Editing Dirac equation (section)

=== Making the Schrödinger equation relativistic ===
The Dirac equation is superficially similar to the Schrödinger equation for a massive [[free particle]]:
<math display="block">-\frac{\hbar^2}{2m}\nabla^2\phi = i\hbar\frac{\partial}{\partial t}\phi ~.</math>

The left side represents the square of the momentum operator divided by twice the mass, which is the non-relativistic kinetic energy. Because relativity treats space and time as a whole, a relativistic generalization of this equation requires that space and time derivatives must enter symmetrically as they do in the [[Maxwell equation]]s that govern the behavior of light – the equations must be differentially of the ''same order'' in space and time. In relativity, the momentum and the energies are the space and time parts of a spacetime vector, the [[four-momentum]], and they are related by the relativistically invariant relation
<math display="block">E^2 = m^2c^4 + p^2c^2 ,</math>
which says that the [[Four-momentum#Minkowski norm|length of this four-vector]] is proportional to the rest mass {{math|''m''}}. Substituting the operator equivalents of the energy and momentum from the Schrödinger theory produces the [[Klein–Gordon equation]] describing the propagation of waves, constructed from relativistically invariant objects,
<math display="block">\left(-\frac{1}{c^2}\frac{\partial^2}{\partial t^2} + \nabla^2\right)\phi = \frac{m^2c^2}{\hbar^2}\phi ,</math>
with the wave function <math>\phi</math> being a relativistic scalar: a complex number that has the same numerical value in all frames of reference. Space and time derivatives both enter to second order. This has a telling consequence for the interpretation of the equation. Because the equation is second order in the time derivative, one must specify initial values both of the wave function itself and of its first time-derivative in order to solve definite problems. Since both may be specified more or less arbitrarily, the wave function cannot maintain its former role of determining the [[probability density function|probability density]] of finding the electron in a given state of motion. In the Schrödinger theory, the probability density is given by the positive definite expression
<math display="block">\rho = \phi^*\phi </math>
and this density is convected according to the probability current vector
<math display="block">J = -\frac{i\hbar}{2m}(\phi^*\nabla\phi - \phi\nabla\phi^*) </math>
with the conservation of probability current and density following from the continuity equation:
<math display="block">\nabla\cdot J + \frac{\partial\rho}{\partial t} = 0~.</math>

The fact that the density is [[Positive-definite function|positive definite]] and convected according to this 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, one must generalize the Schrödinger expression of the density and current so that space and time derivatives again enter symmetrically in relation to the scalar wave function. The Schrödinger expression can be kept for the current, but the probability density must be replaced by the symmetrically formed expression{{explain|reason=Why?|date=November 2021}}
<math display="block">\rho = \frac{i\hbar}{2mc^2} \left(\psi^*\partial_t\psi - \psi\partial_t\psi^* \right) ,</math>
which now becomes the 4th component of a spacetime vector, and the entire [[Probability current|probability 4-current density]] has the relativistically covariant expression
<math display="block">J^\mu = \frac{i\hbar}{2m} \left(\psi^*\partial^\mu\psi - \psi\partial^\mu\psi^* \right) .</math>

The continuity equation is as before. Everything is compatible with relativity now, but the expression for the density is no longer positive definite; the initial values of both {{math|''ψ''}} and {{math|∂<sub>''t''</sub>''ψ''}} may be freely chosen, and the density may thus become negative, something that is impossible for a legitimate probability density. Thus, one cannot get a simple generalization of the Schrödinger equation under the naive assumption that the wave function is a relativistic scalar, and the equation it satisfies, second order in time.

Although it is not a successful relativistic generalization of the Schrödinger equation, this equation is resurrected in the context of [[quantum field theory]], where it is known as the [[Klein–Gordon equation]], and describes a spinless particle field (e.g. [[pi meson]] or [[Higgs boson]]). Historically, Schrödinger himself arrived at this equation before the one that bears his name but soon discarded it. In the context of quantum field theory, the indefinite density is understood to correspond to the ''charge'' density, which can be positive or negative, and not the probability density.