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Commutators and propagators of Moyal star-products and microscopic causality for noncommutative scalar field theory

We study the Moyal commutators and their expectation values between vacuum states and non-vacuum states for noncommutative scalar field theory. Then from the Moyal commutators, we demonstrate that the microscopic causality is satisfied for scalar fields on noncommutative spacetime. We construct the Feynman propagator of Moyal star-product for noncommutative scalar field theory.