Quantum Field Theories on a Noncommutative Euclidean Space: Overview of New Physics

In this talk I briefly review recent developments in quantum field theories on a noncommutative Euclidean space, with Heisenberg-like commutation relations between coordinates. I will be concentrated on new physics learned from this simplest class of non-local field theories, which has applications to both string theory and condensed matter systems, and possibly to particle phenomenology. 1. Noncommutative Field Theories In this talk I will give a brief overview of the new physics recently learned from quantum field theories on a noncommutative Euclidean space. Below I will call them simply as noncommutative field theories (NCFT). (Because of the time limitation, it is regretful that many significant topics and contributions in this field have to be left out.) By definition, a noncommutative Euclidean space is a space with noncommuting spatial coordinates, satisfying the Heisenberg-like commutation relations: [x, x] = iθ , (1) where θ = −θji are real constants. (There are other types of noncommutative spaces, with coordinates satisfying Lie-algebraic or Yang-Baxter relations, or with space-time noncommutativity. In this talk I am restricted only to the case of Eq. (1).) There are two ways of interpreting these commutation relations. The first way is to interpret x as operators in a Hilbert space satisfying Eq. (1). In this interpretation, the noncommutative space can be mathematically viewed as generalization of phase space in usual quantum mechanics. The second is to interpret coordinates x as functions that generate a noncommutative algebra of functions (or fields) on the space. One may develop a classical field theory from this point of view, by realizing the algebra of fields in the set of ordinary functions of commuting variables {xμ} with the Moyal star product [1] (i.e. deformation quantization)