Noncommutative Field Theory

TL;DR

Noncommutative field theory extends standard quantum field theory to spaces with $[x^i,x^j]=i\theta^{ij}$, yielding nonlocal interactions and a rich range of phenomena. The paper develops the formal framework (algebras, star products, symbols, and NC spaces), analyzes kinematics and gauge theories (including the Seiberg-Witten map), and explores solitons, instantons, and quantum-field-theoretic aspects such as UV/IR mixing. It then connects these ideas to the quantum Hall effect and delves into the mathematical structure (Morita equivalence, deformation quantization) and the string/M-theory origins (D-branes, open strings, holography). The review highlights how NC theories reproduce and extend familiar gauge theory phenomena, reveals novel nonlocal observables like open Wilson lines, and elucidates how NC frameworks arise in limits of M-theory and string theory, while also identifying key open questions about renormalization, unitarity with timelike noncommutativity, and the full scope of NC dualities.

Abstract

We review the generalization of field theory to space-time with noncommuting coordinates, starting with the basics and covering most of the active directions of research. Such theories are now known to emerge from limits of M theory and string theory, and to describe quantum Hall states. In the last few years they have been studied intensively, and many qualitatively new phenomena have been discovered, both on the classical and quantum level. To appear in Reviews of Modern Physics.

Figures (6)

• Figure 1: The interaction of two dipoles.
• Figure 2: Double line notation and phase factors.
• Figure 3: The contraction $({\rm Tr~} \phi^m)({\rm Tr~} \phi^n) \leftrightarrow {\rm Tr~} \phi^{m+n-2}$.
• Figure 4: Tadpoles come with no phase factor.
• Figure 5: A non-planar diagram.