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# GEOMETRIC ALGEBRAS IN E.M. - PowerPoint PPT Presentation

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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Manuel Berrondo

Brigham Young University

Provo, UT, 84097

• Electro-magnetostatics

• Dispersion and diffraction E.M.

• Complex Numbers G0,1= C

• Hamilton Quaternions G0,2= H

• Pauli Algebra G3

• Dirac Algebra G1,3

• 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

Algebraic Properties of R

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

Orthonormal basis

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 (?)

SECOND STEP: EULER FORMULAS

In 3-d:

given that

In general, rotating A about θ k:

q

B = B k

THIRD STEP: ANTISYM. PRODUCT

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

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

e2

φ

Rotations in R2

Euler:

a

e1

Inverse of multivector in G2Conjugate:

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

Inverse of multivector in G3defining Clifford conjugate:

• 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┴