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Register a new userElliptic stochastic quantization of Sinh-Gordon QFT
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Added by
Francesco Carlo De Vecchi
Publication date
2021
fields
Physics
and research's language is
English
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The (elliptic) stochastic quantization equation for the (massive) $cosh(beta varphi)_2$ model, for the charged parameter in the $L^2$ regime (i.e. $beta^2 < 4 pi$), is studied. We prove the existence, uniqueness and the properties of the invariant measure of the solution to this equation. The proof is obtained through a priori estimates and a lattice approximation of the equation. For implementing this strategy we generalize some properties of Besov space in the continuum to analogous results for Besov spaces on the lattice. As a final result we show as how to use the stochastic quantization equation to verify the Osterwalder-Schrader axioms for the $cosh (beta varphi)_2$ quantum field theory, including the exponential decay of correlation functions.
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(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-)construction 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.
We study a class of elliptic SPDEs with additive Gaussian noise on $mathbb{R}^2 times M$, with $M$ a $d$-dimensional manifold equipped with a positive Radon measure, and a real-valued non linearity given by the derivative of a smooth potential $V$, convex at infinity and growing at most exponentially. For quite general coefficients and a suitable regularity of the noise we obtain, via the dimensional reduction principle discussed in our previous paper on the topic, the identity between the law of the solution to the SPDE evaluated at the origin with a Gibbs type measure on the abstract Wiener space $L^2 (M)$. The results are then applied to the elliptic stochastic quantization equation for the scalar field with polynomial interaction over $mathbb{T}^2$, and with exponential interaction over $mathbb{R}^2$ (known also as H{o}eg-Krohn or Liouville model in the literature). In particular for the exponential interaction case, the existence and uniqueness properties of solutions to the elliptic equation over $mathbb{R}^{2 + 2}$ is derived as well as the dimensional reduction for the values of the ``charge parameter $sigma = frac{alpha}{2sqrt{pi}} < sqrt{4 left( 8 - 4 sqrt{3} right) pi} simeq sqrt{4.23pi}$, for which the model has an Euclidean invariant probability measure (hence also permitting to get the corresponding relativistic invariant model on the two dimensional Minkowski space).
Using the integrability of the sinh-Gordon equation, we demonstrate the spectral stability of its elliptic solutions. By constructing a Lyapunov functional using higher-order conserved quantities of the sinh-Gordon equation, we show that these elliptic solutions are orbitally stable with respect to subharmonic perturbations of arbitrary period.
We consider space-time quantum fields with exponential/trigonometric interactions. In the context of Euclidean quantum field theory, the former and the latter are called the Hoegh-Krohn model and the Sine-Gordon model, respectively. The main objective of the present paper is to construct infinite dimensional diffusion processes which solve modified stochastic quantization equations for these quantum fields on the two-dimensional torus by the Dirichlet form approach and to prove strong uniqueness of the corresponding Dirichlet operators.
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 methods 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.
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