Quantum motion equation and Poincare translation invariance of noncommutative field theory

Abstract

We study the Moyal commutators and their expectation values between vacuum states and non-vacuum states for noncommutative scalar field theory. For noncommutative $ϕ^{\star4}$ scalar field theory, we derive its energy-momentum tensor from translation transformation and Lagrange field equation. We generalize the Heisenberg and quantum motion equations to the form of Moyal star-products for noncommutative $ϕ^{\star4}$ scalar field theory for the case $θ^{0i}=0$ of spacetime noncommutativity. Then we demonstrate the Poincar{\' e} translation invariance for noncommutative $ϕ^{\star4}$ scalar field theory for the case $θ^{0i}=0$ of spacetime noncommutativity.