Statistical Field Theory University of Cambridge Part III Mathematical Tripos
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Renormalization Group Flows from Holography; Supersymmetry and a C-Theorem
© 1999 International Press Adv. Theor. Math. Phys. 3 (1999) 363-417 Renormalization Group Flows from Holography; Supersymmetry and a c-Theorem D. Z. Freedmana, S. S. Gubser6, K. Pilchc and N. P. Warnerd aDepartment of Mathematics and Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 b Department of Physics, Harvard University, Cambridge, MA 02138, USA c Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484, USA d Theory Division, CERN, CH-1211 Geneva 23, Switzerland Abstract We obtain first order equations that determine a super symmetric kink solution in five-dimensional Af = 8 gauged supergravity. The kink interpo- lates between an exterior anti-de Sitter region with maximal supersymme- try and an interior anti-de Sitter region with one quarter of the maximal supersymmetry. One eighth of supersymmetry is preserved by the kink as a whole. We interpret it as describing the renormalization group flow in J\f = 4 super-Yang-Mills theory broken to an Af = 1 theory by the addition of a mass term for one of the three adjoint chiral superfields. A detailed cor- respondence is obtained between fields of bulk supergravity in the interior anti-de Sitter region and composite operators of the infrared field theory. We also point out that the truncation used to find the reduced symmetry critical point can be extended to obtain a new Af = 4 gauged supergravity theory holographically dual to a sector of Af = 2 gauge theories based on quiver diagrams. e-print archive: http.y/xxx.lanl.gov/abs/hep-th/9904017 *On leave from Department of Physics and Astronomy, USC, Los Angeles, CA 90089 364 RENORMALIZATION GROUP FLOWS We consider more general kink geometries and construct a c-function that is positive and monotonic if a weak energy condition holds in the bulk gravity theory. -
Path Integrals in Quantum Mechanics
Path Integrals in Quantum Mechanics Dennis V. Perepelitsa MIT Department of Physics 70 Amherst Ave. Cambridge, MA 02142 Abstract We present the path integral formulation of quantum mechanics and demon- strate its equivalence to the Schr¨odinger picture. We apply the method to the free particle and quantum harmonic oscillator, investigate the Euclidean path integral, and discuss other applications. 1 Introduction A fundamental question in quantum mechanics is how does the state of a particle evolve with time? That is, the determination the time-evolution ψ(t) of some initial | i state ψ(t ) . Quantum mechanics is fully predictive [3] in the sense that initial | 0 i conditions and knowledge of the potential occupied by the particle is enough to fully specify the state of the particle for all future times.1 In the early twentieth century, Erwin Schr¨odinger derived an equation specifies how the instantaneous change in the wavefunction d ψ(t) depends on the system dt | i inhabited by the state in the form of the Hamiltonian. In this formulation, the eigenstates of the Hamiltonian play an important role, since their time-evolution is easy to calculate (i.e. they are stationary). A well-established method of solution, after the entire eigenspectrum of Hˆ is known, is to decompose the initial state into this eigenbasis, apply time evolution to each and then reassemble the eigenstates. That is, 1In the analysis below, we consider only the position of a particle, and not any other quantum property such as spin. 2 D.V. Perepelitsa n=∞ ψ(t) = exp [ iE t/~] n ψ(t ) n (1) | i − n h | 0 i| i n=0 X This (Hamiltonian) formulation works in many cases. -
Simulating Quantum Field Theory with a Quantum Computer
Simulating quantum field theory with a quantum computer John Preskill Lattice 2018 28 July 2018 This talk has two parts (1) Near-term prospects for quantum computing. (2) Opportunities in quantum simulation of quantum field theory. Exascale digital computers will advance our knowledge of QCD, but some challenges will remain, especially concerning real-time evolution and properties of nuclear matter and quark-gluon plasma at nonzero temperature and chemical potential. Digital computers may never be able to address these (and other) problems; quantum computers will solve them eventually, though I’m not sure when. The physics payoff may still be far away, but today’s research can hasten the arrival of a new era in which quantum simulation fuels progress in fundamental physics. Frontiers of Physics short distance long distance complexity Higgs boson Large scale structure “More is different” Neutrino masses Cosmic microwave Many-body entanglement background Supersymmetry Phases of quantum Dark matter matter Quantum gravity Dark energy Quantum computing String theory Gravitational waves Quantum spacetime particle collision molecular chemistry entangled electrons A quantum computer can simulate efficiently any physical process that occurs in Nature. (Maybe. We don’t actually know for sure.) superconductor black hole early universe Two fundamental ideas (1) Quantum complexity Why we think quantum computing is powerful. (2) Quantum error correction Why we think quantum computing is scalable. A complete description of a typical quantum state of just 300 qubits requires more bits than the number of atoms in the visible universe. Why we think quantum computing is powerful We know examples of problems that can be solved efficiently by a quantum computer, where we believe the problems are hard for classical computers. -
Characterization of a Subclass of Tweedie Distributions by a Property
Characterization of a subclass of Tweedie distributions by a property of generalized stability Lev B. Klebanov ∗, Grigory Temnov † December 21, 2017 Abstract We introduce a class of distributions originating from an exponential family and having a property related to the strict stability property. Specifically, for two densities pθ and p linked by the relation θx pθ(x)= e c(θ)p(x), we assume that their characteristic functions fθ and f satisfy α(θ) fθ(t)= f (β(θ)t) t R, θ [a,b] , with a and b s.t. a 0 b . ∀ ∈ ∈ ≤ ≤ A characteristic function representation for this family is obtained and its properties are investigated. The proposed class relates to stable distributions and includes Inverse Gaussian distribution and Levy distribution as special cases. Due to its origin, the proposed distribution has a sufficient statistic. Besides, it combines stability property at lower scales with an exponential decay of the distri- bution’s tail and has an additional flexibility due to the convenient parametrization. Apart from the basic model, certain generalizations are considered, including the one related to geometric stable distributions. Key Words: Natural exponential families, stability-under-addition, characteristic func- tions. 1 Introduction Random variables with the property of stability-under-addition (usually called just stabil- ity) are of particular importance in applied probability theory and statistics. The signifi- arXiv:1006.2070v1 [stat.ME] 10 Jun 2010 cance of stable laws and related distributions is due to their major role in the Central limit problem and their link with applied stochastic models used in e.g. physics and financial mathematics. -
Conformal Symmetry in Field Theory and in Quantum Gravity
universe Review Conformal Symmetry in Field Theory and in Quantum Gravity Lesław Rachwał Instituto de Física, Universidade de Brasília, Brasília DF 70910-900, Brazil; [email protected] Received: 29 August 2018; Accepted: 9 November 2018; Published: 15 November 2018 Abstract: Conformal symmetry always played an important role in field theory (both quantum and classical) and in gravity. We present construction of quantum conformal gravity and discuss its features regarding scattering amplitudes and quantum effective action. First, the long and complicated story of UV-divergences is recalled. With the development of UV-finite higher derivative (or non-local) gravitational theory, all problems with infinities and spacetime singularities might be completely solved. Moreover, the non-local quantum conformal theory reveals itself to be ghost-free, so the unitarity of the theory should be safe. After the construction of UV-finite theory, we focused on making it manifestly conformally invariant using the dilaton trick. We also argue that in this class of theories conformal anomaly can be taken to vanish by fine-tuning the couplings. As applications of this theory, the constraints of the conformal symmetry on the form of the effective action and on the scattering amplitudes are shown. We also remark about the preservation of the unitarity bound for scattering. Finally, the old model of conformal supergravity by Fradkin and Tseytlin is briefly presented. Keywords: quantum gravity; conformal gravity; quantum field theory; non-local gravity; super- renormalizable gravity; UV-finite gravity; conformal anomaly; scattering amplitudes; conformal symmetry; conformal supergravity 1. Introduction From the beginning of research on theories enjoying invariance under local spacetime-dependent transformations, conformal symmetry played a pivotal role—first introduced by Weyl related changes of meters to measure distances (and also due to relativity changes of periods of clocks to measure time intervals). -
Arxiv:1910.05796V2 [Math-Ph] 26 Oct 2020 5
Towards a conformal field theory for Schramm-Loewner evolutions Eveliina Peltola [email protected] Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, D-53115 Bonn, Germany We discuss the partition function point of view for chordal Schramm-Loewner evolutions and their relation- ship with correlation functions in conformal field theory. Both are closely related to crossing probabilities and interfaces in critical models in two-dimensional statistical mechanics. We gather and supplement previous results with different perspectives, point out remaining difficulties, and suggest directions for future studies. 1. Introduction 2 2. Schramm-Loewner evolution in statistical mechanics and conformal field theory4 A. Schramm-Loewner evolution (SLE) 5 B. SLE in critical models – the Ising model6 C. Conformal field theory (CFT) 7 D. Martingale observables for interfaces 9 3. Multiple SLE partition functions and applications 12 A. Multiple SLEs and their partition functions 12 B. Definition of the multiple SLE partition functions 13 C. Properties of the multiple SLE partition functions 16 D. Relation to Schramm-Loewner evolutions and critical models 18 4. Fusion and operator product expansion 20 A. Fusion and operator product expansion in conformal field theory 21 B. Fusion: analytic approach 22 C. Fusion: systematic algebraic approach 23 D. Operator product expansion for multiple SLE partition functions 26 arXiv:1910.05796v2 [math-ph] 26 Oct 2020 5. From OPE structure to products of random distributions? 27 A. Conformal bootstrap 27 B. Random tempered distributions 28 C. Bootstrap construction? 29 Appendix A. Representation theory of the Virasoro algebra 31 Appendix B. Probabilistic construction of the pure partition functions 32 Appendix C. -
Effective Quantum Field Theories Thomas Mannel Theoretical Physics I (Particle Physics) University of Siegen, Siegen, Germany
Generating Functionals Functional Integration Renormalization Introduction to Effective Quantum Field Theories Thomas Mannel Theoretical Physics I (Particle Physics) University of Siegen, Siegen, Germany 2nd Autumn School on High Energy Physics and Quantum Field Theory Yerevan, Armenia, 6-10 October, 2014 T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Overview Lecture 1: Basics of Quantum Field Theory Generating Functionals Functional Integration Perturbation Theory Renormalization Lecture 2: Effective Field Thoeries Effective Actions Effective Lagrangians Identifying relevant degrees of freedom Renormalization and Renormalization Group T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Lecture 3: Examples @ work From Standard Model to Fermi Theory From QCD to Heavy Quark Effective Theory From QCD to Chiral Perturbation Theory From New Physics to the Standard Model Lecture 4: Limitations: When Effective Field Theories become ineffective Dispersion theory and effective field theory Bound Systems of Quarks and anomalous thresholds When quarks are needed in QCD É. T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Lecture 1: Basics of Quantum Field Theory Thomas Mannel Theoretische Physik I, Universität Siegen f q f et Yerevan, October 2014 T. Mannel, Siegen University Effective Quantum -
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]. -
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).......................... -
Effective Field Theories, Reductionism and Scientific Explanation Stephan
To appear in: Studies in History and Philosophy of Modern Physics Effective Field Theories, Reductionism and Scientific Explanation Stephan Hartmann∗ Abstract Effective field theories have been a very popular tool in quantum physics for almost two decades. And there are good reasons for this. I will argue that effec- tive field theories share many of the advantages of both fundamental theories and phenomenological models, while avoiding their respective shortcomings. They are, for example, flexible enough to cover a wide range of phenomena, and concrete enough to provide a detailed story of the specific mechanisms at work at a given energy scale. So will all of physics eventually converge on effective field theories? This paper argues that good scientific research can be characterised by a fruitful interaction between fundamental theories, phenomenological models and effective field theories. All of them have their appropriate functions in the research process, and all of them are indispens- able. They complement each other and hang together in a coherent way which I shall characterise in some detail. To illustrate all this I will present a case study from nuclear and particle physics. The resulting view about scientific theorising is inherently pluralistic, and has implications for the debates about reductionism and scientific explanation. Keywords: Effective Field Theory; Quantum Field Theory; Renormalisation; Reductionism; Explanation; Pluralism. ∗Center for Philosophy of Science, University of Pittsburgh, 817 Cathedral of Learning, Pitts- burgh, PA 15260, USA (e-mail: [email protected]) (correspondence address); and Sektion Physik, Universit¨at M¨unchen, Theresienstr. 37, 80333 M¨unchen, Germany. 1 1 Introduction There is little doubt that effective field theories are nowadays a very popular tool in quantum physics. -
Scale Without Conformal Invariance in Membrane Theory
Scale without conformal invariance in membrane theory Achille Mauria,∗, Mikhail I. Katsnelsona aRadboud University, Institute for Molecules and Materials, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands Abstract We investigate the relation between dilatation and conformal symmetries in the statistical mechanics of flex- ible crystalline membranes. We analyze, in particular, a well-known model which describes the fluctuations of a continuum elastic medium embedded in a higher-dimensional space. In this theory, the renormalization group flow connects a non-interacting ultraviolet fixed point, where the theory is controlled by linear elas- ticity, to an interacting infrared fixed point. By studying the structure of correlation functions and of the energy-momentum tensor, we show that, in the infrared, the theory is only scale-invariant: the dilatation symmetry is not enhanced to full conformal invariance. The model is shown to present a non-vanishing virial current which, despite being non-conserved, maintains a scaling dimension exactly equal to D − 1, even in presence of interactions. We attribute the absence of anomalous dimensions to the symmetries of the model under translations and rotations in the embedding space, which are realized as shifts of phonon fields, and which protect the renormalization of several non-invariant operators. We also note that closure of a sym- metry algebra with both shift symmetries and conformal invariance would require, in the hypothesis that phonons transform as primary fields, the presence of new shift symmetries which are not expected to hold on physical grounds. We then consider an alternative model, involving only scalar fields, which describes effective phonon-mediated interactions between local Gaussian curvatures.