Fundamental constraints on two-time physics

TL;DR

This paper explores the implications of extending classical and quantum dynamics to two-time frameworks, revealing fundamental constraints on evolution and observability due to generalized uncertainty relations.

Contribution

It introduces a consistent extension of classical and quantum mechanics to two times, showing how evolution is constrained and how observability is limited by new uncertainty relations.

Findings

Classical dynamics with two times reduces to effective single-time evolution.

Quantum two-time extension preserves probability and introduces generalized uncertainty.

Constraints limit the observability of the second time dimension.

Abstract

We show that generalizations of classical and quantum dynamics with two times lead to fundamentally constrained evolution. At the level of classical physics, Newton's second law is extended and exactly integrated in $1+2$ dimensional space, leading to effective single-time evolution for any initial condition. In the domain of quantum mechanics, we follow strictly the hypothesis of probability conservation by extending the Heisenberg picture to unitary evolution with two times. As a result, the observability of two temporal axes is constrained by a generalized uncertainty relation involving level spacings, total duration of the effect and Planck's constant.