✡ ✡✡ ✪ ✪✪ � � � ✱ ✱ ✱✱ ✑ ✑✑ ✟✟ ❡ ❡❡ ❅ ❅❅ ❧ ❧ ❧ ◗ ◗◗ ❍❍��� ❳❳❳ ❤❤❤❤ ✭✭✭✭ ✏✏✟ IFT Instituto de F́ısica Teórica Universidade Estadual Paulista DISSERTAÇÃO DE MESTRADO Gabriel Cozzella Orientador G. E. A. Matsas Julho de 2016 Dissertação de mestrado Orientador G. E. A. Matsas Guilherme Barbosa Barros IFT-D.0004/20 Março de 2020 A classical analog of the quantum vacuum and its connection to the Unruh effectLattice study of Φ4-theory using stochastic quantization Adeilton Dean Marques Valois Gastão Inácio Krein Julho de 2020 Dissertação de mestrado Orientador High-loop expansion in lattice Φ4-theory using stochastic quantization Multi-loop expansion in lattice Φ⁴-theory using stochastic quantization IFT – D.011/ 20 Agosto de 2020 Valois, Adeilton Dean Marques. V198m Multi-loop expansion in lattice Φ⁴-theory using stochastic quantization / Adeilton Dean Marques Valois. – São Paulo, 2020 86 f. : il. Dissertação (mestrado) – Universidade Estadual Paulista (Unesp), Instituto de Física Teórica (IFT), São Paulo Orientador: Gastão Inácio Krein 1. Teoria quântica de campos. 2. Mecânica estatística. 3. Computação de alto desempenho. I. Título Sistema de geração automática de fichas catalográficas da Unesp. Biblioteca do Instituto de Física Teórica (IFT), São Paulo. Dados fornecidos pelo autor(a). i Dedication Há poucos meses, pareceria loucura me imaginar vivendo uma devastadora catástrofe global, onde não se pode abraçar, beijar, ou compartilhar momentos especiais com pessoas queridas. Pareceria insanidade deixar de ver as notícias para não perder as esperanças, com o número de mortes pelo mundo ultrapassando o incontável. Pareceria pesadelo pensar que ficaria meses e meses trancafiado entre quatro paredes sem o calor dos abraços e apertos de mão de meus amigos. Mas a maior insanidade de todas, é que enquanto escrevia estas linhas, tudo isso acontecia simultaneamente. 2020 tem sido um ano marcado por muitas perdas, lágrimas e saudade, e será para sempre lembrado como aquele em que máscaras eram quase tão importantes quanto nossas vestes. Mas também será lembrado como o ano em que sentimos falta das coisas mais corriqueiras da vida. Aquelas que muitas vezes passam inteiramente despercebidas, como um simples sorriso nos rostos familiares do dia-a-dia. Por sorte, sempre fui rodeado por pessoas únicas que me proporcionaram inestimável companhia do início ao fim, e especialmente durante o período de tormentas na pandemia do coronavírus. A essas pessoas dedico o meu trabalho. Ao meu orientador, Gastão Krein, que sempre com muita paciência, dedicação e sabedo- ria me instruiu na teoria de campos, e me abriu novas possibilidades ao me introduzir no universo da quantização estocástica. Ao meu amigo Luis Albino, que em tempos de trevas me presenteou com sua amizade, carisma e senso de humor. Gracias por todas as conversas divertidas sobre las chicas hermosas que víamos no trem à caminho do IFT. Amigo, ¡te voy a extrañar ! Aos meus amigos Carlisson e Rosane por me receberem sempre tão generosamente em sua casa. Fosse para cozinhar uma deliciosa pizza de arroz, ou para iniciar calorosas discussões sobre política, ou sobre as novas aplicações de Machine Learning à física. Ao meu amigo Enzo Solis, por sempre alegrar a casa com suas músicas motivacionais, que levantavam nosso astral a cada dia. Ao meu amigo Gabriel Caro, por sua amizade e seus valiosos ensi- namentos na língua espanhola, que me fizeram alcançar desenvoltura para falar: el pelo del cuero del cuerpo del puerco. À minha amiga Ana Mizher, que prontamente me auxiliou de maneira decisiva na escrita da carta de motivação que mudou a minha vida, e me permitiu lograr um doutorado em Bielefeld, na Alemanha. Ao professor Bruno El-Bennich, por ser tão solícito ao aceitar escrever as cartas de recomendação que impulsionaram minha busca por esse doutorado. E nesse assunto, não tenho como esquecer do meu amigo Jairo Rojas, que me dava sua opinião crítica sempre que eu a pedia. Também graças a ele, minha carta pôde alcançar o nível de profissionalidade necessário para que eu fosse aprovado. ¡Muchas gracias, mi pata! Além de todos os meus queridos companheiros do IFT, minha família tem desempenhado um papel crucial nessa longa jornada que venho trilhando. Obrigado por acreditarem em mim e na educação como instrumento enriquecedor da alma. Ao meu pai, Rubenilson Caldas Valois, à minha mãe, Adelyze Margarida Marques Valois, à minha irmãzinha, Nathalya Marques Valois, à minha avó Maria José Monteiro Marques, à minha madrinha, Rosa Valois, e ao meu padrinho, João Maciel. Vocês bem sabem que apesar das vitórias, sempre há perdas, e que a dor de uma perda pode ser como uma ferida que queima mesmo com o mais suave toque, um sentimento vagamente transmitido por palavras. Somente os corações judiados por ela hão de compreender o sentido desta mensagem. Minha maior perda nesta pandemia não foi tempo, dinheiro ou oportunidade, e sim meu tio, Laércio Demóstenes, que faleceu aos ii 58 anos, vítima da COVID-19. À sua memória dedico estas palavras: Ah se Deus me concedesse um único dia a mais na sua companhia... Pela manhã, deixaria que cortasse meu cabelo uma última vez, ainda que não fosse nada do jeito que gosto, só para ouvir você dizer o quanto aquele corte me caía bem. No almoço, pediria que me recontasse uma das histórias pitorescas de quando eu era criança, e que me mostrasse meus antigos desenhos, que com tanto apreço você os guardava. À tarde, montaria um derradeiro karaokê com sua velha e empoeirada caixa de som para que juntos cantássemos suas favoritas de Raul Seixas. Ao anoitecer, reuniria a família para que víssemos uma daquelas antiguidades do cinema que você gostava de chamar de “clássicos”. E à meia noite, daria o último adeus, repetindo mil vezes o quanto você é importante para mim. Daqui a 50 ou 60 anos, quando o restante de nós houver partido, nossa família estará completa como outrora, já que sem você, há um vazio que consolo algum no mundo pode preencher. Por fim, gostaria de agradecer imensamente a CAPES, instituição que me proporcionou apoio financeiro, sem o qual este trabalho não seria possível. iii Abstract A variety of techniques were devised in the last century to consistently describe the quantum theory of fields and connect it to experiments in the context of particle physics and funda- mental interactions. One of these techniques is the so-called stochastic quantization, which uses the Langevin equation to generate stochastic processes where a noise source plays the role of quantum fluctuations. In this work, we explore stochastic quantization on a lattice framework to approach quantum field theory through numerical simulations. As the only known fully non-perturbative regularization method, lattice simulations offer a promising route to handle the awkward divergencies arising from the ultraviolet spectrum of quantum theories. In this scenario, we carry out many multi-thread numerical simulations in 2, 3 and 4 dimensions to compute non-perturbative correlation functions of the Euclidean Φ4 theory with and without symmetry breaking. We also approach stochastic quantization with multi- loop expansions in powers of ~ to show its order-by-order consistency. In this approach, the quantum corrections result from the solution of an infinite set of coupled Langevin equations. These equations were truncated and numerically solved with a code developed in this work up to large powers of ~. We demonstrate that the results agree in high precision with analytical predictions from standard perturbation theory for several values of the couplings constant within a given interval. Finally, we sketch a numerical renormalization procedure to obtain physical parameters in the continuum limit. Key-words: Quantum field theory - Stochastic Quantization - Numerical Simulations. Fields of knowledge: Quantum field theory - Statistical Mechanics - High-Performance Computing. iv Resumo Uma variedade de técnicas foi desenvolvida no século passado para descrever a teoria quân- tica de campos de forma consistente e conectá-la aos experimentos no contexto da física de partículas e interações fundamentais. Uma dessas técnicas é a chamada quantização es- tocástica, que usa a equação de Langevin para gerar processos estocásticos onde uma fonte de ruído faz o papel das flutuações quânticas. Neste trabalho, exploramos a quantização estocástica na rede para abordar a teoria quântica de campos através de simulações numéri- cas. Sendo o único método conhecido de regularização não-perturbativa, as simulações na rede oferecem uma rota promissora para lidar com as complexas divergências que surgem do espectro ultravioleta de teorias quânticas. Neste cenário, executamos várias simulações numéricas em paralelo em duas, três e quatro dimensões para calcular funções de correlação não-perturbativas da teoria escalar euclidiana Φ4 com e sem quebra espontânea de simetria. Também abordamos a quantização estocástica com expansões em muitos loops em potências de ~ para mostrar sua consistência ordem a ordem. Nessa abordagem, as correções quânti- cas resultam da solução de um conjunto infinito de equações de Langevin acopladas. Essas equações foram truncadas e resolvidas numericamente com um código desenvolvido neste trabalho até altas potências de ~. Nós demonstramos que os resultados concordam em alta precisão com as predições analíticas da teoria de perturbação padrão para alguns valores da constante de acoplamentos dentro de um dado intervalo. Por fim, esboçamos um procedi- mento de renormalização numérica para obter parâmetros físicos no limite do contínuo. Palavras-chaves: Teoria quântica de campos - Quantização estocástica - Simulações na rede Áreas do conhecimento: Teoria quântica de campos - Mecânica estatística - Computação de alta performace. Contents Contents v List of Figures vii 1 Introduction and Motivation xiii 2 Review of Stochastic Processes 3 2.1 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Itô calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Stratonovich calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Review of Stochastic Quantization 3 3.1 The Langevin equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.2 The Fokker-Planck equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 Stochastic quantization in non-relativistic quantum mechanics . . . . . . . . 13 3.4 Stochastic quantization in quantum field theory . . . . . . . . . . . . . . . . 19 3.5 Quantum n-point functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4 Lattice Simulations in Φ4-theory 25 4.1 The scalar field on a lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2 Numerical simulations using LASER code . . . . . . . . . . . . . . . . . . . 28 4.3 Spontaneous symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . 32 v vi CONTENTS 5 Review of Noise Perturbation Theory and New Results 39 5.1 Feyman diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.2 Multi-loop expansion in the stochastic formalism . . . . . . . . . . . . . . . 44 5.3 Perturbative n-point functions . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.4 New results with LASER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.5 Error analysis in loop expansion . . . . . . . . . . . . . . . . . . . . . . . . . 49 6 Review of Lattice Renormalization and New Results 53 6.1 Why renormalization? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.2 From lattice to continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6.3 Mass renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.4 Coupling renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6.5 Numerical renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 7 Conclusions and perspectives 69 Appendices 75 A 77 A.1 Numerical algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 A.2 Remarks on the Monte Carlo approach . . . . . . . . . . . . . . . . . . . . . 78 A.3 Importance of reliable random number generation . . . . . . . . . . . . . . . 79 A.4 Error estimation for mR and λR . . . . . . . . . . . . . . . . . . . . . . . . . 81 Bibliography 83 List of Figures 2.1 At a given instant of time t, we would have a configuration as shown in this figure. The N particles in magenta are colliding with the pollen (green) at time t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.1 Noise probability distributions for samples of different sizes normalized by the amount of random numbers in each sample. The green and the blue histograms have been displaced horizontally for the sake of clarity. Notice how the distribution converges to a Gaussian profile as the sample size increases. 8 3.2 Potential function without spontaneous symmetry breaking. The empty black circle in the origin indicates the single classical vacuum for this potential. The square mass and coupling used in the plot are given in the upper part. . . . 19 4.1 Lattice topologies for periodic boundary conditions in 1 dimension (left) and 2 dimensions (right). Due to the periodic boundary conditions, in one dimension the lattice assumes a ring-like shape, whereas in two it turns into a torus. . . 26 4.2 (a) Several solutions of the Langevin equation as a function of τ , using different initial conditions, reinforcing that the asymptotic solution is always the same and given by the thermalized zone. (b) Stochastic averages of the field ϕ with increasing number of noise realizations, where we notice drastic improvements as we go from 1 to 10, then from 10 to 1000 samples. . . . . . . . . . . . . . 30 vii viii LIST OF FIGURES 4.3 In the left-hand side, the fluctuating continuous lines show the convergence of stochastic averages of the free propagator on a 10 × 10 lattice computed for three distances as a function of τ . The dashed lines represent the mean value of the thermalized zone, using Nr = 60, leading to their physical value as a function of the distance x1 −x2 in the right-hand side. Empty red circles indicate the analytical result and their excellent match with the simulation. . 31 4.4 Numerical 2-point functions in the free case from stochastic quantization (black lines-and-points) for different lattice sizes compared to the exact ana- lytical values (red empty circles). The dashed vertical lines are placed in the middle of each lattice, where it starts to be periodic. . . . . . . . . . . . . . 32 4.5 Numerical (black lines-and-points) and analytical (empty red circles) propa- gators for nine values of λ averaged with 100 noise realizations each. Notice that the 2-loop analytic propagator remains consistent with simulations. . . 33 4.6 Numerical 2-point functions in momentum space (colourful lines-and-points) compared to the analytical perturbation theory (empty black circles) as a function of the quantum number k. The simulations were done on a 16 × 16 lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.7 Potential function in the case of spontaneous symmetry breaking. The empty black circles indicate the two classical vacua for this potential. The square mass and coupling used in the plot are given in the upper part. . . . . . . . 34 4.8 (a) Several solutions of the Langevin equation on a 8× 8 lattice, with spota- neous breaking of symmetry using different initial conditions around each classical vacuum. (b) 10 more solutions, all with initial condition set to the origin (which corresponds to the local maximum of the potential). In this case, the solutions randomly select a vacuum to which they fall. The parameters used in the simulations are indicated above each plot. . . . . . . . . . . . . . 35 LIST OF FIGURES ix 4.9 Three solutions with spontaneous symmetry breaking for large coupling (λ = 5), showing transition between the vacua. The two upper plots have been displaced vertically for clarity, with the dashed lines indicating the local max- imum in each case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.1 Relevant diagrams for perturbative Φ4-theory in the symmetric phase. . . . . 41 5.2 Relevant diagrams for perturbative Φ4-theory in the broken phase. . . . . . . 43 5.3 Comparison between the zero-momentum propagators on a 8× 8 lattice from perturbative stochastic simulations (continuous lines) and analytical perturba- tion theory (dashed lines) up to 2 loops as a function of the coupling constant λ. The dashed purple line represents the full solution of the Langevin equa- tion. On the top, the mass squared and the number of noise realizations used are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.4 Comparison between the zero-momentum propagators on a 4× 4× 4× 4 lat- tice from perturbative stochastic simulations (continuous lines) and analytical perturbation theory (dashed lines) up to 2 loops as a function of the coupling constant λ. On the top, the mass squared and the number of noise realizations used are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.5 Fractional error plots for position-space propagators as a function of distance on a 16 × 16 lattice using Nr = 10000 and m2 = 1. The dashed line in each plot represents the 5% error line taken as reference. . . . . . . . . . . . . . . 51 5.6 Average error as a function of the coupling constant up to O(8), using Nr = 10000 and m2 = 1 in each simulation. The dashed line represents the 5% error taken as reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 x LIST OF FIGURES 6.1 Long range interactions (red wavelength) generate low momentum transfer, while short range ones (blue wavelength) generate high momentum transfer. Notice that because the blue is the highest momentum transfered in this lat- tice, there are no UV divergencies. . . . . . . . . . . . . . . . . . . . . . . . 54 6.2 2-dimensional representation of the infinite volume limite, where the lattice length and the number of sites must gradually increase, while the lattice spac- ing must decrease until we turn a lattice into a continuous Euclidean space. . 55 6.3 Autocorrelations as a function of the number of lattice sites for different sizes L. The dashed line marks the exact value of G(0), in the continuum limit, from analytical calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.4 Momentum-space autocorrelation on a 2-dimensional lattice in the continuum limit as a function of the scale a/L = 1/N . The blue fit is a log function showing the type of divergence within G̃(0). . . . . . . . . . . . . . . . . . . 58 6.5 Three disconnected diagrams arising from the 4-point function that must be discounted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 6.6 Renormalization in 2 dimensions with error bars. Top: renormalized coupling from full Langevin equation. Numbers above the x-axis are the values of N . Bottom: renormalized mass, with the dashed line showing the the bare mass whereby we started the process. Both quantities are plotted as a function of the scale a/L = 1/N . The table on the right gives the input parameters of the simulation and the renormalized outputs. . . . . . . . . . . . . . . . . . 66 6.7 Renormalization in 3 dimensions with error bars. Top: renormalized coupling from full Langevin equation. Numbers above the x-axis are the values of N . Bottom: renormalized mass, with the dashed line showing the the bare mass whereby we started the process. Both quantities are plotted as a function of the scale a/L = 1/N . The table on the right gives the input parameters of the simulation and the renormalized outputs. . . . . . . . . . . . . . . . . . 67 LIST OF FIGURES xi 6.8 Renormalization in 4 dimensions with error bars. Top: renormalized coupling from full Langevin equation. Numbers above the x-axis are the values of N . Bottom: renormalized mass, with the dashed line showing the the bare mass whereby we started the process. Both quantities are plotted as a function of the scale a/L = 1/N . The table on the right gives the input parameters of the simulation and the renormalized outputs. . . . . . . . . . . . . . . . . . 68 xii LIST OF FIGURES Chapter 1 Introduction and Motivation In the last decades, lattice methods have played an essential role in modern problems of High-Energy Physics (HEP). The efficient computation of physical quantities through robust lattice simulations have boosted our understanding of fundamental particles and their interactions. In this scenario, lattice QCD stands as one of the most prominent and rapidly evolving areas within HEP. It uses importance sampling algorithms to compute expectation values from the Euclidean path integral, using field configurations with a Boltzmann weight. Among the achievements of lattice simulations within the Standard Model are the precise determination of the quark masses, and the strong force contribution to the magnetic moment of the muon (for details, we refer to the reviews [Sanfilippo, 2015, Meyer and Wittig, 2019], respectively). Despite their remarkable success in Euclidean space, importance-sampling-based algo- rithms are known to fail in Minkowski-space problems, and for HEP under extreme con- ditions. In these cases, the path integral has a non-positive probability measure due to a complex factor in the exponent, rendering importance sampling meaningless. A situation where the method is said to suffer from a numerical sign problem or, more generically, a complex phase problem [Goy et al., 2017]. Therefore, from the theoretical side, the under- standing of HEP theories under extreme conditions has been hindered by the lack of tools xiii 1 to treat the sign problem. On the other hand, from the experimental side, new hadronic colliders, like FAIR and NICA, which aim to investigate the high chemical potential region of the QCD phase diagram, have triggered the efforts to find a solution to the sign problem. With Monte Carlo hindered by this problem, we need to look for alternative solutions that can shed some light on finite chemical potential systems in QFT. In 1981, another approach, called stochastic quantization, was developed by Parisi and Wu as an attempt to circumvent the problem of Gribov ambiguity in non-Abelian gauge the- ories by avoiding gauge-fixing [Parisi et al., 1981]. In turn, stochastic quantization uses the Langevin equation, from non-equilibrium statistical mechanics, to generate stochastic pro- cesses whose thermal equilibrium gives the correct distribution as dictated by the Euclidean path integral. Since the ’80s, many authors have explored applications of this technique in plenty of physical contexts, which go across Ginzburg-Landau systems [Attanasio, 2013], cos- mological inflation [Kunze, 2006], where the inflaton field is modeled through the stochastic dynamics, fermionic fields [Ridky, 1995], gauge-invariant fields [Kapoor, 2018], semi-classical gravity [dos Reis et al., 2019], etc. Some authors have also argued that stochastic quanti- zation might provide a route towards the solution to the sign problem, and thus lead to a better understanding of finite density QCD [Aarts, 2009]. In this work, we apply stochastic quantization to study the Φ4-theory as a toy model in the lattice framework through a novel simulation code. This code was developed from scratch as part of this investigation to comply with our needs in simulating the Langevin equation. We use this code to compute the exact n-point correlation functions by solving the full Langevin equation with quartic interaction. We also apply the so-called Noise Perturbation Theory to expand the field in powers of ~ within the Langevin equation and derive a set of coupled differential equations providing quantum corrections to the zeroth-order. We truncate and implement this set within our code to obtain its solution up to order ~10. Thence, we compute numerical propagators in the perturbative framework and compare with available analytical expressions up to 2 loops, showing the accordance between the 2 CHAPTER 1. INTRODUCTION AND MOTIVATION two even for large coupling. Even though we stop at 2 loops, we argue that this method can be straightforwardly implemented to go far beyond the analytical perturbation theory. Therefore, this text is organized as follows: in chapter 2, we review stochastic processes, which form the underpinnings of stochastic quantization. In chapter 3, we use the concepts from chapter 2 to discuss the main aspects of stochastic quantization, and review its applications to non-relativistic quantum mechanics and quantum field theory. In chapter 4 we formulate stochastic quantization on the lattice and use our novel lattice code to reproduce some results obtained by others. Afterward, in chapter 5, we present our new results on high-order expansion in 2 and 4-dimensions. In chapter 6, we review a numerical renormalization scheme from previous works and apply it to our case-study, generating new data in 2, 3, and 4 dimensions. In chapter 7, we finally conclude and discuss future perspectives in the field. Chapter 2 Review of Stochastic Processes Natural phenomena driven by statistical fluctuations are among the most common hap- penings in daily life. For centuries, the physics of such phenomena has been studied by plenty of eminent scientists. With the outcome of statistical mechanics, it became possible to describe the stochastic behavior of a huge set of interacting objects. While Newtonian mechanics gives the precise motion of one or two particles interacting, statistical mechanics focus on average properties from myriads of particles. Take for instance the famous two-body problem, like two heavy-weight asteroids orbiting one another in the Kuiper belt. Although to carry out all calculations requires considerable effort, the equations of motion for this problem were shown to be solvable analytically. With a third asteroid in the surroundings, only under special symmetry conditions can they be treated exactly. However, as we continue to jam hundreds, billions, trillions of bodies together, single orbits and velocities start to be- come unimportant, for collective behavior begins to dominate over individual features. In this scenario, determinism gives space to stochastic fluctuations, and Newtonian mechanics gives space to statistical mechanics, reducing uncountable microscopic degrees of freedom to a small set of relevant parameters. Even though statistical mechanics and Newton’s theory bear intimate relationships, pioneers like Ludwig Boltzmann never succeeded in the proof that the former comes from the latter. Instead, they resigned themselves to the use of it, in 3 4 CHAPTER 2. REVIEW OF STOCHASTIC PROCESSES the description of statistically fluctuating systems, since the laws describing a huge number of objects are surprisingly simpler than the ones for just several of them. As a consequence of that, statistical mechanics triggered an enormous breakthrough in the study of many-particle systems and their thermodynamic effects. Therefore, in the next pages, we focus on reviewing the most prominent physical and mathematical aspects of stochastic systems. 2.1 Brownian motion Before rushing to the operational part, we go through a short digression to introduce the historical context. During an eminent experiment by the Scottish botanist Robert Brown in 1827, a grain of pollen was seen, through the lens of a microscope, performing a mysterious motion over the surface of the water. This quirky behavior raised the question of why pollen, and other light objects, had an odd tendency to jiggle while floating on water. Think of a toy boat sailing across the plane waters of a beach on a mild day, without wind, waves, or external impulses, and it seems unnatural to expect the boat to randomly sway sideways. So what is the big deal with an ordinary grain of pollen? The point is that toy boats are usually macroscopic objects, billions of times larger than tiny water molecules, and thus collisions between them do not affect the boat’s trajectory significantly. However, pollen particles are typically only a few micrometers long, and therefore much more susceptible to collisions and changes in the direction of their motion. Due to the homogeneous bath of molecules in the medium, hits happen isotropically and they force the pollen to shake, producing an overall disordered motion. A phenomenon that is known as Brownian motion, and thenceforth the starting point of this review. In mathematical terms, Brownian-moving particles have their stochastic trajectory de- termined by a very simple differential equation: dWt = η(t)dt (2.1.1) 2.1. BROWNIAN MOTION 5 where t is time and η the noise source. In physical terms, the noise source η encodes the information about the haphazard collisions in a medium with several mols of particles1, which makes it a sort of mathematical summary of the random pushes conveyed by the environment to the substrate embedded on it. To illustrate the origin of the noise source, consider a grain of pollen, like in Robert Brown’s experiment, subject to collisions with medium particles in two spatial directions, as in Figure 2.1. The dynamics of this particle may have infinitely Figure 2.1: At a given instant of time t, we would have a configuration as shown in this figure. The N particles in magenta are colliding with the pollen (green) at time t. many patterns depending on the choice of η to plug into Eq.(2.1.1). So what functional shape should a random source have? Very likely, no other but an ill-defined expression is suitable to describe η(t). However, remember that the only important degree of freedom is the average effect of the noise upon the particle, and the probability distribution beneath this average is smooth and well-defined. Pictorially, η(t) can be regarded as vector whose components are proportional to the sum of the forces provided by the medium particles in the x and y directions. Considering F(i) to be the force vector aforementioned, the components of η(t) can be written as an average of forces bestowed by each particle (in magenta) colliding with 1Recall that a mol has roughly 1023 particles. 6 CHAPTER 2. REVIEW OF STOCHASTIC PROCESSES the pollen (in green) at time t: ηx(t) ∝ F (1) x + F (2) x + ...+ F (N) x N ηy(t) ∝ F (1) y + F (2) y + ...+ F (N) y N (2.1.2) According to a recurrent theorem in probability theory, known as the central limit theorem, if the number of particles N in the ensemble is huge (we consider 1023 to be so!) then the probability distribution underlying ηx(t) and ηy(t) is well approximated by a Gaussian distribution. In formal terms, we can state it in several different ways, among which we present the version by Lindeberg and Lévy: Theorem 1. Suppose {F (1) x , F (2) x , ..., F (N) x } is a set of N random numbers with mean µ and variance σ2 < ∞. Then as N approaches infinity, the quantity √ N(ηx(t)− µ) approaches a Gaussian distribution with mean zero and variance σ2. This pictorial view, alongside the central limit theorem, helps us to develop an intuition about why the bell-shaped distribution is a natural choice in the context of Brownian motion. An identical version of this theorem is valid when we switch x by y in the previous statement, and so it would be if we considered extra spatial directions. Notice that we did not need to specify with which probability the numbers F (i) x are selected. Even if we tried, it would be useless spending time on a binge to determine with what force each particle shocks with the pollen. Luckily, Nature makes collective behavior prevail over individual behavior when N is large enough, and the probability dictating the sum of these forces becomes nearly Gaussian anyway. A stochastic process of this type is characterized as a Wiener process and it bears the following properties: • W0 = 0. • Wt evolves with independent steps, i.e., the incrementation step at time t+dt does not depend on the step at time t. • The steps of Wt are selected according to a Gaussian distribution with mean 0 and 2.2. ITÔ CALCULUS 7 variance dt. • It has a non-neglectable square variation dW 2, which is equivalent to time variation inside stochastic integrals: dW 2 ∼ dt. From now on, we focus on the more operational aspects of Wiener processes. These aspects attempt to convey a meaning to calculus in the context of stochastic processes, and they come in two main varieties, each of them is approached next. 2.2 Itô calculus As far as stochastic processes are concerned, it turns out that the usual definitions of Riemannian calculus are not appropriate. Ultimately, we want to solve Eq.(2.1.1) for a given set of Gaussian numbers η(t) and predict the average trajectory of the Brownian particle. Due to the stochastic nature of the noise source η, conventional derivation and integration concepts no longer apply. Therefore, they must be modified in order to encompass the needs of stochastic dynamics. Recall that in ordinary calculus, we define the Riemannian integral of a function f(x) inside the interval [a, b], given a partition {a = x0 < x1 < ... < xN = b}, as the limit of partial sums of rectangular areas: ∫ b a f(x)dx ≡ lim N→∞ N∑ n=1 f(xn−1)(xn − xn−1) (2.2.1) Moreover, we could also extend this idea to a more general concept of integral, for instance, regarding the integration measure not as a variable, but as a smooth function of x. This extension leads to the definition of the Riemann-Stieltjes integral of f(x) with respect to another function ω(x): ∫ ω(b) ω(a) f(x)dω(x) ≡ lim N→∞ N∑ n=1 f(xn−1)[ω(xn)− ω(xn−1)] (2.2.2) 8 CHAPTER 2. REVIEW OF STOCHASTIC PROCESSES In the special case of ω(x) = x, we recover the commonplace definition of Riemannian integral. The two sums in the right-hand sides of Eqs.(2.2.2) and (2.2.2) have the remarkable property of converging to a unique value for their correspondent left-hand sides regardless of the exact point whereby we draw the rectangles. In this sense, we could have replaced f(xn−1) in the definitions above by the rightmost choice f(xn), the midpoint f((xn+xn−1)/2) or any other point within the interval [xn, xn−1] and these integrals would still be unique. Giving a step further, suppose we want to compute an integral similar to Eq.(2.2.2), but with f and α depending not on x, but on the Brownian variable Xt ≡ X(t). Naming this integral Yt, this is how it looks like: Yt = ∫ t 0 f(X)dα(X) ≡ lim N→∞ N∑ n=1 f(Xn−1)[α(Xn)− α(Xn−1)] (2.2.3) Yt is called Itô integral and at first sight it seems identical to Eq.(2.2.2). However, as a consequence of X being a stochastic process, the right-hand side of the Itô integral does not necessarily converge to a unique. Instead, depending on the point of reference, Eq.(2.2.3) may or may not equal the Riemman-Stieltjes integral. Indeed, it only does so when we choose the leftmost point in the interval, i.e. f(xn−1). Notwithstanding this odd property, Itô integrals turn out to be extremely important in the mathematical description of stochastic processes, as we will see throughout this text, for they allow systematic operations with them. Moreover, instead of just analyzing how Xt evolves with time, we might be interested in studying the dynamics of functions of that variable. Consider, for instance, an arbitrary function of the Brownian motion, f(Xt). Once we know from Eq.(2.1.1) the behavior of Xt for all times, without difficulties, we can find out the behavior of f(Xt) by computing its variation through Taylor expansion: df(Xt) = f(Xt + dXt)− f(Xt) =�� ��f(Xt) + f ′(Xt)dXt + 1 2 f ′′(Xt)(dXt) 2 + ...−�� ��f(Xt) 2.3. STRATONOVICH CALCULUS 9 = f ′(Xt)dXt + 1 2 f ′′(Xt)(dXt) 2 +O(3) (2.2.4) Truncating in second order and using Eq.(2.1.1): df(Xt) = f ′(Xt)dXt + 1 2 f ′′(Xt)(dXt) 2 (2.2.5) df(Xt) = f ′(Xt)dXt + 1 2 f ′′(Xt)dt (2.2.6) Eq.(2.2.6) is known as Itô’s lemma and it implies that any function of the Brownian motion f(Xt) also evolves as a stochastic process. Notice that it resembles the familiar chain rule in ordinary calculus, except for the extra term involving the second derivative of f(Xt). Despite the difference, the appearance of this term is the hallmark of Itô calculus and it presents no conceptual flaws whatsoever. Nevertheless, sometimes it might be easier to treat equations if that extra term is absent, not to mention that it would look more familiar given our acquaintance with the chain rule of calculus. Therefore, it is convenient to resort to another very common way of defining stochastic integrals in physics and mathematics. We deal with this alternative in the next section. 2.3 Stratonovich calculus Stratonovich calculus is a reformulation of Itô’s approach that redefines its integral so that the chain rule, as we know it from ordinary calculus, is valid for stochastic processes as well. Since both of them provide consistent mathematical results, whether an Itô or Stratonovich approach should be used is completely dictated by the particular features of the problem. Integration, in the Stratonovich sense, is denoted with an empty circle ◦ and is defined as: St = ∫ t 0 f(W ) ◦ dW ≡ lim N→∞ N∑ n=1 f(Wn) + f(Wn−1) 2 (Wn −Wn−1) (2.3.1) 2.3. STRATONOVICH CALCULUS 1 To make a connection between Itô and Stratonovich integrals, we start separating the sums above: St ≡ 1 2 lim N→∞ N∑ n=1 f(Wn)(Wn −Wn−1) + 1 2 lim N→∞ N∑ n=1 f(Wn−1)(Wn −Wn−1) (2.3.2) St = 1 2 lim N→∞ N∑ n=1 f(Wn)(Wn −Wn−1) + 1 2 Yt (2.3.3) where in the last line we recognized the second term as the definition of Itô’s integral. Now we relate f(Wn) to f(Wn−1) by Taylor expansion: f(Wn) = f(Wn−1 + dW ) = f(Wn−1) + ∂f(Wn−1) ∂W dW (2.3.4) Noticing that dW = Wn −Wn−1 and replacing the expansion above up to order one: St = 1 2 lim N→∞ N∑ n=1 [ f(Wn−1) + ∂f(Wn−1) ∂W dW ] (Wn −Wn−1) + 1 2 Yt (2.3.5) = 1 2 lim N→∞ N∑ n=1 f(Wn−1)(Wn −Wn−1) + 1 2 lim N→∞ N∑ n=1 ∂f(Wn−1) ∂W (Wn −Wn−1) 2 + 1 2 Yt (2.3.6) = 1 2 lim N→∞ N∑ n=1 ∂f(Wn−1) ∂W (Wn −Wn−1) 2 + Yt (2.3.7) Since the first term in the right-hand side is just the definition of Itô’s integral for the derivative of f , this relation can be written as: St = 1 2 ∫ ∂f(W ) ∂W dW 2 + Yt (2.3.8) Recall that Wiener processes have a non-neglectable square measure dW 2, which can be replaced by dt inside the stochastic integral: St = 1 2 ∫ ∂f(W ) ∂W dt+ Yt (2.3.9) 2 CHAPTER 2. REVIEW OF STOCHASTIC PROCESSES This is the basic conversion recipe for swapping between Itô and Stratonovich prescriptions. All the content of this chapter forms the foundation for the following topics to be discussed next. In the coming chapter, we will insert stochastic theory in the context of classical mechanics, and finally, extrapolate the ideas to quantum problems. Chapter 3 Review of Stochastic Quantization In the last chapter, we reviewed the Brownian motion as the simplest of stochastic pro- cesses. To define the concept of stochastic integrals, we used the pure Brownian motion, without any other source of dynamics, and generalized Riemannian integrals to stochastic variables, as originally done by Itô and Stratonovich. In spite of its simplicity, there is a sur- prisingly large number of systems that can be modeled as Wiener processes, either inside or outside physics. However, considerable attention must be paid to other types of systems that require physics beyond Brownian motion to have their behavior precisely understood. In this chapter, we commence by reviewing the Ornstein-Uhlenbeck process, in the form of the so- called Langevin equation, as the first natural generalization to the Wiener process. This pro- cess will be a strong underpinning to prepare the ground to apply stochastic quantization in quantum mechanics, as done by other authors [Houard and Irac-Astaud, 1992, Namiki, 2008]. In simple terms, stochastic quantization stands as a robust tool for deriving the quantized version of classical theories employing stochastic processes. It also provides a useful concep- tual bridge between quantum and statistical physics, playing an important role in numerical simulations. This statement is true in both contexts of quantum mechanics and quantum field theory. We postpone the latter until the next chapter. 3 4 CHAPTER 3. REVIEW OF STOCHASTIC QUANTIZATION 3.1 The Langevin equation Imagine that in the same medium where Brownian motion is taking place, the pollen particle is also subject to a friction force, that tends to dissipate energy. Additionally, suppose we want to access the full dynamics of the particle and predict its trajectory and velocity within this medium. To do so, it requires no more than Newton’s second law of classical mechanics: F = m dv dt (3.1.1) Now, we need to know beforehand which ingredients to plug into the force F . Since the particle is embedded in a fluid, there must be a drift force (Fd) associated with the friction bestowed by the surrounding environment acting against the particle’s motion. Not unlike the familiar case of a pointlike object moving across the air, the drift force can be reasonably modeled as proportional to the first power of velocity: Fd = −λv. Where λ is a medium- dependent constant that measures the strength of this resistance. The other force, which arises from the endless collisions between the pollen and the microscopic molecules of the medium, is stochastic. Hence, from the interpretation given by Eq.(2.1.2), this force can be very suitably regarded as pure Brownian motion. Putting everything together, we end up with a very simple equation of motion: m dv(t) dt = −λv(t) + η(t) (3.1.2) A first-order stochastic differential equation, describing the velocity of a test particle with mass m at each instant of time, called the Langevin equation. It belongs to a class of stochastic processes named Ornstein-Uhlenbeck, which differs from a Wiener process by a drift term −λv(t), accounting for the resisting environment. Notice that in the case of no resistance (λ = 0) we recover the Brownian motion equation, with dWt ≡ mdv(t). Hitherto we are not able to determine the solution to the Langevin equation since no functional form was specified to the noise. The only thing we know about it is what the central 3.1. THE LANGEVIN EQUATION 5 limit theorem taught us, and it might contain a clue on where to seek the missing information about the noise. Intuitively, the absence of external fields (electric, magnetic, etc.) implies that the collisions on a homogeneous isotropic medium can not have any preferential direction. In other words, by averaging over many collisions for long enough the result should be nearly zero. This condition is indeed one of the criteria used to define the Wiener process in section 2.1. Furthermore, time-averages in statistical mechanics are equivalent to ensemble averages by the ergodic hypothesis, where ensemble means a collection of particles. Therefore, instead of “long enough”, we could think of several replicas of the system with which we average over the different noises in each of them. This is like having a large number of independent copies of the situation illustrated in Figure 2.1. This isotropy feature is translated into the compact notation of stochastic average as 〈 η(t) 〉η = 0. In order to guess the second noise property, let us assume the structure of all molecules colliding to be irrelevant, such that they can be approximated by point sources. This as- sumption constraints the collisions to happen locally, lasting for an infinitesimal time interval, implying that two collisions at different times, say t and t′, have no overlap and are thus com- pletely uncorrelated. Therefore, it seems reasonable to have a proportionality between the correlation function of two noises and a Dirac delta function: 〈 η(t)η(t′) 〉η ∝ δ(t− t′), for it must be zero when t 6= t′ and infinite (entirely correlated) when t = t′. Therefore, we guess the second property should have the form: 〈 η(t)η(t′) 〉η = αδ(t− t′) (3.1.3) where α is a proportionality constant to be determined. Any kind of noise satisfying this property is called white noise, referring to the fact that it receives a contribution from all frequencies1. To express α in terms of known physical parameters, we search for solutions to the Langevin equation. Using variation of parameters, the solution of Eq.(3.1.2) can be 1Recall that a Dirac delta can be expressed in Fourier series as an infinite sum of sines and cosines with equal weight. 6 CHAPTER 3. REVIEW OF STOCHASTIC QUANTIZATION expressed as: v(t) = v0e −λt/m + 1 m ∫ t 0 dt1η(t1)e λ(t1−t)/m (3.1.4) And the square velocity is: v(t)2 = v20e −2λt/m + 2v0 m e−λt/m ∫ t 0 dt2η(t2)e λ(t2−t)/m+ + 1 m2 ∫ t 0 ∫ t 0 dt1dt2η(t1)η(t2)e λ(t1+t2−2t)/m (3.1.5) Since the noise function η appearing on the right-hand side depends on the individual motion of each molecule in the medium, in practical terms, it is not possible to know the velocities of each molecule colliding with the test particle. This means it only makes sense to compute the above integrals within average brackets 〈 · 〉η. Using the first property to eliminate the average of the second term, the average square velocity is given by: 〈 v(t)2 〉 η = 〈 v20 〉 η e−2λt/m + 1 m2 ∫ t 0 ∫ t 0 dt1dt2 〈 η(t1)η(t2) 〉η e λ(t1+t2−2t)/m (3.1.6) From Eq.(3.1.3), the equation above becomes: 〈 v(t)2 〉 η = 〈 v20 〉 η e−2λt/m + α m2 ∫ t 0 ∫ t 0 dt1dt2δ(t1 − t2)e λ(t1+t2−2t)/m (3.1.7) = 〈 v20 〉 η e−2λt/m + 1 m2 ∫ t 0 dt1e 2λ(t1−t)/m (3.1.8) = 〈 v20 〉 η e−2λt/m + α 2λm ( 1− e−2λt/m ) (3.1.9) Using this result, the average kinetic energy associated with the motion of the Brownian particle is given by: 〈 E(t) 〉η = 1 2 m 〈 v(t)2 〉 η = 1 2 m 〈 v20 〉 η e−2λt/m + α 4λ ( 1− e−2λt/m ) (3.1.10) At this point, we introduce another fundamental theorem in statistical physics called the 3.1. THE LANGEVIN EQUATION 7 equipartition theorem: Theorem 2. Each degree of freedom of any thermodynamic system in thermal equilibrium at temperature T has average energy equal to kBT/2. Therefore, in thermal equilibrium, when t→ ∞, the equipartition theorem must hold: lim t→∞ 〈 E(t) 〉η = 1 2 kBT = α 4λ ∴ α = 2λkBT (3.1.11) that also gives, for v2: lim t→∞ 〈 v(t)2 〉 η = kBT m (3.1.12) and we are finally able to write the complete relations satisfied by the noise: • Property 1: 〈 η(t) 〉η = 0 (3.1.13) • Property 2: 〈 η(t)η(t′) 〉η = 2λkBTδ(t− t′) (3.1.14) Properties (3.1.13) and (3.1.14) fully specify the Gaussian shape of the noise η(t) since we have the two required moments of the probability distribution. Figure 3.1 shows how the probability distribution of the noise indeed converges to a Gaussian profile for large samples. Even though we have presented the Ornstein-Uhlenbeck process for a special case (when the drift force has the form −λv), the Langevin equation describing this process, and also properties 1 and 2, come from far more general principles. Instead of a resisting force proportional to the velocity, we could think of other potential functions and write a more comprehensive equation: m dv(t) dt = −∂U(v, t) ∂v + η(t) (3.1.15) where U(v, t) is a generic potential function to which the particle is subject. To restore the special case of Eq.(3.1.2), we simply choose a quadratic potential U(v, t) = λv2/2. After 8 CHAPTER 3. REVIEW OF STOCHASTIC QUANTIZATION Figure 3.1: Noise probability distributions for samples of different sizes normalized by the amount of random numbers in each sample. The green and the blue histograms have been displaced horizontally for the sake of clarity. Notice how the distribution converges to a Gaussian profile as the sample size increases. solving the Langevin equation with a given potential U(v, t), our ultimate goal is to compute stochastic averages and draw as much information as we can about the system. How this equation connects statistical mechanics to quantum mechanics we shall see in the next section by looking at the probability distribution underlying v(t) in the Langevin equation. 3.2 The Fokker-Planck equation Alternatively to having a dynamical equation, like the Langevin equation, solving it and then computing expectation values, as we did in the previous section, we could rush directly to the last part if we had at our immediate disposal the probability distribution that governs the system. In the case of statistical mechanical systems, the latter is usually the preferred route since we often have absolutely no knowledge about the equation describing their dynamics before equilibrium in terms of stochastic processes. What we know is the probability content, after equilibrium, which is encoded in a mathematical object called the partition function. For instance, there is no apparent manner to describe the particles of a van der Waals gas as Brownian-moving entities and still, we can write an elegant partition function that carries 3.2. THE FOKKER-PLANCK EQUATION 9 everything to be known about the equilibrium state of the gas. That is why in this section we follow [Namiki, 2008] to interpret the Langevin equation in terms of probability through the concept of partition function. In a more general context, a partition function Z for a 1-dimensional system at temperature T with potential energy E(x) can be roughly expressed as: Z ∼ ∫ dx e−E(x)/kBT (3.2.1) In the quantum realm, as opposed to classical mechanics, we lack the option of describing the complete dynamics of particles, for we never know their exact trajectories. If we did, this would imply a rude violation of Heisenberg’s uncertainty principle. Yet, Nature allows us to have access to their full probabilistic content through the probability amplitudes, or wave functions. In the Feynman picture of quantum mechanics, we even have a sort of partition function, as in statistical mechanics, called the path integral. Without details on its derivation (for which we refer the reader to [Zee, 2010]), we present the Feynman partition function in Euclidean space for a system with action SE[x] as: ZF = ∫ Dx e−SE [x]/~ (3.2.2) with the cursive D representing functional integration. Considering an arbitrary Ornstein- Uhlenbeck process, represented by the variable v(t), a single stochastic trajectory of this process is given by one particular solution of the Langevin equation with external potential U(v) and a noise ensemble2 η(t): m dv(t) dt = −∂U(v(t)) ∂v + η(t) (3.2.3) dv(t) = − 1 m ∂U(v(t)) ∂v dt+ 1 m dW (t) (3.2.4) 2Since the Langevin equation is stochastic, different trajectories would come out with different sets of random numbers η. 10 CHAPTER 3. REVIEW OF STOCHASTIC QUANTIZATION where we have defined dW (t) ≡ η(t)dt, as in chapter 2. Now consider a test function f which depends on the variable v(t) and let us compute the impact on f of performing an infinitesimal variation v(t) −→ v(t) + dv: df(v(t)) = f(v(t) + dv)− f(v(t)) (3.2.5) By Taylor-expanding the right-hand side, we obtain: df(v(t)) =����f(v(t)) + ∂f(v(t)) ∂v dv + 1 2 ∂2f(v(t)) ∂v2 dv2 + ...−����f(v(t)) (3.2.6) Using Eq.(3.2.4) and neglecting terms of order higher than dt: df = − 1 m ∂f ∂v ∂U ∂v dt+ 1 m ∂f ∂v dW + 1 2m2 ∂2f ∂v2 dW 2 (3.2.7) In order to have an equilibrium state at t→ ∞, the variation of average values must vanish at such time: limt→∞ 〈 df 〉η = limt→∞ d 〈 f 〉η = 0. Since we will deal with equilibrium quantities from now on, we replace the notation limt→∞ 〈 · 〉η by simply: 〈 · 〉. Therefore, the right-hand side of the equation above must vanish on average too: 〈 − 1 m ∂f ∂v ∂U ∂v dt+ 1 m ∂f ∂v dW + 1 2m2 ∂2f ∂v2 dW 2 〉 = 0 (3.2.8) From the variation of v(t), combined with Eq.(3.2.4), we have: v(t+ dt) = v(t) + dv (3.2.9) v(t+ dt) = v(t)− 1 m ∂U(v(t)) ∂v dt+ 1 m dW (t) (3.2.10) Performing the time shift t −→ t− dt: v(t) = v(t− dt)− 1 m ∂U(v(t− dt)) ∂v dt+ 1 m dW (t− dt) (3.2.11) 3.2. THE FOKKER-PLANCK EQUATION 11 Because the value of v in the present time t is determined by the noise in the past time t− dt, there must be no correlations between v(t) and dW (t). Therefore, the second term in Eq.(3.2.8) vanishes. Besides that, from Eqs.(3.1.13) and (3.1.14), we can write the correlation of dWt as: 〈 dW (t)dW (t′) 〉η = 2λkBTdt. These facts leave us with: 〈 −∂f ∂v ∂U ∂v dt+ λkBT m ∂2f ∂v2 dt 〉 = 0 (3.2.12) Applying the definition of average value from the probability distribution ρ(v), the equation above can be written as: ∫ dvρ(v) [ −∂f ∂v ∂U ∂v + λkBT m ∂2f ∂v2 ] = 0 (3.2.13)∫ dv [ −∂f ∂v ∂U ∂v ρ(v)− λkBT m ∂f ∂v ∂ρ(v) ∂v ] = 0 (3.2.14)∫ dv ∂f ∂v [ ∂U ∂v ρ(v) + λkBT m ∂ρ(v) ∂v ] = 0 (3.2.15) where in the second equality we have used integration by parts. From the equations above, it turns out that the probability distribution satisfies the following equation: ∂U ∂v ρ(v) + λkBT m ∂ρ(v) ∂v = 0 (3.2.16) This is the so-called Fokker-Planck equation for a system in contact with a thermal bath at temperature T and subject to an external potential U(v). It plays a similar role as the Schrödinger equation, in the Schrödinger picture of quantum mechanics: that of describ- ing how probabilities evolve. Moreover, it can be straightforwardly integrated, yielding the solution: ρ(v) = ρ0e −mU(v)/λkBT = ρ0e −E(v)/kBT (3.2.17) 12 CHAPTER 3. REVIEW OF STOCHASTIC QUANTIZATION where E(v) is the energy of the system. By requiring that the integral of ρ(v) over v be equal to unity, as a normalization condition, we are able to determine ρ0 and recast ρ(v) as: ρ(v(t)) = e−mU(v)/λkBT∫ due−mU(u)/λkBT = 1 Z e−mU(v)/λkBT (3.2.18) Thus, the solution of the Fokker-Planck equation is precisely equal to the Boltzmann weight of statistical mechanics for an arbitrary potential U . Therefore, given a stochastic variable undergoing a process respecting the Langevin dynamics, in the equilibrium limit the average of any function f of v is determined by integrating f over v with weights given by Eq.(3.2.18): 〈 f 〉 = ∫ dvf(v)ρ(v) = 1 Z ∫ dvf(v)e−E(v)/kBT (3.2.19) Applying it to the previous case of U(v) = λv2/2, the partition function Z is: Z = ∫ ∞ −∞ dve−mU(v)/λkBT = ∫ ∞ −∞ dve−mv 2/2kBT = √ 2πkBT m (3.2.20) Let us take f ≡ v2 as an example and compute its average, as in section 3.1: 〈 v2 〉 = 1 Z ∫ ∞ −∞ dv v2e−mv 2/2kBT = √ m 2πkBT 1 2 √ π ( 2kBT m )3/2 = kBT m (3.2.21) Therefore, we see that the average value of v2 as 〈 v2 〉 obtained from an equilibrium dis- tribution equals the one coming from stochastic dynamics: limt→∞ 〈 v(t)2 〉η, in Eq.(3.1.12). This result elegantly connects the stochastic averages that we had from the Langevin equa- tion with the probabilistic content given by the partition function. It means that averaging along a stochastic trajectory for large t is equivalent to weight-average this function with the Boltzmann probability e−E/kBT in equilibrium. To go a bit further, if that probability distribution can be reproduced for an arbitrary potential function U , what could we do to have the same equivalence with Euclidean path integrals in quantum mechanics? The handwavy recipe to that is the following: replace the 3.3. STOCHASTIC QUANTIZATION IN NON-RELATIVISTIC QUANTUM MECHANICS13 thermodynamic measure kBT by the quantum measure ~ and redefine the potential function as the Euclidean action of the theory. That means kBT −→ ~ and U −→ SE. These replacements turn Eq.(3.2.19) into: 〈 f(v) 〉η = 1 ZF ∫ Dvf(v)e−SE [v]/~ (3.2.22) where dv was replaced by Dv to account for the functional nature of SE[v]. The key idea of stochastic quantization is encoded in the relationship between Eq.(3.2.19) and the equation above. We formalize this idea in the following section, after a brief historical perspective of quantum mechanics. 3.3 Stochastic quantization in non-relativistic quantum mechanics The first successful attempt to arrive at quantum mechanics from a classical theory was accomplished by Paul Dirac in 1926. The method, that consists in promoting classical vari- ables to operators, was first presented in Dirac’s doctoral thesis as the “method of classical analogy”. Although this method, known today as canonical quantization, has been replaced by more modern approaches, it is still widely used for pedagogical purposes in quantum me- chanics and quantum field theory textbooks. A few years later, a new idea was conceived by another brilliant mind. The concept of a path integral, shortly discussed in the previous sec- tion, was introduced by Richard Feynman and subsequently used as a powerful quantizing tool. His innovative idea to interpret quantization allowed physicists to build a theoreti- cal bridge connecting quantum mechanics to statistical mechanics in an extremely intuitive fashion. Finally, in 1981, another fascinating technique to perform quantization, this time through stochastic processes, was developed by Parisi and Wu in [Parisi et al., 1981], the so-called stochastic quantization method. 14 CHAPTER 3. REVIEW OF STOCHASTIC QUANTIZATION We saw in the previous section that the Langevin equation obeys a probability law ac- cording to a decreasing exponential, with exponent U , that we readily recognized as the Boltzmann weight. In the same chapter, we also realized that we could obtain an analog of the Euclidean path integral if we replaced U by the action of a given classical theory and kBT by ~. Therefore, we need to recast the Langevin equation, with the derivative of a classical action playing the role of drift force. Since SE is a functional object, for it is defined as an integral, we also replace partial differentiation by functional differentiation. Additionally, we postulate that the evolution of the Langevin equation is parametrized not by conventional time t but by a fictitious parameter called stochastic time3 τ such that the solution of the Langevin equation will be a stochastic function of τ and the actual time t. Therefore, the Langevin equation of stochastic quantization becomes: ∂x(t, τ) ∂τ = −δSE[x] δx + η(t, τ) (3.3.1) where τ is the stochastic time used as a fictitious parameter to lead the system towards thermalization. Due to the drift term, this newly defined process should asymptotically converge to an equilibrium state. Since τ is merely fictitious, the solution to this equation only procures physical meaning when thermalization takes place, i.e. when τ → ∞, and the equation does not depend on τ anymore. Likewise the statistical mechanical case approached in the beginning of this chapter, in the quantum case the noise source η(t, τ) must obey two similar properties: • Property 1: 〈 η(t, τ) 〉η = 0 (3.3.2) • Property 2: 〈 η(t, τ)η(t′, τ) 〉η = 2~δ(t− t′)δ(τ − τ ′) (3.3.3) where τ enters as an additional variable to the noise, introducing an extra Dirac delta func- tion. In summary, given a classical action and a noise satisfying the properties above, the 3Even though we call it time, this new parameter has nothing to do with the real time whatsoever, denoted by t. 3.3. STOCHASTIC QUANTIZATION IN NON-RELATIVISTIC QUANTUM MECHANICS15 solution of the Langevin equation furnishes the quantized version of a classical system when τ → ∞. In this limit, the stochastic averages derived from it will correspond to the quantum expectation values. For instance, quantities of the type 〈 x(t1, τ)x(t2, τ)...x(tN , τ) 〉η equate the vacuum expectation value 〈0|X̂(t1)X̂(t2)...X̂(tN)|0〉, in the operator formalism, for very large τ . This equivalence is a direct consequence of the fact that the Langevin equation obeys Boltzmann-like weights from the Fokker-Planck equation. Another remarkable fact about the Langevin equation is that any initial conditions for τ are acceptable, meaning that the system will always evolve to the same equilibrium regardless of the value of x(t, 0). Similar to the statistical mechanical case, where the noise source carried information about thermal fluctuations, in stochastic quantization it carries information about quantum fluctuations. This is an implication of property 2, whereby we deduce that η(t, τ) is pro- portional to √ ~. Hence, to recover classical mechanics (no quantum fluctuations), we can forcibly turn the noise off. Besides that, due to the asymptotic equilibrium, the first deriva- tive in the left-hand side of Eq.(3.3.1) must vanish for very large τ . Thus, applying both limits, the Langevin equation becomes: δSE[x] δx = 0 (3.3.4) which is simply the Euler-Lagrange equation. This means that stochastic quantization, in the absence of noise, is equivalent to classical mechanics in the equilibrium limit. To exemplify the usefulness of stochastic quantization in non-relativistic quantum me- chanics, we apply the Langevin equation to quantize a classical harmonic oscillator, as done in [Namiki, 2008]. Thence we compute the quantum correlator 〈0|X̂(t)X̂(t′)|0〉 in the stochas- tic formalism and compare it to the path integral result. Consider the classical action of a harmonic oscillating system: S[x] = 1 2 ∫ dt [ m ( ∂x ∂t )2 −mω2 0x 2 ] (3.3.5) 16 CHAPTER 3. REVIEW OF STOCHASTIC QUANTIZATION To make a direct connection with statistical mechanics, we need to perform a Wick rotation in the time variable and work with the Euclidean action. A Wick rotation rotates the time direction by 90◦ in the complex plane: t→ −it. By doing that, we end up with the Euclidean action: SE[x] = 1 2 ∫ dt [ m ( ∂x ∂t )2 +mω2 0x 2 ] (3.3.6) Then, to compute 〈0|X̂(t)X̂(t′)|0〉, in path integral, we need to solve: 〈0|X̂(t)X̂(t′)|0〉 = 1 Z ∫ Dx x(t)x(t′)e−SE [x]/~ (3.3.7) Although dreadful, the calculation can be done analytically and the result is: 〈0|X̂(t)X̂(t′)|0〉 = ~ 2mω0 e−ω0|t−t′| (3.3.8) To turn around such costly hand-work, we can use stochastic quantization instead. Applying Eq.(3.3.1) to this system, by acting with the functional derivative on the action above, yields the following Langevin equation: ∂x(t, τ) ∂τ = m ∂2x(t, τ) ∂t2 −mω2 0x(t, τ) + η(t, τ) (3.3.9) where we now describe the position of the harmonic oscillator as a stochastically fluctuating function of t and τ . Similarly to the case of Brownian motion, step one is to solve Eq.(3.3.9), and step two computing stochastic averages. The first step can be accomplished by Fourier- transforming Eq.(3.3.9): ∂x̃(ω, τ) ∂τ = −m(ω2 + ω2 0)x̃(ω, τ) + η̃(ω, τ) (3.3.10) 3.3. STOCHASTIC QUANTIZATION IN NON-RELATIVISTIC QUANTUM MECHANICS17 and also Fourier-transforming the noise property in Eq.(3.3.3): 〈 η̃(ω, τ)η̃(ω′, τ ′) 〉η = 2~δ(ω + ω′)δ(τ − τ ′) (3.3.11) Defining Ω2(ω) ≡ m(ω2 + ω2 0) to simplify notation, we use variation of parameters to write an explicit solution for Eq.(3.3.10) as: x̃(ω, τ) = x̃(ω, 0)e−Ω2(ω)τ + ∫ τ 0 dτ1e Ω2(ω)(τ1−τ)η̃(ω, τ1) (3.3.12) The analog of the quantum correlator 〈0|X̂(t1)X̂(t2)|0〉 in the stochastic formalism is given by the average 〈 x(t, τ)x(t′, τ ′) 〉η, which we first obtain in frequency space: 〈 x̃(ω, τ)x̃(ω′, τ ′) 〉η = 〈 x̃(ω, 0)x̃(ω′, 0) 〉η e −Ω2(ω)τ−Ω2(ω′)τ ′+ +x̃(ω, 0)e−Ω2(ω)τ ∫ τ ′ 0 dτ2e Ω2(ω′)(τ2−τ ′) 〈 η̃(ω′, τ2) 〉η + +x̃(ω′, 0)e−Ω2(ω′)τ ′ ∫ τ 0 dτ1e Ω2(ω)(τ1−τ) 〈 η̃(ω, τ1) 〉η + + ∫ τ 0 ∫ τ ′ 0 dτ1dτ2e Ω2(ω)(τ1−τ)+Ω2(τ2−τ ′) 〈 η̃(ω, τ1)η̃(ω′, τ2) 〉η (3.3.13) Because of property 1: 〈 η̃(ω, τ1) 〉η = 〈 η̃(ω′, τ2) 〉η = 0, and we are left with: 〈 x̃(ω, τ)x̃(ω′, τ ′) 〉η = 〈 x̃(ω, 0)x̃(ω′, 0) 〉η e −Ω2(ω)τ−Ω2(ω′)τ ′+ + ∫ τ 0 ∫ τ ′ 0 dτ1dτ2e Ω2(ω)(τ1−τ)+Ω2(ω′)(τ2−τ ′) 〈 η̃(ω, τ1)η̃(ω′, τ2) 〉η (3.3.14) Using the Fourier-transform of property 2 in the second term on the right-hand side to solve the double integral, considering the two possible cases (either τ ≤ τ ′ or τ > τ ′), the solution becomes: 〈 x̃(ω, τ)x̃(ω′, τ ′) 〉η = 〈 x̃(ω, 0)x̃(ω′, 0) 〉η e −Ω2(ω)τ−Ω2(ω′)τ ′+ 18 CHAPTER 3. REVIEW OF STOCHASTIC QUANTIZATION + ~ Ω2(ω) δ(ω + ω′) [ e−Ω2(ω)|τ−τ ′| − e−Ω2(ω)(τ+τ ′) ] (3.3.15) Using the relation between the frequency-space correlation and the Green function from [Namiki, 2008]: 〈 x̃(ω, τ)x̃(ω′, τ ′) 〉η = δ(ω + ω′)G̃(ω, τ − τ ′), then integrating over ω′, the equation above can be rewritten in terms of the Green function in frequency space G̃ as: G̃(ω, τ − τ ′) = G̃(ω, 0)e−Ω2(ω)(τ+τ ′) + ~ Ω2(ω) [ e−Ω2(ω)|τ−τ ′| − e−Ω2(ω)(τ+τ ′) ] (3.3.16) Making τ ′ = τ , we have: G̃(ω, 0) = G̃(ω, 0)e−2Ω2(ω)τ + ~ Ω2(ω) [ 1− e−2Ω2(ω)τ ] ∴ G̃(ω, 0) = ~ Ω2(ω) (3.3.17) Replacing it back into Eq.(3.3.16), we have a complete Green’s function for the harmonic oscillator in frequency space: G̃(ω, τ − τ ′) = ~ Ω2(ω) e−Ω2(ω)|τ−τ ′| (3.3.18) Returning to position space through inverse Fourier transform: G(t− t′, τ − τ ′) = ~ 2πm ∫ +∞ −∞ dω 1 ω2 + ω2 0 eiω(t−t ′)−Ω2(ω)|τ−τ ′| (3.3.19) Identifying the existence of complex poles at ω = iω0 and ω = −iω0 in this integral, the equation above can be easily calculated through the residues theorem of complex analysis, which, after the asymptotic limit, results in: lim τ=τ ′→∞ G(t− t′, τ − τ ′) = G(t− t′) = ~ 2mω0 e−ω0|t−t′| (3.3.20) The right-hand side of this equation is precisely the 2-point correlation function of a harmonic oscillating particle as we obtained through the path integral in Eq.(3.3.8). This result reflects 3.4. STOCHASTIC QUANTIZATION IN QUANTUM FIELD THEORY 19 the equivalence between quantum mechanics and stochastic quantization in the asymptotic limit as claimed before. Although we presented this equivalence for a very specific case, the applicability of stochastic quantization does not come to a halt in the territory of quantum mechanics. Therefore, in the next section, we review stochastic quantization in quantum field theory. 3.4 Stochastic quantization in quantum field theory To describe quantum field theory (QFT hereafter) in stochastic quantization, we simply need to adapt the formalism to a more fundamental object, other than quantum mechanical systems or grains of pollen. Throughout this work, we consider the Euclidean Φ4-theory to review and, later on, present our new results in QFT. In 1-dimensional space, the scalar field with symmetric phase (positive mass term) has an action given by: SE[Φ] = ∫ dx ( 1 2 ∂µΦ∂ µΦ + 1 2 m2Φ2 + λ 4! Φ4 ) (3.4.1) The case of spontaneous symmetry breaking (when the mass term is negative) will be discussed in chapter 4. The shape of the interaction potential is given by Figure 3.2, and its expression is given by the equation beside it: U(Φ) = 1 2 m2Φ2 + λ 4! Φ4 (3.4.2) 20 CHAPTER 3. REVIEW OF STOCHASTIC QUANTIZATION Figure 3.2: Potential function without spontaneous symmetry breaking. The empty black circle in the origin indicates the single classical vacuum for this potential. The square mass and coupling used in the plot are given in the upper part. In order to insert the Φ4-theory in the stochastic formalism, we start by defining a new field to be a stochastic function of spacetime and τ : ϕ(x, τ), alongside its correspondent Langevin equation as: ∂ϕ(x, τ) ∂τ = −δSE[Φ] δΦ(x) ∣∣∣∣ Φ→ϕ + η(x, τ) (3.4.3) where the noise source η(x, τ) is required to satisfy analogous properties as well, similar to the case of the previous chapter: • Property 1: 〈 η(x, τ) 〉η = 0 (3.4.4) • Property 2: 〈 η(x, τ)η(x′, τ) 〉η = 2~δ(x− x′)δ(τ − τ ′) (3.4.5) Notice the subtle difference between the physical field Φ = Φ(x), and its stochastic counter- part ϕ = ϕ(x, τ). In the latter case there is an extra depence on the fictitious time τ , since it is needed to describe the stochastic dynamics of ϕ. By acting with the functional derivative on Eq.(3.4.1) and doing Φ → ϕ, the correspondent Langevin equation can be easily obtained as: ∂ϕ ∂τ = (�−m2)ϕ− λ 6 ϕ3 + η (3.4.6) For that equation and the following ones, we consider ~ ≡ 1, unless we explicitly state the opposite. The dependencies on x and τ were omitted for simplicity. Once the Langevin equation is known, the remaining task is to find its solution by following the same procedure of section 3.3. Furthermore, we could work out the Langevin equation in momentum space by simply Fourier-transforming the field ϕ in the variable x: ∫ dPeiPx ∂ϕ̃(P, τ) ∂τ =− ∫ dPeiPx(P 2 +m2)ϕ̃(P, τ)+ 3.5. QUANTUM N -POINT FUNCTIONS 21 − λ 6 ∫ dPdqdkei(P+q+k)xϕ̃(P, τ)ϕ̃(q, τ)ϕ̃(k, τ) + ∫ dPeiPxη̃(P, τ) (3.4.7) Multiplying both sides by e−ipx and integrating with respect to x introduces a Dirac delta δ(P − p) which kills the integral over P and simplifies the equation above to: ∂ϕ̃(p, τ) ∂τ = −(p2 +m2)ϕ̃(p, τ)− λ 6 ∫ dqdkϕ̃(p− q − k, τ)ϕ̃(q, τ)ϕ̃(k, τ) + η̃(p, τ) (3.4.8) Since no approximations were done so far, the solution of Eqs.(3.4.6) and (3.4.8) can be used to obtain fully non-perturbative quantities exactly, either in position or momentum space. However, depending on the shape of the potential, this task might or might not be feasible. Unfortunately, the Φ4 potential falls into the “not” case, so to turn around perturbation theory we need to resort to numerical techniques. These we postpone until section 4, for now we concentrate on n-point functions in quantum field theory. 3.5 Quantum n-point functions The n-point correlation functions are the cornerstones to study physical processes in quantum field theory. Even though they are not observables themselves, they play an important role in obtaining observables from the so-called renormalization procedure, which will be the main topic of chapter 6. From the brief discussion on Feynman path integrals of section 3.2, the Euclidean n-point functions of a theory are computed in the operator formalism as the weighted sum given by a functional integration: 〈0|Φ̂(x1)Φ̂(x2)...Φ̂(xn)|0〉 = 1 ZF ∫ DΦ Φ(x1)Φ(x2)...Φ(xn)e −SE [Φ] (3.5.1) where |0〉 is the vacuum state of the theory. To compute the right-hand side of the equation above in stochastic quantization, we should start from the Langevin equation. Suppose we 22 CHAPTER 3. REVIEW OF STOCHASTIC QUANTIZATION managed to solve Eq.(3.4.6) for a given ensemble of noises η(x, τ) and a fixed initial condition for ϕ(x, 0). The qualitative behavior of the solution would show up as a noisy profile tending to have an asymptotically flat average as τ grows. This flat part is precisely the equilibrium state that we seek to measure. However, since we have solved it only once, the measured value is embedded in statistical fluctuations. Therefore, we should solve it again, with another set of noises, and repeat the task many times. By collecting plenty of independent stochastic solutions of Eq.(3.4.6), say ϕ1, ϕ2, ..., ϕNr , where Nr is called the number of noise realizations, the arithmetic average of these values is a better estimation of the equilibrium state. In the limit of infinite solutions (Nr → ∞), one would find a perfectly smooth curve, which recovers the path integral result as τ → ∞, with decreasingly small errors. Before that limit, we define the n-point correlation function in stochastic quantization for a finite τ as: 〈 ϕ(x1, τ)ϕ(x2, τ)...ϕ(xn, τ) 〉η = lim Nr→∞ 1 Nr Nr∑ j=1 ϕj(x1, τ)ϕj(x2, τ)...ϕ(xn, τ) (3.5.2) According to the Fokker-Planck equation, in the large τ limit the ϕ’s must obey the Boltz- mann distribution e−SE [Φ], thus the expectation values in Eq.(3.5.2) should agree with the path integral: lim τ→∞ 〈 ϕ(x1, τ)ϕ(x2, τ)...ϕ(xn, τ) 〉η = 1 ZF ∫ DΦ Φ(x1)Φ(x2)...Φ(xn)e −SE [Φ] (3.5.3) This is a direct consequence of the central limit theorem presented in chapter 2, which implies that an arithmetic average (left side) and a weighted average (right side) are identical in the infinite sample limit. This was seen in section 3.3 for the case of the 2-point function of a quantum harmonic oscillator, but it is valid in general for n-point functions in QFT. We conclude this section by summarizing how stochastic quantization is applied to calculate these quantities in real problems, which will be important for numerical simulations. • Derive the correspondent Langevin equation for the target theory given its action, as shown in Eq.(3.4.3). 3.5. QUANTUM N -POINT FUNCTIONS 23 • Obtain numerous solutions with different noise sets for all x until τ is large enough for the system to be thermalized in each solution. • Compute the arithmetic average of (products of) these solutions, as in Eq.(3.5.2), for large τ . 24 CHAPTER 3. REVIEW OF STOCHASTIC QUANTIZATION Chapter 4 Lattice Simulations in Φ4-theory Lattice theories are among the most powerful techniques in non-perturbative QFT, for they dismiss the use of perturbative expansions and provide a natural path towards computer sim- ulations. Hence, in this chapter, we use standard methods of lattice QFT from [Smit, 2002] to formulate stochastic quantization in discrete space. Thence, we introduce our new code to obtain numerical solutions of the Langevin equation in Φ4-theory and compute n-points function following the average procedure of [Berges and Stamatescu, 2005]. We commence by using the code to reproduce some of the known results of Φ4-theory and prepare the ground for our new results in the next chapter. To simulate the Langevin equation in the computer we need to discretize the field ϕ on a lattice, which consists of granulating it in a spacetime defined by lattice sites, instead of continuous variables. 4.1 The scalar field on a lattice Everything so far has been done in continuous space, that is with the spacetime components x ∈ R, but in lattice QFT the field is set to exist only in specific points placed a distance a, called lattice spacing, apart from its nearest neighbors. This construction conveys a crystalline structure to spacetime, making it look like a net of sewed strings, with the field 25 26 CHAPTER 4. LATTICE SIMULATIONS IN Φ4-THEORY defined in the nodes. For a d-dimensional hypercubic lattice with N sites1 in each spatial direction, its length is defined as L ≡ aN and its volume as V = Ld. Moreover, we choose our lattices to be periodic, in which case the fields must satisfy periodic boundary conditions: ϕ(x, τ) = ϕ(x + L, τ), ∀τ . Figure 4.1 illustrates two types of lattice topologies for periodic conditions. This new framework requires replacing the original action by a discrete version Figure 4.1: Lattice topologies for periodic boundary conditions in 1 dimension (left) and 2 dimensions (right). Due to the periodic boundary conditions, in one dimension the lattice assumes a ring-like shape, whereas in two it turns into a torus. that encodes the same dynamics and restores its continuous counterpart in the appropriate limit. Usually, there are plenty of actions with this characteristic. In d dimensions, the simplest one resembling its continuous analog is the following: SE = ad ∑ x [∑ µ ∂µΦ(x)∂ µΦ(x) + 1 2 m2Φ2(x) + λ 4! Φ4(x) ] (4.1.1) where the continuous derivatives are redefined on the lattice as: ∂µΦ(x) −→ 1 a [Φ(x+ aµ̂)− Φ(x)] (4.1.2) 1Not to be confused with the number of noise realizations Nr 4.1. THE SCALAR FIELD ON A LATTICE 27 and µ̂ is a unit vector pointing on a specific direction in spacetime. In the non-interacting case (λ = 0), this theory is completely solvable and the momentum space propagator can be obtained explicitly, as in [Montvay and Münster, 1997]: G̃(k) = [ 4 a2 sin2 ( kπ N ) +m2 ]−1 (4.1.3) where k, with −N/2 ≤ k < N/2, is the momentum quantum number. It can be verified that, due to periodic boundary conditions, the propagator satisfies: G̃(k) = G̃(k + N). In the continuum limit (a→ 0 and N → ∞), it can be proven that the equation above restores the free propagator of a scalar theory: (k2+m2)−1. Using the inverse Fourier transformation in descrete space, we obtain the position-space propagator for the free theory: G(0)(x1, x2) = 1 N N−1∑ k=0 e2iπ(x1−x2)k/N 4 sin2 (kπ/N)/a2 +m2 (4.1.4) In the interacting case, obtaining an analytical expression requires perturbative corrections in lattice theory. These corrections arise from expansions in the path integral, giving birth to the so-called quantum loops. Leaving the details for the next chapter, we simply state that by expanding the interacting part within SE[Φ] in the right-hand side of Eq.(3.5.1), namely the one where the coupling λ appears, one gets the desired corrections order by order. In the case of the 2-point function: 〈0|Φ̂(x)Φ̂(x′)|0〉 = G(0)(x, x′) +G(1)(x, x′) +G(2)(x, x′) +O(3) (4.1.5) where the superscript 0 refers to the free contribution and the other terms, which depend on the coupling, bear the interactions. However, in practical terms, we are unable to compute the full sum, for each term gets more and more complicated as the order increases. Therefore, given an order n, by truncating this infinite sum at some order, we define an approximate solution for the 2-point function, denoted by Gn−loop(x, x′), which takes into account terms 28 CHAPTER 4. LATTICE SIMULATIONS IN Φ4-THEORY with proportionality from λ0 up until λn. The explicit formulas of Gn−loop(x, x′) when n = 1 or 2 are provided through lattice perturbation theory in chapter 5 for the symmetric and broken phase cases. 4.2 Numerical simulations using LASER code When the target theory features a complicated interaction term, hampering exact solutions to be obtained, numerical algorithms stand as magnificent resources. The most widely known among the lattice scientific community is called the MCMC, mentioned in the introduction, which consists in selecting relevant field configurations2 and average over them with the Boltzmann weight e−SE (we make a few remarks concerning this topic in appendix A.2). On the other hand, through the stochastic quantization procedure, we can evolve a stochastic Partial Differential Equation (PDE) and obtain field configurations ϕ(x, τ) whose probability distributions ρ[ϕ] are already given by the correct Boltzmann factor, and then compute their arithmetic average instead. A considerable part of the work for this master’s thesis was to build a code from scratch that performed this task consistently. This code was entirely written in the C++ programming language and we call it LASER (an abbreviation of LAngevin SolvER). We use it in this chapter to reproduce quantities in the Φ4-theory that are previously known from perturbation theory. For simplicity, we take the 1-dimensional case and recast Eq.(3.4.6) in discrete form by turning continuous derivatives into discrete ones, with x and τ being replaced by lattice indices j and n: ∂ϕ(x, τ) ∂τ −→ 1 ∆τ (ϕn+1 j − ϕnj ) (4.2.1) ∂2ϕ(x, τ) ∂x2 −→ 1 a2 (ϕnj+1 − 2ϕnj + ϕnj−1) (4.2.2) 2From the minimum action principle, a relevant field configuration signifies one that minimizes the action. 4.2. NUMERICAL SIMULATIONS USING LASER CODE 29 Replacing them into Eq.(3.4.6) yields: ϕn+1 j = ϕnj +∆τ [ ϕnj+1 − 2ϕnj + ϕnj−1 a2 −m2ϕnj − λ 6 (ϕnj ) 3 + ηnj ] (4.2.3) where we have introduced the discrete version of the gaussian noise ηnj in the place of η(x, τ). From the continuous properties 1 and 2, recalling a Dirac delta δ(y) has dimensions of 1/y, the equivalent discrete properties are given by: 〈 ηnj 〉 η = 0 〈 ηnj η m k 〉 η = 2 ∆τa δjkδnm (4.2.4) These properties are the touchstones of a gaussian white noise. They imply that the discrete noise must be of the form ηnj = √ 2/∆τaRn j , where Rn j is a random number to be selected with gaussian probability for given j and n, as we saw in Figure 3.1 of chapter 3. Replacing it into Eq.(4.2.3), we get: ϕn+1 j = ϕnj +∆τ [ ϕnj+1 − 2ϕnj + ϕnj−1 a2 −m2ϕnj − λ 6 (ϕnj ) 3 ] + √ 2∆τ a Rn j (4.2.5) Finding a suitable algorithm, Eq.(4.2.3) can be implemented and efficiently solved with high precision on codes (we briefly remark on algorithms in appendix A.1). For convenience, we eliminate the lattice spacing a from the equation above by naively fixing it as a ≡ 1, temporar- ily purging the Langevin equation of the lattice spacing. Chapter 6 will be entirely dedicated to exploring a more fundamental interpretation of this parameter in renormalization. In 2 or more dimensions, we merely need to add more indices and extra derivatives relative to those indices. In Figures 4.2 below we show solutions of the Langevin equation for different initial conditions and for increasing noise realizations Nr. From the general definition in Eq.(3.5.2) one can readily obtain n-point correlation functions of the field as outputs of simulations by averaging over products of the field. In Figure 4.3, we compute a 2-point function of the type 〈 ϕ(x1, τ)ϕ(x2, τ) 〉η from τ = 0 until τ sufficiently large to thermalize 30 CHAPTER 4. LATTICE SIMULATIONS IN Φ4-THEORY (a) (b) Figure 4.2: (a) Several solutions of the Langevin equation as a function of τ , using different initial conditions, reinforcing that the asymptotic solution is always the same and given by the thermalized zone. (b) Stochastic averages of the field ϕ with increasing number of noise realizations, where we notice drastic improvements as we go from 1 to 10, then from 10 to 1000 samples. and converge to the QFT propagator 〈0|Φ̂(x1)Φ̂(x2)|0〉 in the free case for several lattice distances x1 − x2. Thereby, we analyze its accuracy by also plotting the expected analytical values given by Eq.(4.1.4). Similar results as a function of the stochastic time may be found in previous investigations of non-equilibrium scalar fields in [Berges and Stamatescu, 2005], and for semi-classical gravity fields in [dos Reis et al., 2019]. In Figure 4.4, we reproduce the free propagators of Φ4-theory computed for different lattice sizes using the average procedure illustrated in Figure 4.3. In the same plot, we compare the simulation outputs with the analytical values given by Eq.(4.1.4) for different number of sites, N . Notice that in all cases, the propagators as a function of the lattice distance x1 − x2 stand in accordance with the analytical values. Moreover, we could also estimate the QFT 4-point function 〈0|Φ̂(x1)Φ̂(x2)Φ̂(x3)Φ̂(x4)|0〉 using the large τ limit of the stochastic average of four fields, given by: χ4(x1, x2, x3, x4) = lim τ→∞ 〈 ϕ(x1, τ)ϕ(x2, τ)ϕ(x3, τ)ϕ(x4, τ) 〉η (4.2.6) and also the ones containing odd products of fields, like the 3-point function: χ3(x1, x2, x3) = lim τ→∞ 〈 ϕ(x1, τ)ϕ(x2, τ)ϕ(x3, τ) 〉η (4.2.7) 4.2. NUMERICAL SIMULATIONS USING LASER CODE 31 Figure 4.3: In the left-hand side, the fluctuating continuous lines show the convergence of stochastic averages of the free propagator on a 10× 10 lattice computed for three distances as a function of τ . The dashed lines represent the mean value of the thermalized zone, using Nr = 60, leading to their physical value as a function of the distance x1 − x2 in the right- hand side. Empty red circles indicate the analytical result and their excellent match with the simulation. However, we postpone the calculation of 4-point functions until chapter 6, where they play an essential role to perform the continuum limit of the lattice. Similar to the free case, we can execute simulations with non-zero coupling and solve the Langevin equation to its full extent for arbitrarily large λ. The results can then be compared with the perturbative expansion up to 2 loops as in Eq.(4.1.5), verifying the agreement between theory and simulations. In Figure 4.5, we show numerical and analytical propagators as functions of the distance for finite values of λ in a 10× 10 lattice. For increasing values of the coupling, we barely notice deviations of the 2-loop analytical propagator from the simulation, showing that stochastic quantization remains consistent even for strong λ. As the last proof of the validity of stochas- tic quantization, consider that we are interested in reproducing the momentum-space 2-point function. To accomplish this task, we could proceed in two ways: either solve the Langevin equation directly in momentum space, namely Eq.(3.4.8), or solving it in position space and Fourier-transform the solution before computing averages. However, the former turns out to be extremely costly in terms of CPU time, for we need to solve an integro-differential equa- 32 CHAPTER 4. LATTICE SIMULATIONS IN Φ4-THEORY Figure 4.4: Numerical 2-point functions in the free case from stochastic quantization (black lines-and-points) for different lattice sizes compared to the exact analytical values (red empty circles). The dashed vertical lines are placed in the middle of each lattice, where it starts to be periodic. tion. Therefore, we adopt the latter option and perform a Fourier transform with ϕ(x, τ) to obtain ϕ̃(p, τ) for several values of p = 2πk/N by varying the quantum number k. In Figure 4.6, the results for the numerical propagators in momentum space are shown alongside the analytical perturbative calculations. Hitherto, plenty of successful examples have been added to the stochastic quantization repertoire for the symmetric case, i.e. when the theory has a single vacuum. Now, we deal with a case where the theory has more than one vacuum. 4.3 Spontaneous symmetry breaking In the previous section, we saw that, because the theory had just one vacuum, the solution of the Langevin equation always converged to the same equilibrium, regardless of the initial conditions. In this sense, the theory was symmetric, since we could replace the vacumm Φ0 4.3. SPONTANEOUS SYMMETRY BREAKING 33 Figure 4.5: Numerical (black lines-and-points) and analytical (empty red circles) propagators for nine values of λ averaged with 100 noise realizations each. Notice that the 2-loop analytic propagator remains consistent with simulations. by −Φ0, and nothing would change. On the other hand, if we invert the sign of the mass term in Eq.(4.1.1), such that the new action becomes: SE = ad ∑ x [∑ µ ∂µΦ(x)∂ µΦ(x)− 1 2 m2Φ2(x) + λ 4! Φ4(x) ] (4.3.1) 34 CHAPTER 4. LATTICE SIMULATIONS IN Φ4-THEORY Figure 4.6: Numerical 2-point functions in momentum space (colourful lines-and-points) compared to the analytical perturbation theory (empty black circles) as a function of the quantum number k. The simulations were done on a 16× 16 lattice. the potential acquires a novel shape, as shown in Figure 4.7 below. U(Φ) = −1 2 m2Φ2 + λ 4! Φ4 (4.3.2) Figure 4.7: Potential function in the case of spontaneous symmetry breaking. The empty black circles indicate the two classical vacua for this potential. The square mass and coupling used in the plot are given in the upper part. In this case, there are two available vacua on which the equilibrium can be placed, and the symmetry between Φ0 and −Φ0 no longer exists. By either heading to the positive or to the negative vacuum, the field breaks the vacuum symmetry of the action, and this symmetry is said to be spontaneously broken, as discussed in [Smit, 2002]. This behavior 4.3. SPONTANEOUS SYMMETRY BREAKING 35 should manifest itself through the convergence of the Langevin equation to Φ0 or −Φ0. Thus, equipped with the tools from the previous section, we can apply them to reproduce known results from spontaneous symmetry breaking in lattice Φ4-theory. The location of the two minima indicated in Figure 4.7 can be easily identified, applying simple rules of calculus to the potential in Eq.(4.3.2), as: Φ± 0 = ±m √ 6 λ (4.3.3) Replacing the values used in the plot (m2 = 1 and λ = 0.5), we find Φ± 0 = ± √ 12 ≈ ±3.464, which correspond to the two classical vacua of the theory. Thence we can determine the depth of the potential barrier of the vacua, given by ∆U0 ≡ U(0)− U(Φ± 0 ): ∆U0 = 3m4 2λ (4.3.4) In Figure 4.8 below, we demonstrate how the field bends towards one vacuum or the other depending on the initial condition by solving the Langevin equation with broken symmetry. Compare it with Figure 4.2, without symmetry breaking, where the vacuum at the origin was the only possibility. Even though the bosonic field can choose one vacuum or the other, (a) (b) Figure 4.8: (a) Several solutions of the Langevin equation on a 8×8 lattice, with spotaneous breaking of symmetry using different initial conditions around each classical vacuum. (b) 10 more solutions, all with initial condition set to the origin (which corresponds to the local maximum of the potential). In this case, the solutions randomly select a vacuum to which they fall. The parameters used in the simulations are indicated above each plot. 36 CHAPTER 4. LATTICE SIMULATIONS IN Φ4-THEORY dwelling the rest of its life there, if the coupling is strong enough, quantum fluctuations can induce transitions between the vacua. Since the potential barrier is inversely proportional to λ, from Eq.(4.3.4), the larger the coupling the less effort requires the particle to cross the energy obstacle. Therefore, there must be a critical value of λ for which the field would start to switch between Φ+ 0 and Φ− 0 from time to time. Furthermore, the transition rate (number of times the field crosses the potential barrier per unit of stochastic time) must be proportional to the coupling. Figure 4.9 shows three solutions with λ ten times bigger than in the other cases where we see the field crossing the potential barrier of the vacua and bouncing between them. In the broken symmetry case, since we can have either Φ+ 0 or Φ− 0 , in order to compute Figure 4.9: Three solutions with spontaneous symmetry breaking for large coupling (λ = 5), showing transition between the vacua. The two upper plots have been displaced vertically for clarity, with the dashed lines indicating the local maximum in each case. the n-point functions, the vacuum contribution must be subtracted. Below, we show the new expression for the 2-point function with broken symmetry: G(x1, x2) = lim τ→∞ [ 〈 ϕ(x1, τ)ϕ(x2, τ) 〉η − 〈 ϕ(x1, τ) 〉η 〈 ϕ(x2, τ) 〉η ] (4.3.5) To this expression, we give the name of connected Green function. Notice that without 4.3. SPONTANEOUS SYMMETRY BREAKING 37 breaking of symmetry, the second term in the equation above is zero, since 〈 ϕ(x1, τ) 〉η = 〈 ϕ(x2, τ ′) 〉η = 0, and it reduces to the quantity previously calculated. Furthermore, with the vacuum expectation value being different from zero, in order to calculate the 2-point function, it turns out to be more convenient to shift the field and solve the Langevin equation around one of the classical vacua Φ± 0 . To exemplify, let us choose the positive vacuum Φ0 ≡ Φ+ 0 and define a shifted field as σ ≡ Φ−Φ0. The action of the theory can thus be expressed in terms of σ as: SE[σ] = ∑ x [ 1 2 ∂µσ∂ µσ − 1 2 m2(σ + Φ0) 2 + λ 4! (σ + Φ0) 4 ] (4.3.6) = ∑ x [ 1 2 ∂µσ∂ µσ − 1 2 m2(σ2 + 2σΦ0 + Φ2 0) + λ 4! (σ4 + 4σ3Φ0 + 6σ2Φ2 0 + 4σΦ3 0 + Φ4 0) ] (4.3.7) Replacing the known expression for the classical vacuum Φ0 = √ 6m2/λ inside the equation above, we can simplify and reduce the equation above to: SE[σ] = ∑ x [ 1 2 ∂µσ∂ µσ + 1 2 m2σ2 + λ 4! σ4 + √ λ 6 mσ3 − 3m4 2λ ] (4.3.8) Since constant terms in the action do not contribute to the equations of motion, we can neglect the last term in the equation above. By making σ → Φ again, we end up with: SE[Φ] = ∑ x [ 1 2 ∂µΦ∂ µΦ + 1 2 m2Φ2 + λ 4! Φ4 + √ λ 6 mΦ3 ] (4.3.9) This is the new action for the broken phase case. Notice that a new sort of interaction arises, which intermixes coupling and mass. This novel term, which is proportional to √ λ, produces a phase with broken symmetry. Now, we can translate this system into a stochastic process through the Langevin equation: ∂ϕ ∂τ = (�−m2)ϕ− λ 6 ϕ3 − √ 3λ 2 mϕ2 + η (4.3.10) 38 CHAPTER 4. LATTICE SIMULATIONS IN Φ4-THEORY and in discrete form: ϕn+1 j = ϕnj +∆τ [ ϕnj+1 − 2ϕnj + ϕnj−1 a2 −m2ϕnj − λ 6 (ϕnj ) 3 − √ 3λ 2 m(ϕnj ) 2 ] + √ 2∆τ a Rn j (4.3.11) Notice that in the free case (when λ = 0), both symmetric and non-symmetric cases have the same propagator, for they have the coupling-free content. Besides that, by switching the sign of ϕ in the equation above, the Langevin equation does not remain the same, as before, due to the term proportional to Φ2. The different behavior created by this term will also manifest itself in the analytical calculations presented in the next chapter when we approach stochastic quantization in perturbative expansion. Chapter 5 Review of Noise Perturbation Theory and New Results For most theories in the Standard Model of particle physics we can not find an exact so- lution, and perturbative expansion becomes the common approach to deal with them. In the path integral formalism, for example, we start from a given action and expand the part that depends on the coupling, leaving the free part unchanged, and compute perturbative approximations where each term is proportional to a power of this coupling. Each of these expanded terms corresponds to an intuitive picture, called a Feynman diagram, depicting a sort of interaction. In this chapter, we follow [Montvay and Münster, 1997] to review the main aspects of the perturbative method on the lattice, applied to the Φ4-theory case. com- paring it with another type of perturbation in stochastic quantization. Afterwards, based on [Attanasio, 2013], we discuss how perturbation theory is implemented within stochastic quantization and present the new results with LASER, in 2 and 4 dimensions. 5.1 Feyman diagrams Consider the 1-dimensional Euclidean action of a single scalar theory on the lattice with Φ4 interaction term akin to chapter 4. The Feynman path integral for this theory can be written 39 40CHAPTER 5. REVIEW OF NOISE PERTURBATION THEORY AND NEW RESULTS as: ZF = ∫ Dϕ e− ∑ x ( 1 2 ∂µΦ∂µΦ+ 1 2 m2Φ2+ λ 4! Φ4 ) (5.1.1) Even though we chose the 1-dimensional case, for simplicity, every step taken hereafter is identically mapped into higher dimensional cases with minor modifications. The 2-point function can be obtained from the path integral as: 〈0|Φ̂(x1)Φ̂(x2)|0〉 = 1 ZF ∫ DΦ Φ(x1)Φ(x2)e − ∑ x ( 1 2 ∂µΦ∂µΦ+ 1 2 m2Φ2+ λ 4! Φ4 ) (5.1.2) Expanding the term proportional to λ in the exponent: 〈0|Φ̂(x1)Φ̂(x2)|0〉 = 1 ZF ∑ n 1 n! ∫ DΦ Φ(x1)Φ(x2) [ λ 4! ∑ x Φ4(x) ]n e− ∑ x ( 1 2 ∂µΦ∂µΦ+ 1 2 m2Φ2 ) (5.1.3) Therefore, the 2-point function can be written as a sum of terms that depend on the summing index n with a factor λn: 〈0|Φ̂(x1)Φ̂(x2)|0〉 = ∑ n G(n)(x1, x2) (5.1.4) which is precisely the formula we proposed in Eq.(4.1.5), with the definition: G(n)(x1, x2) ≡ λn ZFn!(4!)n ∫ DΦ Φ(x1)Φ(x2) [∑ x Φ4(x) ]n e− ∑ x ( 1 2 ∂µΦ∂µΦ+ 1 2 m2Φ2 ) (5.1.5) Despite its scarying appearence, the right-hand side of this equation can be computed with a bunch of mathematical tricks involving functional derivatives. Due to the large number of steps, without any deep insights concerning stochastic theory, we choose not to show all of them explicitly. Instead, we refer the readership to this very detaild book on path integrals: [Das, 1993], where these tricks are presented in the continuum, whereby we adapt the steps to the lattice case. Each term generated by a given n produces some Feynman diagrams corresponding to interactions proportional to λn. The sum of all diagrams from n = 0 to 5.1. FEYMAN DIAGRAMS 41 n → ∞ result in the full interactive propagator. In Figure 5.1, we illustrate a few of these diagrams that come out from the loop expansion. First, to compute G1−loop(x1, x2) in the Figure 5.1: Relevant diagrams for perturbative Φ4-theory in the symmetric phase. 1-dimensional case, we need to solve the awful-looking Eq.(5.1.5) for n = 1 and add all contributions in the form of a so-called 1 Particle Irreducible (hereafter 1-PI) diagram, i.e., a diagram that can not be broken into simpler diagrams. All the diagrams shown above are legitimate 1-PI diagrams, for they can not be separated into simpler pieces, whereas if we combined two of them with a propagator line it would not be 1-PI, since it would comprise two fundamental diagrams. At 1 loop, the solution to the right-hand side of Eq.(5.1.5) is given by: G(1)(x1, x2) = −λ 2 ∑ x G(0)(x1, x)G (0)(x, x)G(0)(x, x2) (5.1.6) which in momentum space looks like: G̃(1)(p) = −λ 2 G̃(0)(p) [ 1 N ∑ q G̃(0)(q) ] G̃(0)(p) ≡ −λ 2 G̃(0)(p)J1G̃ (0)(p) (5.1.7) where N is the number of lattice sites1, and we have introduced the function J1 to aid in the following steps. Finally, we need to add up all combinations of 1-PI diagrams at 1-loop. Graphically, we have: 1In the original expression it should be the lattice size L = aN , but since we made a = 1 for convenience, L can be replaced by N in this case. 42CHAPTER 5. REVIEW OF NOISE PERTURBATION THEORY AND NEW RESULTS G̃1−loop(p) = + + + ... (5.1.8) = G̃(0)(p) − λ 2 G̃(0)(p)J1G̃ (0)(p) + λ2 4 G̃(0)(p)J1G̃ (0)(p)J1G̃ (0)(p) + ... (5.1.9) = G̃(0)(p)[1 − λ 2 G̃(0)(p)J1 + λ2 4 G̃(0)(p)2J2 1 +...] (5.1.10) where, in the first equality, the ellipsis (...) contains three bubbles, four bubbles and so on. Furthermore, we stressed through the red-coloured bubbles that the interpretation of the function J1 is that of a quantum loop. Identifying above the alternate geometric series: G̃1−loop(p) = G̃(0)(p) ∞∑ n=0 (−1)n [ λ 2 G̃(0)(p)J1 ]n (5.1.11) = G̃(0)(p) [ 1 1 + λG̃(0)(p)J1/2 ] (5.1.12) In the last equality, we used the formula for the sum of the alternate geometric series. Replacing the free propagator from Eq.(4.1.3), with a ≡ 1, we obtain: G̃1−loop(p) = 1 m2 + 4 sin2 (p/2) 1[ 1 + λ 2 J1 m2+4 sin2 (p/2) ] (5.1.13) = 1 m2 + 4 sin2 (p/2) + λ 2 J1 (5.1.14) the 1-loop propagator in position space is then straightforwardly obtained from an inverse Fourier transform: G1−loop(x1, x2) = 1 N ∑ p eip(x1−x2) m2 + 4 sin2 ( p 2 ) + λ 2 J1 (5.1.15) Recall that p = 2πk/N , where k is the momentum quantum number. Therefore, the sum over p is equivalent to a sum over k from −N/2 to N/2 − 1. To compute the next order in 5.1. FEYMAN DIAGRAMS 43 quantum correction, we introduce the momentum space functions below: Jn = 1 N ∑ p [ G̃(0)(p) ]n (5.1.16) I2(p) = 1 N ∑ q G̃(0)(q)G̃(0)(p− q) (5.1.17) I3(p) = 1 N2 ∑ q ∑ k G̃(0)(q)G̃(0)(k)G̃(0)(p− q − k) (5.1.18) which result directly from solving Eq.(5.1.5) with n = 2, where Jn is a generalization of J1. Repeating the same 1-PI procedure with 2 loops, we can finally jot down the next-order approximation for the momentum-space propagator: G2−loop(x1, x2) = 1 N ∑ p eip(x1−x2) m2 + 4 sin2 ( p 2 ) + λ 2 J1 − λ2 4 J1J2 − λ2 6 I3(p) (5.1.19) Finally, we illustrate in Figure 5.2 the first three relevant diagrams in Φ4-theory with broken symmetry. The 1-loop propagator in this case is given by: Figure 5.2: Relevant