2T Physics, Scale Invariance and Topological Vector Fields

Abstract

We construct, in classical two-time physics, the necessary structure for the most general configuration space formulation of quantum mechanics containing gravity in d + 2 dimensions. This structure is composed of a symmetric Riemannian metric tensor and of a vector field that defines a section of a flat U (1) bundle over space-time. This construction is possible because of the existence of a finite local scale invariance of the Hamiltonian and because two-time physics contains, at the classical level, a local generalization of the discrete duality symmetry between position and momentum that underlies the structure of quantum mechanics.

2T Physics and Quantum Mechanics

“…This paper also gives relevant contributions to the development of 2T physics with vector fields. In addition to the local scale invariance of the classical Hamiltonian equations of motion in the case when A M = A M (X) that was presented in [19], this paper also points out that the classical Hamiltonian equations of motion are invariant under local scale transformations in the case when A M = A M (P ). For each of the symmetries of the 2T action in the case when A M = A M (X) that were presented in [19], this paper also presents the corresponding symmetry with A M = A M (P ).…”

“…In addition to the local scale invariance of the classical Hamiltonian equations of motion in the case when A M = A M (X) that was presented in [19], this paper also points out that the classical Hamiltonian equations of motion are invariant under local scale transformations in the case when A M = A M (P ). For each of the symmetries of the 2T action in the case when A M = A M (X) that were presented in [19], this paper also presents the corresponding symmetry with A M = A M (P ). This existence of the same symmetries of the 2T action in position space and in momentum space is what gives support for the new quantum mechanical equations we write here.…”

“…In this paper we use the results of [19], together with new results that are presented here, to show how one can extend to momentum space the general position space formulation of quantum mechanics described in [20]. As we will see here, in addition to the concept of a momentum dependent metric tensor which was found necessary in [19], this paper evidentiates the need for the concept of a momentum dependent vector field. Combining these two concepts we can write down the basic equations for a formulation of quantum mechanics in total agreement with the wave-particle duality.…”

“…In a previous paper [19], we presented a finite local scale invariance of the 2T physics Hamiltonian and showed how this local invariance can be used to relate the d + 2 dimensional Minkowski space of 2T physics to a Riemannian space of the same dimensionality. Although changing from a flat space to a curved position dependent space using a local invariance is already an interesting observation, it is not the only one.…”

Dual Quantum Mechanics

“…Starting from action (3.4b) we can construct the following action [28] in the presence of a vector field A M (X)…”