[Paper Review] The connection between statistical mechanics and quantum field theory

Barry M. McCoy|arXiv (Cornell University)|Mar 16, 1994

Theoretical and Computational Physics24 citations

TL;DR

This four-lecture series establishes a deep formal and conceptual connection between statistical mechanics and quantum field theory (QFT) by showing that the path integral formulation of Euclidean QFT and equilibrium statistical mechanics share identical mathematical structures, particularly through the partition function and correlation functions. The key contribution is demonstrating that critical phenomena, phase transitions, and non-perturbative effects—such as level crossing in the chiral Potts model—can be understood through this unifying framework, revealing that QFT and statistical mechanics are different facets of the same underlying physics.

ABSTRACT

A four part series of lectures on the connection of statistical mechanics and quantum field theory. The general principles relating statistical mechanics and the path integral formulation of quantum field theory are presented in the first lecture. These principles are then illustrated in lecture 2 by a presentation of the theory of the Ising model for $H=0$, where both the homogeneous and randomly inhomogeneous models are treated and the scaling theory and the relation with Fredholm determinants and Painlev{é} equations is presented. In lecture 3 we consider the Ising model with $H eq 0$, where the relation with gauge theory is used to discuss the phenomenon of confinement. We conclude in the last lecture with a discussion of quantum spin diffusion in one dimensional chains and a presentation of the chiral Potts model which illustrates the physical effects that can occur when the Euclidean and Minkowski regions are not connected by an analytic continuation. (To be published as part of the Proceedings of the Sixth Annual Theoretical Physics Summer School of the Australian National University which was held in Canberra during Jan. 1994.)

Motivation & Objective

• To clarify the formal and conceptual equivalence between Euclidean quantum field theory and equilibrium statistical mechanics, particularly through the path integral formulation.
• To demonstrate that critical phenomena and phase transitions in statistical systems can be understood as analogous to quantum field theoretic phenomena, including confinement and level crossing.
• To investigate non-perturbative effects such as those in the chiral Potts model, where Lorentz invariance and analytic continuation between Minkowski and Euclidean spaces break down.
• To show that the Ising model and its generalizations provide a unifying framework for studying both quantum field theory and statistical mechanics, especially in one-dimensional systems with diffusion and spin dynamics.

Proposed method

• Uses the formal similarity between the Euclidean path integral in QFT and the partition function in statistical mechanics, where the action $ S_E $ corresponds to the energy $ E $, and $ \hbar $ corresponds to $ kT $.
• Applies the scaling theory and Fredholm determinant techniques to analyze critical behavior in the Ising model at $ H = 0 $, linking it to Painlevé transcendents.
• Analyzes the Ising model with $ H \neq 0 $ using gauge theory analogies to study confinement-like phenomena, particularly in the context of spin chains.
• Examines quantum spin diffusion in one-dimensional chains using the chiral Potts model, which breaks Lorentz invariance and shows asymmetric space-time dependence.
• Employs non-perturbative methods such as the Yang-Baxter equation and holonomic systems to compute correlation functions beyond standard QFT assumptions.
• Uses numerical evaluation of eigenvalues and level spacing distributions to detect phase transitions via level crossing, even in the absence of singularities in the ground state energy.

Experimental results

Research questions

• RQ1 How do the path integral formulations of Euclidean quantum field theory and equilibrium statistical mechanics formally relate, and what physical insights arise from this correspondence?
• RQ2 What is the role of the Ising model in unifying concepts from statistical mechanics and quantum field theory, particularly in the context of critical phenomena and scaling behavior?
• RQ3 How can the phenomenon of confinement in gauge theories be analogously understood in the Ising model with $ H \neq 0 $, and what does this reveal about topological and dynamical constraints?
• RQ4 In what ways does the chiral Potts model challenge the standard assumption of analytic continuation between Minkowski and Euclidean spaces, and what new physics emerges from this breakdown?
• RQ5 What are the implications of level crossing transitions—where the ground state energy does not exhibit singularities—for the conventional classification of phase transitions in statistical systems?

Key findings

• The partition functions of Euclidean QFT and classical statistical mechanics are formally identical when the action $ S_E $ is mapped to the energy $ E $, and $ \hbar $ to $ kT $, establishing a deep structural equivalence.
• In the Ising model at $ H = 0 $, the scaling limit leads to correlation functions governed by Fredholm determinants and Painlevé transcendents, linking exactly solvable models to nonlinear differential equations.
• For the Ising model with $ H \neq 0 $, the system exhibits confinement-like behavior analogous to gauge theories, particularly in the context of spin chains and topological constraints.
• The chiral Potts model exhibits level crossing transitions where the ground state energy remains smooth, yet the eigenstate changes due to non-positive eigenvalues in the transfer matrix, indicating a phase transition outside the scope of conventional critical theory.
• Numerical evaluation shows that for $ 0.9013 < \lambda < 1/0.9013 $, the eigenvalue $ e_r(P) $ becomes negative, signaling level crossing and a phase transition not detectable via energy singularities.
• Correlation functions in the chiral Potts model lack Lorentz invariance, and space and time are treated asymmetrically, indicating that the standard QFT assumption of analytic continuation between Minkowski and Euclidean signatures does not hold in general.