(Aka Second Quantization) 1 Quantum Field Theory
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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 -
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 -
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)). -
Supersymmetric Dualities and Quantum Entanglement in Conformal Field Theory
Supersymmetric Dualities and Quantum Entanglement in Conformal Field Theory Jeongseog Lee A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for Acceptance by the Department of Physics Adviser: Igor R. Klebanov September 2016 c Copyright by Jeongseog Lee, 2016. All rights reserved. Abstract Conformal field theory (CFT) has been under intensive study for many years. The scale invariance, which arises at the fixed points of renormalization group in rela- tivistic quantum field theory (QFT), is believed to be enhanced to the full conformal group. In this dissertation we use two tools to shed light on the behavior of certain conformal field theories, the supersymmetric localization and the quantum entangle- ment Renyi entropies. The first half of the dissertation surveys the infrared (IR) structure of the = 2 N supersymmetric quantum chromodynamics (SQCD) in three dimensions. The re- cently developed F -maximization principle shows that there are richer structures along conformal fixed points compared to those in = 1 SQCD in four dimensions. N We refer to one of the new phenomena as a \crack in the conformal window". Using the known IR dualities, we investigate it for all types of simple gauge groups. We see different decoupling behaviors of operators depending on the gauge group. We also observe that gauging some flavor symmetries modifies the behavior of the theory in the IR. The second half of the dissertation uses the R`enyi entropy to understand the CFT from another angle. At the conformal fixed points of even dimensional QFT, the entanglement entropy is known to have the log-divergent universal term depending on the geometric invariants on the entangling surface. -
Quantum Mechanics Propagator
Quantum Mechanics_propagator This article is about Quantum field theory. For plant propagation, see Plant propagation. In Quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. In Feynman diagrams, which calculate the rate of collisions in quantum field theory, virtual particles contribute their propagator to the rate of the scattering event described by the diagram. They also can be viewed as the inverse of the wave operator appropriate to the particle, and are therefore often called Green's functions. Non-relativistic propagators In non-relativistic quantum mechanics the propagator gives the probability amplitude for a particle to travel from one spatial point at one time to another spatial point at a later time. It is the Green's function (fundamental solution) for the Schrödinger equation. This means that, if a system has Hamiltonian H, then the appropriate propagator is a function satisfying where Hx denotes the Hamiltonian written in terms of the x coordinates, δ(x)denotes the Dirac delta-function, Θ(x) is the Heaviside step function and K(x,t;x',t')is the kernel of the differential operator in question, often referred to as the propagator instead of G in this context, and henceforth in this article. This propagator can also be written as where Û(t,t' ) is the unitary time-evolution operator for the system taking states at time t to states at time t'. The quantum mechanical propagator may also be found by using a path integral, where the boundary conditions of the path integral include q(t)=x, q(t')=x' . -
General Relativity 2020–2021 1 Overview
N I V E R U S E I T H Y T PHYS11010: General Relativity 2020–2021 O H F G E R D John Peacock I N B U Room C20, Royal Observatory; [email protected] http://www.roe.ac.uk/japwww/teaching/gr.html Textbooks These notes are intended to be self-contained, but there are many excellent textbooks on the subject. The following are especially recommended for background reading: Hobson, Efstathiou & Lasenby (Cambridge): General Relativity: An introduction for Physi- • cists. This is fairly close in level and approach to this course. Ohanian & Ruffini (Cambridge): Gravitation and Spacetime (3rd edition). A similar level • to Hobson et al. with some interesting insights on the electromagnetic analogy. Cheng (Oxford): Relativity, Gravitation and Cosmology: A Basic Introduction. Not that • ‘basic’, but another good match to this course. D’Inverno (Oxford): Introducing Einstein’s Relativity. A more mathematical approach, • without being intimidating. Weinberg (Wiley): Gravitation and Cosmology. A classic advanced textbook with some • unique insights. Downplays the geometrical aspect of GR. Misner, Thorne & Wheeler (Princeton): Gravitation. The classic antiparticle to Weinberg: • heavily geometrical and full of deep insights. Rather overwhelming until you have a reason- able grasp of the material. It may also be useful to consult background reading on some mathematical aspects, especially tensors and the variational principle. Two good references for mathematical methods are: Riley, Hobson and Bence (Cambridge; RHB): Mathematical Methods for Physics and Engi- • neering Arfken (Academic Press): Mathematical Methods for Physicists • 1 Overview General Relativity (GR) has an unfortunate reputation as a difficult subject, going back to the early days when the media liked to claim that only three people in the world understood Einstein’s theory. -
Path Integral in Quantum Field Theory Alexander Belyaev (Course Based on Lectures by Steven King) Contents
Path Integral in Quantum Field Theory Alexander Belyaev (course based on Lectures by Steven King) Contents 1 Preliminaries 5 1.1 Review of Classical Mechanics of Finite System . 5 1.2 Review of Non-Relativistic Quantum Mechanics . 7 1.3 Relativistic Quantum Mechanics . 14 1.3.1 Relativistic Conventions and Notation . 14 1.3.2 TheKlein-GordonEquation . 15 1.4 ProblemsSet1 ........................... 18 2 The Klein-Gordon Field 19 2.1 Introduction............................. 19 2.2 ClassicalScalarFieldTheory . 20 2.3 QuantumScalarFieldTheory . 28 2.4 ProblemsSet2 ........................... 35 3 Interacting Klein-Gordon Fields 37 3.1 Introduction............................. 37 3.2 PerturbationandScatteringTheory. 37 3.3 TheInteractionHamiltonian. 43 3.4 Example: K π+π− ....................... 45 S → 3.5 Wick’s Theorem, Feynman Propagator, Feynman Diagrams . .. 47 3.6 TheLSZReductionFormula. 52 3.7 ProblemsSet3 ........................... 58 4 Transition Rates and Cross-Sections 61 4.1 TransitionRates .......................... 61 4.2 TheNumberofFinalStates . 63 4.3 Lorentz Invariant Phase Space (LIPS) . 63 4.4 CrossSections............................ 64 4.5 Two-bodyScattering . 65 4.6 DecayRates............................. 66 4.7 OpticalTheorem .......................... 66 4.8 ProblemsSet4 ........................... 68 1 2 CONTENTS 5 Path Integrals in Quantum Mechanics 69 5.1 Introduction............................. 69 5.2 The Point to Point Transition Amplitude . 70 5.3 ImaginaryTime........................... 74 5.4 Transition Amplitudes With an External Driving Force . ... 77 5.5 Expectation Values of Heisenberg Position Operators . .... 81 5.6 Appendix .............................. 83 5.6.1 GaussianIntegration . 83 5.6.2 Functionals ......................... 85 5.7 ProblemsSet5 ........................... 87 6 Path Integral Quantisation of the Klein-Gordon Field 89 6.1 Introduction............................. 89 6.2 TheFeynmanPropagator(again) . 91 6.3 Green’s Functions in Free Field Theory .