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GEOMETRIC ALGEBRAS

Explore the applications and examples of geometric algebras in physics, including mechanics, electromagnetostatics, dispersion, diffraction, quantum mechanics, field theory, and general relativity. Learn about properties, operations, and algebraic properties of geometric algebras. Discover illustrations like electrostatics, Euler formulas, and interactions with magnetic fields. Understand the basics of geometric calculus, Maxwell's equations, Green's functions, and the Dirac Equation in G1,3.

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GEOMETRIC ALGEBRAS

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  1. GEOMETRIC ALGEBRAS Manuel Berrondo Brigham Young University Provo, UT, 84097 berrondo@byu.edu

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

  3. Examples of geometric algebras • Complex Numbers G0,1= C • Hamilton Quaternions G0,2= H • Pauli Algebra G3 • Dirac Algebra G1,3

  4. 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:

  5. Geometric or matrix product

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

  7. Algebras • include metrics: • define geometricproduct FIRST STEP: Inverse of a vector a≠ 0 a-1 0 2 a 1

  8. 3d:ê1 ê2 ê3 êi •êk = i k êi-1= êi Euclidian 4d: 0 1 2 3 μ•ν= gμν (1,-1,-1,-1) Minkowski Orthonormal basis Example: (ê1+ ê2)-1 = (ê1+ ê2)/2 1

  9. Electrostatics: method of images w r • - q • ql • q • q Plane: charge -q atcê1, image (-q) at -cê1 Sphere radius a, charge q atbê1, find image (?)

  10. Choose scales for r and w:

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

  12. Example 1: Lorentz Equation q B = B k

  13. Example 2: Foucault Pendulum l S Coriolis • mg

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

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

  16. Examples of Clifford algebras

  17. G2: R R2 I R C = even algebra = spinors

  18. I e2 • z a C isomorphic to R2 1 e1 • z* z = ê1a ê1z = a Reflection: z  z* a a’ = ê1z* = ê1a ê1 In general, a’ = n a n, with n2 = 1

  19. a’ e2 φ Rotations in R2 Euler: a e1

  20. Inverse of multivector in G2Conjugate: Generalizing

  21. G3 R3 iR R H = R + i R3= = even algebra = spinors iR3

  22. Geometric product in G3 : Rotations: e P/2 generates rotations with respect to planedefined by bivector P

  23. Spinors σ, Pauli matrices

  24. 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

  25. Spin precession with uniform B : In G3, spinor (quaternion) Ψ, Solution in G3: Spin precesses on plane i B withangularfreq. independent of ħ !

  26. Inverse of multivector in G3defining Clifford conjugate: • Generalizing:

  27. Special relativity and paravectors in G3 • Paravector p = p0 + p Examples of paravectors:

  28. Lorentz transformations Transform the paravector p = p0 + p = p0 + p=+ p┴ into: B p B = B2 (p0 + p=) + p┴ R p R† = p0 + p= + R2 p┴

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

  30. First order Green Functions Euclidian spaces : is solution of

  31. Maxwell’s Equation • Maxwell’s multivector F = E +i c B • current density paravector J = (ε0c)-1(cρ + j) • Maxwell’s equation:

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

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

  34. Fundamental Theorem of Calculus • Euclidian case : dkx = oriented volume, e.g. ê1 ê2 ê3 = i dk-1x = oriented surface, e.g. ê1 ê2 = iê3 F(x)– multivector field, V – boundary of V

  35. Example: divergence theorem • d3x = i d, where d = |d3 x| • d2 x = i nda , where da = |d2 x| • F = v(r)

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

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

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

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

  40. 6 bivectors split into two groups: êi = i0 and I êk = ij

  41. Dirac Equation in G1,3 • Vector in Minkowski space : • Quantization:

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

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

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