Mathematical basis, phase transitions and singularities of (3+1)-dimensional phi4 scalar field model

TL;DR

This paper explores the mathematical structure, phase transitions, and singularities of a (3+1)-dimensional phi4 scalar field model, linking it to topological quantum field theories and the 3D Ising model to understand critical phenomena.

Contribution

It introduces a novel framework for analyzing the (3+1)D phi4 scalar field model using topological and complex parameter space methods, connecting quantum field theory with statistical mechanics.

Findings

The model must be set up on the Jordan-von Neumann-Wigner framework.

The ergodic hypothesis is violated at finite temperatures.

A relation between the coupling constants in the phi4 model and the 3D Ising model is established.

Abstract

The lambda phi4 scalar field model can be applied to interpret pion-pion scattering and properties of hadrons. In this work, the mathematical basis, phase transitions and singularities of a (3+1)-dimensional (i.e., (3+1)D) phi4 scalar field model are investigated. It is found that as a specific example of topological quantum field theories, the (3+1)D phi4 scalar field model must be set up on the Jordan-von Neumann-Wigner framework and dealt with the parameter space of complex time (or complex temperature). The use of the time average and the topologic Lorentz transformation representing Reidemeister moves ensure the integrability, which takes into account for the contributions of nontrivial topological structures to physical properties of the many-body interacting system. The ergodic hypothesis is violated at finite temperatures in the (3+1)D phi4 scalar field model. Because the quantum field theories with ultraviolet cutoff can be mapped to the models in statistical mechanics, the (3+1)D phi4 scalar field model with ultraviolet cutoff is studied by inspecting its relation with the three-dimensional (3D) Ising model. Furthermore, the direct relation between the coupling K in the 3D Ising model and the bare coupling lambda0 in the (3+1)D phi4 scalar field model is determined in the strong coupling limit. The results obtained in the present work can be utilized to investigate thermodynamic physical properties and critical phenomena of quantum (scalar) field theories.