Scalar Fields in Particle Physics
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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 ........................... -
Exact Solutions to the Interacting Spinor and Scalar Field Equations in the Godel Universe
XJ9700082 E2-96-367 A.Herrera 1, G.N.Shikin2 EXACT SOLUTIONS TO fHE INTERACTING SPINOR AND SCALAR FIELD EQUATIONS IN THE GODEL UNIVERSE 1 E-mail: [email protected] ^Department of Theoretical Physics, Russian Peoples ’ Friendship University, 117198, 6 Mikluho-Maklaya str., Moscow, Russia ^8 as ^; 1996 © (XrbCAHHeHHuft HHCTtrryr McpHMX HCCJicAOB&Htift, fly 6na, 1996 1. INTRODUCTION Recently an increasing interest was expressed to the search of soliton-like solu tions because of the necessity to describe the elementary particles as extended objects [1]. In this work, the interacting spinor and scalar field system is considered in the external gravitational field of the Godel universe in order to study the influence of the global properties of space-time on the interaction of one dimensional fields, in other words, to observe what is the role of gravitation in the interaction of elementary particles. The Godel universe exhibits a number of unusual properties associated with the rotation of the universe [2]. It is ho mogeneous in space and time and is filled with a perfect fluid. The main role of rotation in this universe consists in the avoidance of the cosmological singu larity in the early universe, when the centrifugate forces of rotation dominate over gravitation and the collapse does not occur [3]. The paper is organized as follows: in Sec. 2 the interacting spinor and scalar field system with £jnt = | <ptp<p'PF(Is) in the Godel universe is considered and exact solutions to the corresponding field equations are obtained. In Sec. 3 the properties of the energy density are investigated. -
An Introduction to Supersymmetry
An Introduction to Supersymmetry Ulrich Theis Institute for Theoretical Physics, Friedrich-Schiller-University Jena, Max-Wien-Platz 1, D–07743 Jena, Germany [email protected] This is a write-up of a series of five introductory lectures on global supersymmetry in four dimensions given at the 13th “Saalburg” Summer School 2007 in Wolfersdorf, Germany. Contents 1 Why supersymmetry? 1 2 Weyl spinors in D=4 4 3 The supersymmetry algebra 6 4 Supersymmetry multiplets 6 5 Superspace and superfields 9 6 Superspace integration 11 7 Chiral superfields 13 8 Supersymmetric gauge theories 17 9 Supersymmetry breaking 22 10 Perturbative non-renormalization theorems 26 A Sigma matrices 29 1 Why supersymmetry? When the Large Hadron Collider at CERN takes up operations soon, its main objective, besides confirming the existence of the Higgs boson, will be to discover new physics beyond the standard model of the strong and electroweak interactions. It is widely believed that what will be found is a (at energies accessible to the LHC softly broken) supersymmetric extension of the standard model. What makes supersymmetry such an attractive feature that the majority of the theoretical physics community is convinced of its existence? 1 First of all, under plausible assumptions on the properties of relativistic quantum field theories, supersymmetry is the unique extension of the algebra of Poincar´eand internal symmtries of the S-matrix. If new physics is based on such an extension, it must be supersymmetric. Furthermore, the quantum properties of supersymmetric theories are much better under control than in non-supersymmetric ones, thanks to powerful non- renormalization theorems. -
Vector Fields
Vector Calculus Independent Study Unit 5: Vector Fields A vector field is a function which associates a vector to every point in space. Vector fields are everywhere in nature, from the wind (which has a velocity vector at every point) to gravity (which, in the simplest interpretation, would exert a vector force at on a mass at every point) to the gradient of any scalar field (for example, the gradient of the temperature field assigns to each point a vector which says which direction to travel if you want to get hotter fastest). In this section, you will learn the following techniques and topics: • How to graph a vector field by picking lots of points, evaluating the field at those points, and then drawing the resulting vector with its tail at the point. • A flow line for a velocity vector field is a path ~σ(t) that satisfies ~σ0(t)=F~(~σ(t)) For example, a tiny speck of dust in the wind follows a flow line. If you have an acceleration vector field, a flow line path satisfies ~σ00(t)=F~(~σ(t)) [For example, a tiny comet being acted on by gravity.] • Any vector field F~ which is equal to ∇f for some f is called a con- servative vector field, and f its potential. The terminology comes from physics; by the fundamental theorem of calculus for work in- tegrals, the work done by moving from one point to another in a conservative vector field doesn’t depend on the path and is simply the difference in potential at the two points. -
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 -
General Relativity and Cosmology: Unsolved Questions and Future Directions
Article General Relativity and Cosmology: Unsolved Questions and Future Directions Ivan Debono 1,∗,† and George F. Smoot 1,2,3,† 1 Paris Centre for Cosmological Physics, APC, AstroParticule et Cosmologie, Université Paris Diderot, CNRS/IN2P3, CEA/lrfu, Observatoire de Paris, Sorbonne Paris Cité, 10, rue Alice Domon et Léonie Duquet, 75205 Paris CEDEX 13, France; [email protected] 2 Physics Department and Lawrence Berkeley National Laboratory, University of California, Berkeley, 94720 CA, USA 3 Helmut and Anna Pao Sohmen Professor-at-Large, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, 999077 Hong Kong, China * Correspondence: [email protected]; Tel.: +33-1-57276991 † These authors contributed equally to this work. Academic Editors: Lorenzo Iorio and Elias C. Vagenas Received: 21 August 2016; Accepted: 14 September 2016; Published: 28 September 2016 Abstract: For the last 100 years, General Relativity (GR) has taken over the gravitational theory mantle held by Newtonian Gravity for the previous 200 years. This article reviews the status of GR in terms of its self-consistency, completeness, and the evidence provided by observations, which have allowed GR to remain the champion of gravitational theories against several other classes of competing theories. We pay particular attention to the role of GR and gravity in cosmology, one of the areas in which one gravity dominates and new phenomena and effects challenge the orthodoxy. We also review other areas where there are likely conflicts pointing to the need to replace or revise GR to represent correctly observations and consistent theoretical framework. Observations have long been key both to the theoretical liveliness and viability of GR. -
SCALAR FIELDS Gradient of a Scalar Field F(X), a Vector Field: Directional
53:244 (58:254) ENERGY PRINCIPLES IN STRUCTURAL MECHANICS SCALAR FIELDS Ø Gradient of a scalar field f(x), a vector field: ¶f Ñf( x ) = ei ¶xi Ø Directional derivative of f(x) in the direction s: Let u be a unit vector that defines the direction s: ¶x u = i e ¶s i Ø The directional derivative of f in the direction u is given as df = u · Ñf ds Ø The scalar component of Ñf in any direction gives the rate of change of df/ds in that direction. Ø Let q be the angle between u and Ñf. Then df = u · Ñf = u Ñf cos q = Ñf cos q ds Ø Therefore df/ds will be maximum when cosq = 1; i.e., q = 0. This means that u is in the direction of f(x). Ø Ñf points in the direction of the maximum rate of increase of the function f(x). Lecture #5 1 53:244 (58:254) ENERGY PRINCIPLES IN STRUCTURAL MECHANICS Ø The magnitude of Ñf equals the maximum rate of increase of f(x) per unit distance. Ø If direction u is taken as tangent to a f(x) = constant curve, then df/ds = 0 (because f(x) = c). df = 0 Þ u· Ñf = 0 Þ Ñf is normal to f(x) = c curve or surface. ds THEORY OF CONSERVATIVE FIELDS Vector Field Ø Rule that associates each point (xi) in the domain D (i.e., open and connected) with a vector F = Fxi(xj)ei; where xj = x1, x2, x3. The vector field F determines at each point in the region a direction. -
9.7. Gradient of a Scalar Field. Directional Derivatives. A. The
9.7. Gradient of a Scalar Field. Directional Derivatives. A. The Gradient. 1. Let f: R3 ! R be a scalar ¯eld, that is, a function of three variables. The gradient of f, denoted rf, is the vector ¯eld given by " # @f @f @f @f @f @f rf = ; ; = i + j + k: @x @y @y @x @y @z 2. Symbolically, we can consider r to be a vector of di®erential operators, that is, @ @ @ r = i + j + k: @x @y @z Then rf is symbolically the \vector" r \times" the \scalar" f. 3. Note that rf is a vector ¯eld so that at each point P , rf(P ) is a vector, not a scalar. B. Directional Derivative. 1. Recall that for an ordinary function f(t), the derivative f 0(t) represents the rate of change of f at t and also the slope of the tangent line at t. The gradient provides an analogous quantity for scalar ¯elds. 2. When considering the rate of change of a scalar ¯eld f(x; y; z), we talk about its rate of change in a direction b. (So that b is a unit vector.) 3. The directional derivative of f at P in direction b is f(P + tb) ¡ f(P ) Dbf(P ) = lim = rf(P ) ¢ b: t!0 t Note that in this de¯nition, b is assumed to be a unit vector. C. Maximum Rate of Change. 1. The directional derivative satis¯es Dbf(P ) = rf(P ) ¢ b = jbjjrf(P )j cos γ = jrf(P )j cos γ where γ is the angle between b and rf(P ). -
Introduction to Supersymmetry
Introduction to Supersymmetry Pre-SUSY Summer School Corpus Christi, Texas May 15-18, 2019 Stephen P. Martin Northern Illinois University [email protected] 1 Topics: Why: Motivation for supersymmetry (SUSY) • What: SUSY Lagrangians, SUSY breaking and the Minimal • Supersymmetric Standard Model, superpartner decays Who: Sorry, not covered. • For some more details and a slightly better attempt at proper referencing: A supersymmetry primer, hep-ph/9709356, version 7, January 2016 • TASI 2011 lectures notes: two-component fermion notation and • supersymmetry, arXiv:1205.4076. If you find corrections, please do let me know! 2 Lecture 1: Motivation and Introduction to Supersymmetry Motivation: The Hierarchy Problem • Supermultiplets • Particle content of the Minimal Supersymmetric Standard Model • (MSSM) Need for “soft” breaking of supersymmetry • The Wess-Zumino Model • 3 People have cited many reasons why extensions of the Standard Model might involve supersymmetry (SUSY). Some of them are: A possible cold dark matter particle • A light Higgs boson, M = 125 GeV • h Unification of gauge couplings • Mathematical elegance, beauty • ⋆ “What does that even mean? No such thing!” – Some modern pundits ⋆ “We beg to differ.” – Einstein, Dirac, . However, for me, the single compelling reason is: The Hierarchy Problem • 4 An analogy: Coulomb self-energy correction to the electron’s mass A point-like electron would have an infinite classical electrostatic energy. Instead, suppose the electron is a solid sphere of uniform charge density and radius R. An undergraduate problem gives: 3e2 ∆ECoulomb = 20πǫ0R 2 Interpreting this as a correction ∆me = ∆ECoulomb/c to the electron mass: 15 0.86 10− meters m = m + (1 MeV/c2) × . -
Conceptual Barriers to a Unified Theory of Physics 1 Introduction
Conceptual barriers to a unified theory of physics Dennis Crossley∗ Dept. of Physics, University of Wisconsin-Sheboygan, Sheboygan, WI 53081 August 31, 2012 Abstract The twin pillars of twentieth-century physics, quantum theory and general relativity, have conceptual errors in their foundations, which are at the heart of the repeated failures to combine these into a single unified theory of physics. The problem with quantum theory is related to the use of the point-particle model, and the problem with general relativity follows from a misinterpretation of the significance of the equivalence principle. Correcting these conceptual errors leads to a new model of matter called the space wave model which is outlined here. The new perspective gained by space wave theory also makes it clear that there are conceptual errors in the two main thrusts of twenty-first- century theoretical physics, string theory and loop quantum gravity. The string model is no more satisfactory than the point-particle model and the notion that space must be quantized is, frankly, nonsensical. In this paper I examine all of these conceptual errors and suggest how to correct them so that we can once again make progress toward a unified theory of physics. 1 Introduction { the challenge of the unification program The goal of theoretical physics is to construct a single unified description of fundamental particles and their interactions, but flaws in the foundations of physics have prevented physi- cists from achieving this goal. In this essay I examine conceptual errors that have led to this impasse and propose alternatives which break this impasse and let us once again move forward toward the goal of a unified theory of physics. -
3. Introducing Riemannian Geometry
3. Introducing Riemannian Geometry We have yet to meet the star of the show. There is one object that we can place on a manifold whose importance dwarfs all others, at least when it comes to understanding gravity. This is the metric. The existence of a metric brings a whole host of new concepts to the table which, collectively, are called Riemannian geometry.Infact,strictlyspeakingwewillneeda slightly di↵erent kind of metric for our study of gravity, one which, like the Minkowski metric, has some strange minus signs. This is referred to as Lorentzian Geometry and a slightly better name for this section would be “Introducing Riemannian and Lorentzian Geometry”. However, for our immediate purposes the di↵erences are minor. The novelties of Lorentzian geometry will become more pronounced later in the course when we explore some of the physical consequences such as horizons. 3.1 The Metric In Section 1, we informally introduced the metric as a way to measure distances between points. It does, indeed, provide this service but it is not its initial purpose. Instead, the metric is an inner product on each vector space Tp(M). Definition:Ametric g is a (0, 2) tensor field that is: Symmetric: g(X, Y )=g(Y,X). • Non-Degenerate: If, for any p M, g(X, Y ) =0forallY T (M)thenX =0. • 2 p 2 p p With a choice of coordinates, we can write the metric as g = g (x) dxµ dx⌫ µ⌫ ⌦ The object g is often written as a line element ds2 and this expression is abbreviated as 2 µ ⌫ ds = gµ⌫(x) dx dx This is the form that we saw previously in (1.4). -
(TEGR) As a Gauge Theory: Translation Or Cartan Connection? M Fontanini, E
Teleparallel gravity (TEGR) as a gauge theory: Translation or Cartan connection? M Fontanini, E. Huguet, M. Le Delliou To cite this version: M Fontanini, E. Huguet, M. Le Delliou. Teleparallel gravity (TEGR) as a gauge theory: Transla- tion or Cartan connection?. Physical Review D, American Physical Society, 2019, 99, pp.064006. 10.1103/PhysRevD.99.064006. hal-01915045 HAL Id: hal-01915045 https://hal.archives-ouvertes.fr/hal-01915045 Submitted on 7 Nov 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Teleparallel gravity (TEGR) as a gauge theory: Translation or Cartan connection? M. Fontanini1, E. Huguet1, and M. Le Delliou2 1 - Universit´eParis Diderot-Paris 7, APC-Astroparticule et Cosmologie (UMR-CNRS 7164), Batiment Condorcet, 10 rue Alice Domon et L´eonieDuquet, F-75205 Paris Cedex 13, France.∗ and 2 - Institute of Theoretical Physics, Physics Department, Lanzhou University, No.222, South Tianshui Road, Lanzhou, Gansu 730000, P R China y (Dated: November 7, 2018) In this paper we question the status of TEGR, the Teleparallel Equivalent of General Relativity, as a gauge theory of translations. We observe that TEGR (in its usual translation-gauge view) does not seem to realize the generally admitted requirements for a gauge theory for some symmetry group G: namely it does not present a mathematical structure underlying the theory which relates to a principal G-bundle and the choice of a connection on it (the gauge field).