Reciprocal relativity of noninertial frames and the quaplectic group

TL;DR

This paper introduces the quaplectic group, a new symmetry framework in extended phase space, unifying relativistic and noninertial transformations, and linking to quantum mechanics through its noncommutative structure.

Contribution

It proposes the quaplectic group as a reciprocal relativity framework that eliminates the need for an absolute inertial frame and connects to quantum mechanics via its unitary representations.

Findings

The quaplectic group generalizes Lorentz transformations to noninertial frames.

It incorporates noncommutative position, momentum, time, and energy operators.

The group provides a foundation for quantum mechanics in noninertial reference frames.

Abstract

Newtonian mechanics has the concept of an absolute inertial rest frame. Special relativity eliminates the absolute rest frame but continues to require the absolute inertial frame. General relativity solves this for gravity by requiring particles to have locally inertial frames on a curved position-time manifold. The problem of the absolute inertial frame for other forces remains. We look again at the transformations of frames on an extended phase space with position, time, energy and momentum degrees of freedom. Under nonrelativistic assumptions, there is an invariant symplectic metric and a line element dt^2. Under special relativistic assumptions the symplectic metric continues to be invariant but the line elements are now -dt^2+dq^2/c^2 and dp^2-de^2/c^2. Max Born conjectured that the line element should be generalized to the pseudo- orthogonal metric -dt^2+dq^2/c^2+ (1/b^2)(dp^2-de^2/c^2). The group leaving these two metrics invariant is the pseudo-unitary group of transformations between noninertial frames. We show that these transformations eliminate the need for an absolute inertial frame by making forces relative and bounded by b and so embodies a relativity that is 'reciprocal' in the sense of Born. The inhomogeneous version of this group is naturally the semidirect product of the pseudo-unitary group with the nonabelian Heisenberg group. This is the quaplectic group. The Heisenberg group itself is the semidirect product of two translation groups. This provides the noncommutative properties of position and momentum and also time and energy that are required for the quantum mechanics that results from considering the unitary representations of the quaplectic group.