Space-Time Position Operators

A formalism is presented giving the development of $\psi \left(\overline{x}\right)$ in time with the interpretation that $\psi \left(\overline{x}\right)$ is the probability amplitude for observing an event at a space-time point $\overline{x}$. No properties other than the four space-time coordinates are associated with an event. A Hilbert space is defined in which $\psi \left(\overline{x}\right)$ is the result of a scalar product. The space-time position operators defined in this Hilbert space have no association with particle properties, such as mass. These operators cannot be defined in the Hilbert space spanned by solutions of a Schrödinger equation, since the operators lead out of the Hilbert subspace belonging to a given mass. It is shown that state vectors in Hilbert space that are eigenvectors of $P_{\mu }{P}^{\mu }$ produce position amplitudes satisfying the Klein-Gordon equation. The relation between this Hilbert space and the one introduced by Dirac is discussed. Spin is not considered.