Contracted Representation of Yang's Space-Time Algebra and Buniy-Hsu-Zee's Discrete Space-Time

TL;DR

This paper explores a contracted representation of Yang's space-time algebra (YSTA) that incorporates finite spatial degrees of freedom, aligning with theories of discrete space-time and enabling divergence-free noncommutative field theories.

Contribution

It introduces a contracted form of YSTA with finite degrees of freedom, connecting discrete space-time concepts to noncommutative field theories and classical limits.

Findings

Contracted YSTA has finite spatial degrees of freedom.

The representation supports divergence-free noncommutative field theories.

Classical quantum mechanics relations emerge as a limit of YSTA.

Abstract

Motivated by the recent proposition by Buniy, Hsu and Zee with respect to discrete space-time and finite spatial degrees of freedom of our physical world with a short- and a long-distance scales, $l_P$ and $L,$ we reconsider the Lorentz-covariant Yang's quantized space-time algebra (YSTA), which is intrinsically equipped with such two kinds of scale parameters, $\lambda$ and $R$. In accordance with their proposition, we find the so-called contracted representation of YSTA with finite spatial degrees of freedom associated with the ratio $R/\lambda$, which gives a possibility of the divergence-free noncommutative field theory on YSTA. The canonical commutation relations familiar in the ordinary quantum mechanics appear as the cooperative Inonu-Wigner's contraction limit of YSTA, $\lambda \to 0$ and $R \to \infty.$