A modest redirection of quantum field theory solves all current problems

TL;DR

This paper proposes a modified quantization approach for quantum field theory by restricting field and momentum variables, leading to a more consistent and finite Hamiltonian, and introduces a novel quantum correction term.

Contribution

It introduces a new representation of momentum as a ratio involving the field, which results in a finite Hamiltonian density and a novel quantization method for field theories.

Findings

Hamiltonian densities become finite with the new representation.

Quantization involves scaled behavior and an unexpected $rac{ ext{ extonehalf} ext{ extonehalf}}{ ext{ extphi}^2}$ term.

The approach offers a potentially valid alternative to standard quantum field theory quantization.

Abstract

Standard quantization using, for example, path integration of field theory models, includes paths of momentum and field reach infinity in the Hamiltonian density, while the Hamiltonian itself remains finite. That fact causes considerable difficulties. In this paper, we represent $\pi(x)$ by $k(x)/\phi(x)$. To insure proper values for $\pi(x)$ it is necessary to restrict $0<|\phi(x)|<\infty$ as well as $0\leq|k(x)|<\infty$. Indeed that leads to Hamiltonian densities in which $\phi(x)^p$, where $p$ can be even integers between $4$ and $\infty$. This leads to a completely satisfactory quantization of field theories using situations that involve scaled behavior leading to an unexpected, $\hbar^2/\hat{\phi}(x)^2$ which arises only in the quantum aspects. Indeed, it is fair to claim that this symbol change leads to valid field theory quantizations.