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Advances in Geometric Algebras for Electromagnetics and Physical Applications

This comprehensive overview explores the applications of geometric algebras in electromagnetic fields, particularly in electrostatics and magnetostatics. Key concepts include multivectors, the geometric product, and methods such as the method of images in solving physical problems. The paper provides examples including complex numbers, Hamilton quaternions, and Dirac algebra, emphasizing their importance in vector space extensions and transformations (like Lorentz transformations). It also covers various algebraic properties essential in defining mathematical frameworks for physical phenomena.

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Advances in Geometric Algebras for Electromagnetics and Physical Applications

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

  2. Physical Applications: • Electro-magnetostatics • Dispersion and diffraction E.M.

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

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

  7. 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

  8. 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 (?)

  9. Choose scales for r and w:

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

  11. Example: Lorentz Equation q B = B k

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

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

  14. Examples of Clifford algebras

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

  16. 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

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

  18. Inverse of multivector in G2Conjugate: Generalizing

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

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

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

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

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

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

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

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

  27. 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

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

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

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

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

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

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

  34. 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┴

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