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

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.

15 pages; Revised version, accepted for publication in Physics Letters B

References in corpus (11)

• (2+1)D Exotic Newton-Hooke Symmetry, Duality and Projective Phase
• The Standard Model of Particles and Forces in the Framework of 2T-physics
• Reciprocal relativity of noninertial frames and the quaplectic group
• World-line Quantisation of a Reciprocally Invariant System
• A Novel View on the Physical Origin of E8
• Relativity group for noninertial frames in Hamilton's mechanics
• Clifford Algebra of Nonrelativistic Phase Space and the Concept of Mass
• Hamilton-Jacobi theory for Hamiltonian systems with non-canonical symplectic structures
• Extra Dimensions: Will their Spinors Play a Role in the Standard Model?
• Born reciprocity and the 1/r potential
• The relativistic phase space and Newman-Penrose basis