Quantum Fields and Probability Antti Kupiainen September 3, 2018 Contents 1 Introduction5 1.1 Background . .5 1.2 Overview . .5 2 Quantum Fields7 2.1 Examples . .7 2.1.1 Harmonic oscillator . .7 2.1.2 Phonon field theory - an infrared divergence . 10 2.1.3 Vibrating string - an ultraviolet divergence . 13 2.2 Second quantization . 15 2.2.1 Examples . 19 2.2.2 Free scalar field . 20 2.3 Interacting Fields . 22 3 Euclidean Quantum Field Theory 25 3.1 Wiener Measure . 26 3.2 Feynman-Kac formula . 27 3.3 Harmonic oscillator - Ornstein-Uhlenbeck process . 28 3.4 The free field . 29 4 Random Fields 31 4.1 The Ginzburg-Landau model . 31 4.2 Gaussian Integrals . 32 4.3 The Gaussian Ginzburg-Landau model . 33 4.4 Measures on s0 .................................. 35 4.4.1 Bochner's Theorem . 35 4.4.2 Cylinder measures . 36 4.4.3 Minlos Theorem . 37 4.5 Measures on S0 .................................. 41 4.6 Gaussian measures . 44 4.6.1 The Euclidean Free Feld . 44 4.6.2 Wick Formula . 45 5 From Random Fields to Quantum Fields 48 5.1 Reflection positivity . 48 5.2 Reflection positive Gaussian measures . 50 5.2.1 Example: harmonic oscillator and free field . 50 5.3 Reconstruction for Free Field . 51 5.3.1 Time zero fields . 51 5.3.2 Harmonic oscillator once more . 51 5.3.3 Free field . 53 5.3.4 Fock space . 53 2 6 Perturbation Theory 55 6.1 Regularization . 55 6.1.1 Momentum space regularization . 55 6.1.2 Scaling limit . 56 6.1.3 Lattice Regularization . 57 6.2 Gaussian critical point . 57 6.2.1 Massless case . 58 6.2.2 Massive case . 59 6.3 Perturbation Theory for the GL model . 60 6.3.1 Measures . 60 6.3.2 Perturbation Theory . 61 6.3.3 Feynman graphs . 61 6.3.4 Momentum space representation . 67 6.4 Infrared and Ultraviolet Divergences . 69 6.4.1 Infrared divergences . 70 6.4.2 Ultraviolet . 72 7 The renormalization group 76 7.1 The Block-spin Transformation . 76 7.1.1 Gaussian fixed point . 78 7.2 Transformations on Measures . 78 7.3 Transformations on Hamiltonians . 79 7.4 Fixed Points of RL ................................ 81 7.5 Linear Analysis around a Fixed Point: Critical Exponents . 83 7.6 Effective Field Theories . 84 7.7 Scaling limit . 86 7.8 Non-Scale Invariant Theories . 87 7.9 The Gaussian Fixed Point . 89 7.9.1 Block spin fixed point . 89 7.9.2 RG in Momentum Space . 90 8 Hierarchical model 93 8.1 Local RG transformation . 93 8.2 Linearization . 94 8.3 Perturbative analysis of the hierarchical RG . 95 8.4 d > 4: Gaussian critical point . 96 8.5 d = 4: Infrared asymptotic freedom . 97 8.5.1 Susceptibility divergence . 98 8.6 d < 4: Wilson-Fisher fixed point . 99 8.7 d < 4: Superrenormalizable QFT . 99 8.7.1 d =2 ................................... 101 8.7.2 d =3 ................................... 101 8.7.3 Infinite Volume Limit . 102 8.7.4 d " 4.................................... 102 8.7.5 d = 4: Triviality . 103 8.8 Nonperturbative analysis . 103 9 RG analysis of the Ginzburg Landau Model 107 9.1 Linear RG . 107 9.2 The space of local Hamiltonians . 109 9.3 Perturbative analysis of RV ........................... 112 9.4 Iteration . 114 9.4.1 d > 4................................... 115 9.4.2 d =4................................... 116 3 9.4.3 d = 4 − : anomalous scaling . 117 9.4.4 Other fixed points for d < 4....................... 119 9.5 Quantum Field Theories . 119 9.5.1 Effective Field Theories . 119 9.5.2 Polchinski equation . 121 9.5.3 Superrenormalizable QFT . 122 9.5.4 d < 4: nontrivial fixed points . 122 9.5.5 Asymptotic Freedom . 123 10 Stochastic PDE's with Rough Noise 125 10.1 Stochastic Differential Equations and Stochastic Quantization . 125 10.2 Linear case: Gaussian Process . 125 10.3 Nonlinear SDE: Local existence . 125 10.4 Perturbation Theory . 125 10.5 Rough SPDE's . 125 10.5.1 Dimensionless variables . 126 10.5.2 Hierarchical SPDE . 127 10.5.3 Effective equations . 127 10.5.4 Dimensionless formulation . 128 10.5.5 RG for SPDE . 128 10.5.6 Linearized RG map . 131 10.5.7 RG iteration, d =2............................ 133 A Some functional analysis 135 B Perron-Frobenius theorem 136 4 Chapter 1 Introduction 1.1 Background Quantum Field Theory (QFT) has become a universal framework to study physical systems with infinite number of degrees of freedom. Originally developed for high energy physics it lead to the explanation of universality in phase transitions using the Renormalization Group (RG), a powerful method to study scale invariant problems. QFT and RG ideas were thereafter applied to noisy systems, dynamical systems, non-equilibrium systems and many other problems. Early on QFT was also acknowledged as an interesting problem for mathematicians. This lead first to axiomatic characterizations of what sort of mathematical object QFT is and in then to Constructive QFT and the use of probabilistic methods to produce concrete examples of QFT. These techniques were then used in rigorous statistical mechanics and disordered systems. Following the pioneering work of Belavin, Polyakov and Zamoldchikov [?] a beautiful class of QFTs was uncovered by physicists, the two dimensional Conformal Field Theories (CFT). They have inspired a lot of new ideas in mathematics: the Schramm-Loewner Evolution (SLE) and Liouville Quantum Gravity and the theory of random surfaces. In Stochastic Partial Differential Equations (SPDE) physicists had also used QFT to study universality, a notable example being the Kardar-Parisi-Zhang (KPZ) equation where exact non-conventional scaling behavior was uncovered. KPZ and related equations posed hard problems for analysis due to the combination of a very singular noise and nonlinearity. Decisive progress was achieved here by Martin Hairer who introduced QFT ideas on renormalization to their mathematical analysis [?]. 1.2 Overview In quantum mechanics, a system is determined by giving all its possible states, observables and the time evolution. The possible states form a Hilbert space H, while the observables are given by self-adjoint operators A on this space. These operators evolve in time as eitH=~Ae−itH=~ for some positive opertaor H, called the Hamiltonian. In quantum field theory, one considers fields '(t; ~x) that are operators1 on the Hilbert d−1 space H. Here t 2 R is the time component and ~x 2 R is a point in space (d = 4 being the physical case). In general, these fields can be spinors, vectors etc., but we will mostly consider scalar fields: '(t; ~x) 2 R. The fields evolve in time as '(t; ~x) = eitH=~'(0; ~x)e−itH=~ (1.1) Additionally, one postulates the existence of a vacuum vector 0 2 H by H 0 = 0. The 1in fact operator valued distribution, see Sect.?? 5 basic objects of an axiomatic approach to QFT are then the Wightman functions: Wn(z1; :::; zn) = ( 0;'(z1):::'(zn) 0) for zi = (ti; ~xi). Finally, H carries a representation of the symmetries of the system: e.g. in a relativistic theory the Poincar´egroup. The positivity of the Hamiltonian and the time evolution (1.1) allow to analytically continue the Wightman functions to `imaginary time'. The substitution τ ! −iτ gives the so-called Schwinger functions Sn: Sn (τ1; ~x1); :::; (τn; ~xn) = Wn (−iτ1; ~x1); :::; (−iτn; ~xn) The point of this substitution is that the Schwinger functions can be written as correlation functions: Sn (τ1; ~x1); :::; (τn; ~xn) = E '(τ1; ~x1):::'(τn; ~xn) where '(τ; ~x) is a random field on some probability space Ω. This viewpoint also leads to constructive QFT: construct nontrivial random fields and the reconstruct the quantum fields by analytic continuation. The latter turns out to be possible under a condition on the random fields, so-called reflection positivity, see Section5. The imaginary time formulation of QFT is called Euclidean QFT. Finally, there is a deep connection between Euclidean quantum field theory and sta- tistical physics. In statistical physics, one often considers.