Noncommutative Model with Spontaneous Time Generation and Planckian bound

TL;DR

This paper explores how noncommutative geometry can lead to spontaneous generation of time, resulting in a noncommutative Schrödinger equation with a Planckian momentum bound and novel dispersion relations.

Contribution

It introduces a noncommutative space model where time emerges spontaneously, deriving a consistent Schrödinger equation and identifying a momentum bound.

Findings

Existence of plane waves for |p|<π/2λ

Derivation of a noncommutative Schrödinger equation with correct classical limit

Identification of a Planckian bound on spatial momentum

Abstract

We illustrate the thesis that if time did not exist, we would have to create it if space is noncommutative, and extend functions by something like Schroedinger's equation. We propose that the phenomenon is a somewhat general mechanism within noncommutative geometry for `spontaneous time generation'. We show in detail how this works for the $su_2$ algebra $[x_i,x_j]=2\imath\lambda \epsilon_{ij}{}^kx_k$ as noncommutative space, by explicitly adjoining the forced time variable. We find the natural induced noncommutative Schroedingers equation and show that it has the correct classical limit for a particle of some mass $m\ne 0$, which is generated as a second free parameter by the theory. We show that plane waves exist provided $|\vec p|< \pi/2\lambda$, i.e. we find a Planckian bound on spatial momentum. We also propose dispersion relations $|{\del p^0\over\del \vec p}|=|\tan({\lambda}|\vec p|)|/m\lambda$ for the model and explore some elements of the noncommutative geometry. The model is complementary to our previous bicrossproduct one.