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GEOMETRIC ALGEBRAS IN E.M.

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Manuel Berrondo

Brigham Young University

Provo, UT, 84097

berrondo@byu.edu

- Electro-magnetostatics
- Dispersion and diffraction E.M.

- Complex Numbers G0,1= C
- Hamilton Quaternions G0,2= H
- Pauli Algebra G3
- Dirac Algebra G1,3

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:

+

closure

commutativity

associativity

zero

negative

•

closure

commutativity

associativity

unit

reciprocal

and + distributivity

Vector spaces :linear combinations

- include metrics:
- define geometricproduct
FIRST STEP: Inverse of a vector a≠ 0

a-1

0

2

a

1

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

Example: (ê1+ ê2)-1 = (ê1+ ê2)/2

1

w

r

•

- q

•

ql

•

q

•

q

Plane: charge -q atcê1, image (-q) at -cê1

Sphere radius a, charge q atbê1, find image (?)

Choose scales for r and w:

In 3-d:

given that

In general, rotating A about θ k:

q

B = B k

b

a

sweep

b

sweep

a

- anticommutative
- associative
- distributive
- absolute value => area

- non commutative
- associative
- distributive
- closure:
- extend vector space
- unit = 1
- inverse (conditional)

G2:

R

R2

I R

C = even algebra = spinors

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

a’

e2

φ

Rotations in R2

Euler:

a

e1

Generalizing

G3

R3

iR

R

H = R + i R3=

= even algebra = spinors

iR3

Rotations:

e P/2 generates rotations with respect to

planedefined by bivector P

- Generalizing:

- is a vector differential operator acting on:
(r) – scalar field

E(r) – vector field

Euclidian spaces :

is solution of

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

Gauss’s law. Solution using first

order Green’s function:

Explicitly:

Ampère’s law. Solution using first

order Green’s function:

Explicitly:

- 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

- d3x = i d, where d = |d3 x|
- d2 x = i nda , where da = |d2 x|
- F = v(r)

- where the first order Green function is:

- particular case : F = f y f = 0
where f (x) is a monogenic multivectorial

field: f = 0

Helmholtz:

Spherical wave (outgoing or incoming)

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

- Paravector p = p0 + p
Examples of paravectors:

Transform the paravector

p = p0 + p = p0 + p=+ p┴ into:

B p B = B2 (p0 + p=) + p┴

R p R† = p0 + p= + R2 p┴