Quantum mechanics with space-time noncommutativity

TL;DR

This paper develops an effective Schrödinger equation in a (1+1)-dimensional noncommutative space-time where time and space are operator-valued, providing a framework for quantum mechanics in such noncommutative geometries.

Contribution

It introduces a novel formalism for quantum mechanics on noncommutative space-time with operator-valued time and space, including an induced inner product for interpretation.

Findings

Derived an effective commutative Schrödinger equation in noncommutative space-time.

Established a quantum mechanical interpretation with a suitable inner product.

Explored applications of the developed formalism in noncommutative quantum mechanics.

Abstract

We construct an effective commutative Schr\"odinger equation in Moyal space-time in $(1+1)$-dimension where both $t$ and $x$ are operator-valued and satisfy $\left[ \hat{t}, \hat{x} \right] = i \theta$. Beginning with a time-reparametrised form of an action we identify the actions of various space-time coordinates and their conjugate momenta on quantum states, represented by Hilbert-Schmidt operators. Since time is also regarded as a configuration space variable, we show how an `induced' inner product can be extracted, so that an appropriate quantum mechanical interpretation is obtained. We then discuss several other applications of the formalism developed so far.