Twisted gauge and gravity theories on the Groenewold-Moyal plane

Abstract

Recent work [1,2] indicates an approach to the formulation of diffeomorphism invariant quantum field theories (qft's) on the Groenewold-Moyal (GM) plane. In this approach to the qft's, statistics gets twisted and the S-matrix in the non-gauge qft's become independent of the noncommutativity parameter θ µν . Here we show that the noncommutative algebra has a commutative spacetime algebra as a substructure: the Poincaré, diffeomorphism and gauge groups are based on this algebra in the twisted approach as is known already from the earlier work of [1]. It is natural to base covariant derivatives for gauge and gravity fields as well on this algebra. Such an approach will in particular introduce no additional gauge fields as compared to the commutative case and also enable us to treat any gauge group (and not just U (N )). Then classical gravity and gauge sectors are the same as those for θ µν = 0, but their interactions with matter fields are sensitive to θ µν . We construct quantum noncommutative gauge theories (for arbitrary gauge groups) by requiring consistency of twisted statistics and gauge invariance. In a subsequent paper (whose results are summarized here), the locality and Lorentz invariance properties of the S-matrices of these theories will be analyzed, and new non-trivial effects coming from noncommutativity will be elaborated. This paper contains further developments of [3] and a new formulation based on its approach.

Noncommutative gauge theory and renormalisability

“…a = 0, the integrand is proportional to (k 2 ) −n , for k 2 → 0, as we have already mentioned. But a ̸ = 0 implies that the integrand behaves like 15) which is independent of the loop order n. Using multiscale analysis, the perturbative renormalisability of this model to all orders could be shown [25].…”

“…This connection is not fully understood at present and studied from different points of view, see e.g. [10,11,12,13,14,15] and references therein for a merely exemplary list of quotations.…”

Interacting Quantum Topologies and the Quantum Hall Effect

“…Let us use the convention that the first slot in the two fold tensor product of single particle states belongs to the first particle and the second slot belongs to the second particle. Next, following [3], we introduce these labels into the Drinfel'd twist element: F …”

Renormalization of noncommutative quantum field theories

“…Twisted field theories involving real scalar field φ θ and having a φ 4 θ, * interactions are discussed in [15] and are shown to be free from UV/IR mixing. Gauge field theories with nonabelian gauge groups are constructed in [16,17]. Construction of thermal field theories is done in [18][19][20] while [21] discusses the twisted bosonization in two dimensional noncommutative spacetime.…”