Relativistic Wave Equations Les Rencontres Physiciens-Mathématiciens De Strasbourg - RCP25, 1973, Tome 18 « Conférences De : J
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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. -
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. -
Introductory Lectures on Quantum Field Theory
Introductory Lectures on Quantum Field Theory a b L. Álvarez-Gaumé ∗ and M.A. Vázquez-Mozo † a CERN, Geneva, Switzerland b Universidad de Salamanca, Salamanca, Spain Abstract In these lectures we present a few topics in quantum field theory in detail. Some of them are conceptual and some more practical. They have been se- lected because they appear frequently in current applications to particle physics and string theory. 1 Introduction These notes are based on lectures delivered by L.A.-G. at the 3rd CERN–Latin-American School of High- Energy Physics, Malargüe, Argentina, 27 February–12 March 2005, at the 5th CERN–Latin-American School of High-Energy Physics, Medellín, Colombia, 15–28 March 2009, and at the 6th CERN–Latin- American School of High-Energy Physics, Natal, Brazil, 23 March–5 April 2011. The audience on all three occasions was composed to a large extent of students in experimental high-energy physics with an important minority of theorists. In nearly ten hours it is quite difficult to give a reasonable introduction to a subject as vast as quantum field theory. For this reason the lectures were intended to provide a review of those parts of the subject to be used later by other lecturers. Although a cursory acquaintance with the subject of quantum field theory is helpful, the only requirement to follow the lectures is a working knowledge of quantum mechanics and special relativity. The guiding principle in choosing the topics presented (apart from serving as introductions to later courses) was to present some basic aspects of the theory that present conceptual subtleties. -
Electromagnetic Field Theory
Electromagnetic Field Theory BO THIDÉ Υ UPSILON BOOKS ELECTROMAGNETIC FIELD THEORY Electromagnetic Field Theory BO THIDÉ Swedish Institute of Space Physics and Department of Astronomy and Space Physics Uppsala University, Sweden and School of Mathematics and Systems Engineering Växjö University, Sweden Υ UPSILON BOOKS COMMUNA AB UPPSALA SWEDEN · · · Also available ELECTROMAGNETIC FIELD THEORY EXERCISES by Tobia Carozzi, Anders Eriksson, Bengt Lundborg, Bo Thidé and Mattias Waldenvik Freely downloadable from www.plasma.uu.se/CED This book was typeset in LATEX 2" (based on TEX 3.14159 and Web2C 7.4.2) on an HP Visualize 9000⁄360 workstation running HP-UX 11.11. Copyright c 1997, 1998, 1999, 2000, 2001, 2002, 2003 and 2004 by Bo Thidé Uppsala, Sweden All rights reserved. Electromagnetic Field Theory ISBN X-XXX-XXXXX-X Downloaded from http://www.plasma.uu.se/CED/Book Version released 19th June 2004 at 21:47. Preface The current book is an outgrowth of the lecture notes that I prepared for the four-credit course Electrodynamics that was introduced in the Uppsala University curriculum in 1992, to become the five-credit course Classical Electrodynamics in 1997. To some extent, parts of these notes were based on lecture notes prepared, in Swedish, by BENGT LUNDBORG who created, developed and taught the earlier, two-credit course Electromagnetic Radiation at our faculty. Intended primarily as a textbook for physics students at the advanced undergradu- ate or beginning graduate level, it is hoped that the present book may be useful for research workers -
(Aka Second Quantization) 1 Quantum Field Theory
221B Lecture Notes Quantum Field Theory (a.k.a. Second Quantization) 1 Quantum Field Theory Why quantum field theory? We know quantum mechanics works perfectly well for many systems we had looked at already. Then why go to a new formalism? The following few sections describe motivation for the quantum field theory, which I introduce as a re-formulation of multi-body quantum mechanics with identical physics content. 1.1 Limitations of Multi-body Schr¨odinger Wave Func- tion We used totally anti-symmetrized Slater determinants for the study of atoms, molecules, nuclei. Already with the number of particles in these systems, say, about 100, the use of multi-body wave function is quite cumbersome. Mention a wave function of an Avogardro number of particles! Not only it is completely impractical to talk about a wave function with 6 × 1023 coordinates for each particle, we even do not know if it is supposed to have 6 × 1023 or 6 × 1023 + 1 coordinates, and the property of the system of our interest shouldn’t be concerned with such a tiny (?) difference. Another limitation of the multi-body wave functions is that it is incapable of describing processes where the number of particles changes. For instance, think about the emission of a photon from the excited state of an atom. The wave function would contain coordinates for the electrons in the atom and the nucleus in the initial state. The final state contains yet another particle, photon in this case. But the Schr¨odinger equation is a differential equation acting on the arguments of the Schr¨odingerwave function, and can never change the number of arguments. -
TASI 2008 Lectures: Introduction to Supersymmetry And
TASI 2008 Lectures: Introduction to Supersymmetry and Supersymmetry Breaking Yuri Shirman Department of Physics and Astronomy University of California, Irvine, CA 92697. [email protected] Abstract These lectures, presented at TASI 08 school, provide an introduction to supersymmetry and supersymmetry breaking. We present basic formalism of supersymmetry, super- symmetric non-renormalization theorems, and summarize non-perturbative dynamics of supersymmetric QCD. We then turn to discussion of tree level, non-perturbative, and metastable supersymmetry breaking. We introduce Minimal Supersymmetric Standard Model and discuss soft parameters in the Lagrangian. Finally we discuss several mech- anisms for communicating the supersymmetry breaking between the hidden and visible sectors. arXiv:0907.0039v1 [hep-ph] 1 Jul 2009 Contents 1 Introduction 2 1.1 Motivation..................................... 2 1.2 Weylfermions................................... 4 1.3 Afirstlookatsupersymmetry . .. 5 2 Constructing supersymmetric Lagrangians 6 2.1 Wess-ZuminoModel ............................... 6 2.2 Superfieldformalism .............................. 8 2.3 VectorSuperfield ................................. 12 2.4 Supersymmetric U(1)gaugetheory ....................... 13 2.5 Non-abeliangaugetheory . .. 15 3 Non-renormalization theorems 16 3.1 R-symmetry.................................... 17 3.2 Superpotentialterms . .. .. .. 17 3.3 Gaugecouplingrenormalization . ..... 19 3.4 D-termrenormalization. ... 20 4 Non-perturbative dynamics in SUSY QCD 20 4.1 Affleck-Dine-Seiberg -
Harmonic Oscillator: Motion in a Magnetic Field
Harmonic Oscillator: Motion in a Magnetic Field * The Schrödinger equation in a magnetic field The vector potential * Quantized electron motion in a magnetic field Landau levels * The Shubnikov-de Haas effect Landau-level degeneracy & depopulation The Schrödinger Equation in a Magnetic Field An important example of harmonic motion is provided by electrons that move under the influence of the LORENTZ FORCE generated by an applied MAGNETIC FIELD F ev B (16.1) * From CLASSICAL physics we know that this force causes the electron to undergo CIRCULAR motion in the plane PERPENDICULAR to the direction of the magnetic field * To develop a QUANTUM-MECHANICAL description of this problem we need to know how to include the magnetic field into the Schrödinger equation In this regard we recall that according to FARADAY’S LAW a time- varying magnetic field gives rise to an associated ELECTRIC FIELD B E (16.2) t The Schrödinger Equation in a Magnetic Field To simplify Equation 16.2 we define a VECTOR POTENTIAL A associated with the magnetic field B A (16.3) * With this definition Equation 16.2 reduces to B A E A E (16.4) t t t * Now the EQUATION OF MOTION for the electron can be written as p k A eE e 1k(B) 2k o eA (16.5) t t t 1. MOMENTUM IN THE PRESENCE OF THE MAGNETIC FIELD 2. MOMENTUM PRIOR TO THE APPLICATION OF THE MAGNETIC FIELD The Schrödinger Equation in a Magnetic Field Inspection of Equation 16.5 suggests that in the presence of a magnetic field we REPLACE the momentum operator in the Schrödinger equation -
Renormalization and Effective Field Theory
Mathematical Surveys and Monographs Volume 170 Renormalization and Effective Field Theory Kevin Costello American Mathematical Society surv-170-costello-cov.indd 1 1/28/11 8:15 AM http://dx.doi.org/10.1090/surv/170 Renormalization and Effective Field Theory Mathematical Surveys and Monographs Volume 170 Renormalization and Effective Field Theory Kevin Costello American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair MichaelA.Singer Eric M. Friedlander Benjamin Sudakov MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 81T13, 81T15, 81T17, 81T18, 81T20, 81T70. The author was partially supported by NSF grant 0706954 and an Alfred P. Sloan Fellowship. For additional information and updates on this book, visit www.ams.org/bookpages/surv-170 Library of Congress Cataloging-in-Publication Data Costello, Kevin. Renormalization and effective fieldtheory/KevinCostello. p. cm. — (Mathematical surveys and monographs ; v. 170) Includes bibliographical references. ISBN 978-0-8218-5288-0 (alk. paper) 1. Renormalization (Physics) 2. Quantum field theory. I. Title. QC174.17.R46C67 2011 530.143—dc22 2010047463 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. -
Regularization and Renormalization of Non-Perturbative Quantum Electrodynamics Via the Dyson-Schwinger Equations
University of Adelaide School of Chemistry and Physics Doctor of Philosophy Regularization and Renormalization of Non-Perturbative Quantum Electrodynamics via the Dyson-Schwinger Equations by Tom Sizer Supervisors: Professor A. G. Williams and Dr A. Kızılers¨u March 2014 Contents 1 Introduction 1 1.1 Introduction................................... 1 1.2 Dyson-SchwingerEquations . .. .. 2 1.3 Renormalization................................. 4 1.4 Dynamical Chiral Symmetry Breaking . 5 1.5 ChapterOutline................................. 5 1.6 Notation..................................... 7 2 Canonical QED 9 2.1 Canonically Quantized QED . 9 2.2 FeynmanRules ................................. 12 2.3 Analysis of Divergences & Weinberg’s Theorem . 14 2.4 ElectronPropagatorandSelf-Energy . 17 2.5 PhotonPropagatorandPolarizationTensor . 18 2.6 ProperVertex.................................. 20 2.7 Ward-TakahashiIdentity . 21 2.8 Skeleton Expansion and Dyson-Schwinger Equations . 22 2.9 Renormalization................................. 25 2.10 RenormalizedPerturbationTheory . 27 2.11 Outline Proof of Renormalizability of QED . 28 3 Functional QED 31 3.1 FullGreen’sFunctions ............................. 31 3.2 GeneratingFunctionals............................. 33 3.3 AbstractDyson-SchwingerEquations . 34 3.4 Connected and One-Particle Irreducible Green’s Functions . 35 3.5 Euclidean Field Theory . 39 3.6 QEDviaFunctionalIntegrals . 40 3.7 Regularization.................................. 42 3.7.1 Cutoff Regularization . 42 3.7.2 Pauli-Villars Regularization . 42 i 3.7.3 Lattice Regularization . 43 3.7.4 Dimensional Regularization . 44 3.8 RenormalizationoftheDSEs ......................... 45 3.9 RenormalizationGroup............................. 49 3.10BrokenScaleInvariance ............................ 53 4 The Choice of Vertex 55 4.1 Unrenormalized Quenched Formalism . 55 4.2 RainbowQED.................................. 57 4.2.1 Self-Energy Derivations . 58 4.2.2 Analytic Approximations . 60 4.2.3 Numerical Solutions . 62 4.3 Rainbow QED with a 4-Fermion Interaction . -
Formulation of Einstein Field Equation Through Curved Newtonian Space-Time
Formulation of Einstein Field Equation Through Curved Newtonian Space-Time Austen Berlet Lord Dorchester Secondary School Dorchester, Ontario, Canada Abstract This paper discusses a possible derivation of Einstein’s field equations of general relativity through Newtonian mechanics. It shows that taking the proper perspective on Newton’s equations will start to lead to a curved space time which is basis of the general theory of relativity. It is important to note that this approach is dependent upon a knowledge of general relativity, with out that, the vital assumptions would not be realized. Note: A number inside of a double square bracket, for example [[1]], denotes an endnote found on the last page. 1. Introduction The purpose of this paper is to show a way to rediscover Einstein’s General Relativity. It is done through analyzing Newton’s equations and making the conclusion that space-time must not only be realized, but also that it must have curvature in the presence of matter and energy. 2. Principal of Least Action We want to show here the Lagrangian action of limiting motion of Newton’s second law (F=ma). We start with a function q mapping to n space of n dimensions and we equip it with a standard inner product. q : → (n ,(⋅,⋅)) (1) We take a function (q) between q0 and q1 and look at the ds of a section of the curve. We then look at some properties of this function (q). We see that the classical action of the functional (L) of q is equal to ∫ds, L denotes the systems Lagrangian. -
Hawking Radiation and the Casimir Effect
Quantum field theory on a quantum space-time: Hawking radiation and the Casimir effect Jorge Pullin Horace Hearne Institute for Theoretical Physics Louisiana State University With Rodolfo Gambini and Javier Olmedo University of the Republic of Uruguay 1 Plan • Review of quantum theory of spherically symmetric vacuum space-8mes. • The maer Hamiltonian as a parameterized Dirac observable. • Quantum vacua. • Hawking radiaon. • Casimir effect. 2 Summary: We have recently found, in closed form, the space of physical states corresponding to spherically symmetric vacuum space-times in loop quantum gravity. We wish to consider the quantization of a test scalar fields on such quantum space-times. The idea will be to represent the matter part of the Hamiltonian constraint as a parameterized Dirac observable for the gravitational variables and we can therefore evaluate its expectation value on states of the physical space of states of vacuum gravity. We choose states very peaked around a Schwarzschild space-time of a given mass. The resulting expectation value of the matter part of the Hamiltonian constraint becomes a classical Hamiltonian, quantum corrected due to the quantum background space time. We proceed to quantize such Hamiltonian in the traditional way, defining modes and creation and annihilation operators and obtain its vacua. We then compute the Hawking radiation. Main result: the quantum background space-time acts as a lattice discretization of the field theory, naturally regulating it and eliminating infinities, but otherwise changing in small but important ways the traditional picture of QFT on CST. 3 The quantum background: vacuum spherically symmetric LQG After a rescaling and combination of the constraints that turns their algebra into a Lie algebra, we were able to solve in closed form for the space of physical states of spherically symmetric vacuum LQG (RG, JP PRL 110, 211301) We use the variables adapted to spherical symmetry developed by Bojowald and Swiderski (CQG23, 2129 (2006)).