Classical Physics with SpaceTime Algebra David Hestenes Arizona State University x x(τ ) x 0 Santalo 2016
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: 691-704 (2003). <http://geocalc.clas.asu.edu/ > Space Time Algebra (Springer: 2015) 2 nd Ed.
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
Differentiation by vectors Vector product: Vector derivative: Names: del (grad) = div + curl One differential operator!: Vector field: E = E(x) Electrostatics: Magnetostatics:
Vector derivatives in Rectangular coordinates: Position vector: Vector derivative: Basic derivatives for routine calculations:?? for constant a
Chain Rule
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)
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
Antiderivatives: x n R R solves electrostatic and magnetostatic problems solid angle pseudoscalar
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
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
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
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
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
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:
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 + 1 2 F jk σ k σ j = E + ib i 2 = 1 [D. Hestenes (2003), Am. J. Phys. 71: 691-714]
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)
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)
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
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
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
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!
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
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
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!
Outline and References <http:\\modelingnts.la.asu.edu> <http://www.mrao.cam.ac.uk> 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).
THE END