Beyond the Relativistic Point Particle: A Reciprocally Invariant System and its Generalisation

TL;DR

This paper explores a reciprocally invariant system with extended symmetries, investigates its classical solutions, and proposes generalizations linking it to theories like Kaluza-Klein and Clifford space for unification.

Contribution

It introduces a generalized reciprocally invariant system incorporating local Sp(2) covariance and connects it to higher-dimensional unification frameworks.

Findings

Classical solutions lack linear evolution terms, indicating non-propagating behavior.

Generalizations lead to models resembling rigid particles with helical worldlines.

Connections established between the generalized system, Kaluza-Klein theories, and Clifford space.

Abstract

We investigate a reciprocally invariant system proposed by Low and Govaerts et al., whose action contains both the orthogonal and the symplectic forms and is invariant under global $O(2,4)\cap Sp(2,4)$ transformations. We find that the general solution to the classical equations of motion has no linear term in the evolution parameter, $\tau$, but only the oscillatory terms, and therefore cannot represent a particle propagating in spacetime. As a remedy, we consider a generalisation of the action by adopting a procedure similar to that of Bars et al., who introduced the concept of a $\tau$ derivative that is covariant under local Sp(2) transformations between the phase space variables $x^\mu(\tau)$ and $p^\mu (\tau)$. This system, in particular, is similar to a rigid particle whose action contains the extrinsic curvature of the world line, which turns out to be helical in spacetime. Another possible generalisation is the introduction of a symplectic potential proposed by Montesinos. We show how the latter approach is related to Kaluza-Klein theories and to the concept of Clifford space, a manifold whose tangent space at any point is Clifford algebra Cl(8), a promising framework for the unification of particles and forces.