Position-dependent noncommutative products: classical construction and field theory

We look in Euclidean $R^4$ for associative star products realizing the commutation relation $[x^\mu,x^\nu]=i\Theta^{\mu\nu}(x)$, where the noncommutativity parameters $\Theta^{\mu\nu}$ depend on the position coordinates $x$. We do this by adopting Rieffel's deformation theory (originally formulated for constant $\Theta$ and which includes the Moyal product as a particular case) and find that, for a topology $R^2 \times R^2$, there is only one class of such products which are associative. It corresponds to a noncommutativity matrix whose canonical form has components $\Theta^{12}=-\Theta^{21}=0$ and $\Theta^{34}=-\Theta^{43}= \theta(x^1,x^2)$, with $\th(x^1,x^2)$ an arbitrary positive smooth bounded function. In Minkowski space-time, this describes a position-dependent space-like or magnetic noncommutativity. We show how to generalize our construction to $n\geq 3$ arbitrary dimensions and use it to find traveling noncommutative lumps generalizing noncommutative solitons discussed in the literature. Next we consider Euclidean $\lambda\phi^4$ field theory on such a noncommutative background. Using a zeta-like regulator, the covariant perturbation method and working in configuration space, we explicitly compute the UV singularities. We find that, while the two-point UV divergences are non-local, the four-point UV divergences are local, in accordance with recent results for constant $\Theta$.Comment: 1+22 pages, no figure

Index theory for locally compact noncommutative geometries

Spectral triples for nonunital algebras model locally compact spaces in noncommutative geometry. In the present text, we prove the local index formula for spectral triples over nonunital algebras, without the assumption of local units in our algebra

Index theory for locally compact noncommutative geometries

Spectral triples for nonunital algebras model locally compact spaces in noncommutative geometry. In the present text, we prove the local index formula for spectral triples over nonunital algebras, without the assumption of local units in our algebra. This formula has been successfully used to calculate index pairings in numerous noncommutative examples. The absence of any other effective method of investigating index problems in geometries that are genuinely noncommutative, particularly in the nonunital situation, was a primary motivation for this study and we illustrate this point with two examples in the text. In order to understand what is new in our approach in the commutative setting we prove an analogue of the Gromov-Lawson relative index formula (for Dirac type operators) for even dimensional manifolds with bounded geometry, without invoking compact supports. For odd dimensional manifolds our index formula appears to be completely new. As we prove our local index formula in the framework of semifinite noncommutative geometry we are also able to prove, for manifolds of bounded geometry, a version of Atiyah's L^2-index Theorem for covering spaces. We also explain how to interpret the McKean-Singer formula in the nonunital case. In order to prove the local index formula, we develop an integration theory compatible with a refinement of the existing pseudodifferential calculus for spectral triples. We also clarify some aspects of index theory for nonunital algebras.Comment: 133 pages. to appear in Memoirs of the American Mathematical Society. Published version will have different pagination, and an inde

Spectral flow for nonunital spectral triples

We prove two results about nonunital index theory left open in a previous paper. The first is that the spectral triple arising from an action of the reals on a C*-algebra with invariant trace satisfies the hypotheses of the nonunital local index formula. The second result concerns the meaning of spectral flow in the nonunital case. For the special case of paths arising from the odd index pairing for smooth spectral triples in the nonunital setting, we are able to connect with earlier approaches to the analytic definition of spectral flow

Probing exciton localization in non-polar GaN/AlN Quantum Dots by single dot optical spectroscopy

We present an optical spectroscopy study of non-polar GaN/AlN quantum dots by time-resolved photoluminescence and by microphotoluminescence. Isolated quantum dots exhibit sharp emission lines, with linewidths in the 0.5-2 meV range due to spectral diffusion. Such linewidths are narrow enough to probe the inelastic coupling of acoustic phonons to confined carriers as a function of temperature. This study indicates that the carriers are laterally localized on a scale that is much smaller than the quantum dot size. This conclusion is further confirmed by the analysis of the decay time of the luminescence