GEOMETRIC ALGEBRAS Manuel Berrondo Brigham Young University Provo, UT,

Physical Applications: Mechanics: Foucault pendulum Electro-magnetostatics Dispersion and diffraction E.M. Quantum Mechanics: spin precession Field Theory: Dirac equation (General Relativity )

Examples of geometric algebras Complex Numbers G 0,1 = C Hamilton Quaternions G 0,2 = H Pauli Algebra G 3 Dirac Algebra G 1,3

Extension of the vector space inverse of a vector: reflections, rotations, Lorentz transform. integrals: Cauchy (≥ 2 d), Stokes’ theorem Based on the idea of MULTIVECTORS: and including lengths and angles:

Geometric or matrix product

Algebraic Properties of R + closure commutativity associativity zero negative closure commutativity associativity unit reciprocal and + distributivity Vector spaces :linear combinations

Algebras include metrics : define geometric product FIRST STEP: Inverse of a vector a ≠ 0 a -1 a 1 2 0

Orthonormal basis 3d: ê 1 ê 2 ê 3 ê i ê k =  i k ê i -1 = ê i Euclidian 4d:  0  1  2  3  μ  ν = g μν  (1,-1,-1,-1) Minkowski 1 Example: (ê 1 + ê 2 ) -1 = (ê 1 + ê 2 )/2

Electrostatics: method of images w r q - q q q l Plane: charge -q at cê 1, image (-q) at -cê 1 Sphere radius a, charge q at bê 1, find image (?)

Choose scales for r and w: rwcharge 0 a V = 0 cbq -c-c?q’ = ?

SECOND STEP: EULER FORMULAS In 3-d: given that In general, rotating A about θ k:

Example 1: Lorentz Equation q B = B k

Example 2: Foucault Pendulum Coriolis S mgmg l

THIRD STEP: ANTISYM. PRODUCT sweep a b a b anticommutative associative distributive absolute value => area

Geometric or matrix product non commutative associative distributive closure: extend vector space unit = 1 inverse (conditional)

Examples of Clifford algebras NotationGeometryDim. G2G2 plane4 G 0,1 complex2 G 0,2 quaternions4 G3G3 Pauli8 G 1,3 Dirac16 G m,n signature (m,n)2 m+n

1scalar vector bivector R I R R2R2 C = even algebra = spinors G2 :G2 :

C isomorphic to R 2 1 e1e1 I e2e2 z z * a z = ê 1 a ê 1 z = a Reflection: z  z * a  a’ = ê 1 z* = ê 1 a ê 1 In general, a’ = n a n, with n 2 = 1

Rotations in R 2 Euler: e1e1 e2e2 a a’ φ

Inverse of multivector in G 2 Conjugate : Generalizing

1 1 scalar ê1ê1 ê2ê2 ê3ê3 3 vector ê2ê3ê2ê3 ê3ê1ê3ê1 ê1ê2ê1ê2 3 bivector ê 1 ê 2 ê 3 = i 1 pseudoscalar R iRiR R3R3 H = R + i R 3 = = even algebra = spinors iR3iR3 G3G3

Geometric product in G 3 : Rotations: e P/2 generates rotations with respect to plane defined by bivector P

Spinors 2-d C 3-d C x C σ, Pauli matrices Pauli G3G3 ( )p 0 + i p = p σ k ( )êk p ê3êk p ê3 j ( )i p ê 3 j σ k ( )i ê k p

Interaction: spin with magnetic field B Pauli’s quantum Hamiltonian with A = vector potential p  p – q A, minimal coupling, magnetic moment : μ =  S = ħ  σ/2

Spin precession with uniform B : In G 3, spinor (quaternion) Ψ, Solution in G 3 : Spin precesses on plane i B with angular freq. independent of ħ !

Inverse of multivector in G 3 defining Clifford conjugate : Generalizing:

Special relativity and paravectors in G 3 Paravector p = p 0 + p Examples of paravectors: xct + r u  (1 + v/c) pE/c + p  φ + cA jcρ + j

Lorentz transformations Transform the paravector p = p 0 + p = p 0 + p = + p ┴ into: B p B = B 2 (p 0 + p = ) + p ┴ R p R † = p 0 + p = + R 2 p ┴

Geometric Calculus (3d)  is a vector differential operator acting on:  (r) – scalar field E(r) – vector field

First order Green Functions Euclidian spaces : is solution of

Maxwell’s Equation Maxwell’s multivector F = E + i c B current density paravector J = (ε 0 c) -1 (cρ + j) Maxwell’s equation:

Electrostatics Gauss’s law. Solution using first order Green’s function: Explicitly:

Magnetostatics Ampère’s law. Solution using first order Green’s function: Explicitly:

Fundamental Theorem of Calculus Euclidian case : d k x = oriented volume, e.g. ê 1 ê 2 ê 3 = i d k-1 x = oriented surface, e.g. ê 1 ê 2 = iê 3 F (x) – multivector field,  V – boundary of V

Example: divergence theorem d 3 x = i d , where d  = |d 3 x| d 2 x = i nda, where da = |d 2 x| F = v(r)

Green’s Theorem (Euclidian case ) where the first order Green function is:

Cauchy’s theorem in n dimensions particular case : F = f y  f = 0 where f (x) is a monogenic multivectorial field:  f = 0

Inverse of differential paravectors: Helmholtz: Spherical wave (outgoing or incoming)

Electromagnetic diffraction First order Helmholtz equation: Exact Huygens’ principle: In the homogeneous case J = 0

1 1 00 11 22 33 4 0 10 1 1 21 2  2  3 0 20 2 0 30 3 1 31 3 6 0 1 20 1 2 1 2 31 2 3  2  3  0 0 1 30 1 3 4 0 1 2 30 1 2 3 1 6 bivectors split into two groups: ê i =  i  0 and I ê k =  i  j

Dirac Equation in G 1,3 Vector in Minkowski space : Quantization:

Propagator – first order Green function The propagator represents the conditional probability amplitude Fourier transform :

Including external field A, Dirac equation is written as: and the propagator: includes the vacuum polarization term!