Chapter 10 Relativistic Wave Equations
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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. -
Statistics of a Free Single Quantum Particle at a Finite
STATISTICS OF A FREE SINGLE QUANTUM PARTICLE AT A FINITE TEMPERATURE JIAN-PING PENG Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, China Abstract We present a model to study the statistics of a single structureless quantum particle freely moving in a space at a finite temperature. It is shown that the quantum particle feels the temperature and can exchange energy with its environment in the form of heat transfer. The underlying mechanism is diffraction at the edge of the wave front of its matter wave. Expressions of energy and entropy of the particle are obtained for the irreversible process. Keywords: Quantum particle at a finite temperature, Thermodynamics of a single quantum particle PACS: 05.30.-d, 02.70.Rr 1 Quantum mechanics is the theoretical framework that describes phenomena on the microscopic level and is exact at zero temperature. The fundamental statistical character in quantum mechanics, due to the Heisenberg uncertainty relation, is unrelated to temperature. On the other hand, temperature is generally believed to have no microscopic meaning and can only be conceived at the macroscopic level. For instance, one can define the energy of a single quantum particle, but one can not ascribe a temperature to it. However, it is physically meaningful to place a single quantum particle in a box or let it move in a space where temperature is well-defined. This raises the well-known question: How a single quantum particle feels the temperature and what is the consequence? The question is particular important and interesting, since experimental techniques in recent years have improved to such an extent that direct measurement of electron dynamics is possible.1,2,3 It should also closely related to the question on the applicability of the thermodynamics to small systems on the nanometer scale.4 We present here a model to study the behavior of a structureless quantum particle moving freely in a space at a nonzero temperature. -
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. -
The Heisenberg Uncertainty Principle*
OpenStax-CNX module: m58578 1 The Heisenberg Uncertainty Principle* OpenStax This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 Abstract By the end of this section, you will be able to: • Describe the physical meaning of the position-momentum uncertainty relation • Explain the origins of the uncertainty principle in quantum theory • Describe the physical meaning of the energy-time uncertainty relation Heisenberg's uncertainty principle is a key principle in quantum mechanics. Very roughly, it states that if we know everything about where a particle is located (the uncertainty of position is small), we know nothing about its momentum (the uncertainty of momentum is large), and vice versa. Versions of the uncertainty principle also exist for other quantities as well, such as energy and time. We discuss the momentum-position and energy-time uncertainty principles separately. 1 Momentum and Position To illustrate the momentum-position uncertainty principle, consider a free particle that moves along the x- direction. The particle moves with a constant velocity u and momentum p = mu. According to de Broglie's relations, p = }k and E = }!. As discussed in the previous section, the wave function for this particle is given by −i(! t−k x) −i ! t i k x k (x; t) = A [cos (! t − k x) − i sin (! t − k x)] = Ae = Ae e (1) 2 2 and the probability density j k (x; t) j = A is uniform and independent of time. The particle is equally likely to be found anywhere along the x-axis but has denite values of wavelength and wave number, and therefore momentum. -
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 -
The S-Matrix Formulation of Quantum Statistical Mechanics, with Application to Cold Quantum Gas
THE S-MATRIX FORMULATION OF QUANTUM STATISTICAL MECHANICS, WITH APPLICATION TO COLD QUANTUM GAS A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Pye Ton How August 2011 c 2011 Pye Ton How ALL RIGHTS RESERVED THE S-MATRIX FORMULATION OF QUANTUM STATISTICAL MECHANICS, WITH APPLICATION TO COLD QUANTUM GAS Pye Ton How, Ph.D. Cornell University 2011 A novel formalism of quantum statistical mechanics, based on the zero-temperature S-matrix of the quantum system, is presented in this thesis. In our new formalism, the lowest order approximation (“two-body approximation”) corresponds to the ex- act resummation of all binary collision terms, and can be expressed as an integral equation reminiscent of the thermodynamic Bethe Ansatz (TBA). Two applica- tions of this formalism are explored: the critical point of a weakly-interacting Bose gas in two dimensions, and the scaling behavior of quantum gases at the unitary limit in two and three spatial dimensions. We found that a weakly-interacting 2D Bose gas undergoes a superfluid transition at T 2πn/[m log(2π/mg)], where n c ≈ is the number density, m the mass of a particle, and g the coupling. In the unitary limit where the coupling g diverges, the two-body kernel of our integral equation has simple forms in both two and three spatial dimensions, and we were able to solve the integral equation numerically. Various scaling functions in the unitary limit are defined (as functions of µ/T ) and computed from the numerical solutions. -
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. -
Finite Quantum Field Theory and Renormalization Group
Finite Quantum Field Theory and Renormalization Group M. A. Greena and J. W. Moffata;b aPerimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada bDepartment of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada September 15, 2021 Abstract Renormalization group methods are applied to a scalar field within a finite, nonlocal quantum field theory formulated perturbatively in Euclidean momentum space. It is demonstrated that the triviality problem in scalar field theory, the Higgs boson mass hierarchy problem and the stability of the vacuum do not arise as issues in the theory. The scalar Higgs field has no Landau pole. 1 Introduction An alternative version of the Standard Model (SM), constructed using an ultraviolet finite quantum field theory with nonlocal field operators, was investigated in previous work [1]. In place of Dirac delta-functions, δ(x), the theory uses distributions (x) based on finite width Gaussians. The Poincar´eand gauge invariant model adapts perturbative quantumE field theory (QFT), with a finite renormalization, to yield finite quantum loops. For the weak interactions, SU(2) U(1) is treated as an ab initio broken symmetry group with non- zero masses for the W and Z intermediate× vector bosons and for left and right quarks and leptons. The model guarantees the stability of the vacuum. Two energy scales, ΛM and ΛH , were introduced; the rate of asymptotic vanishing of all coupling strengths at vertices not involving the Higgs boson is controlled by ΛM , while ΛH controls the vanishing of couplings to the Higgs. Experimental tests of the model, using future linear or circular colliders, were proposed. -
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 ). -
U(1) Symmetry of the Complex Scalar and Scalar Electrodynamics
Fall 2019: Classical Field Theory (PH6297) U(1) Symmetry of the complex scalar and scalar electrodynamics August 27, 2019 1 Global U(1) symmetry of the complex field theory & associated Noether charge Consider the complex scalar field theory, defined by the action, h i h i I Φ(x); Φy(x) = d4x (@ Φ)y @µΦ − V ΦyΦ : (1) ˆ µ As we have noted earlier complex scalar field theory action Eq. (1) is invariant under multiplication by a constant complex phase factor ei α, Φ ! Φ0 = e−i αΦ; Φy ! Φ0y = ei αΦy: (2) The phase,α is necessarily a real number. Since a complex phase is unitary 1 × 1 matrix i.e. the complex conjugation is also the inverse, y −1 e−i α = e−i α ; such phases are also called U(1) factors (U stands for Unitary matrix and since a number is a 1×1 matrix, U(1) is unitary matrix of size 1 × 1). Since this symmetry transformation does not touch spacetime but only changes the fields, such a symmetry is called an internal symmetry. Also note that since α is a constant i.e. not a function of spacetime, it is a global symmetry (global = same everywhere = independent of spacetime location). Check: Under the U(1) symmetry Eq. (2), the combination ΦyΦ is obviously invariant, 0 Φ0yΦ = ei αΦy e−i αΦ = ΦyΦ: This implies any function of the product ΦyΦ is also invariant. 0 V Φ0yΦ = V ΦyΦ : Note that this is true whether α is a constant or a function of spacetime i.e.