Noncommutative Quantization in 2D Conformal Field Theory

The simplest possible noncommutative harmonic oscillator in two dimensions is used to quantize the free closed bosonic string in two flat dimensions. The partition function is not deformed by the introduction of noncommutativity, if we rescale the time and change the compactification radius appropriately. The four point function is deformed, preserving, nevertheless, the sl(2,C) invariance. Finally the first Ward identity of the deformed theory is derived.Comment: 5 pages, The solitonic contribution to the partition function has been computed. The parameter $\theta$ has been analytically continued to $-i\theta

The gravitational chiral anomaly of spin-1/2 field in the presence of twisted boundary conditions for ordinary field theory

We calculate the chiral anomaly in the neighbourhood of the fixed point space M_h which is constructed by the group action of a discrete symmetry h on a compact manifold M. The Feynman diagrams approach for the corresponding supersymmetric quantum mechanical system with twisted boundary conditions is used. The result we derive in this way agrees with the generalization of the ordinary index theorem (the G-index theorem) on the spin complex.Comment: 8 page

Self-similar solutions of R\'enyi's entropy and the concavity of its entropy power

We study the class of self-similar probability density functions with finite mean and variance which maximize R\'{e}nyi's entropy. The investigation is restricted in the Schwartz space $S(\mathbb{R}^d)$ and in the space of $l$-differentiable compactly supported functions $C_c^l(\mathbb{R}^d)$. Interestingly the solutions of this optimization problem do not coincide with the solutions of the usual porous medium equation with a Dirac point source, as it occurs in the optimization of Shannon's entropy. We also study the concavity of the entropy power in $\mathbb{R}^d$ with respect to time using two different methods. The first one takes advantage of the solutions determined earlier while the second one is based on a setting that could be used for Riemannian manifolds.Comment: 18 pages, 2 figure