EPI is by construction a relativistically covariant theory (Sec. 3.5). Thus, non-covariant uses of EPI, specifically with Fisher coordinates whose dimension is less than four, give approximate answers. However, these are often easier to obtain than by a fully covariant EPI approach. We use EPI in this manner to derive the following effects: the one-dimensional, stationary Schrödinger wave equation, Newton's second law of classical mechanics, and the classical virial theorem. All are recognizably approximations in one sense or another.

Schrödinger wave equation

Here, a one-dimensional analysis is given. The derivation runs parallel to the fully covariant EPI derivation in Chap. 4 of the Klein–Gordon equation. We point out corresponding results as they occur.

The position θ of a particle of mass m is measured as a value y = θ + x (see Eq. (2.1)), where x is a random excursion whose probability amplitude law q(x) is sought. Since the time t is ignored, we are in effect seeking a stationary solution to the problem. Notice that the one-dimensional nature of x violates the premise of covariant coordinates as made in Sec. 4.1.2.

Since the approach is no longer covariant, it is being used improperly. However, a benefit of EPI is that it gives approximate answers when used in a projection of fourspace. The approximate answer will be the non-relativistic, Schrödinger wave equation.

Assume that the particle is moving in a conservative field of scalar potential V(x). Then the total energy W is conserved. This is assumed in the derivation below.