Quantum motion equation and Poincare translation invariance of noncommutative field theory

TL;DR

This paper investigates the properties of noncommutative scalar field theory, deriving its energy-momentum tensor and demonstrating Poincaré invariance within the framework of Moyal star-products for specific spacetime noncommutativity.

Contribution

It extends the quantum motion equations and proves Poincaré translation invariance for noncommutative $^{igstar 4}$ scalar field theory with $ heta^{0i}=0$.

Findings

Derivation of energy-momentum tensor from translation symmetry

Generalization of Heisenberg and motion equations using Moyal star-products

Proof of Poincaré invariance for the specified 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 $\phi^{\star4}$ 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 $\phi^{\star4}$ scalar field theory for the case $\theta^{0i}=0$ of spacetime noncommutativity. Then we demonstrate the Poincar{\' e} translation invariance for noncommutative $\phi^{\star4}$ scalar field theory for the case $\theta^{0i}=0$ of spacetime noncommutativity.