Noncommutative Field Theory on Yang's Space-Time Algebra, Covariant Moyal Star Product and Matrix Model

TL;DR

This paper develops a Lorentz-covariant noncommutative field theory on Yang's space-time algebra, reformulating matrix models as quantum mechanics of D0 branes with a unique spectral structure and a novel field equation.

Contribution

It introduces a Lorentz-covariant noncommutative framework on Yang's space-time algebra, connecting matrix models to quantum mechanics of D0 branes with a new spectral and field structure.

Findings

Spectral structure with discrete space and continuous time eigenvalues.

Derivation of a nontrivial field equation beyond Klein-Gordon.

Reformulation of matrix models as quantum mechanics of D0 branes.

Abstract

Noncommutative field theory on Yang's quantized space-time algebra (YSTA) is studied. It gives a theoretical framework to reformulate the matrix model as quantum mechanics of $D_0$ branes in a Lorentz-covariant form. The so-called kinetic term ($\sim {\hat{P_i}}^2)$ and potential term ($\sim {[\hat{X_i},\hat{X_j}]}^2)$ of $D_0$ branes in the matrix model are described now in terms of Casimir operator of $SO(D,1)$, a subalgebra of the primary algebra $SO(D+1,1)$ which underlies YSTA with two contraction- parameters, $\lambda$ and $R$. $D$-dimensional noncommutative space-time and momentum operators $\hat{X_\mu}$ and $\hat{P_\mu}$ in YSTA show a distinctive spectral structure, that is, space-components $\hat{X_i}$ and $\hat{P_i}$ have discrete eigenvalues, and time-components $\hat{X_0}$ and $\hat{P_0}$ continuous eigenvalues, consistently with Lorentz-covariance. According to the method of Lorentz-covariant Moyal star product proper to YSTA, the field equation of $D_0$ brane on YSTA is derived in a nontrivial form beyond simple Klein-Gordon equation, which reflects the noncommutative space-time structure of YSTA.