The Theory of Finite Displacement Operators and Fundamental Length

The general finite displacement operator in an $n$-dimensional complex continuum is defined as an arbitrary superposition of exponential Taylor operators (a Taylor operator yields a Taylor's series). On restriction to the space-time continuum and four-dimensional time-like intervals of constant length $\omega$, the corresponding finite displacement operator may be considered as an ordinary function of the operators $u_{\sigma }$ (the partial derivative with respect to $x_{\sigma }$), and must satisfy a Klein-Gordon type equation in $u$ space. This equation possesses relativistic invariant and four-vector solutions that in the limit $\omega \to 0$ reduce to 1 and $u_{\sigma }$, respectively. These operators are combined with the Compton wavelength $k$ and the Dirac or Duffin ${\gamma }_{\sigma }$, respectively, to produce a relativistically invariant correspondence type finite-displacement operator generalization of the Dirac-Duffin equation. If the fields are charged, the electromagnetic potentials may be introduced in a manner which leaves the mass spectrum unaltered. The relationship to other nonlocal theories and to the reciprocity theory of Born is briefly considered.