Classical Physics. SpaceTime Algebra
|
|
- Clifton Stevenson
- 5 years ago
- Views:
Transcription
1 Classical Physics with SpaceTime Algebra David Hestenes Arizona State University x x(τ ) x 0 Santalo 2016
2 Objectives of this talk To introduce SpaceTime Algebra (STA) as a unified, coordinate-free mathematical language for classical physics: the vector derivative as the fundamental tool of spacetime calculus and demonstrate its effectiveness in electrodynamics vector and spinor particle mechanics with internal spin and clock. the spacetime split to project 4D invariant physics into 3D geometry of an inertial system. References Spacetime physics with geometric algebra, Am. J. Phys. 71: (2003). < > Space Time Algebra (Springer: 2015) 2 nd Ed.
3 GA unifies and coordinates other algebraic systems Multivector: Quaternion (spinor): Cross product: (even subalgebra) Gibb s vector algebra Matrix algebra is subsidiary to and facilitated by GA Matrix: Determinant: Complex numbers: (involves only the inner product) Row vectors Column vectors
4 Differentiation by vectors Vector product: Vector derivative: Names: del (grad) = div + curl One differential operator!: Vector field: E = E(x) Electrostatics: Magnetostatics:
5 Vector derivatives in Rectangular coordinates: Position vector: Vector derivative: Basic derivatives for routine calculations:?? for constant a
6 Chain Rule
7 Electromagnetic Fields One Electromagnetic Field!: One Maxwell s Equation: F = E + ib 1 ( c t + )F = ρ 1 c J 1 c t E + i 1 c t B + E + i B = ρ 1 c J Use E = E + i( E) and separate k-vector parts: 1 Scalar 3 Vector 3 Bivector 1 Pseudoscalar E = ρ 1 c t E + i 2 B = 1 c J i 1 c t B + i E = 0 i B = 0 Energy-momentum density: 1 2 FF = 1 2 (E + ib)(e ib) = 1 2 (E 2 + B 2 ) + E B Invariants: F 2 = (E + ib) 2 = E 2 B 2 + 2i(E B)
8 Redundancy in conventional mathematics Fund. Thm of calculus: Green s Thm: Stokes Thm: Gauss Thm: Generalized Stokes Thm: Unification in a single Fundamental Theorem: k-vector directed measure: Generalized Cauchy Thm: Cauchy-Riemann Eqn. x R x R
9 Antiderivatives: x n R R solves electrostatic and magnetostatic problems solid angle pseudoscalar
10 x n R dim R = n = 3: R x R dx dim R = n = 2: R Generalised Cauchy Integral Formula Complex variable: z = ax dx = adz
11 EM Plane Waves: Solutions of the form constant on moving plane x = x(t) n Shock wave! x v = cn Monochromatic: s = constant E B k s = constant
12 Plane Waves: E Monochromatic Wave: Wave packet: B k Sign of ω = helicity Energy density: z(s) describes the energy, frequency and polarization structure of the plane wave
13 Electromagnetic Fields in Continuous Media 1 ( c t + )(E + ib) = ρ 1 c J = ρ f + ρ b 1 c (J f + J b ) Bound charge density: Bound current: M = (1 µ 1 )B J f = σ E ρ b = P J b = t P + c M Free charge density Free charge density Constitutive equations for linear, homogeneous, isotropic media: P = (ε 1)E ε = permitivity (dielectric constant) μ = (magnetic) permeability Multiply even part of Max. eqn. by µε c t + E + i µε B = σ c σ = conductivity (Ohm s law) µ to get ε µ ε E Gaussian Units ρ f J f Or: ( 1 v t + )G = 0 G = E + n i B n = µε = c v
14 What is free space? Maxwell s equation for a homogeneous, isotropic medium ε = permitivity (dielectric constant) μ = (magnetic) permeability Maxwell s Equation Wave Equation Invariant under Lorentz transformations µ ε
15 SpaceTime Algebra (STA): Generated by a frame of vectors light cone STA (Real) Dirac Algebra Product: Metric: STA basis: Vector: scalar, vector, bivector, pseudovector, pseudoscalar Bivector: Unit pseudoscalar: i Multivector: Dual:
16 SpaceTime Algebra (STA): STA matrix rep (Real) Dirac Algebra Generated by a frame of vectors: {γ µ } Geometric product: γ 0 2 = 1, γ k 2 = 1 (k = 1,2, 3) bivector: γ µ γ ν = γ ν γ µ γ µ γ ν (µ ν) p = p µ γ µ = mv p 2 = m 2 F = 1 2 F νµ γ µ γ ν SpaceTime Split: Inertial system determined by observer γ 0 Spatial vector frame: { σ k = γ k γ 0 } γ 1 γ 0 γ 1 γ 2 light cone γ 2 γ 0 observer history (c =1) Unit pseudoscalar: i = σ 1 σ 2 σ 3 = γ 0 γ 1 γ 2 γ 3 i pγ 0 = (Eγ 0 + p k γ k )γ 0 = E + p F = F 0k σ k F jk σ k σ j = E + ib i 2 = 1 [D. Hestenes (2003), Am. J. Phys. 71: ]
17 EM field: Geometric Calculus & Electrodynamics Spacetime point: x = x µ γ µ Coordinates: x µ = x γ µ Derivative: = x = γ µ µ µ = x µ F = F(x) = 1 2 F νµ γ µ γ ν ST split: = γ µ F = E + ib Current: J = J(x) = J µ γ µ γ 0 J = ρ J Maxwell s equation: F = J γ 0 F = ( t + x )(E + ib) = ρ J = x E = E + E = E + i E Lorentz Force: m v = qf v E = ρ t E + i(i B) = J E + i t B = 0 i( B) = 0 vγ 0 = v γ 0 (1+ v γ 0 / v γ 0 ) = γ (1+ v) ST split: m γ = qe v m v = qγ (E + v B) (c = 1)
18 Summary for rotations in 2D, 3D and beyond Thm. I: Every rotation can be expressed in the canonical form: where and U is even Note: Thm. II: Every rotation in 3D can be expressed as product of two reflections: Generalizations: III. Thm I applies to Lorentz transformations of spacetime IV. Cartan-Dieudonné Thm (Lipschitz, 1880): Every orthogonal transformation can be represented in the form: Advantages over matrix form for rotations: coordinate-free composition of rotations: parametrizations (see NFCM)
19 Lorentz rotations without matrices or coordinates Rotation of a frame: γ µ e µ = Rγ µ R = a µ η γ η Matrix representation: Orthogonality: SpaceTime Split: a η µ = γ η e µ = γ η Rγ µ R R = ± A ( AA) 1/2 Spin representation: A e µ γ µ = a µ η γ η γ µ Rotor R defined by: or: R = e 1 2 B R = e 1 2 B Boost: R R = 1 Spatial rotation: Uγ 0 U = γ 0 Ri = ir e µ e ν = Rγ µ RRγ ν R = Rγ µ γ ν R = γ µ γ ν R = LU e 0 = Rγ 0R = Lγ 0L = L 2 γ 0 L = (e 0 γ 0 ) 1/2 γ 0 e 0 e k Uσ k U = Uγ k γ 0 U = Uγ k Uγ 0 = L e k e 0 L
20 SpaceTime-splits and particle kinematics Inertial observerdefined by a unit timelike vector: γ 0 ST-split of a spacetime point x: γ 2 0 = 1 ct observer s history x xγ 0 = x γ 0 + x γ 0 xγ 0 = ct + x ct = x γ 0 γ 0 x = ct x x = x γ 0 x 2 = (xγ 0 )(γ 0 x) = (ct + x)(ct x) = c 2 t 2 x 2 γ 0 0 x light cone Particle history: x = x(τ ) dx = dτ c = 1 proper velocity: v = x = dx dτ ST-split: vγ 0 = v 0 (1+ v) time dilation factor: relative velocity: v 2 = 1 = v 0 2 (1 v 2 ) v 0 = v γ 0 = dt dτ = (1 v2 ) 1 2 v = dx dt = dτ dx dt dτ = v γ 0 v 0 x particle history x = x(τ ) x 0
21 Relativistic Physics invariant vectors vs. covariant paravectors Recommended exercise: Undo the original paravector treatment of relativistic mechanics, written in 1980, published in 1999 in NFCM (chapter 9 in 2nd Ed.) The chief advocate: W. E. Baylis. Geometry of Paravector Space with Applications to Relativistic Physics (Kluwer: 2004) Electrodynamics (W. E. Baylis, Birkhäuser, 1999) Exercise: Translate from covariant to invariant
22 v 2 = 1 Spinor particle dynamics particle velocityv = x = v(τ ) can only rotate as the particle traverses its history x = x(τ ) v = Rγ 0 R Rotor: R = R(τ ) Rotor dr eqn. of motion: dτ = R = 1 2 ΩR Ω = 2 R R = 2R R Ω = rotational velocity dv dτ = v = Rγ 0R + Rγ 0R 1 = 2 (Ωv + vω) v = Ω v What does the rotor equation buy us? Ω = q m F m v = q F v Lorentz force! 1 Ω = 0 general solution: R = e Comoving frame: 2 Ωτ R 0 Spin: s = 2 e 3 = 2 R γ 3 R s = q m F s g = 2 v x = x(τ ) x 0 particle history γ 0 observer history e 0 = Rγ 0 R = v = x e e µ = Rγ µ R µ = Ω e µ Classical limit of Dirac equation! Real spinors are natural & useful in both classical & quantum theory!
23 Implications of Real Dirac Theory: the geometry of electron motion with Dirac equation determines a congruence of streamlines, each a potential particle history x = x(τ ) with particle velocity x = v(τ ) = R γ 0 R Dirac wave function de Broglie s electron clock in quantum mechanics! Spinning frame picture of electron motion Rotor: comoving frame: e µ = Rγ µ R velocity: Spin: ψ = (ρe iβ ) 1 2 R e 0 = Rγ 0 R = v S = 2 e 2e 1 determines R = R(τ ) = R[x(τ )] = R 0 e 1 2 ϕ γ 2γ 1 phase ϕ / 2 e 1 e 2 e 1 e 2 e1 ϕ e 1 particle history x 0 v e 2 e 1 e 2 e 2 e 2 e 1 = Rγ 2 γ 1 R = R 0 γ 2 γ 1 R 0 R = R 0 e 1 2 ϕ γ 2γ 1 = R 0 e p x γ 2γ 1 Plane wave solution: p = m e c 2 1 v 2 ϕ = p x = m ec 2 v x = ω Bτ τ = τ(x) = v x ω B = m e c2 = 1 2 dϕ dτ
24 Field strength: Maxwell s Equations: Lorentz force: Spinors: Quantum Mechanics: Summary and comparison Tensor form coordinate-dependent None! Dirac matrices superimposed STA coordinate-free (ST-split) rigid body precession! Real spinors natural & useful in both classical & quantum theory! General General covariance Displacement gauge invariance Relativity: Equivalence principle Rotation gauge covariance
25 Current Status of GA & STA Mathematical scope greater than any other system! Linear algebra Multilinear algebra Differential geometry Hypercomplex function theory (unifying and generalizing real and complex analysis) Lie groups as spin groups Crystallographic group theory Projective geometry Computational geometry distance geometry line geometry & screw theory Physics scope covers all major branches! Class./Rel. Mechanics Spacetime Algebra Gauge theory of gravitation on flat spacetime Coordinate-free Lasenby, Gull & Doran E & M Real QM Dirac eqn. Q.E.D. Electroweak Geometric basis for complex wave functions in QM!
26 Outline and References < < I. Intro to GA and non-relativistic applications Oersted Medal Lecture 2002 (Web and AJP) NFCM (Kluwer, 2nd Ed.1999) New Foundations for Mathematical Physics (Web) 1. Synopsis of GA 2. Geometric Calculus II. Relativistic Physics (covariant formulation) NFCM (chapter 9 in 2nd Ed.) Electrodynamics (W. E. Baylis, Birkhäuser, 1999) III. Spacetime Physics (invariant formulation) Spacetime Physics with Geometric Algebra (Web & AJP) Doran, Lasenby, Gull, Somaroo & Challinor, Spacetime Algebra and Electron Physics (Web) Lasenby & Doran, Geometric Algebra for Physicists (Cambridge: The University Press, 2002).
27 THE END
A simple and compact approach to hydrodynamic using geometric algebra. Abstract
A simple and compact approach to hydrodynamic using geometric algebra Xiong Wang (a) Center for Chaos and Complex Networks (b) Department of Electronic Engineering, City University of Hong Kong, Hong Kong
More informationGeometric Algebra 2 Quantum Theory
Geometric Algebra 2 Quantum Theory Chris Doran Astrophysics Group Cavendish Laboratory Cambridge, UK Spin Stern-Gerlach tells us that electron wavefunction contains two terms Describe state in terms of
More informationSPACETIME CALCULUS. David Hestenes Arizona State University
SPACETIME CALCULUS David Hestenes Arizona State University Abstract: This book provides a synopsis of spacetime calculus with applications to classical electrodynamics, quantum theory and gravitation.
More informationGeometric Algebra and Physics
Geometric Algebra and Physics Anthony Lasenby Astrophysics Group, Cavendish Laboratory, Cambridge, UK Oxford, 14 April 2008 Overview I am basically a cosmologist, interested in the Cosmic Microwave Background,
More informationAlgebra of Complex Vectors and Applications in Electromagnetic Theory and Quantum Mechanics
Mathematics 2015, 3, 781-842; doi:10.3390/math3030781 OPEN ACCESS mathematics ISSN 2227-7390 www.mdpi.com/journal/mathematics Article Algebra of Complex Vectors and Applications in Electromagnetic Theory
More informationGEOMETRY OF THE DIRAC THEORY
In: A Symposium on the Mathematics of Physical Space-Time, Facultad de Quimica, Universidad Nacional Autonoma de Mexico, Mexico City, 67 96, (1981). GEOMETRY OF THE DIRAC THEORY David Hestenes ABSTRACT.
More informationOverthrows a basic assumption of classical physics - that lengths and time intervals are absolute quantities, i.e., the same for all observes.
Relativistic Electrodynamics An inertial frame = coordinate system where Newton's 1st law of motion - the law of inertia - is true. An inertial frame moves with constant velocity with respect to any other
More informationMULTIVECTOR DERIVATIVES WITH RESPECT TO FUNCTIONS
MULTIVECTOR DERIVATIVES WITH RESPECT TO FUNCTIONS Note to avoid any confusion: The first paper dealing with multivector differentiation for physics, and in particular the concept of multivector derivatives
More informationProper particle mechanics
In: Journal of Mathematical Physics, 15 1768 1777 (1974). Proper particle mechanics David Hestenes Abstract Spacetime algebra is employed to formulate classical relativistic mechanics without coordinates.
More informationSpacetime physics with geometric algebra
Spacetime physics with geometric algebra David Hestenes a) Department of Physics and Astronomy, Arizona State University, Tempe, Arizona 85287-1504 Received 22 July 2002; accepted 12 March 2003 This is
More informationParticle Notes. Ryan D. Reece
Particle Notes Ryan D. Reece July 9, 2007 Chapter 1 Preliminaries 1.1 Overview of Special Relativity 1.1.1 Lorentz Boosts Searches in the later part 19th century for the coordinate transformation that
More informationPart IB Electromagnetism
Part IB Electromagnetism Theorems Based on lectures by D. Tong Notes taken by Dexter Chua Lent 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures.
More informationSpecial Relativity and Electromagnetism
1/32 Special Relativity and Electromagnetism Jonathan Gratus Cockcroft Postgraduate Lecture Series October 2016 Introduction 10:30 11:40 14:00? Monday SR EM Tuesday SR EM Seminar Four lectures is clearly
More informationGeometric Algebra. 5. Spacetime Algebra. Dr Chris Doran ARM Research
Geometric Algebra 5. Spacetime Algebra Dr Chris Doran ARM Research History L5 S2 A geometric algebra of spacetime Invariant interval of spacetime is The particle physics convention L5 S3 Need 4 generators
More informationSpacetime Algebra and Electron Physics
Spacetime Algebra and Electron Physics AUTHORS Chris Doran Anthony Lasenby Stephen Gull Shyamal Somaroo Anthony Challinor In P. W. Hawkes, editor Advances in Imaging and Electron Physics Vol. 95, p. 271-386
More informationOn Decoupling Probability from Kinematics in Quantum Mechanics
In: P. F. Fougere (ed.), Maximum Entropy and Bayesian Methods, Kluwer Academic Publishers (1990), 161 183. On Decoupling Probability from Kinematics in Quantum Mechanics David Hestenes Abstract.A means
More informationA Generally Covariant Field Equation For Gravitation And Electromagnetism
3 A Generally Covariant Field Equation For Gravitation And Electromagnetism Summary. A generally covariant field equation is developed for gravitation and electromagnetism by considering the metric vector
More informationGravitation: Special Relativity
An Introduction to General Relativity Center for Relativistic Astrophysics School of Physics Georgia Institute of Technology Notes based on textbook: Spacetime and Geometry by S.M. Carroll Spring 2013
More informationElectromagnetic. G. A. Krafft Jefferson Lab Jefferson Lab Professor of Physics Old Dominion University TAADI Electromagnetic Theory
TAAD1 Electromagnetic Theory G. A. Krafft Jefferson Lab Jefferson Lab Professor of Physics Old Dominion University 8-31-12 Classical Electrodynamics A main physics discovery of the last half of the 2 th
More informationNote 1: Some Fundamental Mathematical Properties of the Tetrad.
Note 1: Some Fundamental Mathematical Properties of the Tetrad. As discussed by Carroll on page 88 of the 1997 notes to his book Spacetime and Geometry: an Introduction to General Relativity (Addison-Wesley,
More informationFYS 3120: Classical Mechanics and Electrodynamics
FYS 3120: Classical Mechanics and Electrodynamics Formula Collection Spring semester 2014 1 Analytical Mechanics The Lagrangian L = L(q, q, t), (1) is a function of the generalized coordinates q = {q i
More informationGauge Theory Gravity with Geometric Calculus
Gauge Theory Gravity with Geometric Calculus David Hestenes 1 Department of Physics and Astronomy Arizona State University, Tempe, Arizona 85287-1504 A new gauge theory of gravity on flat spacetime has
More informationAn Introduction to Geometric Algebra
An Introduction to Geometric Algebra Alan Bromborsky Army Research Lab (Retired) brombo@comcast.net November 2, 2005 Typeset by FoilTEX History Geometric algebra is the Clifford algebra of a finite dimensional
More informationIn J. Keller and Z. Oziewicz (Eds.), The Theory of the Electron, UNAM, Facultad de Estudios Superiores, Cuautitlan, Mexico (1996), p
In J. Keller and Z. Oziewicz (Eds.), The Theory of the Electron, UNAM, Facultad de Estudios Superiores, Cuautitlan, Mexico (1996), p. 1 50. REAL DIRAC THEORY David Hestenes Department of Physics & Astronomy
More informationLecture: Lorentz Invariant Dynamics
Chapter 5 Lecture: Lorentz Invariant Dynamics In the preceding chapter we introduced the Minkowski metric and covariance with respect to Lorentz transformations between inertial systems. This was shown
More informationGEOMETRY OF PARAVECTOR SPACE WITH APPLICATIONS TO RELATIVISTIC PHYSICS
GEOMETRY OF PARAVECTOR SPACE WITH APPLICATIONS TO RELATIVISTIC PHYSICS William E. Baylis Physics Dept., University of Windsor Windsor, ON, Canada N9B 3P4 Abstract Clifford s geometric algebra, in particular
More informationGeometric Algebra as a Unifying Language for Physics and Engineering and Its Use in the Study of Gravity
Adv. Appl. Clifford Algebras 27 (2017), 733 759 c 2016 The Author(s). This article is published with open access at Springerlink.com 0188-7009/010733-27 published online July 20, 2016 DOI 10.1007/s00006-016-0700-z
More informationPHYSICAL APPLICATIONS OF GEOMETRIC ALGEBRA
February 9, 1999 PHYSICAL APPLICATIONS OF GEOMETRIC ALGEBRA LECTURE 8 SUMMARY In this lecture we introduce the spacetime algebra, the geometric algebra of spacetime. This forms the basis for most of the
More informationMechanics Physics 151
Mechanics Physics 151 Lecture 15 Special Relativity (Chapter 7) What We Did Last Time Defined Lorentz transformation Linear transformation of 4-vectors that conserve the length in Minkowski space Derived
More informationA GENERALLY COVARIANT FIELD EQUATION FOR GRAVITATION AND ELECTROMAGNETISM. Institute for Advanced Study Alpha Foundation
A GENERALLY COVARIANT FIELD EQUATION FOR GRAVITATION AND ELECTROMAGNETISM Myron W. Evans Institute for Advanced Study Alpha Foundation E-mail: emyrone@aol.com Received 17 April 2003; revised 1 May 2003
More informationPassive Lorentz Transformations with Spacetime Algebra
Passive Lorentz Transformations with Spacetime Algebra Carlos R. Paiva a Instituto de Telecomunicações, Department of Electrical and Computer Engineering, Instituto Superior Técnico Av. Rovisco Pais,,
More informationChapter 1 Mathematical Foundations
Computational Electromagnetics; Chapter 1 1 Chapter 1 Mathematical Foundations 1.1 Maxwell s Equations Electromagnetic phenomena can be described by the electric field E, the electric induction D, the
More informationMechanics Physics 151
Mechanics Physics 151 Lecture 4 Continuous Systems and Fields (Chapter 13) What We Did Last Time Built Lagrangian formalism for continuous system Lagrangian L Lagrange s equation = L dxdydz Derived simple
More informationNew Foundations in Mathematics: The Geometric Concept of Number
New Foundations in Mathematics: The Geometric Concept of Number by Garret Sobczyk Universidad de Las Americas-P Cholula, Mexico November 2012 What is Geometric Algebra? Geometric algebra is the completion
More informationSpacetime Geometry with Geometric Calculus
Spacetime Geometry with Geometric Calculus David Hestenes 1 Department of Physics and Astronomy Arizona State University, Tempe, Arizona 85287-1504 Geometric Calculus is developed for curved-space treatments
More informationINTRODUCTION TO ELECTRODYNAMICS
INTRODUCTION TO ELECTRODYNAMICS Second Edition DAVID J. GRIFFITHS Department of Physics Reed College PRENTICE HALL, Englewood Cliffs, New Jersey 07632 CONTENTS Preface xi Advertisement 1 1 Vector Analysis
More informationBiquaternion formulation of relativistic tensor dynamics
Juli, 8, 2008 To be submitted... 1 Biquaternion formulation of relativistic tensor dynamics E.P.J. de Haas High school teacher of physics Nijmegen, The Netherlands Email: epjhaas@telfort.nl In this paper
More information4 Relativistic kinematics
4 Relativistic kinematics In astrophysics, we are often dealing with relativistic particles that are being accelerated by electric or magnetic forces. This produces radiation, typically in the form of
More informationGeometric Algebra as a unifying language for Physics and Engineering and its use in the study of Gravity
Geometric Algebra as a unifying language for Physics and Engineering and its use in the study of Gravity Anthony Lasenby Astrophysics Group, Cavendish Laboratory, and Kavli Institute for Cosmology Cambridge,
More informationAttempts at relativistic QM
Attempts at relativistic QM based on S-1 A proper description of particle physics should incorporate both quantum mechanics and special relativity. However historically combining quantum mechanics and
More informationBiquaternion formulation of relativistic tensor dynamics
Apeiron, Vol. 15, No. 4, October 2008 358 Biquaternion formulation of relativistic tensor dynamics E.P.J. de Haas High school teacher of physics Nijmegen, The Netherlands Email: epjhaas@telfort.nl In this
More informationSymmetry and Duality FACETS Nemani Suryanarayana, IMSc
Symmetry and Duality FACETS 2018 Nemani Suryanarayana, IMSc What are symmetries and why are they important? Most useful concept in Physics. Best theoretical models of natural Standard Model & GTR are based
More informationLorentz Transformations and Special Relativity
Lorentz Transformations and Special Relativity Required reading: Zwiebach 2.,2,6 Suggested reading: Units: French 3.7-0, 4.-5, 5. (a little less technical) Schwarz & Schwarz.2-6, 3.-4 (more mathematical)
More informationA Brief Introduction to Relativistic Quantum Mechanics
A Brief Introduction to Relativistic Quantum Mechanics Hsin-Chia Cheng, U.C. Davis 1 Introduction In Physics 215AB, you learned non-relativistic quantum mechanics, e.g., Schrödinger equation, E = p2 2m
More informationE & M Qualifier. January 11, To insure that the your work is graded correctly you MUST:
E & M Qualifier 1 January 11, 2017 To insure that the your work is graded correctly you MUST: 1. use only the blank answer paper provided, 2. use only the reference material supplied (Schaum s Guides),
More informationQuantum Field Theory Notes. Ryan D. Reece
Quantum Field Theory Notes Ryan D. Reece November 27, 2007 Chapter 1 Preliminaries 1.1 Overview of Special Relativity 1.1.1 Lorentz Boosts Searches in the later part 19th century for the coordinate transformation
More informationProjective geometry and spacetime structure. David Delphenich Bethany College Lindsborg, KS USA
Projective geometry and spacetime structure David Delphenich Bethany College Lindsborg, KS USA delphenichd@bethanylb.edu Affine geometry In affine geometry the basic objects are points in a space A n on
More informationSPECIAL RELATIVITY AND ELECTROMAGNETISM
SPECIAL RELATIVITY AND ELECTROMAGNETISM MATH 460, SECTION 500 The following problems (composed by Professor P.B. Yasskin) will lead you through the construction of the theory of electromagnetism in special
More informationOld Wine in New Bottles: A new algebraic framework for computational geometry
Old Wine in New Bottles: A new algebraic framework for computational geometry 1 Introduction David Hestenes My purpose in this chapter is to introduce you to a powerful new algebraic model for Euclidean
More informationGravitation: Tensor Calculus
An Introduction to General Relativity Center for Relativistic Astrophysics School of Physics Georgia Institute of Technology Notes based on textbook: Spacetime and Geometry by S.M. Carroll Spring 2013
More informationLecture 7. both processes have characteristic associated time Consequence strong interactions conserve more quantum numbers then weak interactions
Lecture 7 Conserved quantities: energy, momentum, angular momentum Conserved quantum numbers: baryon number, strangeness, Particles can be produced by strong interactions eg. pair of K mesons with opposite
More informationMysteries and Insights of Dirac Theory
Annales de la Fondation Louis de Broglie, Volume 28 no 3-4, 2003 367 Mysteries and Insights of Dirac Theory David Hestenes Department of Physics and Astronomy Arizona State University, Tempe, Arizona 85287-1504
More informationElectric and Magnetic Forces in Lagrangian and Hamiltonian Formalism
Electric and Magnetic Forces in Lagrangian and Hamiltonian Formalism Benjamin Hornberger 1/26/1 Phy 55, Classical Electrodynamics, Prof. Goldhaber Lecture notes from Oct. 26, 21 Lecture held by Prof. Weisberger
More informationMysteries and Insights of Dirac Theory
Mysteries and Insights of Dirac Theory David Hestenes Department of Physics and Astronomy Arizona State University, Tempe, Arizona 85287-1504 The Dirac equation has a hidden geometric structure that is
More informationRelativity SPECIAL, GENERAL, AND COSMOLOGICAL SECOND EDITION. Wolfgang Rindler. Professor of Physics The University of Texas at Dallas
Relativity SPECIAL, GENERAL, AND COSMOLOGICAL SECOND EDITION Wolfgang Rindler Professor of Physics The University of Texas at Dallas OXPORD UNIVERSITY PRESS Contents Introduction l 1 From absolute space
More informationExamples. Figure 1: The figure shows the geometry of the GPS system
Examples 1 GPS System The global positioning system was established in 1973, and has been updated almost yearly. The GPS calcualtes postion on the earth ssurface by accurately measuring timing signals
More informationTopics for the Qualifying Examination
Topics for the Qualifying Examination Quantum Mechanics I and II 1. Quantum kinematics and dynamics 1.1 Postulates of Quantum Mechanics. 1.2 Configuration space vs. Hilbert space, wave function vs. state
More informationSpecial Theory of Relativity
June 17, 2008 1 1 J.D.Jackson, Classical Electrodynamics, 3rd Edition, Chapter 11 Introduction Einstein s theory of special relativity is based on the assumption (which might be a deep-rooted superstition
More informationScott Hughes 12 May Massachusetts Institute of Technology Department of Physics Spring 2005
Scott Hughes 12 May 2005 24.1 Gravity? Massachusetts Institute of Technology Department of Physics 8.022 Spring 2005 Lecture 24: A (very) brief introduction to general relativity. The Coulomb interaction
More informationThe Spin Bivector and Zeropoint Energy in. Geometric Algebra
Adv. Studies Theor. Phys., Vol. 6, 2012, no. 14, 675-686 The Spin Bivector and Zeropoint Energy in Geometric Algebra K. Muralidhar Physics Department, National Defence Academy Khadakwasla, Pune-411023,
More informationMATH 4030 Differential Geometry Lecture Notes Part 4 last revised on December 4, Elementary tensor calculus
MATH 4030 Differential Geometry Lecture Notes Part 4 last revised on December 4, 205 Elementary tensor calculus We will study in this section some basic multilinear algebra and operations on tensors. Let
More informationContravariant and Covariant as Transforms
Contravariant and Covariant as Transforms There is a lot more behind the concepts of contravariant and covariant tensors (of any rank) than the fact that their basis vectors are mutually orthogonal to
More informationSpace-time algebra for the generalization of gravitational field equations
PRAMANA c Indian Academy of Sciences Vol. 80, No. 5 journal of May 2013 physics pp. 811 823 Space-time algebra for the generalization of gravitational field equations SÜLEYMAN DEMİR Department of Physics,
More informationExamples - Lecture 8. 1 GPS System
Examples - Lecture 8 1 GPS System The global positioning system, GPS, was established in 1973, and has been updated almost yearly. The GPS calculates postion on the earth s surface by accurately measuring
More informationarxiv:physics/ v3 [physics.ed-ph] 11 Jul 2004
Relativity in Introductory Physics William E. Baylis Department of Physics, University of Windsor, Windsor, ON, Canada N9B 3P4 arxiv:physics/0406158v3 [physics.ed-ph] 11 Jul 2004 (Date textdate; Received
More information1 Tensors and relativity
Physics 705 1 Tensors and relativity 1.1 History Physical laws should not depend on the reference frame used to describe them. This idea dates back to Galileo, who recognized projectile motion as free
More informationby M.W. Evans, British Civil List (
Derivation of relativity and the Sagnac Effect from the rotation of the Minkowski and other metrics of the ECE Orbital Theorem: the effect of rotation on spectra. by M.W. Evans, British Civil List (www.aias.us)
More informationarxiv:gr-qc/ v1 6 May 2004
GRAVITY, GAUGE THEORIES AND GEOMETRIC ALGEBRA Anthony Lasenby 1, Chris Doran 2 and Stephen Gull 3 arxiv:gr-qc/0405033v1 6 May 2004 Astrophysics Group, Cavendish Laboratory, Madingley Road, Cambridge CB3
More informationMaxwell s equations. based on S-54. electric field charge density. current density
Maxwell s equations based on S-54 Our next task is to find a quantum field theory description of spin-1 particles, e.g. photons. Classical electrodynamics is governed by Maxwell s equations: electric field
More informationQuantum Field Theory
Quantum Field Theory PHYS-P 621 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory 1 Attempts at relativistic QM based on S-1 A proper description of particle physics
More informationProblem 1, Lorentz transformations of electric and magnetic
Problem 1, Lorentz transformations of electric and magnetic fields We have that where, F µν = F µ ν = L µ µ Lν ν F µν, 0 B 3 B 2 ie 1 B 3 0 B 1 ie 2 B 2 B 1 0 ie 3 ie 2 ie 2 ie 3 0. Note that we use the
More informationClifford Algebras and the Dirac-Bohm Quantum Hamilton-Jacobi Equation.
Clifford Algebras and the Dirac-Bohm Quantum Hamilton-Jacobi Equation. B. J. Hiley. TPRU, Birkbeck, University of London, Malet Street, London WC1E 7HX. (2 March 2010) Abstract In this paper we show how
More informationClassical-quantum Correspondence and Wave Packet Solutions of the Dirac Equation in a Curved Spacetime
Classical-quantum Correspondence and Wave Packet Solutions of the Dirac Equation in a Curved Spacetime Mayeul Arminjon 1,2 and Frank Reifler 3 1 CNRS (Section of Theoretical Physics) 2 Lab. Soils, Solids,
More informationInstructor: Sudipta Mukherji, Institute of Physics, Bhubaneswar
Chapter 1 Lorentz and Poincare Instructor: Sudipta Mukherji, Institute of Physics, Bhubaneswar 1.1 Lorentz Transformation Consider two inertial frames S and S, where S moves with a velocity v with respect
More informationLecture 13 Notes, Electromagnetic Theory II Dr. Christopher S. Baird, faculty.uml.edu/cbaird University of Massachusetts Lowell
Lecture 13 Notes, Electromagnetic Theory II Dr. Christopher S. Baird, faculty.uml.edu/cbaird University of Massachusetts Lowell 1. Covariant Geometry - We would like to develop a mathematical framework
More informationZitterbewegung Modeling
In: Found. Physics., Vol. 23, No. 3, (1993) 365 387. Zitterbewegung Modeling David Hestenes Abstract. Guidelines for constructing point particle models of the electron with zitterbewegung and other features
More informationNumber-Flux Vector and Stress-Energy Tensor
Massachusetts Institute of Technology Department of Physics Physics 8.962 Spring 2002 Number-Flux Vector and Stress-Energy Tensor c 2000, 2002 Edmund Bertschinger. All rights reserved. 1 Introduction These
More informationA Physical Electron-Positron Model in Geometric Algebra. D.T. Froedge. Formerly Auburn University
A Physical Electron-Positron Model in Geometric Algebra V0497 @ http://www.arxdtf.org D.T. Froedge Formerly Auburn University Phys-dtfroedge@glasgow-ky.com Abstract This paper is to present a physical
More informationReview and Notation (Special relativity)
Review and Notation (Special relativity) December 30, 2016 7:35 PM Special Relativity: i) The principle of special relativity: The laws of physics must be the same in any inertial reference frame. In particular,
More informationCovariant electrodynamics
Lecture 9 Covariant electrodynamics WS2010/11: Introduction to Nuclear and Particle Physics 1 Consider Lorentz transformations pseudo-orthogonal transformations in 4-dimentional vector space (Minkowski
More informationLecture 9: Macroscopic Quantum Model
Lecture 9: Macroscopic Quantum Model Outline 1. Development of Quantum Mechanics 2. Schrödinger's Equation Free particle With forces With Electromagnetic force 3. Physical meaning of Wavefunction Probability
More informationAnalytical Mechanics for Relativity and Quantum Mechanics
Analytical Mechanics for Relativity and Quantum Mechanics Oliver Davis Johns San Francisco State University OXPORD UNIVERSITY PRESS CONTENTS Dedication Preface Acknowledgments v vii ix PART I INTRODUCTION:
More informationMultiple Integrals and Vector Calculus (Oxford Physics) Synopsis and Problem Sets; Hilary 2015
Multiple Integrals and Vector Calculus (Oxford Physics) Ramin Golestanian Synopsis and Problem Sets; Hilary 215 The outline of the material, which will be covered in 14 lectures, is as follows: 1. Introduction
More informationClassical Field Theory
April 13, 2010 Field Theory : Introduction A classical field theory is a physical theory that describes the study of how one or more physical fields interact with matter. The word classical is used in
More informationModern Geometric Structures and Fields
Modern Geometric Structures and Fields S. P. Novikov I.A.TaJmanov Translated by Dmitry Chibisov Graduate Studies in Mathematics Volume 71 American Mathematical Society Providence, Rhode Island Preface
More informationAn Introduction to Geometric Algebra and Calculus. Alan Bromborsky Army Research Lab (Retired)
An Introduction to Geometric Algebra and Calculus Alan Bromborsky Army Research Lab Retired abrombo@verizon.net April 1, 2011 2 Introduction Geometric algebra is the Clifford algebra of a finite dimensional
More informationRelativistic Quantum Mechanics
Physics 342 Lecture 34 Relativistic Quantum Mechanics Lecture 34 Physics 342 Quantum Mechanics I Wednesday, April 30th, 2008 We know that the Schrödinger equation logically replaces Newton s second law
More informationarxiv: v3 [physics.optics] 2 Jun 2015
Spacetime algebra as a powerful tool for electromagnetism Justin Dressel a,b, Konstantin Y. Bliokh b,c, Franco Nori b,d arxiv:1411.500v3 [physics.optics] Jun 015 a Department of Electrical and Computer
More informationEinstein Toolkit Workshop. Joshua Faber Apr
Einstein Toolkit Workshop Joshua Faber Apr 05 2012 Outline Space, time, and special relativity The metric tensor and geometry Curvature Geodesics Einstein s equations The Stress-energy tensor 3+1 formalisms
More informationMaxwell s Equations. In the previous chapters we saw the four fundamental equations governging electrostatics and magnetostatics. They are.
Maxwell s Equations Introduction In the previous chapters we saw the four fundamental equations governging electrostatics and magnetostatics. They are D = ρ () E = 0 (2) B = 0 (3) H = J (4) In the integral
More informationELE3310: Basic ElectroMagnetic Theory
A summary for the final examination EE Department The Chinese University of Hong Kong November 2008 Outline Mathematics 1 Mathematics Vectors and products Differential operators Integrals 2 Integral expressions
More informationPhysics 523, General Relativity
Physics 53, General Relativity Homework Due Monday, 9 th October 6 Jacob Lewis Bourjaily Problem 1 Let frame O move with speed v in the x-direction relative to frame O. Aphotonwithfrequencyν measured in
More informationNew Transformation Equations and the Electric Field Four-vector
New Transformation Equations and the Electric Field Four-vector Copyright c 1999 2008 David E. Rutherford All Rights Reserved E-mail: drutherford@softcom.net http://www.softcom.net/users/der555/newtransform.pdf
More informationOn the existence of magnetic monopoles
On the existence of magnetic monopoles Ali R. Hadjesfandiari Department of Mechanical and Aerospace Engineering State University of New York at Buffalo Buffalo, NY 146 USA ah@buffalo.edu September 4, 13
More informationNew Transformation Equations and the Electric Field Four-vector
New Transformation Equations and the Electric Field Four-vector c Copyright 1999 2003 David E. Rutherford All Rights Reserved E-mail: drutherford@softcom.net http://www.softcom.net/users/der555/newtransform.pdf
More informationFigure 1: Grad, Div, Curl, Laplacian in Cartesian, cylindrical, and spherical coordinates. Here ψ is a scalar function and A is a vector field.
Figure 1: Grad, Div, Curl, Laplacian in Cartesian, cylindrical, and spherical coordinates. Here ψ is a scalar function and A is a vector field. Figure 2: Vector and integral identities. Here ψ is a scalar
More informationSuper Yang-Mills Theory in 10+2 dims. Another Step Toward M-theory
1 Super Yang-Mills Theory in 10+2 dims. Another Step Toward M-theory Itzhak Bars University of Southern California Talk at 4 th Sakharov Conference, May 2009 http://physics.usc.edu/~bars/homepage/moscow2009_bars.pdf
More informationA New Formalism of Arbitrary Spin Particle Equations. Abstract
A New Formalism of Arbitrary Spin Particle Equations S.R. Shi Huiyang Radio and TV station,daishui,huiyang,huizhou,guangdong,china,56 (Dated: October 4, 6) Abstract In this paper, a new formalism of arbitrary
More informationClass 3: Electromagnetism
Class 3: Electromagnetism In this class we will apply index notation to the familiar field of electromagnetism, and discuss its deep connection with relativity Class 3: Electromagnetism At the end of this
More informationGeometric algebra and particle dynamics
This is page 1 Printer: Opaque this arxiv:math/05005v [math.gm] 1 Jul 005 Geometric algebra and particle dynamics José B. Almeida ABSTRACT In a recent publication [1] it was shown how the geometric algebra
More information