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

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

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

Gaussian transform of the Weil representation

A description is given of the image of the Weil representation of the symplectic group in the Schwartz space and in the space of tempered distributions under the Gaussian integral transform. We also discuss the problem of infinite dimensional generalization of the Weil representation in the Schwartz space, in order to construct appropriate quantization of free scalar field.Comment: 21 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

Quantum Langlands duality and conformal field theory

V. Drinfeld proposed conjectures on geometric Langlands correspondence and its quantum deformation. We refine these conjectures and propose their relationship with algebraic conformal field theory.Comment: 7 page

Maslov's complex germ and the Weyl--Moyal algebra in quantum mechanics and in quantum field theory

The paper is a survey of some author's results related with the Maslov--Shvedov method of complex germ and with quantum field theory. The main idea is that many results of the method of complex germ and of perturbative quantum field theory can be made more simple and natural if instead of the algebra of (pseudo)differential operators one uses the Weyl algebra (operators with Weyl symbols) with the Moyal *-product. Section 1, devoted to quantum mechanics, contains a closed mathematical description of the Maslov--Shvedov method in the theory of Schrodinger equation, including the method of canonical operator. In particular, it contains a new simple definition of the Maslov index modulo 4. Section 2, devoted to quantum field theory, contains a logically self-consistent exposition of the main results of perturbative quantum field theory not using the subtraction of infinities from the quantum Hamiltonian of free field and normal ordering of operators. It also contains a result (dynamical evolution in quantum field theory in quasiclassical approximation) close to the Maslov--Shvedov quantum field theory complex germ.Comment: 36 page

Binomial theorem and exponent for variables commuting as yx=qxyyx=qxy

We state analogs of the binomial theorem and the exponential function for variables $x$, $y$ commuting as $yx=qxy$.Comment: 2 pages, references adde