Geometric algebras in e m
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GEOMETRIC ALGEBRAS IN E.M. Manuel Berrondo Brigham Young University Provo, UT, 84097 [email protected] Physical Applications:. Electro-magnetostatics Dispersion and diffraction E.M. Examples of geometric algebras . Complex Numbers G 0,1 = C Hamilton Quaternions G 0,2 = H

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Geometric algebras in e m

GEOMETRIC ALGEBRAS IN E.M.

Manuel Berrondo

Brigham Young University

Provo, UT, 84097

[email protected]


Physical applications

Physical Applications:

  • Electro-magnetostatics

  • Dispersion and diffraction E.M.


Examples of geometric algebras

Examples of geometric algebras

  • Complex Numbers G0,1= C

  • Hamilton Quaternions G0,2= H

  • Pauli Algebra G3

  • Dirac Algebra G1,3


Geometric algebras in e m

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:


Algebraic properties of r

+

closure

commutativity

associativity

zero

negative

closure

commutativity

associativity

unit

reciprocal

and + distributivity

Algebraic Properties of R

Vector spaces :linear combinations


Algebras

Algebras

  • include metrics:

  • define geometricproduct

    FIRST STEP: Inverse of a vector a≠ 0

a-1

0

2

a

1


Orthonormal basis

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


Electrostatics method of images

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 (?)


Geometric algebras in e m

Choose scales for r and w:


Second step euler formulas

SECOND STEP: EULER FORMULAS

In 3-d:

given that

In general, rotating A about θ k:


Example lorentz equation

Example: Lorentz Equation

q

B = B k


Third step antisym product

THIRD STEP: ANTISYM. PRODUCT

b

a

sweep

b

sweep

a

  • anticommutative

  • associative

  • distributive

  • absolute value => area


Geometric or matrix product

Geometric or matrix product

  • non commutative

  • associative

  • distributive

  • closure:

  • extend vector space

  • unit = 1

  • inverse (conditional)


Examples of clifford algebras

Examples of Clifford algebras


Geometric algebras in e m

G2:

R

R2

I R

C = even algebra = spinors


Geometric algebras in e m

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


Geometric algebras in e m

a’

e2

φ

Rotations in R2

Euler:

a

e1


Inverse of multivector in g 2 conjugate

Inverse of multivector in G2Conjugate:

Generalizing


Geometric algebras in e m

G3

R3

iR

R

H = R + i R3=

= even algebra = spinors

iR3


Geometric product in g 3

Geometric product in G3 :

Rotations:

e P/2 generates rotations with respect to

planedefined by bivector P


Inverse of multivector in g 3 defining clifford conjugate

Inverse of multivector in G3defining Clifford conjugate:

  • Generalizing:


Geometric calculus 3d

Geometric Calculus (3d)

  • is a vector differential operator acting on:

     (r) – scalar field

    E(r) – vector field


First order green functions

First order Green Functions

Euclidian spaces :

is solution of


Maxwell s equation

Maxwell’s Equation

  • Maxwell’s multivector F = E +i c B

  • current density paravector J = (ε0c)-1(cρ + j)

  • Maxwell’s equation:


Electrostatics

Electrostatics

Gauss’s law. Solution using first

order Green’s function:

Explicitly:


Magnetostatics

Magnetostatics

Ampère’s law. Solution using first

order Green’s function:

Explicitly:


Fundamental theorem of calculus

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


Example divergence theorem

Example: divergence theorem

  • d3x = i d, where d = |d3 x|

  • d2 x = i nda , where da = |d2 x|

  • F = v(r)


Green s theorem euclidian case

Green’s Theorem (Euclidian case )

  • where the first order Green function is:


Cauchy s theorem in n dimensions

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

Inverse of differential paravectors:

Helmholtz:

Spherical wave (outgoing or incoming)


Electromagnetic diffraction

Electromagnetic diffraction

  • First order Helmholtz equation:

  • Exact Huygens’ principle:

    In the homogeneous case J = 0


Special relativity and paravectors in g 3

Special relativity and paravectors in G3

  • Paravector p = p0 + p

    Examples of paravectors:


Lorentz transformations

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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