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

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 ($\sim {\widehat{P_i}}^2)$ 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 $\widehat{X_{\mu }}$ and $\widehat{P_{\mu }}$ in YSTA show a distinctive spectral structure, that is, space-components $\widehat{X_i}$ and $\widehat{P_i}$ have discrete eigenvalues, and time-components $\widehat{X_0}$ and $\widehat{P_0}$ 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.