Proper time operator and its uncertainty relation

TL;DR

This paper explores the quantum properties of a proper time oscillator, establishing an uncertainty relation for proper time and demonstrating its connection to the Klein-Gordon equation, offering a more symmetrical time-space treatment.

Contribution

It introduces a proper time operator for oscillators and derives a novel uncertainty relation, linking proper time to quantum field properties.

Findings

Proper time can be treated as a self-adjoint operator.

Displaced proper time obeys an uncertainty relation similar to position-momentum.

Proper time oscillators satisfy the Klein-Gordon equation.

Abstract

We study the quantum properties of an oscillator in proper time. This proper time oscillator is a particle model with mass that is on shell. Its internal time can be treated as a self-adjoint operator. The displaced time and displaced time rate of the oscillator obey an uncertainty relation resembling the one between position and momentum, which is different from the usual energy-time uncertainty relation. In addition, we demonstrate that a matter field with proper time oscillators satisfies the Klein-Gordon equation. It has properties of a zero-spin quantum field. The formulations adopted permits a more symmetrical treatment between time and space in a matter field.