Mathematical Foundations of Field Theory

TL;DR

This paper develops a rigorous mathematical Hamiltonian framework for classical and quantum field theories, clarifying the structure of linear and nonlinear fields using locally square-integrable functions to address foundational issues.

Contribution

It introduces a new formulation of field configuration space using locally square-integrable functions, resolving key problems in defining field operators and dynamics.

Findings

Clarifies the structure of linear fields.

Proposes a plausible formulation for nonlinear fields.

Addresses the field multiplication problem with dense domain for field products.

Abstract

A mathematically rigorous Hamiltonian formulation for classical and quantum field theories is given. New results include clarifications of the structure of linear fields, and a plausible formulation for nonlinear fields. Many mathematical formulations of field theory suffer greatly from either a failure to explicitly define the field configuration space, or else from the choice to define field operators as distributions. A solution to such problems is given by instead using locally square-integrable functions, and by paying close attention to this space's topology. One benefit of this is a clarification of the field multiplication problem: The pointwise product of fields is still not defined for all states, but it is densely defined, and this is shown to be sufficient for specifying dynamics. Significant progress is also made, through this choice of configuration space, in appropriately representing field states with `infinitely many particles', or those which do not go to zero at infinity.