Uncertainty relation and non-dispersive states in Finite Quantum Mechanics

TL;DR

This paper explores a classical sector within Finite Quantum Mechanics, identifying states with zero position-momentum uncertainty and constructing non-dispersive bases, revealing unique short-distance and deterministic properties.

Contribution

It introduces a classical subset of states in FQM with zero uncertainty bound and explicitly constructs non-dispersive bases aligned with deterministic interpretations.

Findings

Classical states with vanishing uncertainty bound identified.

Wave packets decorrelate rapidly under long-period linear maps.

Explicit non-dispersive bases constructed consistent with deterministic FQM.

Abstract

In this letter, we provide evidence for a classical sector of states in the Hilbert space of Finite Quantum Mechanics (FQM). We construct a subset of states whose the minimum bound of position -momentum uncertainty (equivalent to an effective $\hbar$) vanishes. The classical regime, contrary to standard Quantum Mechanical Systems of particles and fields, but also of strings and branes appears in short distances of the order of the lattice spacing. {}For linear quantum maps of long periods, we observe that time evolution leads to fast decorrelation of the wave packets, phenomenon similar to the behavior of wave packets in t' Hooft and Susskind holographic picture. Moreoever, we construct explicitly a non - dispersive basis of states in accordance with t' Hooft's arguments about the deterministic behavior of FQM.