arXiv subjects
Publications and source records attributed to A. V. Stoyanovsky.
We construct a new version of infinite Grassmannian and infinite dimensional analog of the Weil representation of the affine symplectic group in the space of distributions. We give definition of a mathematical solution of the quantum field theory Schrödinger equation in the constructed space, and give examples of solutions of this equation.
We rigorously define renormalized evolution operator of the Schrödinger equation in the infinite dimensional Weyl-Moyal algebra for any time interval for arbitrary Hamiltonian depending on time. We state that for renormalizable field theories, in the interaction representation, and for the time interval being the full real axis, our construction yields standard renormalized $S$-matrix and Green functions of perturbative quantum field theory.
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 Schrödinger 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.
This paper is a survey of author's mathematical and logical study of the problem of quantization of fields.
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.
We conjecture that the renormalized perturbative $S$-matrix of quantum field theory coincides with the evolution operator of the standard functional differential Schrodinger equation whose right hand side (quantum local Hamiltonian) is understood as an element of an appropriate quantization of the Poisson algebra of classical field theory Hamiltonians. We show how to construct a quantization of this algebra, close to the algebra of differential operators on infinite dimensional space, but seemingly not appropriate for 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 $ϕ^4$ 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 $d\ge 6$).
We give a mathematical definition of quantum field theory on a manifold, and definition of quantization of a classical field theory given by a variational principle.
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.
We construct a mathematical model analogous to quantum field theory, but without the notion of vacuum and without measurable physical quantities. This model is a direct mathematical generalization of scattering theory in quantum mechanics to path integrals with multidimensional trajectories (whose mathematical interpretation has been given in a previous paper). In this model the normal ordering of operators in the Fock space is replaced by the Weyl-Moyal algebra. This model shows to be useful in proof of various results in quantum field theory: one first proves these results in the mathematical model and then "translates" them into the usual language of quantum field theory by more or less "ugly" procedures.
We state analogs of the binomial theorem and the exponential function for variables $x$, $y$ commuting as $yx=qxy$.
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.
Using the notion of distribution on an infinite dimensional space defined in our previous paper, we give definition of a version of dynamical evolution in quantum field theory, motivated by heuristic formulas involving path integrals.
We define the notion of distribution on an infinite dimensional space motivated by the notion of Feynman path integral and by construction of probability measures for generalized random fields. This notion of distribution turns out to be mathematically equivalent to the notion of generating functional of Green functions.
Using the Maslov--Shvedov method of complex germ, we show that quantum field theory S-matrix can exist outside perturbation theory in the principal order of quasiclassical approximation only under the condition that the tangent symplectic transformation to the evolution operator of non-linear classical field equation is unitarily implementable in the Fock space. However, the results of Maslov--Shvedov's book imply that this condition is seemingly always satisfied.
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.
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.
If we develop into perturbation series the evolution operator of the Heisenberg equation in the infinite dimensional Weyl algebra, say, for the $ϕ^4$ 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 $S(g)$.