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

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, $λ$ 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 $R/λ$, 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, $λ\to 0$ and $R \to \infty.$