Analyticity and Forward Dispersion Relations in Noncommutative Quantum Field Theory

TL;DR

This paper establishes the analytical properties and dispersion relations for elastic forward scattering amplitudes in noncommutative quantum field theory, focusing on space-space noncommutativity and extending to general cases.

Contribution

It proves dispersion relations in noncommutative QFT with space-space noncommutativity using microcausality and reduction formalisms, extending the understanding of analyticity in such theories.

Findings

Dispersion relation similar to commutative QFT for space-space noncommutativity.

Existence of LSZ and BMP formalisms in noncommutative QFT.

Remarks on noncommutative cases with time-space noncommutativity and nonforward amplitudes.

Abstract

We derive the analytical properties of the elastic forward scattering amplitude of two scalar particles from the axioms of the noncommutative quantum field theory. For the case of only space-space noncommutativity, i.e. $\theta_{0i}=0$, we prove the dispersion relation which is similar to the one in commutative quantum field theory. The proof in this case is based on the existence of the analog of the usual microcausality condition and uses the Lehmann-Symanzik-Zimmermann (LSZ) or equivalently the Bogoliubov-Medvedev-Polivanov (BMP) reduction formalisms. The existence of the latter formalisms is also shown. We remark on the general noncommutative case, $\theta_{0i}\neq0$, as well as on the nonforward scattering amplitude and mention their peculiarities.