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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. -
Hamiltonian Perturbation Theory on a Lie Algebra. Application to a Non-Autonomous Symmetric Top
Hamiltonian Perturbation Theory on a Lie Algebra. Application to a non-autonomous Symmetric Top. Lorenzo Valvo∗ Michel Vittot Dipartimeno di Matematica Centre de Physique Théorique Università degli Studi di Roma “Tor Vergata” Aix-Marseille Université & CNRS Via della Ricerca Scientifica 1 163 Avenue de Luminy 00133 Roma, Italy 13288 Marseille Cedex 9, France January 6, 2021 Abstract We propose a perturbation algorithm for Hamiltonian systems on a Lie algebra V, so that it can be applied to non-canonical Hamiltonian systems. Given a Hamil- tonian system that preserves a subalgebra B of V, when we add a perturbation the subalgebra B will no longer be preserved. We show how to transform the perturbed dynamical system to preserve B up to terms quadratic in the perturbation. We apply this method to study the dynamics of a non-autonomous symmetric Rigid Body. In this example our algebraic transform plays the role of Iterative Lemma in the proof of a KAM-like statement. A dynamical system on some set V is a flow: a one-parameter group of mappings associating to a given element F 2 V (the initial condition) another element F (t) 2 V, for any value of the parameter t. A flow on V is determined by a linear mapping H from V to itself. However, the flow can be rarely computed explicitly. In perturbation theory we aim at computing the flow of H + V, where the flow of H is known, and V is another linear mapping from V to itself. In physics, the set V is often a Lie algebra: for instance in classical mechanics [4], fluid dynamics and plasma physics [20], quantum mechanics [24], kinetic theory [18], special and general relativity [17]. -
1 Introduction 2 Classical Perturbation Theory
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Classical and Quantum Perturbation Theory for two Non–Resonant Oscillators with Quartic Interaction Luca Salasnich Dipartimento di Matematica Pura ed Applicata, Universit`a di Padova, Via Belzoni 7, I–35131 Padova, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Padova, Via Marzolo 8, I–35131 Padova, Italy Istituto Nazionale per la Fisica della Materia, Unit`a di Milano, Via Celoria 16, I–20133 Milano, Italy Abstract. We study the classical and quantum perturbation theory for two non–resonant oscillators coupled by a nonlinear quartic interaction. In particular we analyze the question of quantum corrections to the torus quantization of the classical perturbation theory (semiclassical mechanics). We obtain up to the second order of perturbation theory an explicit analytical formula for the quantum energy levels, which is the semiclassical one plus quantum corrections. We compare the ”exact” quantum levels obtained numerically to the semiclassical levels studying also the effects of quantum corrections. Key words: Classical Perturbation Theory; Quantum Mechanics; General Mechanics 1 Introduction Nowadays there is considerable renewed interest in the transition from classical mechanics to quantum mechanics, a powerful motivation behind that being the problem of the so–called quantum chaos [1–3]. An important aspect is represented by the semiclassical quantization formula of the (regular) energy levels for quasi–integrable systems [4–6], the so–called torus quantization, initiated by Einstein [7] and completed by Maslov [8]. It has been recently shown [9,10] that, for perturbed non–resonant harmonic oscillators, the algorithm of classical perturbation theory can be used to formulate the quantum mechanical perturbation theory as the semiclassically quantized classical perturbation theory equipped with the quantum corrections in powers of ¯h ”correcting” the classical Hamiltonian that appears in the classical algorithm. -
Perturbation Theory and Exact Solutions
PERTURBATION THEORY AND EXACT SOLUTIONS by J J. LODDER R|nhtdnn Report 76~96 DISSIPATIVE MOTION PERTURBATION THEORY AND EXACT SOLUTIONS J J. LODOER ASSOCIATIE EURATOM-FOM Jun»»76 FOM-INST1TUUT VOOR PLASMAFYSICA RUNHUIZEN - JUTPHAAS - NEDERLAND DISSIPATIVE MOTION PERTURBATION THEORY AND EXACT SOLUTIONS by JJ LODDER R^nhuizen Report 76-95 Thisworkwat performed at part of th«r«Mvchprogmmncof thcHMCiattofiafrccmentof EnratoniOTd th« Stichting voor FundtmenteelOiutereoek der Matctk" (FOM) wtihnnmcWMppoft from the Nederhmdie Organiutic voor Zuiver Wetemchap- pcigk Onderzoek (ZWO) and Evntom It it abo pabHtfMd w a the* of Ac Univenrty of Utrecht CONTENTS page SUMMARY iii I. INTRODUCTION 1 II. GENERALIZED FUNCTIONS DEFINED ON DISCONTINUOUS TEST FUNC TIONS AND THEIR FOURIER, LAPLACE, AND HILBERT TRANSFORMS 1. Introduction 4 2. Discontinuous test functions 5 3. Differentiation 7 4. Powers of x. The partie finie 10 5. Fourier transforms 16 6. Laplace transforms 20 7. Hubert transforms 20 8. Dispersion relations 21 III. PERTURBATION THEORY 1. Introduction 24 2. Arbitrary potential, momentum coupling 24 3. Dissipative equation of motion 31 4. Expectation values 32 5. Matrix elements, transition probabilities 33 6. Harmonic oscillator 36 7. Classical mechanics and quantum corrections 36 8. Discussion of the Pu strength function 38 IV. EXACTLY SOLVABLE MODELS FOR DISSIPATIVE MOTION 1. Introduction 40 2. General quadratic Kami1tonians 41 3. Differential equations 46 4. Classical mechanics and quantum corrections 49 5. Equation of motion for observables 51 V. SPECIAL QUADRATIC HAMILTONIANS 1. Introduction 53 2. Hamiltcnians with coordinate coupling 53 3. Double coordinate coupled Hamiltonians 62 4. Symmetric Hamiltonians 63 i page VI. DISCUSSION 1. Introduction 66 ?. -
Bessel's Differential Equation Mathematical Physics
R. I. Badran Bessel's Differential Equation Mathematical Physics Bessel's Differential Equation: 2 Recalling the DE y n y 0 which has a sinusoidal solution (i.e. sin nx and cos nx) and knowing that these solutions can be treated as power series, we can find a solution to the Bessel's DE which is written as: 2 2 2 x y xy (x p )y 0 . The solution is represented by a series. This series very much look like a damped sine or cosine. It is called a Bessel function. To solve the Bessel' DE, we apply the Frobenius method by mn assuming a series solution of the form y an x . n0 Substitute this solution into the DE equation and after some mathematical steps you may find that the indicial equation is 2 2 m p 0 where m= p. Also you may get a1=0 and hence all odd a's are zero as well. The recursion relation can be found as a a n n2 (n m 2)2 p 2 with (n=0, 1, 2, 3, ……etc.). Case (1): m= p (seeking the first solution of Bessel DE) We get the solution 2 4 p x x y a x 1 2(2p 2) 2 4(2p 2)(2p 4) This can be rewritten as: x p2n J (x) y a (1)n p 2n n0 2 n!( p n)! 1 Put a 2 p p! The Bessel function has the factorial form x p2n J (x) (1)n p 2n p , n0 n!( p n)!2 R. -
Modifications of the Method of Variation of Parameters
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector International Joumal Available online at www.sciencedirect.com computers & oc,-.c~ ~°..-cT- mathematics with applications Computers and Mathematics with Applications 51 (2006) 451-466 www.elsevier.com/locate/camwa Modifications of the Method of Variation of Parameters ROBERTO BARRIO* GME, Depto. Matem~tica Aplicada Facultad de Ciencias Universidad de Zaragoza E-50009 Zaragoza, Spain rbarrio@unizar, es SERGIO SERRANO GME, Depto. MatemAtica Aplicada Centro Politdcnico Superior Universidad de Zaragoza Maria de Luna 3, E-50015 Zaragoza, Spain sserrano©unizar, es (Received February PO05; revised and accepted October 2005) Abstract--ln spite of being a classical method for solving differential equations, the method of variation of parameters continues having a great interest in theoretical and practical applications, as in astrodynamics. In this paper we analyse this method providing some modifications and generalised theoretical results. Finally, we present an application to the determination of the ephemeris of an artificial satellite, showing the benefits of the method of variation of parameters for this kind of problems. ~) 2006 Elsevier Ltd. All rights reserved. Keywords--Variation of parameters, Lagrange equations, Gauss equations, Satellite orbits, Dif- ferential equations. 1. INTRODUCTION The method of variation of parameters was developed by Leonhard Euler in the middle of XVIII century to describe the mutual perturbations of Jupiter and Saturn. However, his results were not absolutely correct because he did not consider the orbital elements varying simultaneously. In 1766, Lagrange improved the method developed by Euler, although he kept considering some of the orbital elements as constants, which caused some of his equations to be incorrect. -
The Standard Form of a Differential Equation
Fall 06 The Standard Form of a Differential Equation Format required to solve a differential equation or a system of differential equations using one of the command-line differential equation solvers such as rkfixed, Rkadapt, Radau, Stiffb, Stiffr or Bulstoer. For a numerical routine to solve a differential equation (DE), we must somehow pass the differential equation as an argument to the solver routine. A standard form for all DEs will allow us to do this. Basic idea: get rid of any second, third, fourth, etc. derivatives that appear, leaving only first derivatives. Example 1 2 d d 5 yx()+3 ⋅ yx()−5yx ⋅ () 4x⋅ 2 dx dx This DE contains a second derivative. How do we write a second derivative as a first derivative? A second derivative is a first derivative of a first derivative. d2 d d yx() yx() 2 dx dxxd Step 1: STANDARDIZATION d Let's define two functions y0(x) and y1(x) as y0()x yx() and y1()x y0()x dx Then this differential equation can be written as d 5 y1()x +3y ⋅ 1()x −5y ⋅ 0()x 4x dx This DE contains 2 functions instead of one, but there is a strong relationship between these two functions d y1()x y0()x dx So, the original DE is now a system of two DEs, d d 5 y1()x y0()x and y1()x +3y ⋅ 1()x −5y ⋅ 0()x 4x⋅ dx dx The convention is to write these equations with the derivatives alone on the left-hand side. d y0()x y1()x dx This is the first step in the standardization process. -
Conformal Invariance and Quantum Integrability of Sigma Models on Symmetric Superspaces
Physics Letters B 648 (2007) 254–261 www.elsevier.com/locate/physletb Conformal invariance and quantum integrability of sigma models on symmetric superspaces A. Babichenko a,b a Racah Institute of Physics, the Hebrew University, Jerusalem 91904, Israel b Department of Particle Physics, Weizmann Institute of Science, Rehovot 76100, Israel Received 7 December 2006; accepted 2 March 2007 Available online 7 March 2007 Editor: L. Alvarez-Gaumé Abstract We consider two dimensional nonlinear sigma models on few symmetric superspaces, which are supergroup manifolds of coset type. For those spaces where one loop beta function vanishes, two loop beta function is calculated and is shown to be zero. Vanishing of beta function in all orders of perturbation theory is shown for the principal chiral models on group supermanifolds with zero Killing form. Sigma models on symmetric (super) spaces on supergroup manifold G/H are known to be classically integrable. We investigate a possibility to extend an argument of absence of quantum anomalies in nonlocal current conservation from nonsuper case to the case of supergroup manifolds which are asymptotically free in one loop. © 2007 Elsevier B.V. All rights reserved. 1. Introduction super coset PSU(2, 2|4)/SO(1, 4) × SO(5). Hyperactivity in at- tempts to exploit integrability and methods of Bethe ansatz as a Two dimensional (2d) nonlinear sigma models (NLSM) on calculational tool in checks of ADS/CFT correspondence (see, supermanifolds, with and without WZ term, seemed to be ex- e.g., reviews [12] and references therein), also supports this in- otic objects, when they appeared in condensed matter physics terest, since both spin chains (on the gauge theory side) and 2d twenty years ago as an elegant calculational tool in problems NLSM (on the ADS side) appearing there, usually have a su- of self avoiding walks [1] and disordered metals [2]. -
Perturbation Theory in Celestial Mechanics
Perturbation Theory in Celestial Mechanics Alessandra Celletti Dipartimento di Matematica Universit`adi Roma Tor Vergata Via della Ricerca Scientifica 1, I-00133 Roma (Italy) ([email protected]) December 8, 2007 Contents 1 Glossary 2 2 Definition 2 3 Introduction 2 4 Classical perturbation theory 4 4.1 The classical theory . 4 4.2 The precession of the perihelion of Mercury . 6 4.2.1 Delaunay action–angle variables . 6 4.2.2 The restricted, planar, circular, three–body problem . 7 4.2.3 Expansion of the perturbing function . 7 4.2.4 Computation of the precession of the perihelion . 8 5 Resonant perturbation theory 9 5.1 The resonant theory . 9 5.2 Three–body resonance . 10 5.3 Degenerate perturbation theory . 11 5.4 The precession of the equinoxes . 12 6 Invariant tori 14 6.1 Invariant KAM surfaces . 14 6.2 Rotational tori for the spin–orbit problem . 15 6.3 Librational tori for the spin–orbit problem . 16 6.4 Rotational tori for the restricted three–body problem . 17 6.5 Planetary problem . 18 7 Periodic orbits 18 7.1 Construction of periodic orbits . 18 7.2 The libration in longitude of the Moon . 20 1 8 Future directions 20 9 Bibliography 21 9.1 Books and Reviews . 21 9.2 Primary Literature . 22 1 Glossary KAM theory: it provides the persistence of quasi–periodic motions under a small perturbation of an integrable system. KAM theory can be applied under quite general assumptions, i.e. a non– degeneracy of the integrable system and a diophantine condition of the frequency of motion. -
CORSO ESTIVO DI MATEMATICA Differential Equations Of
CORSO ESTIVO DI MATEMATICA Differential Equations of Mathematical Physics G. Sweers http://go.to/sweers or http://fa.its.tudelft.nl/ sweers ∼ Perugia, July 28 - August 29, 2003 It is better to have failed and tried, To kick the groom and kiss the bride, Than not to try and stand aside, Sparing the coal as well as the guide. John O’Mill ii Contents 1Frommodelstodifferential equations 1 1.1Laundryonaline........................... 1 1.1.1 Alinearmodel........................ 1 1.1.2 Anonlinearmodel...................... 3 1.1.3 Comparingbothmodels................... 4 1.2Flowthroughareaandmore2d................... 5 1.3Problemsinvolvingtime....................... 11 1.3.1 Waveequation........................ 11 1.3.2 Heatequation......................... 12 1.4 Differentialequationsfromcalculusofvariations......... 15 1.5 Mathematical solutions are not always physically relevant . 19 2 Spaces, Traces and Imbeddings 23 2.1Functionspaces............................ 23 2.1.1 Hölderspaces......................... 23 2.1.2 Sobolevspaces........................ 24 2.2Restrictingandextending...................... 29 2.3Traces................................. 34 1,p 2.4 Zero trace and W0 (Ω) ....................... 36 2.5Gagliardo,Nirenberg,SobolevandMorrey............. 38 3 Some new and old solution methods I 43 3.1Directmethodsinthecalculusofvariations............ 43 3.2 Solutions in flavours......................... 48 3.3PreliminariesforCauchy-Kowalevski................ 52 3.3.1 Ordinary differentialequations............... 52 3.3.2 Partial differentialequations............... -
Differential Equations and Linear Algebra
Chapter 1 First Order Equations 1.1 Four Examples : Linear versus Nonlinear A first order differential equation connects a function y.t/ to its derivative dy=dt. That rate of change in y is decided by y itself (and possibly also by the time t). Here are four examples. Example 1 is the most important differential equation of all. dy dy dy dy 1/ y 2/ y 3/ 2ty 4/ y2 dt D dt D dt D dt D Those examples illustrate three linear differential equations (1, 2, and 3) and a nonlinear differential equation. The unknown function y.t/ is squared in Example 4. The derivative y or y or 2ty is proportional to the function y in Examples 1, 2, 3. The graph of dy=dt versus y becomes a parabola in Example 4, because of y2. It is true that t multiplies y in Example 3. That equation is still linear in y and dy=dt. It has a variable coefficient 2t, changing with time. Examples 1 and 2 have constant coefficient (the coefficients of y are 1 and 1). Solutions to the Four Examples We can write down a solution to each example. This will be one solution but it is not the complete solution, because each equation has a family of solutions. Eventually there will be a constant C in the complete solution. This number C is decided by the starting value of y at t 0, exactly as in ordinary integration. The integral of f.t/ solves the simplest differentialD equation of all, with y.0/ C : D dy t 5/ f.t/ The complete solution is y.t/ f.s/ds C : dt D D C Z0 1 2 Chapter 1.