[Paper Review] Quantum restoration of broken symmetries

V. V. Belokurov, E. T. Shavgulidze|arXiv (Cornell University)|Mar 14, 2013

Quantum Mechanics and Applications4 references3 citations

TL;DR

This paper demonstrates that a non-linear, non-local transformation maps a self-interacting quantum field theory with spontaneously broken symmetry into a free field theory, preserving functional integral equivalence. Despite classically broken symmetry, the quantum theory's classical limit restores the original symmetry due to integration over singular function spaces, revealing a mechanism of quantum symmetry restoration via path integral structure and Ito calculus corrections.

ABSTRACT

A certain non-linear non-local substitution is shown to transform the action of the self-interacting quantum field to the free one. The functional integrals in both theories are equal to each other. However, the integrations are performed over different functional spaces. The classical action and the classical limit of the corresponding quantum theory turn out to be different. And the symmetry originally broken in the classical theory is restored in the classical limit of the quantum theory.

Motivation & Objective

• To demonstrate that a non-linear, non-local field redefinition transforms an interacting quantum field theory into a free theory while preserving functional integral equivalence.
• To show that the classical limit of the quantum theory restores the symmetry spontaneously broken in the classical action.
• To clarify the role of singular function spaces (X) in defining valid quantum measures for interacting fields.
• To establish a connection between Ito stochastic integrals and symmetry restoration in quantum field theory.
• To challenge conventional understanding of symmetry breaking by showing quantum dynamics can restore symmetries absent in classical solutions.

Proposed method

• Introduce a non-local field redefinition: $\chi(t) = \varphi(t) + a\int_{-T}^{t}(\varphi^2(\tau) - \beta^2)d\tau$, mapping the interacting theory to a free theory.
• Use Ito calculus to rewrite boundary and linear terms in the action as $\int \varphi^2 d\varphi = \frac{1}{3}(\varphi^3(T) - \varphi^3(-T)) - \int \varphi dt$, enabling exact functional integral equivalence.
• Show that the functional integral over the singular space $X^+$ (containing functions with $ (t - t_i^*)^{-1} $-type singularities) yields the same result as the free theory over continuous functions $C[-T,T]$.
• Demonstrate that the classical limit $\hbar \to 0$ of the quantum theory leads to $\ddot{\chi} = 0$, implying $\dot{\chi} = \text{const} = 0$, which translates to $\dot{\varphi} + a(\varphi^2 - \beta^2) = 0$ in the interacting picture.
• Identify the classical limit solution $\tilde{\varphi}^+(t)$ as a solution of the symmetric Euler-Lagrange equation (7), not the broken-symmetry equation (9), due to the singular integration domain.
• Construct the mirror case $A_-$ with opposite sign in the integral term, leading to $X^-$ and symmetric restoration via $\tilde{\varphi}^-(t)$, confirming full symmetry recovery in quantum theory.

Experimental results

Research questions

• RQ1 Can a non-local field redefinition transform a self-interacting quantum field theory into a free theory while preserving the functional integral?
• RQ2 Why does the classical limit of a quantum theory with a broken-symmetry action restore the original symmetry?
• RQ3 How do Ito stochastic integrals contribute to symmetry restoration in quantum field theories?
• RQ4 What is the structure of the functional space over which the quantum measure is defined in interacting theories?
• RQ5 Does quantum mechanics inherently restore symmetries broken in the classical action, and if so, through what mechanism?

Key findings

• The functional integral of the interacting $\phi^4$ theory on the singular space $X^+$ is exactly equal to the free field functional integral on $C[-T,T]$ via the non-local transformation.
• The classical limit of the quantum theory yields $\dot{\chi}(t) = 0$, implying $\chi(t)$ is constant, which corresponds to $\dot{\varphi}(t) + a(\varphi^2(t) - \beta^2) = 0$ in the interacting picture.
• The solution $\tilde{\varphi}^+(t) = \beta \tanh(bt + c)$ or $\beta \coth(bt + c)$ satisfies the symmetric Euler-Lagrange equation (7), not the broken-symmetry equation (9), proving symmetry restoration.
• The measure $Z^{-1} \exp\left\{-\frac{1}{2}\int (\dot{\varphi})^2 dt - \frac{1}{2}\int \varphi^4 dt - \frac{1}{3}(\varphi^3(T) - \varphi^3(-T)) + \int \varphi dt \right\} d\varphi$ is well-defined on $X^+$, while the standard Gaussian measure fails on $C$.
• The space $X^+$ contains functions with isolated $ (t - t_i^*)^{-1} $ singularities, and the measure is not defined on continuous functions $C$ due to divergence in perturbative expansion.
• The mirror theory $A_-$ with reversed sign in the linear term leads to symmetric restoration via $X^-$, confirming that both branches of the symmetric solution are recovered in the quantum path integral.