Nonlinear generalized functions and the Heisenberg-Pauli foundations of Quantum Field Theory

TL;DR

This paper introduces a mathematical framework using nonlinear generalized functions to rigorously interpret the foundational calculations of quantum field theory, aiming to address longstanding issues like infinities and domain problems.

Contribution

It proposes applying nonlinear generalized functions to quantum field theory, providing a new mathematical basis for understanding and improving the Heisenberg-Pauli calculations.

Findings

Mathematically justified Heisenberg-Pauli calculations using generalized functions

Eliminates infinities in matrix element calculations of the S operator

Suggests a connection with renormalization theory for small coupling constants

Abstract

In 1929 Heisenberg and Pauli laid the foundations of QFT by quantizing the fields (method of canonical quantization). This general theory of quantized fields has remained undisputed up to now. We show how the unmodified Heisenberg-Pauli calculations make sense mathematically by using a theory of generalized functions adapted to nonlinear operations. By providing an appropriate mathematical setting, nonlinear generalized functions open doors for their understanding but there remains presumably very hard technical problems. (i) Domains of the interacting field operators: a priori the H-P calculations give time dependent dense domains, what is not very convenient; (ii) Calculations of the resulting matrix elements of the S operator: from the unitarity of the S operator as a whole there are no longer ``infinities,'' but a priori there is no other hope than heavy computer calculations; (iii) Connection with renormalization theory: it should provide an approximation when the coupling constant is small. The aim of this paper is to present, on the grounds of a standard mathematical model of QFT (a self interacting scalar boson field), a basis for improvement without significant prerequisites in mathematics and physics. It is an attempt to use nonlinear generalized functions in QFT, starting directly from the calculations made by physicists, in the same way as they have already been used in classical mechanics and general relativity.