اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدAn SPDE approach to perturbation theory of $Phi^4_2$: asymptoticity and short distance behavior
68
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In this paper we study the perturbation theory of $Phi^4_2$ model on the whole plane via stochastic quantization. We use integration by parts formula (i.e. Dyson-Schwinger equations) to generate the perturbative expansion for the $k$-point correlation functions, and prove bounds on the remainder of the truncated expansion using PDE estimates; this in particular proves that the expansion is asymptotic. Furthermore, we derive short distance behaviors of the $2$-point function and the connected $4$-point function, also via suitable Dyson-Schwinger equations combined with PDE arguments.
قيم البحث
اقرأ أيضاً
(Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here. See the abstract in the paper.) We study the construction of the $Phi^3_3$-measure and complete the program on the (non-)construc
tion of the focusing Gibbs measures, initiated by Lebowitz, Rose, and Speer (1988). This problem turns out to be critical, exhibiting the following phase transition. In the weakly nonlinear regime, we prove normalizability of the $Phi^3_3$-measure and show that it is singular with respect to the massive Gaussian free field. Moreover, we show that there exists a shifted measure with respect to which the $Phi^3_3$-measure is absolutely continuous. In the strongly nonlinear regime, by further developing the machinery introduced by the authors (2020), we establish non-normalizability of the $Phi^3_3$-measure. Due to the singularity of the $Phi^3_3$-measure with respect to the massive Gaussian free field, this non-normalizability part poses a particular challenge as compared to our previous works. In order to overcome this issue, we first construct a $sigma$-finite version of the $Phi^3_3$-measure and show that this measure is not normalizable. Furthermore, we prove that the truncated $Phi^3_3$-measures have no weak limit in a natural space, even up to a subsequence. We also study the dynamical problem. By adapting the paracontrolled approach, in particular from the works by Gubinelli, Koch, and the first author (2018) and by the authors (2020), we prove almost sure global well-posedness of the hyperbolic $Phi^3_3$-model and invariance of the Gibbs measure in the weakly nonlinear regime. In the globalization part, we introduce a new, conceptually simple and straightforward approach, where we directly work with the (truncated) Gibbs measure, using the variational formula and ideas from theory of optimal transport.
(Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here. See the abstract in the paper.) Lebowitz, Rose, and Speer (1988) initiated the study of focusing Gibbs measures, which was contin
ued by Brydges and Slade (1996), Bourgain (1997, 1999), and Carlen, Frohlich, and Lebowitz (2016) among others. In this paper, we complete the program on the (non-)construction of the focusing Hartree Gibbs measures in the three-dimensional setting. More precisely, we study a focusing $Phi^4_3$-model with a Hartree-type nonlinearity, where the potential for the Hartree nonlinearity is given by the Bessel potential of order $beta$. We first construct the focusing Hartree $Phi^4_3$-measure for $beta > 2$, while we prove its non-normalizability for $beta < 2$. Furthermore, we establish the following phase transition at the critical value $beta = 2$: normalizability in the weakly nonlinear regime and non-normalizability in the strongly nonlinear regime. We then study the canonical stochastic quantization of the focusing Hartree $Phi^4_3$-measure, namely, the three-dimensional stochastic damped nonlinear wave equation (SdNLW) with a cubic nonlinearity of Hartree-type, forced by an additive space-time white noise, and prove almost sure global well-posedness and invariance of the focusing Hartree $Phi^4_3$-measure for $beta > 2$ (and $beta = 2$ in the weakly nonlinear regime). In view of the non-normalizability result, our almost sure global well-posedness result is sharp. In Appendix, we also discuss the (parabolic) stochastic quantization for the focusing Hartree $Phi^4_3$-measure. We also construct the defocusing Hartree $Phi^4_3$-measure for $beta > 0$.
A new construction of non-Gaussian, rotation-invariant and reflection positive probability measures $mu$ associated with the $varphi ^4_3$-model of quantum field theory is presented. Our construction uses a combination of semigroup methods, and metho
ds of stochastic partial differential equations (SPDEs) for finding solutions and stationary measures of the natural stochastic quantization associated with the $varphi ^4_3$-model. Our starting point is a suitable approximation $mu_{M,N}$ of the measure $mu$ we intend to construct. $mu_{M,N}$ is parametrized by an $M$-dependent space cut-off function $rho_M: {mathbb R}^3rightarrow {mathbb R}$ and an $N$-dependent momentum cut-off function $psi_N: widehat{mathbb R}^3 cong {mathbb R}^3 rightarrow {mathbb R}$, that act on the interaction term (nonlinear term and counterterms). The corresponding family of stochastic quantization equations yields solutions $(X_t^{M,N}, tgeq 0)$ that have $mu_{M,N}$ as an invariant probability measure. By a combination of probabilistic and functional analytic methods for singular stochastic differential equations on negative-indices weighted Besov spaces (with rotation invariant weights) we prove the tightness of the family of continuous processes $(X_t^{M,N},t geq 0)_{M,N}$. Limit points in the sense of convergence in law exist, when both $M$ and $N$ diverge to $+infty$. The limit processes $(X_t; tgeq 0)$ are continuous on the intersection of suitable Besov spaces and any limit point $mu$ of the $mu_{M,N}$ is a stationary measure of $X$. $mu$ is shown to be a rotation-invariant and non-Gaussian probability measure and we provide results on its support. It is also proven that $mu$ satisfies a further important property belonging to the family of axioms for Euclidean quantum fields, it is namely reflection positive.
This work offers a new prospective on asymptotic perturbation theory for varying self-adjoint extensions of symmetric operators. Employing symplectic formulation of self-adjointness we obtain a new version of Krein formula for resolvent difference wh
ich facilitates asymptotic analysis of resolvent operators via first order expansion for the family of Lagrangian planes associated with perturbed operators. Specifically, we derive a Riccati-type differential equation and the first order asymptotic expansion for resolvents of self-adjoint extensions determined by smooth one-parameter families of Lagrangian planes. This asymptotic perturbation theory yields a symplectic version of the abstract Kato selection theorem and Hadamard-Rellich-type variational formula for slopes of multiple eigenvalue curves bifurcating from an eigenvalue of the unperturbed operator. The latter, in turn, gives a general infinitesimal version of the celebrated formula equating the spectral flow of a path of self-adjoint extensions and the Maslov index of the corresponding path of Lagrangian planes. Applications are given to quantum graphs, periodic Kronig-Penney model, elliptic second order partial differential operators with Robin boundary conditions, and physically relevant heat equations with thermal conductivity.
We consider a quantum field model with exponential interactions on the two-dimensional torus, which is called the $exp (Phi)_{2}$-quantum field model or H{o}egh-Krohns 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.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
