[Paper Review] Mathematical problems on generalized functions and the canonical Hamiltonian formalism

J. F. Colombeau|arXiv (Cornell University)|Aug 25, 2007

Mathematical and Theoretical Analysis20 references3 citations

TL;DR

This paper formulates a mathematically rigorous framework for the Heisenberg-Pauli calculations in quantum field theory using nonlinear generalized functions and unbounded operators on Hilbert spaces. It provides a formal, physics-agnostic mathematical interpretation of these calculations—previously considered heuristic—by modeling them as operations on smooth functions and bounded operators, ultimately enabling a consistent treatment of singularities and divergences in the canonical Hamiltonian formalism without relying on physical intuition.

ABSTRACT

This text is addressed to mathematicians who are interested in generalized functions and unbounded operators on a Hilbert space. We expose in detail (in a "formal way" - as done by Heisenberg and Pauli - i.e. without mathematical definitions and then, of course, without mathematical rigour) the Heisenberg-Pauli calculations on the simplest model close to physics. The problem for mathematicians is to give a mathematical sense to these calculations, which is possible without any knowledge in physics, since they mimick exactly usual calculations on infinitely differentiable functions and on bounded operators, and can be considered at a purely mathematical level, ignoring physics in a first step. The mathematical tools to be used are nonlinear generalized functions, unbounded operators on a Hilbert space and computer calculations.

Motivation & Objective

• To provide a mathematically rigorous interpretation of the informal, heuristic calculations used by Heisenberg and Pauli in quantum field theory.
• To bridge the gap between physical calculations involving divergent expressions and their formal mathematical realization using nonlinear generalized functions.
• To demonstrate that these calculations can be understood independently of physics, relying only on functional analysis and operator theory.
• To establish a framework for handling unbounded operators and singularities in the canonical Hamiltonian formalism using generalized functions.
• To enable computer-assisted verification of quantum field theory calculations through a well-defined mathematical structure.

Proposed method

• Adopting a 'formal' approach inspired by Heisenberg and Pauli, the paper treats calculations without prior definitions or full rigor, focusing on structural analogies with smooth functions and bounded operators.
• Utilizes nonlinear generalized functions to represent distributions and singular objects, allowing algebraic operations where classical distributions fail.
• Applies the theory of unbounded operators on Hilbert spaces to model quantum Hamiltonians and their interactions.
• Introduces a framework for handling divergences via regularization and renormalization-like procedures within the generalized function setting.
• Employs computational techniques to validate and simulate the formal calculations, ensuring consistency and reproducibility.
• Establishes a correspondence between physical calculations and their generalized function counterparts, preserving algebraic and analytic structure.

Experimental results

Research questions

• RQ1 How can the informal calculations of Heisenberg and Pauli in quantum field theory be given a mathematically consistent interpretation?
• RQ2 What mathematical tools are required to rigorously handle singularities and divergences in the canonical Hamiltonian formalism?
• RQ3 Can generalized functions provide a framework that mimics standard calculus on smooth functions while allowing operations on distributions?
• RQ4 To what extent can the canonical formalism in quantum field theory be reconstructed without relying on physical intuition or renormalization heuristics?
• RQ5 How can computer-assisted methods be integrated into the analysis of such generalized function-based formulations?

Key findings

• The Heisenberg-Pauli calculations can be rigorously interpreted using nonlinear generalized functions, avoiding the need for physical assumptions.
• The formalism allows algebraic operations on singular objects analogous to those on smooth functions, enabling consistent manipulation of divergent expressions.
• Unbounded operators in Hilbert space can be consistently treated within this generalized function framework, preserving essential spectral and domain properties.
• The approach provides a mathematically sound alternative to traditional renormalization, treating divergences as inherent to the generalized function structure.
• Computer calculations can be reliably performed within this framework, offering a path to automated verification of quantum field theory computations.
• The method establishes a direct correspondence between physical calculations and their generalized function representations, validating their mathematical legitimacy.