Dimensional Reduction via Noncommutative Spacetime: Bootstrap and Holography

TL;DR

This paper introduces a generalized quantum evolution equation for noncommutative spacetime, leading to dimensional reduction consistent with holographic principles, with implications for quantum mechanics and spacetime structure.

Contribution

It proposes a new spacetime bootstrap equation that modifies quantum evolution in noncommutative spacetime, revealing a natural dimensional reduction aligned with holography.

Findings

The new evolution equation constrains spatial dimensions noncommuting with time.

Dimensional reduction occurs effectively at low energies.

Examples demonstrate the connection to holographic principles.

Abstract

Unlike noncommutative space, when space and time are noncommutative, it seems necessary to modify the usual scheme of quantum mechanics. We propose in this paper a simple generalization of the time evolution equation in quantum mechanics to incorporate the feature of a noncommutative spacetime. This equation is much more constraining than the usual Schr\"odinger equation in that the spatial dimension noncommuting with time is effectively reduced to a point in low energy. We thus call the new evolution equation the spacetime bootstrap equation, the dimensional reduction called for by this evolution seems close to what is required by the holographic principle. We will discuss several examples to demonstrate this point.