjugate of the Dirac equation and multiplying from the right by γ⁰, the adjoint equation:

where ∂_μ is understood to act to the left. Multiplying the Dirac equation by ψ from the left, and the adjoint equation by ψ from the right, and subtracting, produces the law of conservation of the Dirac current:

Now we see the great advantage of the first-order equation over the one Schrödinger had tried - this is the conserved current density required by relativistic invariance, only now its 4th component is positive definite and thus suitable for the role of a probability density:

Because the probability density now appears as the fourth component of a relativistic vector, and not a simple scalar as in the Schrödinger equation, it will be subject to the usual effects of the Lorentz transformations such as time dilation. Thus for example atomic processes that are observed as rates, will necessarily be adjusted in a way consistent with relativity, while those involving the measurement of energy and momentum, which themselves form a relativistic vector, will undergo parallel adjustment which preserves the relativistic covariance of the observed values.