Correlation Functions and Diagrams
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An Introduction to Quantum Field Theory
AN INTRODUCTION TO QUANTUM FIELD THEORY By Dr M Dasgupta University of Manchester Lecture presented at the School for Experimental High Energy Physics Students Somerville College, Oxford, September 2009 - 1 - - 2 - Contents 0 Prologue....................................................................................................... 5 1 Introduction ................................................................................................ 6 1.1 Lagrangian formalism in classical mechanics......................................... 6 1.2 Quantum mechanics................................................................................... 8 1.3 The Schrödinger picture........................................................................... 10 1.4 The Heisenberg picture............................................................................ 11 1.5 The quantum mechanical harmonic oscillator ..................................... 12 Problems .............................................................................................................. 13 2 Classical Field Theory............................................................................. 14 2.1 From N-point mechanics to field theory ............................................... 14 2.2 Relativistic field theory ............................................................................ 15 2.3 Action for a scalar field ............................................................................ 15 2.4 Plane wave solution to the Klein-Gordon equation ........................... -
Quantum Field Theory*
Quantum Field Theory y Frank Wilczek Institute for Advanced Study, School of Natural Science, Olden Lane, Princeton, NJ 08540 I discuss the general principles underlying quantum eld theory, and attempt to identify its most profound consequences. The deep est of these consequences result from the in nite number of degrees of freedom invoked to implement lo cality.Imention a few of its most striking successes, b oth achieved and prosp ective. Possible limitation s of quantum eld theory are viewed in the light of its history. I. SURVEY Quantum eld theory is the framework in which the regnant theories of the electroweak and strong interactions, which together form the Standard Mo del, are formulated. Quantum electro dynamics (QED), b esides providing a com- plete foundation for atomic physics and chemistry, has supp orted calculations of physical quantities with unparalleled precision. The exp erimentally measured value of the magnetic dip ole moment of the muon, 11 (g 2) = 233 184 600 (1680) 10 ; (1) exp: for example, should b e compared with the theoretical prediction 11 (g 2) = 233 183 478 (308) 10 : (2) theor: In quantum chromo dynamics (QCD) we cannot, for the forseeable future, aspire to to comparable accuracy.Yet QCD provides di erent, and at least equally impressive, evidence for the validity of the basic principles of quantum eld theory. Indeed, b ecause in QCD the interactions are stronger, QCD manifests a wider variety of phenomena characteristic of quantum eld theory. These include esp ecially running of the e ective coupling with distance or energy scale and the phenomenon of con nement. -
2D Potts Model Correlation Lengths
KOMA Revised version May D Potts Mo del Correlation Lengths Numerical Evidence for = at o d t Wolfhard Janke and Stefan Kappler Institut f ur Physik Johannes GutenbergUniversitat Mainz Staudinger Weg Mainz Germany Abstract We have studied spinspin correlation functions in the ordered phase of the twodimensional q state Potts mo del with q and at the rstorder transition p oint Through extensive Monte t Carlo simulations we obtain strong numerical evidence that the cor relation length in the ordered phase agrees with the exactly known and recently numerically conrmed correlation length in the disordered phase As a byproduct we nd the energy moments o t t d in the ordered phase at in very go o d agreement with a recent large t q expansion PACS numbers q Hk Cn Ha hep-lat/9509056 16 Sep 1995 Introduction Firstorder phase transitions have b een the sub ject of increasing interest in recent years They play an imp ortant role in many elds of physics as is witnessed by such diverse phenomena as ordinary melting the quark decon nement transition or various stages in the evolution of the early universe Even though there exists already a vast literature on this sub ject many prop erties of rstorder phase transitions still remain to b e investigated in detail Examples are nitesize scaling FSS the shap e of energy or magnetization distributions partition function zeros etc which are all closely interrelated An imp ortant approach to attack these problems are computer simulations Here the available system sizes are necessarily -
Full and Unbiased Solution of the Dyson-Schwinger Equation in the Functional Integro-Differential Representation
PHYSICAL REVIEW B 98, 195104 (2018) Full and unbiased solution of the Dyson-Schwinger equation in the functional integro-differential representation Tobias Pfeffer and Lode Pollet Department of Physics, Arnold Sommerfeld Center for Theoretical Physics, University of Munich, Theresienstrasse 37, 80333 Munich, Germany (Received 17 March 2018; revised manuscript received 11 October 2018; published 2 November 2018) We provide a full and unbiased solution to the Dyson-Schwinger equation illustrated for φ4 theory in 2D. It is based on an exact treatment of the functional derivative ∂/∂G of the four-point vertex function with respect to the two-point correlation function G within the framework of the homotopy analysis method (HAM) and the Monte Carlo sampling of rooted tree diagrams. The resulting series solution in deformations can be considered as an asymptotic series around G = 0 in a HAM control parameter c0G, or even a convergent one up to the phase transition point if shifts in G can be performed (such as by summing up all ladder diagrams). These considerations are equally applicable to fermionic quantum field theories and offer a fresh approach to solving functional integro-differential equations beyond any truncation scheme. DOI: 10.1103/PhysRevB.98.195104 I. INTRODUCTION was not solved and differences with the full, exact answer could be seen when the correlation length increases. Despite decades of research there continues to be a need In this work, we solve the full DSE by writing them as a for developing novel methods for strongly correlated systems. closed set of integro-differential equations. Within the HAM The standard Monte Carlo approaches [1–5] are convergent theory there exists a semianalytic way to treat the functional but suffer from a prohibitive sign problem, scaling exponen- derivatives without resorting to an infinite expansion of the tially in the system volume [6]. -
Feynman Diagrams, Momentum Space Feynman Rules, Disconnected Diagrams, Higher Correlation Functions
PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Dr. Anosh Joseph, IISER Mohali LECTURE 05 Tuesday, January 14, 2020 Topics: Feynman Diagrams, Momentum Space Feynman Rules, Disconnected Diagrams, Higher Correlation Functions. Feynman Diagrams We can use the Wick’s theorem to express the n-point function h0jT fφ(x1)φ(x2) ··· φ(xn)gj0i as a sum of products of Feynman propagators. We will be interested in developing a diagrammatic interpretation of such expressions. Let us look at the expression containing four fields. We have h0jT fφ1φ2φ3φ4g j0i = DF (x1 − x2)DF (x3 − x4) +DF (x1 − x3)DF (x2 − x4) +DF (x1 − x4)DF (x2 − x3): (1) We can interpret Eq. (1) as the sum of three diagrams, if we represent each of the points, x1 through x4 by a dot, and each factor DF (x − y) by a line joining x to y. These diagrams are called Feynman diagrams. In Fig. 1 we provide the diagrammatic interpretation of Eq. (1). The interpretation of the diagrams is the following: particles are created at two spacetime points, each propagates to one of the other points, and then they are annihilated. This process can happen in three different ways, and they correspond to the three ways to connect the points in pairs, as shown in the three diagrams in Fig. 1. Thus the total amplitude for the process is the sum of these three diagrams. PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 ⟨0|T {ϕ1ϕ2ϕ3ϕ4}|0⟩ = DF(x1 − x2)DF(x3 − x4) + DF(x1 − x3)DF(x2 − x4) +DF(x1 − x4)DF(x2 − x3) 1 2 1 2 1 2 + + 3 4 3 4 3 4 Figure 1: Diagrammatic interpretation of Eq. -
Notes on Statistical Field Theory
Lecture Notes on Statistical Field Theory Kevin Zhou [email protected] These notes cover statistical field theory and the renormalization group. The primary sources were: • Kardar, Statistical Physics of Fields. A concise and logically tight presentation of the subject, with good problems. Possibly a bit too terse unless paired with the 8.334 video lectures. • David Tong's Statistical Field Theory lecture notes. A readable, easygoing introduction covering the core material of Kardar's book, written to seamlessly pair with a standard course in quantum field theory. • Goldenfeld, Lectures on Phase Transitions and the Renormalization Group. Covers similar material to Kardar's book with a conversational tone, focusing on the conceptual basis for phase transitions and motivation for the renormalization group. The notes are structured around the MIT course based on Kardar's textbook, and were revised to include material from Part III Statistical Field Theory as lectured in 2017. Sections containing this additional material are marked with stars. The most recent version is here; please report any errors found to [email protected]. 2 Contents Contents 1 Introduction 3 1.1 Phonons...........................................3 1.2 Phase Transitions......................................6 1.3 Critical Behavior......................................8 2 Landau Theory 12 2.1 Landau{Ginzburg Hamiltonian.............................. 12 2.2 Mean Field Theory..................................... 13 2.3 Symmetry Breaking.................................... 16 3 Fluctuations 19 3.1 Scattering and Fluctuations................................ 19 3.2 Position Space Fluctuations................................ 20 3.3 Saddle Point Fluctuations................................. 23 3.4 ∗ Path Integral Methods.................................. 24 4 The Scaling Hypothesis 29 4.1 The Homogeneity Assumption............................... 29 4.2 Correlation Lengths.................................... 30 4.3 Renormalization Group (Conceptual).......................... -
Quantum Vacuum Energy Density and Unifying Perspectives Between Gravity and Quantum Behaviour of Matter
Annales de la Fondation Louis de Broglie, Volume 42, numéro 2, 2017 251 Quantum vacuum energy density and unifying perspectives between gravity and quantum behaviour of matter Davide Fiscalettia, Amrit Sorlib aSpaceLife Institute, S. Lorenzo in Campo (PU), Italy corresponding author, email: [email protected] bSpaceLife Institute, S. Lorenzo in Campo (PU), Italy Foundations of Physics Institute, Idrija, Slovenia email: [email protected] ABSTRACT. A model of a three-dimensional quantum vacuum based on Planck energy density as a universal property of a granular space is suggested. This model introduces the possibility to interpret gravity and the quantum behaviour of matter as two different aspects of the same origin. The change of the quantum vacuum energy density can be considered as the fundamental medium which determines a bridge between gravity and the quantum behaviour, leading to new interest- ing perspectives about the problem of unifying gravity with quantum theory. PACS numbers: 04. ; 04.20-q ; 04.50.Kd ; 04.60.-m. Key words: general relativity, three-dimensional space, quantum vac- uum energy density, quantum mechanics, generalized Klein-Gordon equation for the quantum vacuum energy density, generalized Dirac equation for the quantum vacuum energy density. 1 Introduction The standard interpretation of phenomena in gravitational fields is in terms of a fundamentally curved space-time. However, this approach leads to well known problems if one aims to find a unifying picture which takes into account some basic aspects of the quantum theory. For this reason, several authors advocated different ways in order to treat gravitational interaction, in which the space-time manifold can be considered as an emergence of the deepest processes situated at the fundamental level of quantum gravity. -
10 Basic Aspects of CFT
10 Basic aspects of CFT An important break-through occurred in 1984 when Belavin, Polyakov and Zamolodchikov [BPZ84] applied ideas of conformal invariance to classify the possible types of critical behaviour in two dimensions. These ideas had emerged earlier in string theory and mathematics, and in fact go backto earlier (1970) work of Polyakov [Po70] in which global conformal invariance is used to constrain the form of correlation functions in d-dimensional the- ories. It is however only by imposing local conformal invariance in d =2 that this approach becomes really powerful. In particular, it immediately permitted a full classification of an infinite family of conformally invariant theories (the so-called “minimal models”) having a finite number of funda- mental (“primary”) fields, and the exact computation of the corresponding critical exponents. In the aftermath of these developments, conformal field theory (CFT) became for some years one of the most hectic research fields of theoretical physics, and indeed has remained a very active area up to this date. This chapter focusses on the basic aspects of CFT, with a special emphasis on the ingredients which will allow us to tackle the geometrically defined loop models via the so-called Coulomb Gas (CG) approach. The CG technique will be exposed in the following chapter. The aim is to make the presentation self- contained while remaining rather brief; the reader interested in more details should turn to the comprehensive textbook [DMS87] or the Les Houches volume [LH89]. 10.1 Global conformal invariance A conformal transformation in d dimensions is an invertible mapping x x′ → which multiplies the metric tensor gµν (x) by a space-dependent scale factor: gµ′ ν (x′)=Λ(x)gµν (x). -
An Introduction to Supersymmetry
An Introduction to Supersymmetry Ulrich Theis Institute for Theoretical Physics, Friedrich-Schiller-University Jena, Max-Wien-Platz 1, D–07743 Jena, Germany [email protected] This is a write-up of a series of five introductory lectures on global supersymmetry in four dimensions given at the 13th “Saalburg” Summer School 2007 in Wolfersdorf, Germany. Contents 1 Why supersymmetry? 1 2 Weyl spinors in D=4 4 3 The supersymmetry algebra 6 4 Supersymmetry multiplets 6 5 Superspace and superfields 9 6 Superspace integration 11 7 Chiral superfields 13 8 Supersymmetric gauge theories 17 9 Supersymmetry breaking 22 10 Perturbative non-renormalization theorems 26 A Sigma matrices 29 1 Why supersymmetry? When the Large Hadron Collider at CERN takes up operations soon, its main objective, besides confirming the existence of the Higgs boson, will be to discover new physics beyond the standard model of the strong and electroweak interactions. It is widely believed that what will be found is a (at energies accessible to the LHC softly broken) supersymmetric extension of the standard model. What makes supersymmetry such an attractive feature that the majority of the theoretical physics community is convinced of its existence? 1 First of all, under plausible assumptions on the properties of relativistic quantum field theories, supersymmetry is the unique extension of the algebra of Poincar´eand internal symmtries of the S-matrix. If new physics is based on such an extension, it must be supersymmetric. Furthermore, the quantum properties of supersymmetric theories are much better under control than in non-supersymmetric ones, thanks to powerful non- renormalization theorems. -
Introduction to Two-Dimensional Conformal Field Theory
Introduction to two-dimensional conformal field theory Sylvain Ribault CEA Saclay, Institut de Physique Th´eorique [email protected] February 5, 2019 Abstract We introduce conformal field theory in two dimensions, from the basic principles to some of the simplest models. From the representations of the Virasoro algebra on the one hand, and the state-field correspondence on the other hand, we deduce Ward identities and Belavin{Polyakov{Zamolodchikov equations for correlation functions. We then explain the principles of the conformal bootstrap method, and introduce conformal blocks. This allows us to define and solve minimal models and Liouville theory. We also introduce the free boson with its abelian affine Lie algebra. Lecture notes for the \Young Researchers Integrability School", Wien 2019, based on the earlier lecture notes [1]. Estimated length: 7 lectures of 45 minutes each. Material that may be skipped in the lectures is in green boxes. Exercises are in green boxes when their statements are not part of the lectures' text; this does not make them less interesting as exercises. 1 Contents 0 Introduction2 1 The Virasoro algebra and its representations3 1.1 Algebra.....................................3 1.2 Representations.................................4 1.3 Null vectors and degenerate representations.................5 2 Conformal field theory6 2.1 Fields......................................6 2.2 Correlation functions and Ward identities...................8 2.3 Belavin{Polyakov{Zamolodchikov equations................. 10 2.4 Free boson.................................... 10 3 Conformal bootstrap 12 3.1 Single-valuedness................................ 12 3.2 Operator product expansion and crossing symmetry............. 13 3.3 Degenerate fields and the fusion product................... 16 4 Minimal models 17 4.1 Diagonal minimal models........................... -
A Final Cure to the Tribulations of the Vacuum in Quantum Theory Joseph Jean-Claude
A Final Cure to the Tribulations of the Vacuum in Quantum Theory Joseph Jean-Claude To cite this version: Joseph Jean-Claude. A Final Cure to the Tribulations of the Vacuum in Quantum Theory. 2019. hal-01991699 HAL Id: hal-01991699 https://hal.archives-ouvertes.fr/hal-01991699 Preprint submitted on 24 Jan 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A Final Cure to the Tribulations of the Vacuum in Quantum Theory Joseph J. JEAN-CLAUDE April 16, 2017 - Revision I (Jan-2019) © Copyright [email protected] Abstract Quantum Field Theory seems to be of late anything but unraveling, apparently shifting from crisis to crisis. Undeniably, there seems to be running, for many different reasons, a rampant discomfort with the Standard Model, an extension of QFT aiming at complementing it. Prior to the latest supersymmetry upset, what has been known as the “vacuum castastrophe” or the “cosmological constant problem” resoundingly shook the very foundations of the Theory. Effectively the enormous numeric disparity between its predicted value of the vacuum energy density, an element of utmost significance in the Theory, and the measured value of this quantity from a Relativistic approach has prompted some to qualify this mishap as the biggest predictive failure of any Theory. -
10. Interacting Matter 10.1. Classical Real
10. Interacting matter The grand potential is now Ω= k TVω(z,T ) 10.1. Classical real gas − B We take into account the mutual interactions of atoms and (molecules) p Ω The Hamiltonian operator is = = ω(z,T ) kBT −V N p2 N ∂ω(z,T ) H(N) = i + v(r ), r = r r . ρ = = z . 2m ij ij | i − j | V ∂z i=1 i<j X X Eliminating z we can write the equation of state as For example, for noble gases a good approximation of the p = k T ϕ(ρ,T ). interaction potential is the Lennard-Jones 6–12 -potential B Expanding ϕ as the power series of ρ we end up with the σ 12 σ 6 v(r)=4ǫ . virial expansion. r − r Ursell-Mayer graphs V ( r ) Let’s write Q = dr r e−βv(rij ) N 1 · · · N i<j Z Y r 0 s r 6 = dr r (1 + f ), e 1 / r 1 · · · N ij Z i<j We evaluate the partition sums in the classical phase Y where space. The canonical partition function is f = f(r )= e−βv(rij ) 1 ij ij − −βH(N) ZN (T, V )= Z(T,V,N) = Tr N e is Mayer’s function. 1 V ( r ) dΓ e−βH . classical−→ N! limit, Maxwell- Z f ( r ) Boltzman Because the momentum variables appear only r quadratically, they can be integrated and we obtain - 1 1 1 The function f is bounded everywhere and it has the ZN = dp1 dpN dr1 drN same range as the potential v.