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Towards Three-Dimensional Conformal Probability
Abstract
In this outline of a program, based on rigorous renormalization group theory,
we introduce new definitions which allow one to formulate precise mathematical
conjectures related to conformal invariance as studied by physicists in the
area known as higher-dimensional conformal bootstrap which has developed at a
breathtaking pace over the last five years. We also explore a second theme,
intimately tied to conformal invariance for random distributions, which can be
construed as a search for a very general first and second-quantized
Kolmogorov-Chentsov Theorem. First-quantized refers to regularity of sample
paths. Second-quantized refers to regularity of generalized functionals or Hida
distributions and relates to the operator product expansion. Finally, we
present a summary of progress made on a p-adic hierarchical model and point
out possible connections to number theory.
... Since then, hierarchical and p-adic models have attracted a great deal of interest throughout mathematical and theoretical physics, with Dyson's paper having over 1000 citations. They are particularly popular in the context of rigorous renormalization group 2 analyses of critical phenomena [1,2,[24][25][26]57], a topic we discuss in more detail in Section 7.2. We refer the reader to [47,48] for comprehensive overviews of the use of hierarchical and p-adic models in physics, to [23,42] for detailed overviews of the rigorous renormalization group analysis of hierarchical spin systems, and to Tao's blog post [96] for a broad informal discussion of the use of hierarchical models in other parts of mathematics. ...
... When L = p is prime one can think of H 1 p as a discrete analogue of d-dimensional p-adic space Q p just as Z is a discrete analogue of R. See e.g. [1] for background on this perspective in the context of statistical mechanics. ...
... We now state our main theorems, which we describe separately in the three cases d < d c , d > d c , and d = d c . We recall that the critical exponents δ and η, if they exist, are defined by the relations P βc (|K| ≥ n) = n −1/δ±o (1) as n → ∞ and ...
We consider long-range Bernoulli bond percolation on the d-dimensional hierarchical lattice in which each pair of points x and y are connected by an edge with probability , where is fixed and is a parameter. We study the volume of clusters in this model at its critical point , proving precise estimates on the moments of all orders of the volume of the cluster of the origin inside a box. We apply these estimates to prove up-to-constants estimates on the tail of the volume of the cluster of the origin, denoted K, at criticality, namely In particular, we compute the critical exponent to be when d is below the upper-critical dimension and establish the precise order of polylogarithmic corrections to scaling at the upper-critical dimension itself. Interestingly, we find that these polylogarithmic corrections are not those predicted to hold for nearest-neighbour percolation on by Essam, Gaunt, and Guttmann (J. Phys. A 1978). Our work also lays the foundations for the study of the scaling limit of the model: In the high-dimensional case we prove that the sized-biased distribution of the volume of the cluster of the origin inside a box converges under suitable normalization to a chi-squared random variable, while in the low-dimensional case we prove that the suitably normalized decreasing list of cluster sizes in a box is tight in if and only if .
... when x 1 → x 2 while the three points x 2 , x 3 , and x 4 are fixed. This is the simplest instance of Wilson's OPE which here would say that, "inside correlations", one has ...
... The is only needed in order to account for eventual logarithmic corrections. In a conformal field theory (CFT) as discussed in [3], one can take = 0. The k allows more generality for our main theorem and is natural in the setting of probability theory on spaces of temperate distributions, but it is not needed in usual QFT models. ...
... 1.10 relies on a functional-analytic part done in Sect. 3 using tools from what one may call Schwartz-Grothendieck-Fernique theory in view of the foundational works [30,40,79,80,82]. It also relies on Proposition 1 which is a combinatorial estimate in the pure tradition of the École Polytechnique school of constructive QFT founded by Roland Sénéor. ...
We establish a direct connection between two fundamental topics: one in probability theory and one in quantum field theory. The first topic is the problem of pointwise multiplication of random Schwartz distributions which has been the object of recent progress thanks to Hairer’s theory of regularity structures and the theory of paracontrolled distributions introduced by Gubinelli, Imkeller and Perkowski. The second topic is Wilson’s operator product expansion which is a general property of models of quantum field theory and a cornerstone of the bootstrap approach to conformal field theory. Our main result is a general theorem for the almost sure construction of products of random distributions by mollification and suitable additive as well as multiplicative renormalizations. The hypothesis for this theorem is the operator product expansion with precise bounds for pointwise correlations. We conjecture these bounds to be universal features of quantum field theories with gapped dimension spectrum. Our theorem can accommodate logarithmic corrections, anomalous scaling dimensions and even lack of translation invariance. However, it only applies to fields with short distance singularities that are milder than white noise. As an application, we provide a detailed treatment of a scalar conformal field theory of mean field type, i.e., the fractional massless free field also known as the fractional Gaussian field.
... A proof that η = 2 − α for small ǫ has been announced by Mitter for a 1-component continuum model [71]. The long-range model has also recently been studied in connection with conformal invariance of the critical theory for d = d c − ǫ with ǫ small [2,76]. From a mathematical point of view, the long-range O(n) model has the advantage that it can be defined in integer dimension d with α = 1 2 (d+ǫ) chosen so that d is just slightly below the upper critical dimension: d = d c − ǫ = 2α − ǫ. ...
... None of these papers address the computation of critical exponents. Critical correlation functions were studied in a hierarchical version of the model in [45,46], and the recent paper [3] carries out a computation of critical exponents in a different hierarchical setting; see also [2]. ...
... We also need a lower bound on A N for n = 4. By (5.5), (5.10), and Lemma 5.5, apart from a bounded number of scales near 0 and near j m , for j ≤ j m we have δ j [w (2) ] ≍ L ǫj . By restricting the sum in (9.11) to avoid those few scales, the desired lower bound on A N follows similarly. ...
We consider the critical behaviour of long-range O(n) models () on , with interaction that decays with distance r as , for . For , we study the n-component lattice spin model. For , we study the weakly self-avoiding walk via an exact representation as a supersymmetric spin model. These models have upper critical dimension . For dimensions d=1,2,3 and small , we choose , so that is below the upper critical dimension. For small and weak coupling, to order we prove existence of and compute the values of the critical exponent for the susceptibility () and the critical exponent for the specific heat (for ). For the susceptibility, , and a similar result is proved for the specific heat. Expansion in for such long-range models was first carried out in the physics literature in 1972, by Fisher, Ma and Nickel. Our proof adapts and applies a rigorous renormalisation group method developed in previous papers with Bauerschmidt and Brydges for the nearest-neighbour models in the critical dimension d=4, and is based on the construction of a non-Gaussian renormalisation group fixed point. Some aspects of the method simplify below the upper critical dimension, while some require different treatment, and new ideas and techniques with potential future application are introduced.
... These methods led to exact solutions of several CFTs, in particular exact predictions for critical exponents 2 There are a few notable exceptions: I refer to the recent results obtained by this approach on non-exactly solved models and/or models away from the free Fermi point and, more specifically on: crossing probabilities for critical percolation on triangular lattice [41]; Pfaffian nature of boundary spin correlations in interacting Ising models [3]; limit shapes and surface tension for the 5V model [21]; height (de)localization transition in the 6V model [33]. 3 A couple of exceptions are the results of [2] on a p-adic hierarchial version of ϕ 4 3 with long-range interactions (see also [1] for a review on the subject, trying to establish a bridge between RG and CFT, very much in the spirit of the present paper), and the results of [4] on the scaling limit of 2D Ising models with finite range interactions in cylindrical geometry, reviewed below, in Section 2). and closed formulas (or closed equations) for correlation functions of any order. ...
Conformal field theory (CFT) is an extremely powerful tool for explicitly computing critical exponents and correlation functions of statistical mechanics systems at a second order phase transition, or of condensed matter systems at a quantum critical point. Conformal invariance is expected to be a feature of the fixed point theory obtained from a microscopic model at criticality, under appropriate averaging and rescaling operations: the action of the Wilsonian Renormalization Group (RG). Unfortunately, an explicit connection between critical microscopic models and their conformally invariant scaling limit is still lacking in general. Nevertheless, the last decades witnessed significant progress on this topic, both from the mathematical and physics sides, where several new tools have been introduced and their ranges of applications have constantly and significantly increased: I refer here, e.g., to discrete holomorphicity, SLE, the use of lattice Ward Identities in constructive RG, the conformal bootstrap program and its recent applications to 3D CFT. In an effort to make further progress on these problems, the one-day workshop "Emergent CFTs in statistical mechanics" was organized: the goal was to bring together probabilists, mathematical physicists and theoretical physicists, working on various aspects of critical statistical mechanics systems with complementary tools, both at the discrete and the continuum level, in the hope of creating new connections between the different approaches. This paper is based on an introductory talk given at the workshop: after a summary of the main topics discussed in the meeting, I illustrate the approach to the problem based on constructive RG methods, by reviewing recent results on the existence and the explicit characterization of the scaling limit of critical 2D Ising models with finite range interactions in cylindrical geometry.
... It has the advantage of allowing the construction of a broad class of random processes, including many instances that do not admit a pointwise representation. Generalized random processes are typically used as a general framework for the scaling limits of statistical models in conformal field theory [1,2], where the continuous-domain limit fields are typically too irregular to admit a pointwise representation [9,11]. The framework also lends itself to the construction of the d-dimensional Gaussian white noise, as is exploited in white noise analysis [24,25] or for more general classes of Gaussian processes [30,39,47]. ...
There exists several ways of constructing L\'evy white noise, for instance are as a generalized random process in the sense of I.M. Gelfand and N.Y. Vilenkin, or as an independently scattered random measure introduced by B.S. Rajput and J. Rosinski. In this article, we unify those two approaches by extending the L\'evy white noise, defined as a generalized random process, to an independently scattered random measure. We are then able to give general integrability conditions for L\'evy white noises, thereby maximally extending their domain of definition. Based on this connection, we provide new criteria for the practical determination of this domain of definition, including specific results for the subfamilies of Gaussian, symmetric--stable, Laplace, and compound Poisson noises. We also apply our results to formulate a general criterion for the existence of generalized solutions of linear stochastic partial differential equations driven by a L\'evy white noise.
We consider long‐range Bernoulli bond percolation on the dd‐dimensional hierarchical lattice in which each pair of points xx and yy are connected by an edge with probability 1−exp(−β∥x−y∥−d−α), where 0<α<d is fixed and β⩾0 is a parameter. We study the volume of clusters in this model at its critical point β=βc, proving precise estimates on the moments of all orders of the volume of the cluster of the origin inside a box. We apply these estimates to prove up‐to‐constants estimates on the tail of the volume of the cluster of the origin, denoted as KK, at criticality, namely, Pβc(|K|⩾n)≍n−(d−α)/(d+α)d<3αn−1/2(logn)1/4d=3αn−1/2d>3α.\begin{equation*} \mathbb {P}_{\beta _c}(|K|\geqslant n) \asymp {{\left\lbrace \def\eqcellsep{&}\begin{array}{ll}n^{-(d-\alpha)/(d+\alpha)} & d < 3\alpha \\[3pt] n^{-1/2}(\log n)^{1/4} & d=3\alpha \\[3pt] n^{-1/2} & d>3\alpha . \end{array} \right.}} \end{equation*}In particular, we compute the critical exponent δ to be (d+α)/(d−α) when dd is below the upper‐critical dimension dc=3α and establish the precise order of polylogarithmic corrections to scaling at the upper‐critical dimension itself. Our work also lays the foundations for the study of the scaling limit of the model: In the high‐dimensional case d⩾3α, we prove that the sized‐biased distribution of the volume of the cluster of the origin inside a box converges under suitable normalization to a chi‐squared random variable, while in the low‐dimensional case d<3α, we prove that the suitably normalized decreasing list of cluster sizes in a box is tight in ℓp∖{0} if and only if p>2d/(d+α).
This book provides a broad, interdisciplinary overview of non-Archimedean analysis and its applications. Featuring new techniques developed by leading experts in the field, it highlights the relevance and depth of this important area of mathematics, in particular its expanding reach into the physical, biological, social, and computational sciences as well as engineering and technology.
In the last forty years the connections between non-Archimedean mathematics and disciplines such as physics, biology, economics and engineering, have received considerable attention. Ultrametric spaces appear naturally in models where hierarchy plays a central role – a phenomenon known as ultrametricity. In the 80s, the idea of using ultrametric spaces to describe the states of complex systems, with a natural hierarchical structure, emerged in the works of Fraunfelder, Parisi, Stein and others. A central paradigm in the physics of certain complex systems – for instance, proteins – asserts that the dynamics of such a system can be modeled as a random walk on the energy landscape of the system. To construct mathematical models, the energy landscape is approximated by an ultrametric space (a finite rooted tree), and then the dynamics of the system is modeled as a random walk on the leaves of a finite tree. In the same decade, Volovich proposed using ultrametric spaces in physical models dealing with very short distances. This conjecture has led to a large body of research in quantum field theory and string theory. In economics, the non-Archimedean utility theory uses probability measures with values in ordered non-Archimedean fields. Ultrametric spaces are also vital in classification and clustering techniques. Currently, researchers are actively investigating the following areas: p-adic dynamical systems, p-adic techniques in cryptography, p-adic reaction-diffusion equations and biological models, p-adic models in geophysics, stochastic processes in ultrametric spaces, applications of ultrametric spaces in data processing, and more.
This contributed volume gathers the latest theoretical developments as well as state-of-the art applications of non-Archimedean analysis. It covers non-Archimedean and non-commutative geometry, renormalization, p-adic quantum field theory and p-adic quantum mechanics, as well as p-adic string theory and p-adic dynamics. Further topics include ultrametric bioinformation, cryptography and bioinformatics in p-adic settings, non-Archimedean spacetime, gravity and cosmology, p-adic methods in spin glasses, and non-Archimedean analysis of mental spaces. By doing so, it highlights new avenues of research in the mathematical sciences, biosciences and computational sciences.
We present a quick nontechnical introduction to quantum field theory and Wilson’s theory of the renormalization group from the point of view of mathematical analysis. The presentation is geared primarily towards a probability theory, harmonic analysis and dynamical systems theory audience. We also emphasize the use of p-adics in order to set up hierarchical versions of the renormalization group. The latter provide an ideal stepping stone towards the more involved Euclidean space setting.
Hermann Weyl is quoted with saying in Philosophie der Mathematik und Naturwissenschaft 1927, p. 36: As a matter of fact, it is by no means impossible to build up a consistent “non-Archimedean” theory of magnitudes in which the axiom of Eudoxus (usually named after Archimedes) does not hold. Indeed the Axiom of Eudoxus/Archimedes is the main difference between the real and p-adic/ultrametric space; however the axiom is more of a physical one which concerns the process of measurement: exchanging the real numbers field with the p-adic number field is tantamount to exchanging axiomatics in quantum physics.
We present a new construction of the Euclidean Φ 4 quantum field theory on R 3 based on PDE arguments. More precisely, we consider an approximation of the stochastic quantization equation on R 3 defined on a periodic lattice of mesh size ε and side length M . We introduce a new renormalized energy method in weighted spaces and prove tightness of the corresponding Gibbs measures as ε → 0 , M → ∞ . Every limit point is non-Gaussian and satisfies reflection positivity, translation invariance and stretched exponential integrability. These properties allow to verify the Osterwalder–Schrader axioms for a Euclidean QFT apart from rotation invariance and clustering. Our argument applies to arbitrary positive coupling constant, to multicomponent models with O ( N ) symmetry and to some long-range variants. Moreover, we establish an integration by parts formula leading to the hierarchy of Dyson–Schwinger equations for the Euclidean correlation functions. To this end, we identify the renormalized cubic term as a distribution on the space of Euclidean fields.
It is possible to construct Lévy white noises as generalized random processes in the sense of Gel’fand and Vilenkin, or as an independently scattered random measures introduced by Rajput and Rosinski. In this article, we unify those two approaches by extending the Lévy white noise Ẋ, defined as a generalized random process, to an independently scattered random measure. We are then able to give general integrability conditions for Lévy white noises, thereby maximally enlarging their domain of definition. Based on this connection, we provide new criteria for the practical determination of the domain of definition, including specific results for the subfamilies of Gaussian, symmetric-α-stable, generalized Laplace, and compound Poisson white noises. We also apply our results to formulate a general criterion for the existence of generalized solutions of linear stochastic partial differential equations driven by a Lévy white noise.
We define a hierarchical Gaussian field in a way that is motivated by the finite-range decomposition of the Gaussian free field. The hierarchical Gaussian free field is a hierarchical field that has comparable large distance behaviour to the lattice Gaussian free field. We explicitly construct a version of the hierarchical Gaussian field and verify that it has the desired properties. We define the hierarchical |φ|⁴ model and state the main result proved in this book, which gives the critical behaviour of the susceptibility of the 4-dimensional hierarchical |φ|⁴ model. In preparation for the proof of the main result, we reformulate the hierarchical |φ|⁴ model as a perturbation of a Gaussian integral.
We prove a very general theorem for the pointwise and pathwise multiplication of random Schwartz distributions. The hypothesis is Wilson's operator product expansion. Due to the urgency of the matter, we will release this article in three successive installments. In the first installment, this one, the complete proof of our theorem is given. It partly relies on the general theory of distribution kernels in combination with probability theory which we call the Schwartz-Grothendieck-Fernique theory. Section 3, to be supplied in the second installment, will give a very elementary and self-contained presentation of this theory in the particular case of temperate distributions. The third installment will add Section 7 dedicated to the special study of conformal field theories in various dimensions.
We consider the question of conformal invariance of the long-range Ising
model at the critical point. The continuum description is given in terms of a
nonlocal field theory, and the absence of a stress tensor invalidates all of
the standard arguments for the enhancement of scale invariance to conformal
invariance. We however show that several correlation functions, computed to
second order in the epsilon expansion, are nontrivially consistent with
conformal invariance. We proceed to give a proof of conformal invariance to all
orders in the epsilon expansion, based on the description of the long-range
Ising model as a defect theory in an auxiliary higher-dimensional space. A
detailed review of conformal invariance in the d-dimensional short-range Ising
model is also included and may be of independent interest.
We give a concise overview of the theory of regularity structures as first
exposed in [Hai14]. In order to allow to focus on the conceptual aspects of the
theory, many proofs are omitted and statements are simplified. In order to
provide both motivation and focus, we concentrate on the study of solutions to
the stochastic quantisation equations for the Euclidean quantum
field theory which can be obtained with the help of this theory. In particular,
we sketch the proofs of how one can show that this model arises quite naturally
as an idealised limiting object for several classes of smooth models.
Let M be a smooth manifold, a closed embedded submanifold of
M and U an open subset of M. In this paper, we find conditions using a
geometric notion of scaling for to
admit an extension in . We give microlocal conditions on
t which allow to control the wave front set of the extension generalizing a
previous result of Brunetti--Fredenhagen. Furthermore, we show that there is a
subspace of extendible distributions for which the wave front of the extension
is minimal which has applications for the renormalization of quantum field
theory on curved spacetimes.
The goal of this Lecture Note is to prove a new type of limit
theorems for normalized sums of strongly dependent random variables
that play an important role in probability theory or in statistical
physics. Here non-linear functionals of stationary Gaussian fields
are considered, and it is shown that the theory of Wiener-Ito
integrals provides a valuable tool in their study. More precisely, a
version of these random integrals is introduced that enables us to
combine the technique of random integrals and Fourier analysis. The
most important results of this theory are presented together with
some non-trivial limit theorems proved with their help.
This work is a new, revised version of a previous volume written
with the goal of giving a better explanation of some of the details and
the motivation behind the proofs. It does not contain essentially
new results; it was written to give a better insight to the old ones.
In particular, a more detailed explanation of generalized fields
is included to show that what is at the first sight a rather
formal object is actually a useful tool for carrying out heuristic
arguments.
We develop a Renormalization Group (RG) approach to the study of existence
and uniqueness of solutions to stochastic partial differential equations driven
by space-time white noise. As an example we prove well-posedness and
independence of regularization for the model in three dimensions
recently studied by Hairer. Our method is "Wilsonian": the RG allows to
construct effective equations on successive space time scales. Renormalization
is needed to control the parameters in these equations. In particular no theory
of multiplication of distributions enters our approach.
We survey the properties of the log-correlated Gaussian field (LGF), which is
a centered Gaussian random distribution (generalized function) h on , defined up to a global additive constant. Its law is determined by the
covariance formula
which holds for mean-zero test functions . The LGF belongs to
the larger family of fractional Gaussian fields obtained by applying fractional
powers of the Laplacian to a white noise W on . It takes the
form . By comparison, the Gaussian free field (GFF)
takes the form in any dimension. The LGFs with coincide with the 2D GFF and its restriction to a line. These objects
arise in the study of conformal field theory and SLE, random surfaces, random
matrices, Liouville quantum gravity, and (when d=1) finance. Higher
dimensional LGFs appear in models of turbulence and early-universe cosmology.
LGFs are closely related to cascade models and Gaussian branching random walks.
We review LGF approximation schemes, restriction properties, Markov properties,
conformal symmetries, and multiplicative chaos applications.
We discuss a family of random fields indexed by a parameter
which we call the fractional Gaussian fields, given by
\[
\mathrm{FGF}_s(\mathbb{R}^d)=(-\Delta)^{-s/2} W,
\] where W is a white noise on and is the
fractional Laplacian. These fields can also be parameterized by their Hurst
parameter . In one dimension, examples of processes
include Brownian motion () and fractional Brownian motion (). Examples in arbitrary dimension include white noise (), the
Gaussian free field (), the bi-Laplacian Gaussian field (), the
log-correlated Gaussian field (), L\'evy's Brownian motion (), and multidimensional fractional Brownian motion ().
These fields have applications to statistical physics, early-universe
cosmology, finance, quantum field theory, image processing, and other
disciplines.
We present an overview of fractional Gaussian fields including covariance
formulas, Gibbs properties, spherical coordinate decompositions, restrictions
to linear subspaces, local set theorems, and other basic results. We also
define a discrete fractional Gaussian field and explain how the
with can be understood as a long range Gaussian
free field in which the potential theory of Brownian motion is replaced by that
of an isotropic 2s-stable L\'evy process.
It is well know that systems with an interaction decaying as a power of the
distance may have critical exponents that are different from those of
short-range systems. The boundary between long-range and short-range is known,
however the behavior in the crossover region is not well understood. In this
paper we propose a general form for the crossover function and we compute it in
a particular limit. We compare our predictions with the results of numerical
simulations for two-dimensional long-range percolation.
We study the conformal bootstrap for systems of correlators involving
non-identical operators. The constraints of crossing symmetry and unitarity for
such mixed correlators can be phrased in the language of semidefinite
programming. We apply this formalism to the simplest system of mixed
correlators in 3D CFTs with a global symmetry. For the leading
-odd operator and -even operator
, we obtain numerical constraints on the allowed dimensions
assuming that and are
the only relevant scalars in the theory. These constraints yield a small closed
region in space compatible with the known
values in the 3D Ising CFT.
The subject of elliptic curves is one of the jewels of nineteenth-century mathematics, originated by Abel, Gauss, Jacobi, and Legendre. This 1997 book presents an introductory account of the subject in the style of the original discoverers, with references to and comments about more recent and modern developments. It combines three of the fundamental themes of mathematics: complex function theory, geometry, and arithmetic. After an informal preparatory chapter, the book follows an historical path, beginning with the work of Abel and Gauss on elliptic integrals and elliptic functions. This is followed by chapters on theta functions, modular groups and modular functions, the quintic, the imaginary quadratic field, and on elliptic curves. Requiring only a first acquaintance with complex function theory, this book is an ideal introduction to the subject for graduate students and researchers in mathematics and physics, with many exercises with hints scattered throughout the text.
The majority of the "memorable" results of relativistic quantum theory were obtained within the framework of the local quantum field approach. The explanation of the basic principles of the local theory and its mathematical structure has left its mark on all modern activity in this area. Originally, the axiomatic approach arose from attempts to give a mathematical meaning to the quantum field theory of strong interactions (of Yukawa type). The fields in such a theory are realized by operators in Hilbert space with a positive Poincare-invariant scalar product. This "classical" part of the axiomatic approach attained its modern form as far back as the sixties. * It has retained its importance even to this day, in spite of the fact that nowadays the main prospects for the description of the electro-weak and strong interactions are in connection with the theory of gauge fields. In fact, from the point of view of the quark model, the theory of strong interactions of Wightman type was obtained by restricting attention to just the "physical" local operators (such as hadronic fields consisting of ''fundamental'' quark fields) acting in a Hilbert space of physical states. In principle, there are enough such "physical" fields for a description of hadronic physics, although this means that one must reject the traditional local Lagrangian formalism. (The connection is restored in the approximation of low-energy "phe nomenological" Lagrangians.
We prove a very general theorem for the pointwise and pathwise multiplication of random Schwartz distributions. The hypothesis is Wilson's operator product expansion. Due to the urgency of the matter, we will release this article in three successive installments. In the first installment, this one, the complete proof of our theorem is given. It partly relies on the general theory of distribution kernels in combination with probability theory which we call the Schwartz-Grothendieck-Fernique theory. Section 3, to be supplied in the second installment, will give a very elementary and self-contained presentation of this theory in the particular case of temperate distributions. The third installment will add Section 7 dedicated to the special study of conformal field theories in various dimensions.
We review recent results with D. Chelkak and K. Izyurov [10], where we rigorously prove existence and conformal invariance of scaling limits of magnetization and multi-point spin correlations in the critical Ising model on an arbitrary simply connected planar domain. This solves a number of conjectures coming from physical and mathematical literatures. The proof is based on convergence results for discrete holomorphic spinor observables. © 2014 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.
These notes describe an application of the renormalization group to a continuous time weakly self-avoiding walk on a four dimensional lattice. The description includes
the connection between the local time of continuous time random walk and the massless free field;
one of the several formalisms by which the Wilson Renormalisation Group has become the basis for mathematical proof;
parts of a proof that there are corrections in the susceptibility for the four dimensional Edwards model on a lattice with weak coupling.
We investigate stochastic quantization as a mathematical tool for quantum field theory. We test the method for the free scalar field. We find that the usual method of stochastic quantization is incompatible with establishing a Hilbert-space interpretation for transition probabilities in quantum theory. In particular, we prove that for any finite stochastic time, the standard probability measure violates reflection positivity. As a consequence, if one desires to use stochastic quantization in constructive quantum field theory, one needs to find a more robust procedure than the standard one.
These lecture notes grew out of a series of lectures given by the second
named author in short courses in Toulouse, Matsumoto, and Darmstadt. The main
aim is to explain some aspects of the theory of "Regularity structures"
developed recently by Hairer in arXiv:1303.5113 . This theory gives a way to
study well-posedness for a class of stochastic PDEs that could not be treated
previously. Prominent examples include the KPZ equation as well as the dynamic
model. Such equations can be expanded into formal perturbative
expansions. Roughly speaking the theory of regularity structures provides a way
to truncate this expansion after finitely many terms and to solve a fixed point
problem for the "remainder". The key ingredient is a new notion of "regularity"
which is based on the terms of this expansion.
The aim of the paper is to present numerical results supporting the presence
of conformal invariance in three dimensional statistical mechanics models at
criticality and to elucidate the geometric aspects of universality. As a case
study we study three dimensional percolation at criticality in bounded domains.
Both on discrete and continuous models of critical percolation, we test by
numerical experiments the invariance of quantities in finite domains under
conformal transformations focusing on crossing probabilities. Our results show
clear evidence of the onset of conformal invariance in finite realizations
especially for the continuum percolation models. Finally we propose a simple
analytical function approximating the crossing probability among two spherical
caps on the surface of a sphere and confront it with the numerical results.
We introduce SDPB: an open-source, parallelized, arbitrary-precision
semidefinite program solver, designed for the conformal bootstrap. SDPB
significantly outperforms less specialized solvers and should enable many new
computations. As an example application, we compute a new rigorous
high-precision bound on operator dimensions in the 3d Ising CFT,
, .
We show global well-posedness of the dynamic model in the plane. The
model is a non-linear stochastic PDE that can only be interpreted in a
"renormalised" sense. Solutions take values in suitable weighted Besov spaces
of negative regularity.
Using Wilson renormalization group, we show under commonly accepted
assumptions that scale invariance implies conformal invariance in dimension
three for the Ising and O(N) models.
We provide an explicit isomorphism between the space of smooth functions and its sequence space representation which isomorphically maps various spaces of smooth functions onto their sequence-space representation, including the space , of test functions, the space of Schwartz functions and the space of “p-integrable smooth functions” . By restriction and transposition, this isomorphism yields an isomorphism between the space of distributions and its sequence space representation which analogously maps various spaces of distributions isomorphically onto their sequence space representation. We use this isomorphism to construct both a common Schauder basis for these spaces of smooth functions and a common Schauder basis for the corresponding spaces of distributions.
In this review article, we discuss the distinction and possible equivalence between scale invariance and conformal invariance in relativistic quantum field theories. Under some technical assumptions, we can prove that scale invariant quantum field theories in d=2 space–time dimensions necessarily possess the enhanced conformal symmetry. The use of the conformal symmetry is well appreciated in the literature, but the fact that all the scale invariant phenomena in d=2 space–time dimensions enjoy the conformal property relies on the deep structure of the renormalization group. The outstanding question is whether this feature is specific to d=2 space–time dimensions or it holds in higher dimensions, too. As of January 2014, our consensus is that there is no known example of scale invariant but non-conformal field theories in d=4 space–time dimensions under the assumptions of (1) unitarity, (2) Poincaré invariance (causality), (3) discrete spectrum in scaling dimensions, (4) existence of scale current and (5) unbroken scale invariance in the vacuum. We have a perturbative proof of the enhancement of conformal invariance from scale invariance based on the higher dimensional analogue of Zamolodchikov’s cc-theorem, but the non-perturbative proof is yet to come. As a reference we have tried to collect as many interesting examples of scale invariance in relativistic quantum field theories as possible in this article. We give a complementary holographic argument based on the energy-condition of the gravitational system and the space–time diffeomorphism in order to support the claim of the symmetry enhancement. We believe that the possible enhancement of conformal invariance from scale invariance reveals the sublime nature of the renormalization group and space–time with holography.
This review is based on a lecture note on scale invariance vs conformal invariance, on which the author gave lectures at Taiwan Central University for the 5th Taiwan School on Strings and Fields.
We extend and apply a rigorous renormalisation group method to study critical
correlation functions, on the 4-dimensional lattice , for the
weakly coupled n-component spin model for all , and
for the continuous-time weakly self-avoiding walk. For the model,
we prove that the critical two-point function has (Gaussian) decay
asymptotically, for . We also determine the asymptotic decay of the
critical correlations of the squares of components of , including the
logarithmic corrections to Gaussian scaling, for . The above extends
previously known results for to all , and also observes new
phenomena for , all with a new method of proof. For the continuous-time
weakly self-avoiding walk, we determine the decay of the critical generating
function for the "watermelon" network consisting of p weakly mutually- and
self-avoiding walks, for all , including the logarithmic corrections.
This extends a previously known result for , for which there is no
logarithmic correction, to a much more general setting. In addition, for both
models, we study the approach to the critical point and prove existence of
logarithmic corrections to scaling for certain correlation functions. Our
method gives a rigorous analysis of the weakly self-avoiding walk as the case of the model, and provides a unified treatment of both
models, and of all the above results.
Let me begin by describing one of the gems of classical mathematics which first stirred my own enthusiasm for inversive geometry. It illustrates the elegance of the subject and provides a point of interest which we shall glimpse again in the closing chapters of this account.
Nous développons dans cet article les principaux arguments constructifs utilisés en théorie quantique des champs, en nous cantonnant aux théories bosoniques, pour lesquelles il n'existe pas de présentation générale récente. L'article s'adresse d'abord et avant tout à des mathématiciens ou physiciens mathématiciens connaissant les arguments de base de la théorie perturbative des champs, et souhaitant connaître un cadre général dans lequel ils peuvent être rendus rigoureux. Il fournit également un aperçu d'une série d'articles récents [50, 51] visant à donner une définition constructive des chemins rugueux et du calcul stochastique fractionnaire. We develop in this article the principal constructive arguments used in quantum field theory, limiting us to bosonic theories, for which there does not exist any recent general presentation. The article is primarily written for mathematicians or mathematical physicists knowing the basic arguments of quantum field theory, and desiring to discover a general framework in which they can be made rigorous. It also provides a glimpse of a recent series of articles [50, 51] whose aim is to give a constructive definition of rough paths and fractionary stochastic calculus.
Preface.- 1. Basic analysis.- 2. Littlewood-Paley theory.- 3. Transport and transport-diffusion equations.- 4. Quasilinear symmetric systems.- 5. Incompressible Navier-Stokes system.- 6. Anisotropic viscosity.- 7. Euler system for perfect incompressible fluids.- 8. Strichartz estimates and applications to semilinear dispersive equations.- 9. Smoothing effect in quasilinear wave equations.- 10.- The compressible Navier-Stokes system.- References. - List of notations.- Index.
This paper was written several years ago, but no part of it has been published previously. A preprint was distributed to selected experts and seems to have been favorably received. For some time I had hoped to improve on the results of the paper, but as years went by my research took a different direction, and it became implausible that I would add anything significant to the paper as it stands.
L'article introduit le type de la loi de Cauchy-conforme comme l'ensemble des probabilités sur l'espace euclidien n défini par où (p, a) décrit +n + 1 = ]0, + ∞[× n et démontre que si F: n → n est une fonction mesurable, alors F préserve le type de la loi Cauchy-conforme si et seulement si F coïncide presque partout avec une similitude ou une similitude-inversion de n, dans le cas où n ⩾ 2. Ce résultat prolonge l'étude faite auparavant (Letac, Proc. Amer. Math. Soc. 67 (1977), 277–286) du même problème pour n = 1.
