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Stochastic quantization of $λφ_2^4$- theory in 2-d Moyal space

Chunqiu Song, Hendrik Weber, Raimar Wulkenhaar

TL;DR

This work develops a stochastic-quantization framework for the 2D λφ^4 model on non-commutative Moyal space to construct the Moyal λφ^4_2 measure for all λ≥0. It adapts the Da Prato-Debussche trick to a matrix-basis formulation with Ω=1, establishing local well-posedness for the renormalized stochastic quantization equation and deriving a second-order a priori estimate that yields global existence. An invariant measure is obtained via Krylov-Bogoliubov arguments, demonstrating a non-perturbative construction of the NCQFT measure in 2D. The results pave a path toward a rigorous treatment of the four-dimensional planar model, potentially contributing to a program for asymptotic safety in non-commutative QFTs through SPDE techniques.

Abstract

There is strong evidence for the conjecture that the $λφ^4$ QFT- model on 4-dimensional non-commutative Moyal space can be non-perturbatively constructed. As preparation, in this paper we construct the 2-dimensional case with the method of stochastic quantization. We show the local well-posedness and global well-posedness of the stochastic quantization equation, leading to a construction of the Moyal $λφ^4_2$ measure for any non-negative coupling constant $λ$.

Stochastic quantization of $λφ_2^4$- theory in 2-d Moyal space

TL;DR

This work develops a stochastic-quantization framework for the 2D λφ^4 model on non-commutative Moyal space to construct the Moyal λφ^4_2 measure for all λ≥0. It adapts the Da Prato-Debussche trick to a matrix-basis formulation with Ω=1, establishing local well-posedness for the renormalized stochastic quantization equation and deriving a second-order a priori estimate that yields global existence. An invariant measure is obtained via Krylov-Bogoliubov arguments, demonstrating a non-perturbative construction of the NCQFT measure in 2D. The results pave a path toward a rigorous treatment of the four-dimensional planar model, potentially contributing to a program for asymptotic safety in non-commutative QFTs through SPDE techniques.

Abstract

There is strong evidence for the conjecture that the QFT- model on 4-dimensional non-commutative Moyal space can be non-perturbatively constructed. As preparation, in this paper we construct the 2-dimensional case with the method of stochastic quantization. We show the local well-posedness and global well-posedness of the stochastic quantization equation, leading to a construction of the Moyal measure for any non-negative coupling constant .

Paper Structure

This paper contains 15 sections, 36 theorems, 260 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. $\lambda \phi^4$ Model in 2-d Moyal Space
  4. Da Prato-Debussche Trick
  5. Fixed Point Map
  6. Local Existence for Stochastic Quantization Equation
  7. A Priori Estimate
  8. Existence of Invariant Measure
  9. Outlook
  10. Gaussian Hypercontractivity
  11. Moyal Product and Matrix Basis
  12. Spaces of Matrices
  13. Inequalities Related to Correlation Functions
  14. Construction of $: z^2 :$ and $: z^3 :$
  15. 105 Terms Verification

Key Result

Theorem 1.1

For any initial value $v (0) \in H^0$, there exists a random time $T$, which depends on the initial data $\| v (0)\|_{H^0}$ and $z$, such that the renormalized remainder equation has a unique solution up to time $T$ in the space $K_T^{\frac{1}{2} -}$ almost surely.

Theorems & Definitions (64)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 54 more