Hermitian Analyticity, IR/UV Mixing and Unitarity of Noncommutative Field Theories

Abstract

The IR/UV mixing and the violation of unitarity are two of the most intriguing aspects of noncommutative quantum field theories. In this paper the relation between these two phenomena is explained and established in an explicit form. We start out by showing that the S-matrix of noncommutative field theories is hermitian analytic. As a consequence, a noncommutative field theory is unitary if the discontinuities of its Feynman diagram amplitudes agree with the expressions calculated using the Cutkosky formulae. These unitarity constraints relate the discontinuities of amplitudes with physical intermediate states and allow us to see how the IR/UV mixing may lead to a breakdown of unitarity. Specifically, we show that the IR/UV singularity does not lead to the violation of unitarity in the space–space noncommutative case, but it does lead to its violation in a space–time noncommutative field theory. As a corollary, noncommutative field theory without IR/UV mixing will be unitary in both the space–space and space–time noncommutative case. To illustrate this, we introduce and analyse the noncommutative Lee model—an exactly solvable quantum field theory. We show that the model is free from the IR/UV mixing in both the space–space and space–time noncommutative cases. Our analysis is exact. Due to absence of the IR/UV mixing one can expect that the theory is unitary. We present some checks supporting this claim. Our analysis provides a counter example to the generally held belief that field theories with space–time noncommutativity are nonunitary.