Canonical Quantization of Noncommutative Field Theory

A simple method to canonically quantize noncommutative field theories is proposed. As a result, the elementary excitations of a (2n + 1)-dimensional scalar field theory are shown to be bilocal objects living in an (n + 1)-dimensional space-time. Feynman rules for their scattering are derived canonically. They agree, upon suitable redefinitions, with the rules obtained via star-product methods. The IR/UV connection is interpreted within this framework. Introduction and Summary Noncommutative field theories [1] are interesting, nonlocal but most probably consistent, extensions of the usual ones. They also arise as a particular low energy limit of string theory [2, 3]. The fields are defined over a base space which is noncommutative [1], often obeying relations of the type [xμ, xν ] = iθμν . At the classical level, new physical features appear in these theories. For instance, one encounters solitonic excitations in higher dimensions [4], superluminal propagation [5], or waves propagating on discrete spaces [6]. At the quantum level, one has two superimposed structures: the coordinate space, where [x̂μ, x̂ν ] 6= 0, and the dynamical fields’ (fiber) space, where canonically conjugate variables do not commute, [φ̂(t, ~x), π̂(t, ~x)] 6= 0. This two-level structure hampered the canonical quantization of noncommutative (NC) field theories. Consequently, their perturbative quantum dynamics has been studied via star-product techniques [1], i.e. by replacing ∗On leave from: Institute of Atomic Physics P.O. Box MG-6, 76900 Bucharest, Romania; e-mail: acatrine@physics.uoc.gr.