188 research outputs found
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 , 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 of the Frobenius morphism is
given by the formula (the natural logarithm). In particular, it
does not depend on . 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
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
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
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