> <\body> 1 \V notes: Lecture1, ŸŸ1\U2> |<\hidden> \; <\itemize> construct, with full rigour, relativistic quantum fields, i.e. models satisfying axioms encoding quantum mechanics, special relativity and causality. <\itemize> : |)>>, Yukawa>, sine-Gordon, ... : >, ... : no interacting example; critical lattice > has Gaussian scaling limits (Aizenman\UDuminil-Copin 2021). work in . A QFT becomes a probability measure on |)>>, which looks like the continuum limit of a classical spin system at a critical point. (this course): stochastic quantisation, singular SPDEs, variational methods; RG and cluster expansions as complements. |<\hidden> \; |||||||| <\enumerate> Wightman axioms Why operator-valued distributions Källén\ULehmann representation The quantum wave equation |<\cell> <\enumerate> A lattice Hamiltonian on a circle Positivity of energy > Wick rotation Feynman\UKac: quantum > classical Statistical mechanics and criticality Osterwalder\USchrader axioms >>>> Conventions: =c=1>, spacetime dimension , |f,g|\>=>g>, =\k\x>f\x>. |<\hidden> \; 1> \; |<\hidden> \; |||||||| <\itemize> Minkowski space >, ,|)>>, y=x*y-\>. if \0>. Forward cone +>=\|\|>|}>>, closure >+>>. Restricted Lorentz group >>; Poincaré group >=\\\>> acting by \*x+a>. Wigner: a relativistic quantum system carries a (projective) unitary representation of >>. |<\cell> \\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} \\definecolor{navy}{HTML}{1B2A41} \\definecolor{accent}{HTML}{B5402E} \\begin{document} \\begin{tikzpicture}[scale=0.95,\>=Stealth] \ \ \\fill[navy!10] (0,0) -- (2,2) -- (-2,2) -- cycle; \ \ \\draw[-\>] (-2.3,0) -- (2.4,0) node[right] {{}$\\mathbf{x}$}; \ \ \\draw[-\>] (0,-2.2) -- (0,2.4) node[above] {{}$x^0$}; \ \ \\draw[dashed,navy] (-2,-2) -- (2,2); \ \ \\draw[dashed,navy] (2,-2) -- (-2,2); \ \ \\node[navy] at (0,1.45) {{}$V_{\\!+}$}; \ \ \\fill[accent!70] (-1.6,0.15) ellipse (0.35 and 0.18); \ \ \\fill[accent!70] (1.55,-0.1) ellipse (0.35 and 0.18); \ \ \\node[accent] at (-1.6,-0.35) {{}$\\operatorname{supp} f$}; \ \ \\node[accent] at (1.55,0.35) {{}$\\operatorname{supp} g$}; \\end{tikzpicture} \\end{document} |ps>|0.350par|||>>> >>>> |<\hidden> \; <\myframed> Separable Hilbert space >; strongly continuous unitary representation of >>; |)>=\P\a>>, >. \>+>>, so |\|>\0>. unique (up to phase) unit vector > with |)>*\=\>. <\myframed> Dense \\>; for \|)>> an operator > on >, *\\\>, \\>; |\,\*\|\>> tempered; |\,\*\|\>=|\>|)>*\,\|\>>. |<\hidden> \; <\myframed> |)>*\*U|)>=\*|)>|)>>;,\|]>=0> if supports are spacelike separated;polynomials in > applied to > are dense. Formally =\*f\x>, *,|)>=\H*x>\*|)>\H*x>>. ,\,f|)>=|\,\|)>*\*\|)>*\|\>\\|)>>: <\itemize> positivity W*>\g|)>\0>;Poincaré invariance; spectral condition: Fourier support in >+>n-1>> (difference variables); locality; cluster property > unique vacuum. |<\hidden> \; <\theorem*> A sequence |)>> of tempered distributions with =1> satisfying positivity, invariance, spectral condition and locality is the sequence of Wightman functions of a Wightman theory, unique up to unitary equivalence. <\itemize> To construct a QFT it suffices to construct its Wightman functions. positivity involves all > at once and refers to real time, while what we can construct lives in imaginary time. The Euclidean approach replaces it by , checkable on lattice approximations and stable under limits. The axioms say nothing about dynamics: no Lagrangian; the interaction is hidden in the higher >. |<\hidden> \; |)>=|\|>+m|)>>,\|)>=||)>*2*\|)>>> (invariant measure on ==m,p\0|}>>), Fock space over \|)>>: <\equation*> \=\\|)>*|)>\p\x>+a>|)>\p\x>|]>=\|)>>, with |)>,a>|)>|]>=|)>*2*\|)>*\*-|)>>, |)>*\=0>. Then <\equation*> W*=\\|)>\p\>,|\*\|\<\|\|\>>=\\|\|>\\, but <\equation*> |\*\|\<\|\|\>>=W=\\=+\*1|)>. UV divergence: logarithmic for , power-like for 2>. |<\hidden> \; <\proposition*> Assume (W0)\U(W2) and *\=f*U|)>*\\x> for some \\> (\P*\=\> exists\Q). Then \\*\>, hence *\=cf|)>*\>. <\lemma*> For 2>, a >>-invariant measure on >+>> is \>. <\itemize> Combined with locality and Reeh\USchlieder: > is a multiple of >. Quantum fields are : nonlinear expressions in > need a definition > renormalisation. |<\hidden> \; A boost of rapidity > acts as \p\\\>\p|)>>. The sets <\equation*> S>=>+>:2\p\p\2|}>,j\\, are mapped to each other by the boost with >=2>; they are disjoint and cover >+>\>. Invariance and finiteness force >|)>=0>. <\enumerate> Covariance gives f*U**x|)>*|)>*\-\|)>\x=0> for all , so |)>*\=\>. The spectral measure > of > for translations is finite, supported in >+>> and Lorentz invariant, so =|\|\<\|\|\>>*\>. Hence *\=\> for all , and \\*\> by uniqueness of the vacuum. |<\hidden> \; Let >> be the free two-point function of mass 0> and |\,\*\|\>=v*f>. <\theorem*> Under (W0)\U(W2) there is a positive, polynomially bounded measure > on |)>> (the of the field) such that <\equation*> W*=v+|)>>\M|)>*W>*. (|)>=0> if .) =\*\-vf|)>*\> gives -v> of positive type > (Bochner\USchwartz) Fourier transform of a positive measure >, supported in >+>>, |)>=0>, Lorentz invariant. Orbits are the hyperboloids > (and the cone for ), each with a unique invariant measure \>. Disintegrate along . |<\hidden> >> \; <\itemize> =\>>. Conversely =\>>, 0>, forces -v> to be free, by locality (Jost\USchroer). =c*\>> have =c*\>> and no interaction: \\>> is necessary, not sufficient. an isolated atom >> is a particle of mass ; multi-particle continuum from > (or > with \-\> symmetry). Weakly coupled |)>>: Glimm\UJaffe\USpencer 1974. =\M|)>*\\=\>. CCR at sharp times > \=1>, so 1>. In perturbatively \=\> (wave-function renormalisation). |<\hidden> \; The free field solves, as an operator-valued distribution, <\equation*> +m|)>*\=0,\=\-\>, and its sharp-time fields *>, *>, =\*\>, satisfy the CCR <\equation*> *|)>,\*|)>|]>=\\*-|)>,,\|]>=,\|]>=0. <\myframed> A *|)>,\|)>|)>>, \Pdistributed\Q according to the vacuum. Interacting version: +m|)>*\+\*\=0> together with the CCR. |<\hidden> \; , space =\/L*\>, momenta |L>*\>, =+m|)>>: <\equation*> \*=\k*x>+a>\k*x>|>>,,a>>|]>=\>,H=\*a>*a. <\itemize> An infinite family of independent harmonic oscillators, one per mode. With a UV cutoff \R>: <\equation*> |\,\**\|\>=\R>>=>*log +O\\. Wick ordering \\-|\,\*\|\>> removes exactly this divergence (Exercise1.7). |<\hidden> \; <\enumerate> > is meaningless and must be renormalised: Wick ordering suffices in ; more counterterms in ; triviality in . In infinite volume the interacting field at fixed time is not unitarily equivalent to the free one: there is no interaction picture. Even on >, +\*:\*:\x> is positive plus unbounded below. Nelson 1966: bounded below (hypercontractivity). Volume cutoff removed by Glimm\UJaffe; Wightman axioms for weak coupling |)>> by Glimm\UJaffe\USpencer 1974. The Euclidean approach (Symanzik, Nelson, Osterwalder\USchrader, Guerra\URosen\USimon) simplifies this enormously. |<\hidden> \; 1> \; |<\hidden> \; Lattice >=\*\/L*\>, > sites, >=L*>>,\\|)>>, > multiplication, =-\\*\>>, so ,\|]>=\\*>. <\equation*> H>=\*\+>*\|)>+m*\+V|)>|]>=->*\>+U>|)>. <\itemize> Heisenberg fields solve the <\equation*> \*\=>*\|)>-m*\-V|)> \V the quantum wave equation, now rigorous. Schrödinger operator with confining potential: self-adjoint, compact resolvent, >>> trace class with positive kernel. Perron\UFrobenius: simple ground state >\0>, energy >. >\H>-E\0>. |<\hidden> > Wick rotation> \; |||||||| For bounded > and =\t*H>F\t*H>>: <\equation*> W,\,t|)>=|\>,F\-t|)>*>>F*\*F*\>|\>. <\proposition*> extends analytically to -t|)>\0>. At =-\\>, \\\\>, it gives the <\equation*> S,\,\|)>=|\>,F\-\|)>*>>F*\*F*\>|\>, which determine . |<\cell> \\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} \\definecolor{navy}{HTML}{1B2A41} \\definecolor{accent}{HTML}{B5402E} \\begin{document} \\begin{tikzpicture}[scale=0.9,\>=Stealth] \ \ \\fill[navy!10] (-2,0) rectangle (2,-2); \ \ \\draw[-\>] (-2.2,0) -- (2.3,0) node[right] {{}$\\operatorname{Re}\\zeta$}; \ \ \\draw[-\>] (0,-2.2) -- (0,1.2) node[above] {{}$\\operatorname{Im}\\zeta$}; \ \ \\draw[very thick,navy] (0.15,0) -- (1.9,0); \ \ \\node[navy,above] at (1.1,0.05) {{}real time}; \ \ \\draw[very thick,accent] (0,-0.15) -- (0,-1.9); \ \ \\node[accent,anchor=west] at (0.1,-1.85) {{}$\\zeta=-\\mathrm{i}\\tau$}; \ \ \\draw[-\>,thick,gray] (1.3,-0.1) arc (0:-80:1.3); \ \ \\node[navy] at (-1.15,-1.6) {{}$H\\geq0$}; \\end{tikzpicture} \\end{document} |ps>|0.355par|||>>> >>>> |<\hidden> \; <\enumerate> \*>>=\\*\>\E|)>> is a contraction for \\0>, since \0>: analytic in the interior, continuous up to the boundary. An analytic function on the half plane vanishing on \> vanishes; iterate variable by variable. *>>> is a contraction semigroup, which can be represented as a path integral (Feynman\UKac). >>\|\>|\|\|>> as \>. Relativistic version: Wightman functions are boundary values of functions analytic in the forward tube; Lorentz covariance extends them to all non-coincident Euclidean points (Schwinger functions are and ). |<\hidden> <\with|font-size|0.9> \; Kinetic semigroup: |2*\>*\>,\|)>=*\/\|)>*exp |2*\>*-\|\|>|)>>. (symmetric Trotter step): <\equation*> T>=\*U/2>\|2*\>*\>\*U/2>,T>*,\|)>\exp |2*\>*-\|)>-|2>*|)>+U|)>|)>|)>, and >|\/\|\>>\\*H>>> (Trotter). Space-time lattice =*\/T*\|)>\*\/L*\|)>>, >: <\equation*> >|)>\\,\>|)>>\>\\, <\equation*> S,\>|)>=,x|)>>\*\*>>*\|)>+>*\|)>+m*\+V|)>|]>. |<\hidden> \; Lattice Gibbs measure ,\>\|)>=Z\,\>|)>>\\>: <\equation*> *T>>*F*\*F*T>>|)>|>|)>>=F>|)>*\*F>|)>*\,\>\|)>. <\proposition*> <\equation*> |\>,F\-\|)>*>>F*\*F*\>|\>=lim\> lim\0> F>|)>*\*F>|)>\\,\>. Gaussian factor Brownian loops with diffusion >; *>U>|)>> is a Riemann sum for U>|)>\\>; then \> uses the spectral gap. |<\hidden> \; |||||||| *\+*m*q>,\\/2>>,=>. <\itemize> Ground-state transform: *>*\=-\+m*q*\>, minus the generator of X=-m*X\t+\B>. Schwinger function |\,q\*>>q*\|\>=>|2*m>>. This is >+m|)>>: the >. |<\cell> |ou1d.png>|0.460par|||> A sample of the stationary OU process, . >>>> 2.2>>> |<\hidden> \; \\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} \\definecolor{navy}{HTML}{1B2A41} \\definecolor{accent}{HTML}{B5402E} \\begin{document} \\begin{tikzpicture}[scale=0.48,\>=Stealth] \ \ \\foreach \\j in {0,...,6} \\draw[gray!55] (-0.5,\\j) -- (9.5,\\j); \ \ \\foreach \\i in {0,...,9} \\draw[gray!55] (\\i,-0.5) -- (\\i,6.5); \ \ \\foreach \\i in {0,...,9} \\foreach \\j in {0,...,6} \\fill[black!80] (\\i,\\j) circle (2.2pt); \ \ \\draw[very thick,navy] (-0.5,5) -- (9.5,5); \ \ \\node[navy,anchor=west] at (9.9,5) {{}time slice $\\varphi_\\tau=(\\varphi_{\\tau,x})_{x}$}; \ \ \\draw[thick,dashed,accent] (-0.9,3) -- (9.9,3); \ \ \\node[accent,anchor=west] at (9.9,3) {{}$\\Theta$: $\\tau\\mapsto-\\tau$}; \ \ \\draw[decorate,decoration={brace,amplitude=4pt,mirror}] (9.8,0) -- (9.8,1) \ \ \ \ \ node[midway,right=5pt] {{}$T_\\varepsilon\\approx \\mathrm{e}^{-\\varepsilon H_\\varepsilon}$}; \ \ \\draw[-\>] (-1.6,-0.5) -- (-1.6,6.7) node[above] {{}$\\tau$}; \ \ \\draw[-\>] (-0.5,-1.3) -- (9.9,-1.3) node[right] {{}$x\\in\\varepsilon\\mathbb{Z}/L\\mathbb{Z}$}; \\end{tikzpicture} \\end{document} |ps>|0.683par|||>>> Rows are quantum configurations at fixed imaginary time; the transfer matrix connects consecutive rows. As \>, the rows form a stationary Markov process whose one-time marginal is >|)>\\>. |<\hidden> \; Isotropic lattice =\>: <\equation*> S>|)>=|u,v|\>>-\|)>+\*m*\+V|)>|]>. <\itemize> >\|u,v|\>>\*-\|)>>\>\|)>>, >a|)>=\**m*a+V|)>>\a>. A with unbounded spins: short-range classical statistical mechanics. In dimension : *|u,v|\>>> and *>. Nearest-neighbour > : the Euclidean shadow of *H>\0>. Conversely, RP > Hilbert space and 0> (Exercises2.5, 2.7). |<\hidden> \; ||>|0>>||>|*H>>>|>>|>|>>||>|>>||>|-E>>||>||\0>>|>|||>>>>> |<\hidden> critical point> \; <\itemize> In lattice units the mass is *m\0>: correlation length \*m|)>\\> lattice spacings. Continuum QFT (Wilson). Conversely, RP critical models with nontrivial scaling limits give candidate QFTs. Griffiths\USimon: > spins as limits of normalised sums of ferromagnetically coupled Ising spins; critical 4D Ising and > have Gaussian scaling limits (ADC 2021). <\myframed> For *a> and fixed |)>> the limit is trivial: >=\|]>=>*log *m>+O\\>. Take >=\*-6*c>*a|)>>, i.e. :a>: up to a constant. Enough in ; not in . |<\hidden> \; Formally, as \0> and \>: <\equation*> \\|)>*`*`="*Z*exp >*\|\|>+m*\+V|)>|]>\x|)>*\*\. <\itemize> Neither *\> nor the action make sense: typical > are distributions. The Gaussian part defines the > (next lecture). Then \\V>|)>>\>: meaningful in finite volume for (and, with more work, ). Which measures on |)>> correspond to relativistic QFTs? |<\shown> \; |^>=\\>\\|)>>,*,|)>=,|)>>. <\description> |^>z*f|)>> entire. |^>|\|>\exp |f|\<\|\|\>>>+|f|\<\|\|\>>>|)>|)>>, >, complex . for real > supported in \0|}>>, |^>*-\*f|)>|)>\0>. under time translations. <\theorem*> OS0\UOS3 > the moments of > are the Schwinger functions of a unique Wightman theory; OS4 > unique vacuum. |<\hidden> \; <\itemize> => functions of |\|>\0|}>>>, with =\>*F|\>G|]>\0> by OS3. Quotient by null vectors and complete: physical Hilbert space >. Time translations by 0> preserve > and descend to a self-adjoint contraction semigroup >: . Vacuum class of ; analytic continuation \\t> gives the Wightman functions. <\myframed> (1) regularise (lattice, torus); (2) renormalise; (3) uniform estimates, remove cutoffs; (4) verify OS axioms (RP survives limits, Euclidean invariance recovered, regularity, clustering); (5) prove non-Gaussianity. |<\hidden> \; |||||||| (numbering of the notes) <\itemize> 1.3 Which restrictions of the free field make sense? 1.5\U1.6 Canonical sum rule; generalised free fields. 2.2 Harmonic oscillator and Ornstein\UUhlenbeck. 2.4 Correlation length and criticality. 2.5, 2.7 Reflection positivity on the lattice; toy OS reconstruction. |<\cell> <\itemize> Streater\UWightman, . Glimm\UJaffe, , Euclidean chapters. Simon, . Osterwalder\USchrader, CMP 31 (1973), 42 (1975). >>>> the Gaussian free field. > <\initial> <\collection> >>