Random non-Hermitian action theory for stochastic quantum dynamics: from canonical to path integral quantization

TL;DR

We introduce a random non-Hermitian action (RNH) to describe stochastic nonlinear quantum dynamics and develop both canonical and path-integral quantizations for fermionic fields, proving their equivalence. The formalism yields a linear stochastic evolution for a prenormalized state while the normalized physical state obeys a nonlinear stochastic equation, with the dynamics organized into deterministic Hermitian and random parts plus second-order corrections. In the single-particle sector, Gaussian wave packets can localize due to non-Hermiticity, with a long-time width $\sigma^2(\infty)=1/[2m(-\lambda_I)]$ and a random center whose variance grows with the random strength $\gamma$. The path-integral approach provides a compact propagator and reveals how randomness enters through a single-particle action $\tilde{S}$, illustrating a transition from quantum to classical transport and offering a practical route to many-body extensions.

Abstract

We develop a theory of random non-Hermitian action that, after quantization, describes the stochastic nonlinear dynamics of quantum states in Hilbert space. Focusing on fermionic fields, we propose both canonical quantization and path integral quantization, demonstrating that these two approaches are equivalent. Using this formalism, we investigate the evolution of a single-particle Gaussian wave packet under the influence of non-Hermiticity and randomness. Our results show that specific types of non-Hermiticity lead to wave packet localization, while randomness affects the central position of the wave packet, causing the variance of its distribution to increase with the strength of the randomness.