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

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

Quantization on space-like surfaces

We give a mathematical definition of dynamical evolution in quantum field theory, including evolution on space-like surfaces, and show its relationship with the axiomatic and perturbative approaches to QFT.Comment: 4 page

Some additive relations in the Pascal triangle

We derive some, seemingly new, curious additive relations in the Pascal triangle. They arise in summing up the numbers in the triangle along some vertical line up to some place.Comment: 2 page

A-hypergeometric functions in transcendental questions of algebraic geometry

We generalize the known constructions of A-hypergeometric functions. In particular, we show that periods of middle dimension on affine or projective complex algebraic varieties are A-hypergeometric functions of coefficients of polynomial equations of these varieties.Comment: 4 pages, revised versio

No-Counterterm approach to quantum field theory

We give a conjectural way for computing the $S$-matrix and the correlation functions in quantum field theory beyond perturbation theory. The basic idea seems universal and naively simple: to compute the physical quantities one should consider the functional differential Schrodinger equation (without normal orderings), regularize it, consider the regularized evolution operator in the Fock space from $t=T_1$ to $t=T_2$, where the interval $(T_1,T_2)$ contains the support of the interaction cutoff function, remove regularization (without adding counterterms), and tend the interaction cutoff function to a constant. We call this approach to QFT the No-Counterterm approach. We show how to compute the No-Counterterm perturbation series for the $\phi^4$ model in $R^{d+1}$. We give rough estimates which show that some summands of this perturbation series are finite without renormalization (in particular, one-loop integrals for $d=3$ and all integrals for $d\ge 6$).Comment: 8 pages, revised version, title change