> <\body> 2 \V notes: Lecture2, Ÿ3> |<\hidden> \; Positivity of the energy > imaginary time; vacuum expectations are moments of \|)>*`*`="*Z\*\|\|>+*m*\+V|)>|]>>\*\>; relativistic QFTs > measures on |)>> satisfying the OS axioms. The Gaussian part, rigorously. <\enumerate> Construction on > and >; regularity. Reflection positivity; the free field; Källén\ULehmann again. Local operators: Wick powers. Gaussian integration by parts: the Dyson\USchwinger (DS) equation. |<\hidden> \; 2> \; |<\hidden> \; =\=/L*\|)>> or >;|)>> smooth, resp. Schwartz, functions;|)>> with the cylinder ( Borel) >-algebra. <\equation*> A\m-\,C\A,m\0. , are continuous bijections of |)>> (Fourier multipliers +m|)>1>>), and |C|\<\|\|\>>\L>\m>. <\definition*> The > is the centred Gaussian measure on |)>> with <\equation*> \*\*\\|)>=|f,C*g|\>,\\>\\|)>=\*|f,C*f|\>>. Formally \|)>*`*`="*Z\*|\,A*\|\>>\*\>: the Gaussian part of the Euclidean action. |<\hidden> \; |||||||| has kernel : <\equation*> G=k\x>|+m>*k||)>> on >, and \>G*> on >. <\itemize> : =\>/>. : \>*log >>. 3>: \*|4*\>>. Decay \>> at infinity. 2>>: the field is not a function. |<\cell> |gff2d.png>|0.350par|||> A sample of the massive GFF on > (, > Fourier modes). >>>> |<\hidden> \; <\theorem*> Let |)>> be a real orthonormal basis of eigenfunctions of > on >, *e=\*e>, =m+\>, and |)>> i.i.d. >. For -1>, <\equation*> \=\*g*e converges a.s. and in ;H|)>>, and |)>=\>. <\equation*> \|\|\<\|\|\>>>=|)>*\>\\>|)>\\\2*s+2\d. >-bounded >-valued martingale > convergence. And =\*g*|e,f|\>> is Gaussian with variance \*|e,f|\>=|f,C*f|\>>. |<\hidden> > and regularity> \; <\myframed> :\|)>\\> continuous, positive definite, =1> > there is a unique probability measure on |)>> with characteristic functional >. For =\*|f,C*f|\>>>: continuity from |f,C*f|\>\m|f|\<\|\|\>>>>; positive definiteness from *-f|)>=a*a\|f,C*f|\>>> and the Schur product theorem. (sharp, Exercise3.3): >|>|>>||>>, \\0>; not a function>|>>>>>>> The Cameron\UMartin space > (finite action) has . |<\hidden> \; 2> \; |<\hidden> \; =\\N>, > or >; *,|)>=,|)>>; >+B>, -\>, =B>. <\proposition*> (a) ,\|)>=\-\|\|>*\>|)>> (operator-valued kernel).(b) If f\\0|}>>: <\equation*> |\*f,C*f|\>=||)>>\*\>f,\|)>\\|\<\|\|\>>>\0. (c) > on > satisfies OS0\UOS4. (a) \k*t>+w|)>*k|2*\>=>|2*w>> and spectral calculus. (b) For \0\\>, -\|\|>=|\|>+\>: the kernel factorises. (c) OS3: |^>*-\*f|)>=a*a\|\*f,C*f|\>>>; Schur. |<\hidden> \; <\itemize> OS reconstruction of > gives the >: Fock space over >, \|)>> (Nelson's free Markov field). The |)>> exists and is Gaussian with covariance |)>>: the ground state measure *\*\> of the free Hamiltonian in the Schrödinger representation. The GFF is the continuum limit of the lattice spin system of Lecture1 with (Exercise3.10). Markov property (Nelson): rooted in the >. |<\hidden> \; <\itemize> for >: inside and outside are conditionally independent given the field near *\>. >> (Nelson 1973): \||]>=\**f|)>>, and *f=A*u> is supported on *\> -\> is local>. Given the outside: harmonic extension independent Dirichlet GFF (next slide). sharp > > RP:> with > the time-zero field, > >-invariant, > trivial on >: <\equation*> \**F|\>*F|]>=\\|\|]>|\|>|]>\0,\\\|\|]>=\L|)>. Nelson's reconstruction (1973), before Osterwalder\USchrader. preserved by local perturbations >V>> > |)>> as classical statistical mechanics (Guerra\URosen\USimon). Not stable under limits; OS use RP instead: not comparable (generalised free fields are RP, not Markov; the > field is Markov, not RP), but closed under weak limits. |<\hidden> \; |markov_gff.png>|0.86par|||> Lattice GFF, \10>, > a disc. Inside >, <\equation*> \=h+\,-\|)>*h=0*\,h=\*\*\, \||]>> depends only on the field ; > is a Dirichlet GFF, . |<\hidden> \; <\itemize> Symanzik (1966, 1969): QFT through Schwinger functions as moments of a classical Markov field \V the Euclidean programme. =>\*t>p\t> is the Green function of a killed Brownian motion. For V>\>: a gas of paths and loops interacting through local times; for >, intersections are penalised (lattice: Brydges\UFröhlich\USpencer 1982; triviality for 4>). (Lawler\UWerner 2004): Poisson cloud of Brownian loops of intensity >; on a graph \|\,L|\>>=det |)>>>. for => the occupation field is *\>; in the renormalised one is :\>:. Lupu 2016: loop clusters sign clusters of the cable-graph GFF. Werner 2016: spatial Markov property of loop soups. Qian\UWerner 2019. Survey: Werner\UPowell. |<\hidden> \; Wick rotation of >>: \\|)>\|)>|\|>+\\>=G*,|)>>. Hence for any theory satisfying the axioms <\equation*> S=v+|)>>\M|)>*G,=M|)>|+M>. The GFF is =\>>. <\itemize> > decreases in and /G\1> at (2>), so 0> S/G\\|)>|)>\0>: never less singular than free. Scale invariance > unitarity bound \> (Exercise3.2). \\,\|)>> > decay at least like >>: mass gap clustering. |<\hidden> \; =c*\>+c*\>>, m\m>:>=c*-\|)>+c*-\|)>>, the law of *\+c*\> (independent GFFs). RP for each term > OS axioms. <\myframed> =\>|)>\\>|)>>,=c*\>\+c*\\>>,=c*W|)>>+c*W|)>>>.Two kinds of free particles, created by the same field; no scattering (truncated 3>=0>). <\itemize> +m|)>*+m|)>*\=0>, but no Klein\UGordon equation; ,\|)>> are not complete Cauchy data. DS: **F|]>=\**>*h|)>|]>> with >=-\|)>*-\|)>|c*-\|)>+c*-\|)>>> ; > is not Markov. |<\hidden> \; The fourth-order action *|\,-\|)>*-\|)>*\|\>> gives a Markov field with <\equation*> +m|)>*+m|)>>=-m>*+m>-+m>|)>, so =>-\>|)>/-m|)>> has a . <\itemize> By Euclidean Källén\ULehmann it is : the mass > is a ghost of negative norm (Pais\UUhlenbeck, Lee\UWick). Same for any => polynomial of degree 2>: \0>, while M|)>|s+M>*\*\|)>|)>>. <\myframed> For Gaussian fields, (OS) and of the Euclidean field equation (> a differential operator) together single out the free field: covariance -\|)>> (or white noise). |<\hidden> \; <\itemize> all Wightman axioms hold, including locality (W3); positive energy, unique vacuum. Two free species: trivial and redundant, not inconsistent. no Markov property > > alone is not a complete set of local degrees of freedom (no Cauchy problem, no Lagrangian in > only). Cure: the pair ,\|)>> is local and Markov. >> (a continuum of free species) is unphysical: no dynamical law of finite order, and too many localised states. Buchholz\UWichmann nuclearity (cf. Haag\USwieca) needs n\*m>\\>: no sensible thermodynamics. also have a continuum in >, but produced by a interaction: the Euclidean field equation of >, with -\|)>*\+4*\:\>: in the DS equation, is local. |<\hidden> \; 2> \; |<\hidden> \; Interactions V|)>\x> need local functions of >: regularise, subtract, remove the regularisation. <\equation*> \/2>=0>|n!>*H:H=a-c,H=a-3*c*a,H=a-6*c*a+3*c. <\lemma*> > centred Gaussian, *X=a>, *Y=b>: <\equation*> \**H|]>=\*n!*|)>. *a/2>\*b/2>|]>=\\*>>; compare coefficients of *t>. |<\hidden> \; Mollifier >>, >=\>\\>, ,\>=\>\\>\G>, >=G,\>>, <\equation*> :\>:\H>;c>|)>,:\>:=f:\>:. <\theorem*> If \d> (all in ; 2> in ), then :=lim\0>:\>:> exists in |)>>, is independent of >, and <\equation*> \*::\:|]>=\*n!f*G**g\x\y. If \d> and f\0>: >:|]>\\>. |<\hidden> \; =1+log >> (), > (3>). With ,\|)>>: <\itemize> \4*r>: \ ,\>\sup\/2> G\C*K>; \4*r>: \ ,\>\|\|\<\|\|\>>>*r*>G\C*|\|>>*>K\C*K\C*K>. By the lemma, <\equation*> \>:-:\>:|)>|]>=n!f*f*,\>-2*G,\>+G,\>|)>*\0 by dominated convergence: \L>\n*\d>. Otherwise \>\L>>; Fatou. |<\hidden> \; <\itemize> >> has no limit; subtracting >\\> (via >) is the simplest counterterm. A different mollifier changes >: by lower Wick powers with coefficients (Exercise3.4). :> lies in the -th Wiener chaos; |)>\> Fock space over |)>>. Hypercontractivity |X|\<\|\|\>>>\|X|\<\|\|\>>>> gives convergence in all > (and in >> for ). >:> on >, :\:\L> and (Nelson) *:\:>\L|)>>. >*\>; in the Wick exponentials \*\>>:, in > iff \4*\> (sine-Gordon; Exercise3.6). |<\hidden> \; 2> \; |<\hidden> >: Wick powers do not exist> \; <\itemize> If \d>: >:|]>\\>. Worse: there is at all, in any topology or on any space. Albeverio\ULiang 2007: :>, 0>, in (and 5>) is a Hida distribution, not a random variable. >=:\>:\\>, >=|X>|\<\|\|\>>\\>, *X>\9*\>>. Paley\UZygmund, *\*\*Z|)>\|)>**Z|)>/\*Z>, with >>: <\equation*> \*>|\|>\\>|)>\9\ =|X|\<\|\|\>>>.> Upper tails |X|\<\|\|\>>\*\> (hypercontractivity); *\*\|)>\c> (Paley\UZygmund); *\t*\|)>\C*n*t> (Carbery\UWright). So: tight > bounded in >; in probability > in >. Renormalising by >> typically gives Gaussian limits (fourth moment theorem; Albeverio\UZhou 1996): the -th power is lost. |<\hidden> \; Cylinder functions *C>\F|)>=g*|)>,\,\|)>|)>>, C>>: <\equation*> D*F|)>=|\t>*F*+t*h|)>|\|>=\*g|)>|f,h|\>=|D*F,h|\>,D*F=*F|\*\>. <\lemma*> ,\,X|)>> centred Gaussian:*g|]>=\*|]>*\**g|]>>. For |u,\|\>>>: |u,X|\>>|]>=\>*\\*Y+\|u,X|\>>|]>|\|>=0>=\u*\*|]>*\|u,X|\>>|]>>; then Fourier inversion and approximation. |<\hidden> \; <\proposition*> For all \*C>>, \|)>>: <\equation*> \>**F|]>=\>*\*|]>,\*F|\*\>|]>=\*-\|)>*\|]>. ***\|)>|]>=|A*h,C*f|\>=|h,f|\>>; apply the lemma. With |)>*\*\|)>> one gets the hierarchy <\equation*> -\>|)>*S,\,x|)>=\*-x|)>*S|^>\|)>, which with =1>, =0> gives Wick's theorem (Exercise3.7). |<\hidden> Euclidean field equation> \; <\itemize> Minkowski: +m|)>*\=0> and CCR.Euclidean: +m|)>*\> \P\Q /\*\> inside expectations; the contact terms come from the CCR (time ordering). For a formal Gibbs measure \\V|)>>\>: <\equation*> \>**F|]>=\>*-\|)>*\+V|)>|)>|]>, the Euclidean version of +m|)>*\+V|)>=0>. |)>> must be a well-defined local operator: >: =4*\:\>:; field equation for |)>>: Dimock\UGlimm 1974, Albeverio\URöckner 1991. For > the renormalised field equation goes back to Feldman\UR¡czka 1977; renormalised cubic term as a random distribution: Gubinelli\UHofmanová 2021. |<\hidden> |)>>: point splitting and the OPE> \; <\itemize> >: product at one point. : regular part of |)>*\|)>*\|)>> as \x>. Under the GFF (Wick): <\equation*> \|)>*\|)>*\|)>=:\|)>*\|)>*\|)>:+j>G*-y|)>*\|)> regular part :\:>; mollified: \lim>>-3*c>*\>|)>>. (Wilson 1969, Zimmermann 1973): products at nearby points local operators > singular coefficients; composite operator regular part. Field equation +m|)>*\=-4*\*N|]>>. The coefficients are those of the theory. >: Gaussian subtraction suffices (Schrader 1974). >: an extra >*\>>, >\\*log |)>>: *\>> is the sunset mass counterterm (Gubinelli\UHofmanová 2021). Lecture3: OPE of stochastic observables, explicit at every scale. |<\hidden> \; <\theorem*> Let > be a probability measure on |)>> with \L|)>> for all and <\equation*> \>**F|]>=\>*\*|]>\F\\*C>,h\\|)>. Then =\>. Fix , take \|)>> and =\s*\>>, so *F=\s*|f,C*f|\>*F>. With =\>*F>: <\equation*> \s*|f,C*f|\>*\=\>\s*\>|]>=-\\\\=\|2>*|f,C*f|\>>. So |^>=\*|f,C*f|\>>> for all ; linearity of \> and Cramér\UWold give all finite-dimensional laws. |<\hidden> \; <\itemize> The integrability assumption is exactly what is needed for DS to make sense: *> ranges over all >. DS for characteristic functions is a . on the torus: probability measure solves DS (take ); work modulo constants (Exercise3.9). Finite dimensions: DS says \|)>=0>, so uniqueness is easy for any (Exercise3.8). Interacting theories: DS is a singular nonlinear relation; uniqueness is hard and largely open. Results for the exponential model in : De Vecchi\UGubinelli\UTurra. |<\hidden> \; <\itemize> Lecture3: stochastic quantisation; the GFF along a scale parameter; stochastic observables, products and OPE. Lecture4: Polchinski equation and its FBSDE; Wetterich equation; variational structure. Lecture5: observables; the field equation at every resolution and its sub-Gaussian noise. Lectures6\U7: a construction of > on the torus via the FBSDE. \; <\myframed> Gaussian measure |>> renormalised DS equation / scale dynamics |>> Euclidean QFT |>> relativistic QFT. |<\hidden> \; |||||||| (numbering of the notes) <\itemize> 3.2 Unitarity bound. 3.3 Regularity is sharp. 3.6 Wick exponentials, \4*\>. 3.7 Wick's theorem from DS. 3.8 DS in finite dimensions. 3.11 Computer experiments. 3.12 Two masses: positivity vs. locality. |<\cell> <\itemize> Sheffield, , PTRF 2007. Janson, . Glimm\UJaffe; Simon, |)>>. >>>> > <\initial> <\collection> >>