Exact Solution of Quantum Field Theory on Noncommutative Phase Spaces

Abstract

We present the exact solution of a scalar field theory defined with noncommuting position and momentum variables. The model describes charged particles in a uniform magnetic field and with an interaction defined by the Groenewold-Moyal star-product. Explicit results are presented for all Green's functions in arbitrary even spacetime dimensionality. Various scaling limits of the field theory are analysed non-perturbatively and the renormalizability of each limit examined. A supersymmetric extension of the field theory is also constructed in which the supersymmetry transformations are parametrized by differential operators in an infinite-dimensional noncommutative algebra.

Non-commutative Renormalization

“…Using direct space methods, we have provided a new proof of the BPHZ theorem for vulcanized non commutative φ 4 4 [47]. We have also extended the Grosse-Wulkenhaar results to theφ φ φ φ LSZ model [43]. Our proof relies on a multiscale analysis analogous to [42] but in direct space.…”

“…This is just wrong but for a more subtle reason. We have shown that the Grosse-Wulkenhaar method can be extended to renormalize theories such as the Langmann-Szabo-Zaremboφ φ φ φ model [43,44,45] in four dimensions or the Gross-Neveu model in two dimensions. In these theories the ordinary Mehler kernel is replaced by a related kernel which governs propagation of charged particles in a constant background field.…”

“…Let us note that for Ω = 1, the model is invariant. [64,65] and the LSZ models [43,44,45]. These last ones are of the type…”

“…Nevertheless they either are super-renormalizable (Φ 4 2 [39]) or (and) studied at a special point in the parameter space where they are solvable (Φ 3 2 , Φ 3 4 [64,65], the LSZ models [43,44,45]). Although only logarithmically divergent for parity reasons, the non-commutative Gross-Neveu model is a just renormalizable quantum field theory as Φ 4 4 .…”

Symmetries and Renormalization of Noncommutative Field Theory

“…The key to the renormalization of the duality-covariant model is the use of a "matrix basis" for the expansion of fields given by [17,21] …”