GEOMETRIC ALGEBRAS IN E.M.

Manuel BerrondoBrigham Young University

Provo, UT, 84097berrondo@byu.edu

Physical Applications:

• Electro-magnetostatics

• Dispersion and diffraction E.M.

Examples of geometric algebras

• 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:

...321 aaa 21 aa

11 , A

Algebraic Properties of R+

• closure• commutativity• associativity• zero• negative

•• closure• commutativity• associativity• unit• reciprocal

• and + distributivity

Vector spaces : linear combinations

Algebras

• include metrics:

• define geometric product

FIRST STEP: Inverse of a vector a ≠ 0

ba

aaa

a

a

a

aa 2

221 ,

ˆa

a

a-1

a 1 20

Orthonormal basis3d: ê1 ê2 ê3

êi • êk = i k

êi-1

= êi

Euclidian

4d: 0 1 2 3

μ • ν = gμν (1,-1,-1,-1)

Minkowski

1

Example: (ê1 + ê2)-1 = (ê1 + ê2)/2

Electrostatics: method of images wr

2

ˆ

ˆ1

22

ˆ

ˆ1

ˆ1

ˆ 1

1

1

111

e

er

e

erererr

• q

•- q

•q

•ql

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

Sphere radius a, charge q at bê1, find image (?)

Choose scales for r and w:r w charge

0 a V = 0

c b q

-c ? q’ = ?

2

ˆ

ˆ/

12 1

1

e

erw

ca

ba

q

b

a

b

qV

/ˆˆ4

1)(

12

10 eweww

SECOND STEP: EULER FORMULAS

AkAk )ˆ sin(cos

ˆ e

AAkk )ˆ(ˆ

1ˆ sinhˆcosh

1)ˆ( sinˆcos2

2

bb

aab

a

bbe

iaiaei

In 3-d:

'ˆ

AAk e

given that

In general, rotating A about θ k:

Example: Lorentz Equation

ˆ :solves

)(ˆ)( :Derivative

)( 0

ˆ

vkv

vkv

vv k

qBm

tt

ett t

q

B = B k

m

qB

B ,ˆ B

k

000 )(sin)(cos)( vkvvv

ttt

THIRD STEP: ANTISYM. PRODUCT

ab

ba sweep

sweep

a

b

a

b

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

Geometric or matrix product

21

2

a

a

aa

aaaa

bababa

babaab

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

Examples of Clifford algebras

Notation Geometry Dim.

G2 plane 4

G0,1 complex 2

G0,2 quaternions 4

G3 Pauli 8

G1,3 Dirac 16

Gm,n signature (m,n) 2m+n

1 scalar

vector

bivector

1e 2e

I 2121 ˆˆˆˆ eeee

III

ikikkiki

aa

eeeeee

1

2ˆˆˆˆˆ,ˆ2

R

I R

R2

C = even algebra = spinors

G2 :

C isomorphic to R2

1e1

I e2

• z

• z*

a

z = ê1aê1z = a

Reflection: z z*

a a’ = ê1z* = ê1a ê1

In general, a’ = n a n, with n2 = 1

Rotations in R2

Euler:

e1

e2

a

a’

φ

2/2/'

'

II

II

ee

ee

aa

aaa

sinˆˆcos 21)ˆˆ( 21 eeee e

Inverse of multivector in G2

Conjugate:

AA

A

AA

AA

AAAA

IAIA

~

~

~

~

scalar ~~

~

1

aa

*

*1

21 and

zz

zz

a a

aGeneralizing

1 1 scalar

ê1 ê2 ê3 3 vector

ê2ê3 ê3ê1 ê1ê2 3 bivector

ê1ê2ê3= i 1 pseudoscalar

iii

ikikkiki

aa

eeeeee

1

2ˆˆˆˆˆ,ˆ2

RiR

R3

H = R + i R3 == even algebra = spinors

iR3

G3

Geometric product in G3 :

babaab i

babababa ii ,

nn

nn

aa

aaa

aaanna

ˆˆ

ˆˆ

21

21

'

'

)ˆ(ˆ

ii

ii

ee

ee

Rotations:

e P/2 generates rotations with respect to plane defined by bivector P

Inverse of multivector in G3

defining Clifford conjugate:

• Generalizing:

AA

A

AA

AA

AAAA

iiAiiA

~

~

~

~

scalar ~~

~

1

baba

21

a

aa

Geometric Calculus (3d)

is a vector differential operator acting on:

(r) – scalar field

E(r) – vector field

EE

EEE

i

)(

First order Green Functions

21

)(4

12 r

r

Euclidian spaces :

ngyx

yxyx

1

),( is solution of

)(),( )( yxyx ng

Maxwell’s Equation

• Maxwell’s multivector F = E + i c B

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

• Maxwell’s equation:

JF~~

jBE

c

cic

tc 0

11

Electrostatics

02

0

1

0

111

1EE

Gauss’s law. Solution using firstorder Green’s function:

Explicitly:

' with ')'(4

1)(

30

xxrxr

xE

dr

Magnetostatics

jjBjB20200

11

ii

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

Explicitly:

' with ')'(

4)(

30 xxr

rxjxB

dr

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

V V

kkk

V V

kkk

xdxd

xdxd

11

11

)1(

)1(

GG

FF

Example: divergence theorem

• d3 x = i d, where d = |d3 x|

• d2 x = i nda , where da = |d2 x|

• F = v(r)

dad

daiid

V V

V V

ˆ :partscalar

ˆ

vnv

vnv

Green’s Theorem (Euclidian case )

• where the first order Green function is:

)'(')',(

)(

)(

)'('')',()(

)(

1

V

nn

V

nn

xdgI

xdgI

xxxx

xxxx

x

F

F)F(

)2/(

2 with

'

'1)',(

2/

ng

n

nnn

xx

xxxx

Cauchy’s theorem in n dimensions

• particular case : F = f y f = 0

where f (x) is a monogenic multivectorial

field: f = 0

)'('

'

'1

)(

)( 1

V

nn

n

n

fdI

f xxxx

xx

xx)(

Inverse of differential paravectors:

22

21 ,

t

tt

Helmholtz:

1

,41

' ,)(

2

221

j

re

Gk

jkjk

rkj

xx

Spherical wave (outgoing or incoming)

Electromagnetic diffraction

• First order Helmholtz equation:

• Exact Huygens’ principle:

In the homogeneous case J = 0

')'(')'()(

BE for

xnxxx

ic

daGjkS

F

F)F(

FJF jk~

Special relativity and paravectors in G3

• Paravector p = p0 + p

Examples of paravectors:

x ct + r

u (1 + v/c)

p E/c + p

φ + cA

j cρ + j

22 /1

1

cv

Lorentz transformations

)(sinˆ)(cos

)(sinhˆ)(cosh

21

21

21

21

21

21

θiθeR

wweBi θ

wθ

w

Transform the paravector p = p0 + p = p0 + p= + p┴ into:

B p B = B2 (p0 + p=) + p┴

R p R† = p0 + p= + R2 p┴