Quantum Time

TL;DR

This paper explores the quantization of time alongside space using a generalized path integral approach, deriving a four-dimensional Schrödinger equation and proposing experimental tests for quantum time effects.

Contribution

It introduces a novel framework for quantizing time with path integrals, extending quantum theory to include time as a quantum variable.

Findings

Derived a four-dimensional Schrödinger equation from the path integral.

Recovered standard quantum mechanics in classical limits.

Proposed experimental modifications to test quantum time effects.

Abstract

Normally we quantize along the space dimensions but treat time classically. But from relativity we expect a high level of symmetry between time and space. What happens if we quantize time using the same rules we use to quantize space? To do this, we generalize the paths in the Feynman path integral to include paths that vary in time as well as in space. We use Morlet wavelet decomposition to ensure convergence and normalization of the path integrals. We derive the Schr\"odinger equation in four dimensions from the short time limit of the path integral expression. We verify that we recover standard quantum theory in the non-relativistic, semi-classical, and long time limits. Quantum time is an experiment factory: most foundational experiments in quantum mechanics can be modified in a way that makes them tests of quantum time. We look at single and double slits in time, scattering by time-varying electric and magnetic fields, and the Aharonov-Bohm effect in time.