Noncommutative (supersymmetric) electrodynamics in the Yang-Feldman formalism

Abstract

We study quantum electrodynamics on the noncommutative Minkowski space in the Yang-Feldman formalism. Local observables are defined by using covariant coordinates. We compute the two-point function of the interacting field strength to second order and find the infrared divergent terms already known from computations using the so-called modified Feynman rules. It is shown that these lead to nonlocal renormalization ambiguities. Also new nonlocal divergences stemming from the covariant coordinates are found. Furthermore, we study the supersymmetric extension of the model. For this, the supersymmetric generalization of the covariant coordinates is introduced. We find that the nonlocal divergences cancel. At the one-loop level, the only effect of noncommutativity is then a momentum-depenent field strength normalization. We interpret it as an acausal effect and show that its range is independent of the noncommutativity scale.1 Strictly speaking, one should replace Θ µν in (1.1) by a central operator Q µν whose joint spectrum is Σ. But for the present purposes, it suffices to consider a fixed Θ ∈ Σ, as long as one keeps in mind that it transforms as a tensor under Lorentz transformations.2 Here, we do not consider the theories obtained by adding a Grosse-Wulkenhaar potential [5]. As shown in [6], their Minkowski space version is badly divergent. We also do not consider the so-called 1/p 2 theories [7], and the models inspired by the twist approach of Wess and coworkers [8].3 In general, planarity is defined by the genus of the Riemann surface onto which the graph may be drawn [10].

Ultraviolet–Infrared Mixing on the Noncommutative Minkowski Space in the Yang–Feldman Formalism

“…with h as above. This is very similar to the expression for u(x) given in (12), the difference beeing that∆ + is replaced by ∂ m 2∆ + and x by σk. As above, the first replacement does not change the asymptotic behavior.…”

“…instead of (5), where {·, ·, ·} stands for complete symmetrization 3 . Such a symmetrization was already proposed and used in [12] in order to cure certain inconsistencies in the quantization of gauge fields. Using this product in (3), we find…”

Quantum Spacetime and Algebraic Quantum Field Theory

“…We do not comment on gauge theories here, see however, e.g. [20] and the comments on covariant coordinates and gauge invariant quantities in [10].…”

Noncommutative time in quantum field theory

“…In this approach the interaction picture is avoided entirely and the elements of the S-matrix are calculated directly in the Heisenberg picture with the help of the n-point correlation functions (Green's functions). In theories with time-space noncommutativity, the Yang-Feldman-Källén approach has been considered for example in [20,21,48].…”