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Stochastic quantization associated with the $\exp(Φ)_2$-quantum field model driven by space-time white noise on the torus

Masato Hoshino, Hiroshi Kawabi, Seiichiro Kusuoka

TL;DR

This work addresses the stochastic quantization of the $\exp(\Phi)_2$-quantum field on $\mathbb{T}^2$ driven by space-time white noise by developing a robust singular SPDE framework. The authors implement a Da Prato–Debussche-type approach to obtain a time-global, pathwise-unique solution to $\partial_t\Phi_t=\tfrac{1}{2}(\Delta-1)\Phi_t-\tfrac{\alpha}{2}\exp^{\diamond}(\alpha\Phi_t)+\dot W_t$ for $|\alpha|<\sqrt{4\pi}$ and connect this dynamics to the corresponding Gibbs measure $\mu^{(\alpha)}$ via Dirichlet-form methods. They prove global well-posedness of the shifted equation, establish tightness and convergence of stationary approximations to the invariant measure, and show that the diffusion from the Dirichlet form coincides with the strong SPDE solution, thus unifying probabilistic and analytic constructions. The results illuminate the role of Gaussian multiplicative chaos and Wick renormalization in a rigorous stochastic quantization of exponential interactions, with implications for two-dimensional Euclidean QFT and related stochastic dynamics.

Abstract

We consider a quantum field model with exponential interactions on the two-dimensional torus, which is called the $\exp (Φ)_{2}$-quantum field model or Høegh-Krohn's model. In the present paper, we study the stochastic quantization of this model by singular stochastic partial differential equations, which is recently developed. By the method, we construct a unique time-global solution and the invariant probability measure of the corresponding stochastic quantization equation, and identify with an infinite-dimensional diffusion process, which has been constructed by the Dirichlet form approach.

Stochastic quantization associated with the $\exp(Φ)_2$-quantum field model driven by space-time white noise on the torus

TL;DR

This work addresses the stochastic quantization of the -quantum field on driven by space-time white noise by developing a robust singular SPDE framework. The authors implement a Da Prato–Debussche-type approach to obtain a time-global, pathwise-unique solution to for and connect this dynamics to the corresponding Gibbs measure via Dirichlet-form methods. They prove global well-posedness of the shifted equation, establish tightness and convergence of stationary approximations to the invariant measure, and show that the diffusion from the Dirichlet form coincides with the strong SPDE solution, thus unifying probabilistic and analytic constructions. The results illuminate the role of Gaussian multiplicative chaos and Wick renormalization in a rigorous stochastic quantization of exponential interactions, with implications for two-dimensional Euclidean QFT and related stochastic dynamics.

Abstract

We consider a quantum field model with exponential interactions on the two-dimensional torus, which is called the -quantum field model or Høegh-Krohn's model. In the present paper, we study the stochastic quantization of this model by singular stochastic partial differential equations, which is recently developed. By the method, we construct a unique time-global solution and the invariant probability measure of the corresponding stochastic quantization equation, and identify with an infinite-dimensional diffusion process, which has been constructed by the Dirichlet form approach.

Paper Structure

This paper contains 19 sections, 25 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Settings and main theorems
  4. Wick exponential of the Ornstein-Uhlenbeck process
  5. Infinite-dimensional OU process
  6. Wick exponential of GFF
  7. $\exp(\Phi)_2$-quantum field
  8. Wick exponential of the OU process
  9. Global well-posendess of the strong solution
  10. Products of continuous functions and nonnegative distributions
  11. Global well-posedness of $\Upsilon$
  12. Proof of Theorem \ref{['mainthm1']}
  13. Stationary solution
  14. Tightness of stationary solutions
  15. Uniqueness of the limit
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $|\alpha|<\sqrt{4\pi}$ and $\varepsilon>0$. Let $N\in\mathbb{N}$ and consider the initial value problem where $\phi\in\mathcal{D}'(\Lambda)$ and Then for $\mu_0$-a.e. $\phi\in\mathcal{D}'(\Lambda)$, the unique time-global classical solution $\Phi^N$ converges to an $H^{-\varepsilon}$-valued stochastic process $\Phi$ in the space $C([0,T],H^{-\varepsilon}(\Lambda))$ for any $T>0$$\mathbb P$-a

Theorems & Definitions (47)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • ...and 37 more