Analyticity and forward dispersion relations in noncommutative quantum field theory

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. θ 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 LehmannSymanzik-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, θ 0i = 0, as well as on the nonforward scattering amplitude and mention their peculiarities.

Wightman axiomatic approach in noncommutative field theory

“…In the same version of the theory, analytic properties of the elastic scattering amplitude with respect to the energetic and angular variables were investigated [18]- [22]. Forward dispersion relations were derived in [19], [20]. The validity of the Froissart-Martin inequality was proved in [22].…”

Jost-Lehmann-Dyson representation, analyticity in the angular variable, and upper bounds in noncommutative quantum field theory

“…Rigorously speaking, it is necessary that the DRs hold not only for forward scattering but also for an arbitrarily small negative value of t, which is a usual variable. The DRs were proved in the NC case only for the forward scattering amplitude [20], [21]. But in accordance with the analyticity of F (s, cos ϑ) proved for the Lehmann ellipse, the DRs also hold in some domain of negative values of t.…”

“…In [20] and [21], we showed that the forward elastic scattering amplitude in these NC field theories has the same analytic properties as in the commutative theory. Such questions were first considered in [22].…”

“…As shown in [20], the usual Lehmann-Symanzik-Zimmermann formulas are valid in the version of NC field theory under consideration if the related interacting fields have in-and out-fields as their limits. Therefore, up to numerical factors and a polynomial in the energy, we have…”

“…It was proved in [20] and [21] that satisfaction of condition (8) suffices for deriving the usual forward dispersion relations. This fact plays a fundamental role in extending the analyticity domain in the cos ϑ plane for the NC case.…”