On the Electric and Magnetic Field Generation in Expanding Plasmas
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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 -
Magnetohydrodynamics
Magnetohydrodynamics J.W.Haverkort April 2009 Abstract This summary of the basics of magnetohydrodynamics (MHD) as- sumes the reader is acquainted with both fluid mechanics and electro- dynamics. The content is largely based on \An introduction to Magneto- hydrodynamics" by P.A. Davidson. The basic equations of electrodynam- ics are summarized, the assumptions underlying MHD are exposed and a selection of aspects of the theory are discussed. Contents 1 Electrodynamics 2 1.1 The governing equations . 2 1.2 Faraday's law . 2 2 Magnetohydrodynamical Equations 3 2.1 The assumptions . 3 2.2 The equations . 4 2.3 Some more equations . 4 3 Concepts of Magnetohydrodynamics 5 3.1 Analogies between fluid mechanics and MHD . 5 3.2 Dimensionless numbers . 6 3.3 Lorentz force . 7 1 1 Electrodynamics 1.1 The governing equations The Maxwell equations for the electric and magnetic fields E and B in vacuum are written in terms of the sources ρe (electric charge density) and J (electric current density) ρ r · E = e Gauss's law (1) "0 @B r × E = − Faraday's law (2) @t r · B = 0 No monopoles (3) @E r × B = µ J + µ " Ampere's law (4) 0 0 0 @t with "0 the electric permittivity and µ0 the magnetic permeability of free space. The electric field can be thought to consist of a static curl-free part Es = @A −∇V and an induced or in-stationary part Ei = − @t which is divergence-free, with V the electric potential and A the magnetic vector potential. The force on a charge q moving with velocity u is given by the Lorentz force f = q(E+u×B). -
(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. -
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. -
Astrophysical Fluid Dynamics: II. Magnetohydrodynamics
Winter School on Computational Astrophysics, Shanghai, 2018/01/30 Astrophysical Fluid Dynamics: II. Magnetohydrodynamics Xuening Bai (白雪宁) Institute for Advanced Study (IASTU) & Tsinghua Center for Astrophysics (THCA) source: J. Stone Outline n Astrophysical fluids as plasmas n The MHD formulation n Conservation laws and physical interpretation n Generalized Ohm’s law, and limitations of MHD n MHD waves n MHD shocks and discontinuities n MHD instabilities (examples) 2 Outline n Astrophysical fluids as plasmas n The MHD formulation n Conservation laws and physical interpretation n Generalized Ohm’s law, and limitations of MHD n MHD waves n MHD shocks and discontinuities n MHD instabilities (examples) 3 What is a plasma? Plasma is a state of matter comprising of fully/partially ionized gas. Lightening The restless Sun Crab nebula A plasma is generally quasi-neutral and exhibits collective behavior. Net charge density averages particles interact with each other to zero on relevant scales at long-range through electro- (i.e., Debye length). magnetic fields (plasma waves). 4 Why plasma astrophysics? n More than 99.9% of observable matter in the universe is plasma. n Magnetic fields play vital roles in many astrophysical processes. n Plasma astrophysics allows the study of plasma phenomena at extreme regions of parameter space that are in general inaccessible in the laboratory. 5 Heliophysics and space weather l Solar physics (including flares, coronal mass ejection) l Interaction between the solar wind and Earth’s magnetosphere l Heliospheric -
Magnetohydrodynamics
MAGNETOHYDRODYNAMICS International Max-Planck Research School. Lindau, 9{13 October 2006 {4{ Bulletin of exercises n◦4: The magnetic induction equation. 1. The induction equation: Starting from Faraday's law and Ampere's law (neglecting the displacement current) and making use of Ohm's constitutive relation, eliminate the electric field and the current density to obtain the inducion equation for a plasma in the MHD approximation: @B c2 = rot (v B) rot rot B : (1) @t ^ − 4πσe ! Show that if the electrical conductivity is uniform, the induction equation can be cast into the form @B = rot (v B) + η 2 B ; (2) @t ^ r def 2 where η = c =(4πσe) is the \magnetic diffusivity." 2. Combined form of the induction and the continuity equations. Show that the induction equation can be combined with the equation of continuity into one single differential equation which gives the time evolution of B/ρ following a fluid element, viz: D B B 1 = v rot (η rot B) : (3) D t ρ ! ρ · r! − ρ In the case of an ideal MHD-plasma, Eq. (3) simplifies to a form known as Wal´en's equation. 3. Strength of a magnetic flux tube. A magnetic flux tube is the region enclosed by the surface determined by all magnetic field lines passing through a given material circuit Ct . The \strength" of a flux tube is defined as the circulation of the potential vector A along the material circuit Ct , viz. ξ def def 2 Φm(t) = A d α = A[α(ξ; t)] α0(ξ; t) d ξ ; (4) ICt · Zξ1 · where α = α(ξ; t) is the parametric expression of the circuit Ct . -
Turbulent Dynamos: Experiments, Nonlinear Saturation of the Magnetic field and field Reversals
Turbulent dynamos: Experiments, nonlinear saturation of the magnetic field and field reversals C. Gissinger & C. Guervilly November 17, 2008 1 Introduction Magnetic fields are present at almost all scales in the universe, from the Earth (roughly 0:5 G) to the Galaxy (10−6 G). It is commonly believed that these magnetic fields are generated by dynamo action i.e. by the turbulent flow of an electrically conducting fluid [12]. Despite this space and time disorganized flow, the magnetic field shows in general a coherent part at the largest scales. The question arising from this observation is the role of the mean flow: Cowling first proposed that the coherent magnetic field could be due to coherent large scale velocity field and this problem is still an open question. 2 MHD equations and dimensionless parameters The equations describing the evolution of a magnetohydrodynamical system are the equation of Navier-Stokes coupled to the induction equation: @v 1 + (v )v = π + ν∆v + f + (B )B ; (1) @t · r −∇ µρ · r @B = (v B) + η ∆B : (2) @t r × × where v is the solenoidal velocity field, B the solenoidal magnetic field, π the pressure r gradient in the fluid, ν the kinematic viscosity, f a forcing term, µ the magnetic permeabil- ity, ρ the fluid density and η the magnetic permeability. Dealing with the geodynamo, the minimal set of parameters for the outer core are : ρ : density of the fluid • µ : magnetic permeability • 54 ν : kinematic diffusivity • σ: conductivity • R : radius of the outer core • V : typical velocity • Ω : rotation rate • The problem involves 7 independent parameters and 4 fundamental units (length L, time T , mass M and electic current A). -
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. -
Introduction to Gauge Theories and the Standard Model*
INTRODUCTION TO GAUGE THEORIES AND THE STANDARD MODEL* B. de Wit Institute for Theoretical Physics P.O.B. 80.006, 3508 TA Utrecht The Netherlands Contents The action Feynman rules Photons Annihilation of spinless particles by electromagnetic interaction Gauge theory of U(1) Current conservation Conserved charges Nonabelian gauge fields Gauge invariant Lagrangians for spin-0 and spin-g Helds 10. The gauge field Lagrangian 11. Spontaneously broken symmetry 12. The Brout—Englert-Higgs mechanism 13. Massive SU (2) gauge Helds 14. The prototype model for SU (2) ® U(1) electroweak interactions The purpose of these lectures is to give an introduction to gauge theories and the standard model. Much of this material has also been covered in previous Cem Academic Training courses. This time I intend to start from section 5 and develop the conceptual basis of gauge theories in order to enable the construction of generic models. Subsequently spontaneous symmetry breaking is discussed and its relevance is explained for the renormalizability of theories with massive vector fields. Then we discuss the derivation of the standard model and its most conspicuous features. When time permits we will address some of the more practical questions that arise in the evaluation of quantum corrections to particle scattering and decay reactions. That material is not covered by these notes. CERN Academic Training Programme — 23-27 October 1995 OCR Output 1. The action Field theories are usually defined in terms of a Lagrangian, or an action. The action, which has the dimension of Planck’s constant 7i, and the Lagrangian are well-known concepts in classical mechanics. -
Classical-Field Description of the Quantum Effects in the Light-Atom Interaction
Classical-field description of the quantum effects in the light-atom interaction Sergey A. Rashkovskiy Institute for Problems in Mechanics of the Russian Academy of Sciences, Vernadskogo Ave., 101/1, Moscow, 119526, Russia Tomsk State University, 36 Lenina Avenue, Tomsk, 634050, Russia E-mail: [email protected], Tel. +7 906 0318854 March 1, 2016 Abstract In this paper I show that light-atom interaction can be described using purely classical field theory without any quantization. In particular, atom excitation by light that accounts for damping due to spontaneous emission is fully described in the framework of classical field theory. I show that three well-known laws of the photoelectric effect can also be derived and that all of its basic properties can be described within classical field theory. Keywords Hydrogen atom, classical field theory, light-atom interaction, photoelectric effect, deterministic process, nonlinear Schrödinger equation, statistical interpretation. Mathematics Subject Classification 82C10 1 Introduction The atom-field interaction is one of the most fundamental problems of quantum optics. It is believed that a complete description of the light-atom interaction is possible only within the framework of quantum electrodynamics (QED), when not only the states of an atom are quantized but also the radiation itself. In quantum mechanics, this viewpoint fully applies to iconic phenomena, such as the photoelectric effect, which cannot be explained in terms of classical mechanics and classical electrodynamics according to the prevailing considerations. There were attempts to build a so-called semiclassical theory in which only the states of an atom are quantized while the radiation is considered to be a classical Maxwell field [1-6].