SL(2,R)-geometric phase space and (2+2)-dimensions

TL;DR

This paper introduces an SL(2,R)-symmetric geometric framework for phase space, revealing its implications for spacetime signatures and potential applications in various physical theories.

Contribution

It presents a novel geometric structure for phase space emphasizing SL(2,R)-symmetry and explores its consequences for spacetime signatures in physical theories.

Findings

SL(2,R)-symmetry is inherent in symplectic structures

Spacetime signatures are constrained by SL(2,R)-symmetry

Potential applications in different physical models

Abstract

We propose an alternative geometric mathematical structure for arbitrary phase space. The main guide in our approach is the hidden SL(2,R)-symmetry which acts on the phase space changing coordinates by momenta and vice versa. We show that the SL(2,R)-symmetry is implicit in any symplectic structure. We also prove that in any sensible physical theory based on the SL(2,R)-symmetry the signature of the flat target "spacetime" must be associated with either one-time and one-space or at least two-time and two-space coordinates. We discuss the consequences as well as possible applications of our approach on different physical scenarios.