On integral of exponent of a homogeneous polynomial

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.Comment: 4 pages, minor corrections, a reference adde

The Poisson algebra of classical Hamiltonians in field theory and the problem of its quantization

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.Comment: 8 page

Heuristic formula for logarithm of the Frobenius morphism

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.Comment: 2 pages, last remark adde

An attempt to construct dynamical evolution in quantum field theory

If we develop into perturbation series the evolution operator of the Heisenberg equation in the infinite dimensional Weyl algebra, say, for the $\phi^4$ model of field theory, then the arising integrals almost coincide with the usual Feynman diagram integrals. This fact leads to some mathematical definitions which, as it seemed to the author, defined dynamical evolution in quantum field theory in a mathematically rigorous way using the Weyl algebra. In fact the constructions of the paper are well defined in perturbation theory only in one-loop (quasiclassical) approximation. A variation of the construction is related with the Bogolyubov $S$-matrix $S(g)$.Comment: 5 page

On the mathematical sense of renormalization

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.Comment: 5 page

Integral of exponent of a polynomial is a generalized hypergeometric function of the coefficients of the polynomial

We show that the integral \int e^{S(x_1,...,x_n)}dx_1...dx_n, for an arbitrary polynomial S, satisfies a generalized hypergeometric system of differential equations in the sense of I. M. Gelfand et al.Comment: 3 pages, a reference added, minor correction

Generalized Schrodinger equation for free field

We give a logically and mathematically self-consistent procedure of quantization of free scalar field, including quantization on space-like surfaces. A short discussion of possible generalization to interacting fields is added.Comment: 18 page

Excitations Propagating Along Surfaces

A number of equations is deduced which describe propagation of excitations along $n$-dimensional surfaces in $R^N$. Usual excitations in wave theory propagate along 1-dimensional trajectories. The role of the medium of propagation of excitations considered in this paper is played by the infinite dimensional space of $(n-1)$-dimensional surfaces in $R^N$. The role of rays is played by $n$-dimensional solution surfaces of the variational problem. Such a generalization of wave theory can be useful in quantum field theory. Among these equations are the generalized Hamilton--Jacobi equation (known in particular cases in the literature), generalized canonical Hamilton equations, and generalized Schrodinger equation. Besides that, a theory of integration of the generalized Hamilton--Jacobi equation is developed.Comment: 12 pages; formulation and solution of the Cauchy problem for the generalized Hamilton--Jacobi equation adde

Generalizations of wave equations to multidimensional variational problems

This is a survey paper based on previous results of the author. In the paper, we define and discuss the generalizations of linear partial differential equations to multidimensional variational problems. We consider two examples of such equations: first, the generalized Schr\"odinger equation which is a natural candidate for the mathematical equation of quantum field theory, and second, the quantum Plato problem which is a natural candidate for a simplest mathematical equation of string theory and, more generally, theory of $D$-branes. We propose a way to give a mathematical sense to these equations.Comment: 8 pages. Revised version: incorrect renormalization on finite interval remove

A mathematical approach to quantum field theory

We develop a mathematical theory of quantization of multidimensional variational principles, and compare it with traditional constructions of quantum field theory. We conjecture that mathematical realization of quantum field theory axioms, in general, does not exist.Comment: 20 page