Hopf-algebraic Deformations of Products and Wick Polynomials

We present an approach to cumulant–moment relations and Wick polynomials based on extensive use of convolution products of linear functionals on a coalgebra. This allows, in particular, to understand the construction of Wick polynomials as the result of a Hopf algebra deformation under the action of linear automorphisms induced by multivariate moments associated to an arbitrary family of random variables with moments of all orders. We also generalize the notion of deformed product in order to discuss how these ideas appear in the recent theory of regularity structures.

CANONICAL QUANTIZATION, THE DIRAC PROBLEM AND THE ANOMALIES

The quantization a la Dirac coincides with the canonical one if and only if the anomalies are absent. It is shown that in general this quantization must be modified to be the Lie algebra deformation instead of the homomorphism.