Field Theory with Coordinate Dependent Noncommutativity

Abstract

We discuss the formulation of classical field theoretical models on n-dimensional noncommutative space-time defined by a generic associative star product. A simple procedure for deriving conservation laws is presented and applied to field theories in noncommutative space-time to obtain local conservation laws (for the electric charge and for the energy-momentum tensor of free fields) and more generally an energy-momentum balance equation for interacting fields. For free field models an analogy with the damped harmonic oscillator in classical mechanics is pointed out, which allows us to get a physical understanding for the obtained conservation laws. To conclude, the formulation of field theories on curved noncommutative space is addressed.Manfred Schweda passed away in 2017 before the present work which he initiated was finished. His coauthors dedicate this paper to his memory with deep gratitude for his enthusiastic collaboration and friendship as well as for the example he has given to all of us over many years.

Noncommutative geometry and deformation quantization in the quantum Hall fluids with inhomogeneous magnetic fields

“…with {i, j} = {X, Y}, is space-dependent and identified as a space-varying magnetic length. In noncommutative geometry, space-dependent noncommutative parameters have been analyzed in a large number of works, see, for instance, a recent review paper [50]. At the same time, the above expression for the quantum Hall effect has been already derived by Maraner [55] through the operatorial approach by introducing generalized guiding center operators with a symmetric gauge.…”

“…We start by noticing that the Weyl symbol of θij ( X) (i.e. a special kind of phase-space function associated to the operator θij ( X), see [3] for a general definition of Weyl symbol) at leading order can be identified with the real function θ ij (X), which represents a Poisson tensor because it trivially satisfies the Poisson-Jacobi identity [50]…”

“…We will employ symplectic geometry and Fedosov's deformation quantization, which require the introduction of symplectic connections together with a covariant generalization of the Moyal star-product known as Fedosov star-product [42,43] (see also [44,45], for a recent application of symplectic connections to condensed matter systems and [46][47][48] where a start-product, similar to that one by Fedosov, is introduced in the context of the quantum Hall effect). Although Fedosov's deformation quantization has been already employed in the high-energy-physics context to study (curved) noncommutative geometries with space-dependent deforming parameters [49,50], in our case, this parameter will be simply interpreted as a spacially-varying magnetic length.…”