Differential operators on infinite dimensional space and quantum field theory

TL;DR

This paper explores the connection between the renormalized perturbative S-matrix in quantum field theory and the evolution operator of a functional differential Schrödinger equation, proposing a new approach to quantization of classical Hamiltonians.

Contribution

It introduces a novel quantization method of the classical field theory Hamiltonian algebra, linking it to differential operators on infinite dimensional spaces.

Findings

Proposes that the S-matrix coincides with the evolution operator of a functional differential Schrödinger equation.

Constructs a quantization of the classical Hamiltonian algebra related to differential operators.

Highlights challenges in applying this quantization directly to quantum field theory.

Abstract

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.