Field theories in nanocommutative spaces

Abstract

The recent interest in noncommutative spaces has its roots in the study of quantum gravity and string theory. At length scales close to the Planck scale, the usual notion of continuous spacetime is expected to break down due to strong gravity effects [1]. In this regime, the smooth spacetime structure should therefore be replaced. Heuristic arguments by [1] show that this leads to an uncertainty in the spacetime measurements, and the spacetime is described by a noncommutative space. There are various techniques to deform the commutative algebra of functions on the manifold M? to the noncommutative algebra. We consider one such noncommutative deformation: the Groenewold-Moyal plane. The Groenewold-Moyal (GM) plane is a deformation of the usual algebra of functions to a noncommutative algebra by replacing the pointwise product of functions with a *-product: f(x)g(x)becomesf(x)?g(x)=f(x)ei2????????g(x).f(x)g(x) \quad \text{becomes} \quad f(x) * g(x) = f(x)e^{\frac{i}{2}\theta^{\mu\nu}\partial_\mu \partial'_\nu}g(x).f(x)g(x)becomesf(x)?g(x)=f(x)e2i???????????g(x). The usual action of the Lorentz group is not compatible with the above *-product. But it is possible to define a new twisted coproduct to enforce the compatibility of the *-product with the Lorentz group [2]. Thus Lorentz-covariant field theories on the GM plane can be constructed. In the quantum theory, this new coproduct changes the notion of (anti-)symmetrization, which in turn deforms the algebra of the creation-annihilation operators. This leads to the twisting of the statistics, which can be accounted for by addressing transformation of the quantum fields on the ordinary spacetime [3]. While the above motivation for the noncommutativity of the spacetime is set in a very general context, tractable problems involving both noncommutativity and gravity are difficult to formulate. The technical tools necessary for dealing with quantum fields on arbitrary noncommutative manifolds are still in the process of being developed (see for example [4]) and consequently only a handful of specific computations in gravity physics have been done till date. We would like to pose a problem in quantum field theory that is simple to formulate, and though does not involve general relativity, is perhaps the closest that one can get to discussing the nature of quantum theory in curved space. The Unruh effect, which is related to the quantum theory of an observer undergoing a uniform proper acceleration in Minkowski spacetime, is one such example. A uniformly accelerated observer in ordinary spacetime has hyperbolic worldlines. The time translations for such an observer are generated by Lorentz boosts, and the quantum field in the frame of such an observer can be expanded in the basis of the eigenfunctions of the generators of Lorentz boosts [5]. Further, it is well known that the inertial vacuum state is a KMS thermal state for the accelerated observer. The temperature of the state is proportional to the acceleration, and the accelerated observer sees a Bose-Einstein or Fermi-Dirac distribution in the inertial vacuum state. The Lorentz group is an automorphism of the GM plane. In this thesis, we define an accelerated observer on this GM plane, taking advantage of the aforementioned automorphism property. Specifically, a sequence of instantaneous Lorentz boosts relates an accelerated observer to an inertial observer in the GM plane. The twisted field for this observer can be described using the dressing transformation. We adapt the Bisognano-Wichmann theorem appropriately to the field theories in the GM plane and compute the n-point functions. The two-point function gets modified, and in the GM plane the inertial vacuum state is no longer a thermal state [6]. It is, however, still a stationary state. Another way to deform the commutative algebra of functions to a noncommutative algebra leads to the fuzzy spaces. For instance, replacing algebra of the complex numbers Za(a=1,2,…,n)Z_a (a = 1, 2, \dots, n)Za?(a=1,2,…,n) describing Cn\mathbb{C}^nCn by n independent oscillators a?a_\alphaa?? gives the algebra of the fuzzy space Cn\mathbb{C}^nCn. The a?a_\alphaa?? act on the Hilbert space of n independent oscillators. A fuzzy 3-sphere S3S^3S3 can be described by the algebra of X?=a?†a?X_\alpha = a_\alpha^\dagger a_\alphaX??=a?†?a?? where ?=1,2\alpha = 1,2?=1,2 and N=a?†a?N = a_\alpha^\dagger a_\alphaN=a?†?a??. The noncommutative version of the Hopf fibration is given by the Schwinger construction: Xi=a†?iaX_i = a^\dagger \sigma_i aXi?=a†?i?a. The total number operator N takes the fixed value n in the subspace of the two oscillators. The algebra of XiX_iXi? restricted to N is the algebra of the fuzzy 2-sphere S2S^2S2. The sections of the complex line bundle are described by ?\phi?. The topological charge is given by kkk. These ?\phi?’s are elements of a left i-module and a right n-module. The rotations in this bimodule are generated by Li=Ad[Xi]L_i = \text{Ad}[X_i]Li?=Ad[Xi?]. The LiL_iLi?’s generate an SU(2) algebra in the representations j?j?=?j?j??,…,j+j?j \otimes j' = |j-j'|, \dots, j+j'j?j?=?j?j??,…,j+j?. The complex scalar fields ?\phi? can be expanded in the basis of the eigenfunctions of LzL_zLz? and LiLiL_i L_iLi?Li? belonging to the representations j=?j?j??,…,j+j?j = |j-j'|, \dots, j+j'j=?j?j??,…,j+j? (for details, see [7]). We show that we can extend the idea of fuzzy spaces to a class of manifolds called conifolds. Recall that a (2n ? 2)-dimensional conifold Y2n?2Y_{2n-2}Y2n?2? is described by a quadratic ?Za2=0,Za?Cn\sum Z_a^2 = 0, Z_a \in \mathbb{C}^n?Za2?=0,Za??Cn. We show that the algebra of a?a_\alphaa?? (? = 1,2,...,n) restricted to the kernel of ?Za2\sum Z_a^2?Za2? describes the (2n ? 2)-dimensional fuzzy conifold. The base of the fuzzy conifold is described by the intersection with S2n?1S^{2n-1}S2n?1. Further, using the generators of SO(3) and SO(4), we construct Hopf-like maps X:Sp?SqX: S^p \to S^qX:Sp?Sq and X:Sp×SpX: S^p \times S^pX:Sp×Sp [8,9]. These are noncommutative U(1) fibrations. We show that the sections of the complex line bundles can again be expressed in terms of SU(2) and SO(4) harmonics respectively. These fuzzy spaces appear naturally in matrix models describing D-branes. In a three-matrix model in Ramond-Ramond 4-form background, a set of noncommuting N × N matrices satisfying the fuzzy 2-sphere’s algebra extremizes the action. These solutions describe N D0-branes attached to a spherical D2-brane. This fuzzy sphere algebra can be in irreducible or reducible representation. It has been shown [10-12] that the classical energy of such a reducible fuzzy sphere is greater than that of the irreducible one and the system condenses to the irreducible minima by tachyon condensation. We construct these fuzzy spheres in the reducible representation by Schwinger construction using the Brandt-Greenberg oscillators [13]. Further, we construct the quantum states of the fuzzy spheres using GNS construction. The quantum state in the reducible representation is mixed, while the irreducible one is pure. This prevents condensation by any unitary evolution. Also, the quantum state being impure carries intrinsic quantum entropy, which can be macroscopically large, and we compute it explicitly.