> <\body> >|2: The Gaussian free field>|>|> We study the first example of a Euclidean field theory: the Gaussian free field (GFF) on the torus > and on >. We construct it, prove reflection positivity and recover the free relativistic field, discuss the Euclidean Källén\ULehmann representation, and define its local (Wick) operators. We prove that the Dyson\USchwinger equation characterises the GFF among probability measures on distributions. Stochastic quantisation is treated in Lecture. Exercises are collected at the end. These notes follow the slides of Lecture2. > <\table-of-contents|toc> |.>>>>|> The Gaussian free field> |.>>>>|> Definition and construction |.>>>>|> > Reflection positivity and the free field |.>>>>|> > |.>>>>|> > |.>>>>|> > Local operators: Wick products |.>>>>|> > >: no random distribution |.>>>>|> > Gaussian integration by parts and the Dyson\USchwinger equation |.>>>>|> > Outlook |.>>>>|> > |.>>>>|> > |.>>>>|> |.>>>>|> > In Lecture we saw that, after the passage to imaginary time, a relativistic quantum field is encoded in a probability measure on distributions satisfying the Osterwalder\USchrader axioms, formally \|)>\exp *\|\|>+*m*\+V|)>|]>|)>*\*\> (Eq.). In this lecture we treat its Gaussian part rigorously. Conventions are those of Lecture. Sections, equations and exercises keep the numbering of the notes, so that ź1\U2 refer to the notes of Lecture. Throughout, > is either the torus =/L*\|)>> or >, 1>. We write |)>> for >|)>>, resp. the Schwartz space, and |)>> for its dual. We equip |)>> with the >-algebra generated by the maps \\>; it coincides with the Borel >-algebra. Test functions are real unless stated otherwise. Let 0> and <\equation> A\m-\,C\A. is a positive bounded operator on |)>> with |C|\<\|\|\>>\m>. Both and are continuous bijections of |)>>, being Fourier multipliers by +m|)>1>>. <\definition> The on > is the centred Gaussian probability measure =\> on |)>> with covariance : <\equation> \*\*\\|)>=|f,C*g|\>=f*G**g\x\y, equivalently \\>\\|)>=\*|f,C*f|\>>> for all \|)>>. Here is the Green function of . On >, =G*> with =k\x>|+m>*k||)>>>. On >, =\>G*> (method of images). Formally \|)>*`*`="*Z\*|\,A*\|\>>\*\>, the Gaussian part of . The Green function satisfies (Exercise): \>*log >> as 0> if ; \*|4*\>> if 3>; =\>/> if ; and > decays like /2>\>> at infinity. Uniqueness of > is clear, since a probability measure on |)>> is determined by its characteristic functional. |gff2d.png>|0.38par|||> |green_functions.pdf>|0.58par|||>>|Left: a sample of the massive GFF on > (, Fourier cutoff >, zero mode removed). It is not a function: refining the cutoff adds fluctuations at all scales. Right: the Green function > in : bounded, logarithmic and > singularities at the origin, and exponential decay at infinity.> <\theorem> (a) Let =\>, let |)>0>> be a real orthonormal basis of |)>> of eigenfunctions of >, *e=\*e>, \m+\>, and let |)>> be i.i.d. >. For every -1> the series <\equation> \=0>\*g*e converges a.s. and in ;H|)>|)>>, and its law is >. In particular |)>=1> for -1>. (b) On >, > exists by the Bochner\UMinlos theorem. <\proof> (a) With |\|\<\|\|\>>>=|)>*|e,\|\>>, <\equation*> \|\|\<\|\|\>>>=|)>*\>\\>|)>\\\2*s+2\d. The partial sums form an >-bounded martingale with values in the Hilbert space >, so they converge a.s. and in >. For \|)>>, =\*g*|e,f|\>> is centred Gaussian with variance \*|e,f|\>=|f,C*f|\>>. (b) Bochner\UMinlos: if :\|)>\\> is continuous, positive definite and =1>, there is a unique probability measure on |)>> with characteristic functional >. For =\*|f,C*f|\>>>, continuity follows from |f,C*f|\>\m|f|\<\|\|\>>>>. For positive definiteness, write *-f|)>=a*a\>> with =\*Q>> and =|f,C*f|\>> positive semidefinite. Then >|)>> is positive semidefinite by the Schur product theorem. <\remark> The Cameron\UMartin space of > is |)>> with norm |h,A*h|\>>: the fields of finite action. It has measure zero, and typical samples are much rougher (Exercise). In the GFF on > is the stationary Ornstein\UUhlenbeck process. In it lies in >> for every \0> but is not a function. In it lies in >>. On > the samples live in weighted Sobolev spaces. Let =\\N> with > or >, points ,|)>>, *,|)>=,|)>>. Then >+B> with m-\> on >, and we set \B\m>. <\proposition> (a) As an operator-valued kernel in the time variables, ,\|)>=\-\|\|>*\>|)>>. (b) For \|)>> with f\\0|}>>, <\equation> |\*f,C*f|\>=||)>>\*\>f,\|)>\\|\<\|\|\>>>\0. (c) > on > satisfies (OS0)\U(OS4). <\proof> (a) follows from the partial Fourier transform in >, the spectral theorem for , and the scalar identity >\k*t>+w|)>*k|2*\>=\>/>, 0>. (b) Since *f> is supported in \0|}>>, <\align*> |\*f,C*f|\>>|\0\\>|f*|)>,\-\|)>*\>|)>*f|)>|\>\\\\>>||>>|\>f,|)>\*\>f|)>|\>\s\\,>>>> which is. (c) (OS0)\U(OS2) and (OS4) follow from the explicit form of |^>> and the decay of >. For (OS3), commutes with >, so |^>*-\*f|)>=a*a\|\*f,C*f|\>>> with =\*|f,C*f|\>>>. The matrix |\*f,C*f|\>|)>> is positive semidefinite by (b), hence so is its entrywise exponential (Schur). \\documentclass[tikz]{standalone} \\usepackage[T1]{fontenc} \\usepackage[sc]{mathpazo} \\usepackage{amsmath,amssymb,mathtools,bm} \\usepackage[mathscr]{eucal} \\usetikzlibrary{decorations.pathreplacing,arrows.meta,positioning} \\usepackage[dvipsnames]{xcolor} \; \\begin{document} \\begin{tikzpicture}[scale=0.8,\>=Stealth] \ \ \\fill[MidnightBlue!8] (-3,0) rectangle (3,2.2); \ \ \\draw[-\>] (-3.2,0) -- (3.4,0) node[right] {\\small $\\mathbf{x}\\in N$}; \ \ \\draw[-\>] (0,-2.3) -- (0,2.5) node[above] {\\small $\\tau$}; \ \ \\draw[thick,dashed,BrickRed] (-3.2,0) -- (3.2,0); \ \ \\fill[MidnightBlue!60] (1.2,1.2) ellipse (0.7 and 0.35); \ \ \\node at (1.2,1.75) {\\small $\\operatorname{supp} f$}; \ \ \\fill[MidnightBlue!25] (1.2,-1.2) ellipse (0.7 and 0.35); \ \ \\node at (1.2,-1.75) {\\small $\\operatorname{supp}\\Theta f$}; \ \ \\draw[\<-\>,gray] (2.1,1.2) -- (2.1,-1.2) node[midway,right] {\\small $\\Theta$}; \ \ \\node[BrickRed,anchor=east] at (-0.2,0.25) {\\small $\\tau=0$}; \ \ \\node at (-1.7,1.3) {\\small $\\{\\tau\>0\\}$}; \\end{tikzpicture} \\end{document} |ps>||||>>>|Reflection positivity (Proposition). For supported in \0|}>> the covariance between and its mirror image factorises through the time-zero plane: |\*f,C*f|\>=||)>*>\*\>f|)>\\|\<\|\|\>>\0>.> By the OS reconstruction theorem the GFF yields a Wightman theory, the free field of mass of Section. Concretely, the reconstructed Hilbert space is the Fock space over > with \|)>> (Nelson's free Markov field). The time-zero restriction of the GFF is Gaussian with covariance |)>>, the ground state measure of the free Hamiltonian in the Schrödinger representation (Exercise of Lecture). The GFF is also the continuum limit of the lattice model with (for ; in general see Exercise and Exercise). <\remark> For \> open let > be the >-algebra generated by the > with f\U>, and for closed let =K>\> (the >-algebra of ). A random field is for an open set > if |\>>> and >> are conditionally independent given *\>>: the field inside and the field outside interact only through the field near the boundary. (i) >>. For the GFF, > is generated by the > with H> supported in . In Gaussian terms, conditional expectations are then orthogonal projections in the Hilbert space of the >, with **\|]>=|f,C*g|\>>: \|\>|]>=\**f|)>>, where > projects > onto the distributions supported in >. Let f\\> and *f>. Orthogonality gives on the interior of >, hence there, while *f=0> in >. So *f=A*u> is supported on *\>: this is where the of -\> enters. Equivalently, given the outside, the field in > is the -harmonic extension of its boundary values plus an independent GFF with Dirichlet conditions on *\> (the domain Markov property of). The generalised free fields of the two-mass example below are not Markov, because their > is nonlocal. (ii) >-invariant field, Markov with a sharp time-zero >-algebra implies reflection positivity.> Take =\0|}>> and let > be the germ >-algebra of =0|}>>. For the GFF it is generated by the time-zero field ( Markov property), so > acts trivially on it. If is \0|}>>>-measurable, then *F> is \0|}>>>-measurable and **F\|\|]>=\\|\|]>>, so the Markov property gives <\equation*> \**F|\>*F|]>=\**F\|\|]>|\>\\|\|]>|]>=\\|\|]>|\|>|]>\0. Sharpness is essential: the fourth-order field with covariance > in the discussion of positivity versus locality below is Markov but not reflection positive, because its > also contains >*\>, which > reverses. Moreover =\U\\|\|]>>, with > the time translation, is a semigroup on |)>>: by the Markov property for \t|}>>, *P=P>. It is self-adjoint (by >-invariance), positive (=P>), contractive and strongly continuous, so =\>, and the physical Hilbert space is > of the time-zero field, i.e. the Schrödinger representation. This is Nelson's reconstruction of quantum fields from Euclidean invariant Markov fields, which preceded the Osterwalder\USchrader axioms, and it underlies the Feynman\UKac\UNelson formula (Proposition). (iii) A local perturbation >V|)>>> of > factorises over regions and preserves the Markov property. This makes the |)>> measures in finite volume Markov fields, and it gives the classical statistical mechanics picture of: conditional distributions given the outside, DLR equations, correlation inequalities. The Markov property is not stable under weak limits, however, and is hard to establish directly for infinite-volume or renormalised theories. Osterwalder and Schrader replaced it by reflection positivity, which neither implies nor is implied by the Markov property (generalised free fields are reflection positive but not Markov, the > field is Markov but not reflection positive), but is preserved under weak limits. See for more. <\remark> The Euclidean programme goes back to Symanzik, who proposed to study QFT through its Schwinger functions, viewed as the moments of a classical (formally, Markov) random field, before Nelson and Osterwalder\USchrader made this rigorous. In he also introduced a representation of the field in terms of Brownian paths. Since <\equation*> C*=>\*t>p\t, with > the heat kernel of >, the two-point function of the GFF is the Green function of a Brownian motion (with generator >) killed at rate >. For V|)>>\>, the correlations become averages over a gas of Brownian paths and loops with an interaction that depends on their local times. For > the interaction penalises intersections. On the lattice this is the random walk representation of Brydges, Fröhlich and Spencer, the basis of one of the proofs of triviality of > for 4>. Heuristically, the interaction becomes ineffective in the continuum for 4> because two independent Brownian paths do not intersect there. The modern continuum version is the of Lawler and Werner, a Poisson process of unrooted Brownian loops with intensity > times the loop measure. On a finite graph the occupation field of the random-walk loop soup satisfies |\,L|\>>|]>=det |)>>>, while *|\,\|\>>|]>=det |)>>. So for =>, has the law of *\>: this is , a loop version of Dynkin's isomorphism. In the continuum, in , the renormalised occupation field of the soup corresponds to :\>:, the Wick square of Section. Lupu showed that on the cable graph the clusters of the => soup are exactly the sign clusters of the GFF, which is recovered from the soup by tossing independent signs. Qian and Werner decomposed the clusters of the two-dimensional continuum soup, and Werner proved a spatial Markov property for loop soups that parallels Nelson's Markov property of the GFF. For a survey see. <\remark> The computation in Proposition(a) says that \\|)>\|)>|\|>+\\>=G*,|)>>: the Wick rotation of the free two-point function >> is the Green function > of -\>. Rotating term by term, the Schwinger two-point function of any theory satisfying the Wightman (or OS) axioms is <\equation> S=v+|)>>\M|)>*G,x\0,=|)>>M|)>|+M>, a Stieltjes transform in >. (The Fourier form holds up to contact terms, i.e. a polynomial in > invisible at 0>, and provided \M|)>*|)>\\>; otherwise subtractions are needed.) The GFF is the case =\>>. Three consequences follow. (i) > is decreasing in , and for 2> we have /G\1> as 0>, for every . Hence, for every , \\|]>|)>*G>, and, if \0>, <\equation*> liminf0> |G>\\|)>|)>\|]>. The two-point function of any nontrivial theory is at least as singular as the free one. In particular a scalar field cannot have scaling dimension below the free value > (the , Exercise). (ii) If \\,\|)>>, then > decays at least like >>, up to subexponential factors, and exactly at this rate if =inf \>. Conversely, exponential decay at rate > forces \\,\|)>>. So a gap in the spectral measure of > is the same as exponential decay of >; a mass gap of corresponds to exponential clustering of all truncated correlations (compare Exercise). (iii) For any positive measure > on |)>> with \M|)>*|)>\\> and \M|)>*M\\>, the Gaussian measure on |)>> with covariance >=\M|)>*-\|)>> satisfies the OS axioms: >\>=\M|)>*M> is bounded, which gives (OS1) with , and reflection positivity follows from Proposition for each mass 0> and \0>. (If only the first condition holds, the measure may still exist and satisfy the other axioms, but (OS1) in the form of Lecture can fail: for the massless field in 3>, g> with 0> supported in a ball of radius gives |^>|\|>=\*|g,C*g|\>>> with |g,C*g|\>\R>.) It reconstructs the generalised free field of Exercise. Its Dyson\USchwinger equation **F|]>=\**>*h|)>|]>> still characterises it: under mild conditions on > (e.g. > finite with a mass gap), >> is a bijection of |)>>, which is all the proof of Theorem uses. However >> is unless > is a single atom, as the following example shows. Let m\m>, ,c\0>, and <\equation*> \=c*\>+c*\>,C>=c*-\|)>+c*-\|)>. The Gaussian measure >> with covariance >> is the law of *\+c*\>, where ,\> are independent GFFs on > of masses ,m>. It satisfies (OS0)\U(OS4); reflection positivity follows from Proposition applied to each term. Since a Wightman theory is determined up to unitary equivalence by its Wightman functions, which are in turn determined by the Schwinger functions, >> reconstructs to exactly the following theory. Let =L,\\|)>> be the one-particle space of mass and spin , carrying the irreducible positive-energy representation > of >>. Then <\equation*> \=\*>\\>|)>\\>|)>\\>|)>,U=\*>\u>|)>,\=\\\, <\equation*> \=c*\>\+c*\\>, with > the free field of mass . The Wightman functions are given by Wick's theorem with =c*W|)>>+c*W|)>>>, and all truncated functions of order 3> vanish. Physically, these are , of masses > and >, both created from the vacuum by the same field: *\=c*a>\|>>|)>*\+c*a>\|>>|)>*\>. They do not scatter. The mass spectrum consists of the two shells >, > and of the multi-particle continuum from >. If \2*m>, the second shell lies inside the two-particle continuum, but the heavy particle is stable because there is no interaction. This is the generalised free field of Exercise. (1) There is no field equation of second order: +m|)>*+m|)>*\=0>, but > solves no Klein\UGordon equation. The time-zero fields *|)>>, *|)>> satisfy *|)>,\*|)>|]>=\+c|)>*\*-|)>>, which is canonical iff +c=1> (the sum rule of Exercise). Even when they are canonical, they are not a complete set of Cauchy data: *|)>> depends on *\>*|)>> and *\>*|)>> separately, not only on their sum. (2) In Euclidean terms, the DS equation **F|]>=\**>*h|)>|]>> involves the nonlocal operator <\equation*> C>=P*R,P\-\|)>*-\|)>,R\c*-\|)>+c*-\|)>. Replacing by gives the equivalent local identity **F|]>=\**|]>>: a fourth-order Euclidean field equation whose contact term is itself differentiated, the Euclidean form of +m|)>*+m|)>*\=0>. Correspondingly, > has no Markov property: for stationary Gaussian fields with a rational spectral density, such as >=/>, the Markov property holds iff the inverse covariance is a differential operator, i.e. iff > is a polynomial, which is not the case here. The pair ,\|)>> is Markov, but its sum is not: this is the Euclidean counterpart of(1). Conversely, the local fourth-order action *|\,P*\|\>> defines a Markov Gaussian field with covariance <\equation*> P=-\|)>--\|)>|m-m>,\=>-\>|m-m>, with a weight at >. By and the uniqueness of the Stieltjes transform it is not reflection positive (Exercise). It reconstructs at best a theory with an indefinite metric, in which the particle of mass > is a ghost of negative norm. More generally, call the Euclidean field equation if > is a differential operator, i.e. =1/p|)>> with a polynomial; local field equations give Markov fields, and for rational spectral densities the converse holds. The Fourier transform of a covariance satisfying the OS axioms is |)>+\M|)>*+M|)>>, with \0> and a polynomial coming from contact terms invisible to >. If 1>, then , since \0> and the Stieltjes term tends to as \>. Moreover \M|)>*|)>> increases to |)>|)>> as \>, while \0> if 2>; hence =0>, a contradiction. If , > is constant. So a centred Gaussian measure satisfying the OS axioms has a local Euclidean field equation iff its covariance is -\|)>> with 0>, or a multiple of the identity (white noise, whose reconstructed Hilbert space is *\>). Note that it is the covariance, not >, that matters: +c*-\|)>> with 0> is reflection positive and has =c*\>>, but its field equation is nonlocal. (up to the trivial white noise); locality is also behind Nelson's Markov property of the GFF. In Minkowski space the two-mass theory is a perfectly good one. It satisfies all the Wightman axioms, including locality (W3): ,\|]>> is a sum of free-field commutators and vanishes at spacelike separation. The energy is positive and the vacuum unique. It is trivial (no interaction) and redundant, but not inconsistent. \PNonlocal\Q refers only to the Euclidean field equation for > alone, equivalently (for rational spectral densities, as here) to the absence of the Markov property. On the Minkowski side this means that > alone is not a complete set of local degrees of freedom. Its time-zero data do not determine its evolution, and there is no Lagrangian or Hamiltonian written in terms of > only. The cure is to enlarge the field content: the pair ,\|)>> has a local action and is Markov, and > is a linear combination of its components. Generalised free fields become genuinely unphysical when > has a continuous part, i.e. (heuristically) infinitely many free species with masses in a bounded interval. With finitely many masses, > solves a hyperbolic equation of finite order, and the fields in an arbitrarily thin time slice generate all the observables: the , or primitive causality. With a continuum of masses there is no dynamical law of finite order, and this property is not expected to hold. More decisively, the theory violates the phase-space requirements that express \Pfinitely many degrees of freedom per unit volume of phase space\Q, namely the compactness criterion of Haag and Swieca and its sharpening, the nuclearity condition of Buchholz and Wichmann. For a system of free fields the nuclearity condition holds iff n\*m>\\> for all \0>, where > is the multiplicity of the mass >, and this fails when infinitely many species have masses in a bounded interval. Since the nuclearity index of a bounded region plays the role of its partition function, the theory has no sensible thermodynamics. This does not affect interacting theories, whose spectral measure also has a continuous part, the multi-particle continuum. There the continuum is produced by a interaction, and the Euclidean field equation remains local, e.g. **F|]>=\-\|)>*\+4*\:\:|)>|]>> for >, although the inverse two-point function is nonlocal. Thus, for a single field: positive and nonlocal means an incomplete description, which is unphysical for continuous >; local and not positive means ghosts; local, positive and with a nontrivial continuum in > requires interactions. Interactions V|)>\x> and nonlinear field equations require local functions of >, but > is not a function for 2>. We regularise, subtract, and remove the regularisation. The subtraction is encoded in Hermite polynomials with variance 0>, <\equation> \/2>=0>|n!>*H,H=a-c,H=a-3*c*a,H=a-6*c*a+3*c. <\lemma> If > is a centred Gaussian vector with *X=a> and *Y=b>, then **H|]>=\*n!*|)>>. <\proof> *a/2>\*b/2>|]>=\*\*-*s*a-*t*b>=\*>>. Expand both sides in powers of using (the exchange of sum and expectation is justified by Gaussian integrability) and compare coefficients. Fix an even mollifier \\C>|)>> with \=1>, >=\*\*/\|)>> (periodised on the torus), and let >=\>\\>, a smooth function. Then *>*\>|]>=G,\>*> with ,\>=\>\\>\G>. Set >=G,\>> and <\equation> :\>:\H>;c>|)>,:\>:\f:\>:\x. <\key-result> <\theorem> Let 1> with \d>, i.e. every if 2>, 2> if , if 4>. Then for every \|)>>, :\lim\0>:\>:> exists in |)>>, is independent of >, and <\equation> \*::\:|]>=\*n!f*G**g\x\y. If instead \d> and f\0>, then >:|]>\\>. <\proof> 1: a uniform bound.> Let =1+log|)>> if , =1+> if 3>, 1> if . Then G\C*K>, and we claim ,\>\C*K> uniformly in ,\\>. Let ,\|)>>. Then >\\>|)>\B>, \>\\>=1> and |\>\\>|\<\|\|\>>>\r|\|\<\|\|\>>>>. If \4*r>, then ,\>\sup\/2> G\C*K*\C*K>. If \4*r>, then <\equation*> G,\>\|\|\<\|\|\>>>*r*>G\C*|\|>>*>K\C*K\C*K. 2: convergence.> By Lemma and Fubini, <\equation*> \*>::\>:|]>=\*n!f*g*G,\>*\x\y. As ,\\0>, ,\>\G> for 0>. Moreover \L>> iff \d>, and is bounded on \1|}>>. Since L\L>>, dominated convergence gives >:-:\>:|)>|]>=n!f*f*,\>-2*G,\>+G,\>|)>\0>: the sequence is Cauchy in >. The limit covariance is. Running the same computation with two different mollifiers shows independence of >. 3: divergence.> If \d>, then \c>> is not locally integrable, and Fatou's lemma gives >:|]>\n!f*f*G=\>. <\remark> (i) The subtraction of >\\> is the simplest instance of renormalisation: >> itself has no limit when 2>. (ii) The variables :> belong to the -th Wiener chaos, and |)>\\> is the symmetric Fock space over |)>>. By Nelson's hypercontractivity, |X|\<\|\|\>>>\|X|\<\|\|\>>>> for \>, so the convergence holds in every >. A Kolmogorov-type argument gives convergence in >> for . (iii) In on >, the interaction :\:\x> lies in every |)>>, and Nelson's estimate gives *:\:>\L|)>>. This defines the > measure on the torus. (iv) Other local operators: derivatives >*\>, and in the Wick exponentials \*\>>: for \4*\> (in >, Exercise), relevant for the sine-Gordon model, and the real exponentials *\>>: of the Hřegh-Krohn model. >: no random distribution> When \d> the divergence in Theorem is not an artefact of the > norm: no other topology, and no other choice of probability space, can rescue the limit. Albeverio and Liang showed this for the Wick square in , and remarked that the argument extends to 5>: :> is well defined as a Hida distribution, i.e. as a generalised functional of white noise, but for 0> the characteristic functions of its Fourier-cutoff approximations do not converge to a characteristic function (the natural candidate vanishes), so it is not a random variable. For 3>, instead, it is a random variable, and an infinitely divisible one. The following argument is independent and covers all Wick powers. <\proposition> Let \d> and f\0>. Then the family >:|)>\>> is not tight. In particular there is no random distribution >, on any probability space, such that >:\\> in law in > along some sequence \0>. <\proof> Let >=:\>:\\> and >=\*X>>, so that >\\> by Theorem. The states that for 0> with *Z\\> and \>, <\equation*> \*\*\*Z|)>\|)>**Z|)>|\*Z>. (Proof: *Z\\*\*Z+\*\*\*Z>|]>\\*\*Z+*Z|)>*\*\*\*Z|)>> by Cauchy\USchwarz.) Apply it to >> with =>. Hypercontractivity with gives *X>\9*\>>, so <\equation*> \*>|\|>\\>|)>\9=:c\0. Given 0>, for > small enough *\>\R> and hence *>|\|>\R|)>\c>: the family is not tight. If >:\\> in law in >, then >\\> in law, and a sequence converging in law is tight, a contradiction. <\remark> > norm>Let be in the bounded chaos n>\>, e.g. a polynomial of degree n> in finitely many Gaussian variables. Three inequalities control its law in terms of =|X|\<\|\|\>>>> alone, with constants that depend only on : <\itemize> (hypercontractivity): |X|\<\|\|\>>>\*\>, hence *\t*\|)>\exp t|)>> for t>; (Paley\UZygmund): *\*\|)>\c\0>; (Carbery\UWright): *\t*\|)>\C*n*t> for 0>. Consequently, for sequences in a fixed bounded chaos, tightness is equivalent to boundedness in >, and convergence in probability is equivalent to convergence in > (and in every >): apply the Paley\UZygmund bound to -X>. So for Wick powers there is a sharp dichotomy: either convergence in >, or not even tightness. Nontrivial limits of the divergent objects require a change of normalisation: >:\>>: is bounded in >, and typically converges in law to a white noise, which retains no information on the -th power. The fourth moment theorem characterises such Gaussian limits within a fixed chaos, and a central limit theorem of this kind for the lattice >: in 3> is proved in. In particular, in 4> no Wick power with 2> exists as a random distribution, so under the GFF the interaction :\>: cannot even be written down. I thank Ilya Chevyrev and Emilio Ferrucci for discussions clarifying this point. Let *C>> be the space of cylinder functions |)>=g*|)>,\,\|)>|)>> with C>|)>> and \\|)>>. For \|)>> define <\equation> D*F|)>\|\t>*F*+t*h|)>|\|>=\*g*|)>,\,\|)>|)>|f,h|\>=|D*F|)>,h|\>, with \*g|)>*f>, formally =\*F/\*\>. <\lemma> If ,\,X|)>> is a centred Gaussian vector and C>|)>>, then *|]>=\*|]>*\**g|]>>. <\proof> For |u,\|\>>>: |u,X|\>>|]>=|\\>*\\*Y+\|u,X|\>>|]>|\|>=0>=\u*\*|]>*\|u,X|\>>|]>>, which is the claim. The general case follows by Fourier inversion for \|)>> and then by approximation. <\proposition> For all \*C>> and \|)>>, <\equation> \>**F|]>=\>*\*|]>,\*F|\*\>|]>=\*-\|)>*\|]>. <\proof> Under >, *,\|)>,\,\|)>|)>> is centred Gaussian with ***\|)>|]>=|A*h,C*f|\>=|h,f|\>>. Apply Lemma. <\remark> Euclidean field equation> (i) Choosing |)>*\*\|)>> (allowed by approximation, since all moments are finite) gives the hierarchy for the Schwinger functions ,\,x|)>=\*|)>*\*\|)>|]>>: <\equation> -\>|)>*S,\,x|)>=\*-x|)>*S,\,|^>,\,x|)>, which together with =1>, =0> determines all moments (Wick's theorem, Exercise). Compare with : in imaginary time, +m> becomes +m>, and the contact terms on the right come from the CCR, i.e. from time ordering. (ii) For a formal Gibbs measure \\V|)>>\> the same computation gives <\equation> \>**F|]>=\>*-\|)>*\+V|)>|)>|]>, the Euclidean counterpart of the nonlinear field equation (here *\>, so the > of equals > here). Here |)>> must be a well-defined local operator: for >, =4*\:\>:, and the Euclidean field equation for |)>> measures was proved by Dimock and Glimm; see also the Dirichlet form approach of Albeverio and Röckner. For > the renormalised (relativistic) field equation, and with it the DS equations, were first derived by Feldman and Rˇczka, using the constructions of Glimm, Jaffe, Feldman and Osterwalder. The renormalised cubic term is not a random variable under the > measure; a description of it as a random distribution, and a proof of for all \0> in infinite volume, are given in. (iii) The nonlinear term in is a product of distributions at the same point, and the natural way to define it is by point splitting: > is the regular part of |)>*\|)>*\|)>> as \x>. Under the GFF, the Wick decomposition of products (the multi-point version of Section, with |)>*\|)>*\|)>>: the product minus all its contractions) gives <\align*> |)>*\|)>*\|)>>||)>*\|)>*\|)>:>>||+G*-y|)>*\|)>+G*-y|)>*\|)>+G*-y|)>*\|)>,>>>> and the first term tends to :>. This is the beginning of an (OPE): a product of fields at nearby points is a sum of local operators at with coefficients that are singular as the points merge, and the composite operator is what remains after the singular terms are subtracted. With a mollifier the same subtraction reads \lim>>-3*c>*\>|)>>, >=\*\>>. For > in finite volume the measure is absolutely continuous with respect to the GFF and the Gaussian subtraction suffices. In an interacting theory, however, the OPE coefficients are those of the interacting measure. For > the cube needs an additional subtraction >*\>>, with >> of order *log |)>>, so that *\>\\*log |)>> is the second-order (sunset) mass counterterm, i.e. the counterterm of the action reappears as an OPE coefficient of the interacting theory. Composite operators defined by the OPE are Wilson's short-distance expansion and Zimmermann's normal products, in terms of which the field equation of perturbative > reads +m|)>*\=-4*\*N|]>>. Constructively, local products and field equations for |)>> were obtained by point splitting by Schrader; the renormalised cube of is approximated in the spirit of the OPE; convergence of the OPE in perturbation theory is proved in. Lecture describes the OPE at the level of stochastic observables, where the singular coefficients are explicit at every scale. The DS equation is not only satisfied by the GFF; it characterises it. <\key-result> <\theorem> Let > be a probability measure on |)>> such that \L|)>> for all \|)>> and <\equation> \>**F|]>=\>*\*|]>*F\\*C>,h\\|)>. Then =\>. <\proof> Fix a real \|)>> and let \|)>>, so that . For \> apply to the real and imaginary parts of |)>=\s*\>>, which belong to *C>>, using *F=\s*|f,C*f|\>*F>. With \\>|]>> this gives <\equation*> \s*|f,C*f|\>*\=\>\s*\>|]>=-\\, where we used \L|)>> to differentiate under the expectation. Hence =-s*|f,C*f|\>*\> with =1>, so =\|2>*|f,C*f|\>>>. Taking : \\>\\=\*|f,C*f|\>>> for every . Since u*\|)>=\u*f|)>>, this determines the characteristic function of every vector |)>,\,\|)>|)>>, hence all finite-dimensional distributions, hence > on the cylinder >-algebra. So =\>. <\remark> The integrability assumption is the minimal one needed for to make sense, since *> ranges over all >, \|)>>. Only test functions s*\>> were used: DS turns into a first-order linear ODE for the characteristic function. For on the torus no probability measure solves (Exercise). For interacting theories, is a singular nonlinear relation, and uniqueness of its solutions is a hard and largely open problem. In finite dimensions it is easy (Exercise). For a result in this direction for the exponential interaction in see. In Lecture we obtain the GFF as the stationary marginal of the parabolic and elliptic stochastic quantisation equations, and we decompose it along a scale parameter: the field at resolution is a Gaussian martingale, whose terminal value is the GFF. In Lecture interactions are introduced through the Polchinski equation for the effective potential, and through its probabilistic form, a forward\Ubackward stochastic differential equation for the field at increasing resolution. These tools can be used to construct > and >. On > the Wick-ordered interaction *:\:\x> suffices, and the measure satisfies the renormalised DS equation. In Wick ordering no longer suffices. <\exercise> (a) Using =>\*t>p\t>, with > the heat kernel, show that =|)>*|)>*K|)>>, and derive the asymptotics stated after Definition. (b) Check =\>/> for and =\>/|)>> for . (c) Show that on > the difference > is smooth near the diagonal. <\exercise> (a) Show that <\equation*> \\|)>\|)>|\|>+\\>=G*,|)>, and derive from . (b) Show that > is decreasing in and, for 2>, that /G\1> as 0>. Prove the lower bound in Remark(i). (c) Let 3> and suppose =>>> (a scale invariant theory; the scaling dimension >> is not the Laplacian). Using =M*G*>, show that M|)>=c|)>>-d/2>\M> reproduces > for some 0>, for \\>\>, and that this is a locally finite measure iff >\> (for >\>, > is not locally integrable). Using uniqueness of the Stieltjes transform, deduce the unitarity bound >\>, with equality iff \\>, the massless free field. <\exercise> (a) For \0> and |)>> i.i.d. >, show that \a*g>=|)>>. Deduce that a*g\\> a.s. iff a\\>. (b) Conclude that |)>|)>=1> iff -1>. In particular |)>=0> for 2>, and the Cameron\UMartin space > is >-null in every dimension. (c>>) Using hypercontractivity and a Kolmogorov argument, show that \C>> a.s. for every \0>. <\exercise> (a) Show that =n/2>!>*|)>*a>, *H=n*H> and =H+n*c*H>. (b) Show that |)>=H*H|)>>. (c) Show that |)>=!!|)>*H>. Interpret this as the effect of changing the mollifier on >:: Wick powers are defined up to lower order Wick powers with coefficients. <\exercise> (a) Show that >=>*log >+O> for and >\\> for 3>. (b) In , show that >:|]>\log >> for f\0>: the cube is borderline divergent. (c) In , show that >:|]>\log >> as well. <\exercise> In let \*\>>:\\*c>/2>\\*\>>>. Show that <\equation*> \*\*\>>:\*\>>:|\>|]>=\*G,\>*>, and deduce that \*\>>:> converges in |)>> for every if \4*\>, while its second moment diverges for f\0> if \4*\>. What changes for the real exponential *\>>:? (Gaussian multiplicative chaos exists in > for \8*\>, see.) <\exercise> (a) Complete the proof of Lemma. (b) Derive from, justifying the use of polynomial . (c) Deduce Isserlis\UWick: =0> and ,\,x|)>=P>\P>G*-x|)>>. <\exercise> Let =*|x,Q*x|\>+V> on >, with 0> and smooth with > bounded, and x|)>=Z\>\x>. (a) Show that >**F|]>=\>*|\*S,h|\>|]>> for C>>. (b) Let > be a probability measure with *S\L|)>> satisfying the same identity. Show that =\>. the identity says that the distribution \> has zero gradient. (c) Why does this argument fail in infinite dimensions? What replaces it in Theorem? <\exercise> (a) Show that for on > no probability measure on |)>> satisfies with >. take . (b) Show that the massless GFF exists on the quotient |)>/\> (test functions with f=0>), and characterise it by DS there. (c) On >, show that |f,|)>*f|\>\\> iff =0>: the infrared problem of the two-dimensional massless field. <\exercise> Let >> be the centred Gaussian measure on *\/L*\|)>>> with covariance -\>|)>> (with respect to *>), embedded in |)>> via =\*\*f>. Show that >\\> weakly. Relate >> to with (for ; in general see Exercise). <\exercise> (a) Sample the lattice GFF of Exercise on > with =1/N> (i.e. on /N*\|)>> with mass in lattice units) by FFT, for , and check that the empirical variance of > grows like >*log N>. (b) Implement an explicit Euler scheme for the lattice version of in Lecture with (mind the stability constraint *t\\>), and check that the variance of the -th Fourier mode relaxes to > at rate >, with > the eigenvalues of -\>>. (c) Add |)>=4*\*-3*c>*\|)>>, i.e. :\>:, and observe that the solutions have a nontrivial limit as \>, while without the counterterm *c>*\> they converge to zero. <\exercise> In the setting of the example after Remark: (a) Show that >> satisfies (OS0)\U(OS4), and that the theory described there has Schwinger functions equal to the moments of >>. (b) Show that *|)>,\*\*|)>|]>=\+c|)>*\*-|)>>, and that *|)>> is not a linear function of *|)>> and *\*|)>>. (c) Verify **F|]>=\**|]>>. (d) For a single spatial Fourier mode > (on a torus, say, with real test functions ,|)>=g|)>*cos \|)>>), with =|\|>+m>, show that the time kernel of > is |\|>>/2*\-\|\|>>/2*\|)>/-m|)>>, and find supported in \0|}>> with |\*f,P*f|\>\0>. choose with >\*\>g|)>\\=0>. (e) Let be a real polynomial of degree 2>, positive on |)>>. Show that |)>> is not of the form \M|)>*+M|)>> with \0>. \M|)>*|)>> increases to |)>|)>> as \> (monotone convergence), while \0>. . > and chaos inequalities:>. . . ):> reviews; parabolic; singular SPDEs; construction of > via dynamics and variational methods; Euclidean field equations and DS equations for |)>>, for >, and for the exponential model in ; operator product expansion and composite operators; elliptic stochastic quantisation and dimensional reduction. \ <\bib-list|CCHS22> S.Albeverio, F.C.DeVecchi, M.Gubinelli. Elliptic stochastic quantization. 48 (2020), 1693\U1741. S.Albeverio, F.C.DeVecchi, M.Gubinelli. The elliptic stochastic quantization of some two dimensional Euclidean QFTs. 57 (2021), 2372\U2414. S.Albeverio, S.Liang. A note on the renormalized square of free quantum fields in space-time dimension 4>. 131 (2007), 1\U11. S.Albeverio, S.Liang, B.Zegarlinski. Remark on the integration by parts formula for the >-quantum field model. 9 (2006), 149\U154. 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Normal products and the short distance expansion in the perturbation theory of renormalizable interactions. 77 (1973), 570\U601. <\initial> <\collection> <\references> <\collection> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > <\auxiliary> <\collection> <\associate|bib> GV64 Bog98 GV64 GJ87 Nel73a Nel73b Nel73a Nel73a She07 Nel73b GRS75 Sim74 GJ87 Gue06 Sym66 Sym69 Sym69 BFS82 LW04 LJ10 LJ11 Dyn84 Lup16 QW19 Wer16 WP21 Roz82 PU50 LW69 Roz82 Nel73a Haa96 HSw65 BW86 Haa96 Jan97 Nel66 Sim74 GJ87 AL07 PZ32 Jan97 CW01 NP05 AZ96 DG74 AR91 FR77 ALZ06 GH21 Wil69 Zim73 Sch74 GH21 HK12 DGT22 BP25 HRW12 GV64 Bog98 Jan97 She07 LSSW16 BP25 AL07 PZ32 CW01 NP05 AZ96 Sym66 Sym69 Nel73a Nel73b GRS75 GJ87 Sim74 Gue06 Sym69 BFS82 Dyn84 LW04 LJ10 LJ11 Lup16 Wer16 QW19 WP21 DH87 DH88 Gub25 PW81 JLM85 AR91 DPD03 DPZ14 Hai14 GIP15 CC18 MW17 HRW12 GH21 BG20 DG74 AR91 FR77 ALZ06 GH21 DGT22 Wil69 Zim73 Sch74 HK12 PS79 KLP84 ADG20 ADG21 GH19 <\associate|figure> |3.1>||Left: a sample of the massive GFF on |\> (|m=4>, Fourier cutoff |512>, zero mode removed). It is not a function: refining the cutoff adds fluctuations at all scales. Right: the Green function |G> in |d=1,2,3>: bounded, logarithmic and |> singularities at the origin, and exponential decay at infinity.>|> |3.2>||Reflection positivity (Proposition |->|-0.3em|>|0em||0em|>>). For |f> supported in |\0|}>> the covariance between |f> and its mirror image factorises through the time-zero plane: ||\*f,C*f|\>=||)>*>\*\>f|)>\\|\<\|\|\>>\0>.>|> <\associate|toc> |math-font-series||Introduction> |.>>>>|> |math-font-series||3The Gaussian free field> |.>>>>|> |3.1Definition and construction |.>>>>|> > |3.2Reflection positivity and the free field |.>>>>|> > |Example: two masses |.>>>>|> > |Physical meaning of the nonlocal theory |.>>>>|> > |3.3Local operators: Wick products |.>>>>|> > |Beyond |L>: no random distribution |.>>>>|> > |3.4Gaussian integration by parts and the Dyson\USchwinger equation |.>>>>|> > |3.5Outlook |.>>>>|> > |Exercises |.>>>>|> > |math-font-series||References and further reading> |.>>>>|> |Guide |.>>>>|> >