On Laplace-Runge-Lenz Vector as Symmetry Breaking order parameter in Kepler Orbit and Goldstone Boson

Abstract

We introduce a type of symmetry breaking and associated order parameter in connection with Laplace-Runge-Lenz vector of Kepler orbit through an extended spatial dimension and Ensemble view. By implementation of a small extra spatial dimension and embedded infinitesimal toral manifold, it has been shown that emerging of LRL vector under SO(4)symmetry is in analogy with a variety of explicit and spontaneous symmetry breaking situations and related Goldstone bosons such as phonons and spin waves. A theorem introduced to generalize this concept of breaking symmetry. The diffeomorphism of circular orbit(geodesic)to elliptic one proved to be equivalent with a covariant derivative and related parallel displacement in this extended four dimensional spatial space.Respect to ensemble definition this diffeomorphism breaks the O(2) symmetry of initial orbit and Hamiltonian to Z2 resulting in broken generators in quotient space and associated Goldstone boson as perturbing Hamiltonian term leading to a perpetual circular motion on 2-torus comparable to the perpetual motion idea in Time Crystal of Wilczek. This leads to an introduction of gravitational gauge potential under the symmetry of Cartan sub algebra of su(2).