109 research outputs found
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
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 -matrix .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 -dimensional surfaces in . 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 -dimensional surfaces in . The role of rays is
played by -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
-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
We state analogs of the binomial theorem and the exponential function for
variables , commuting as .Comment: 2 pages, references adde
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