arXiv subjects
Publications and source records attributed to A. Stoyanovsky.
We show that the logarithm $\log_q$ of the Frobenius morphism $x\to x^q$ is given by the formula $x\to x\log x$ (the natural logarithm). In particular, it does not depend on $q$. This is the explicit (although heuristical) formula for the operator conjectured by Hilbert whose eigenvalues coincide with the zeroes of the zeta function.
We place the renormalization procedure in quantum field theory into the familiar mathematical context of quantization of Poisson algebras. The Poisson algebra in question is the algebra of classical field theory Hamiltonians constructed in a previous paper (arXiv:1008.3333). Its quantum deformations presumably contain (non-canonically) the algebra of functional differential operators. We explain that this picture contains renormalization as a natural ingredient.
We introduce the notion of G-hypergeometric function, where G is a complex Lie group. In the case when G is a complex torus, this notion amounts to the notion of Gelfand's A-hypergeometric function. We show that the integral $\int e^{P(x_1,...,x_n)}dx_1...dx_n$, where P is a homogeneous polynomial, is a GL(n)-hypergeometric function of algebraic SL(n)-invariants of the polynomial.
We construct the commutative Poisson algebra of classical Hamiltonians in field theory. We pose the problem of quantization of this Poisson algebra. We also make some interesting computations in the known quadratic part of the quantum algebra.
This paper has been withdrawn due to non-clearness of some technical points, as well as lack of a reasonable statement of quantization conjecture.