geometric algebras manuel berrondo brigham young university provo, ut, 84097 [email protected]
Post on 18-Dec-2015
224 views
TRANSCRIPT
Physical Applications:
• Mechanics: Foucault pendulum
• Electro-magnetostatics
• Dispersion and diffraction E.M.
• Quantum Mechanics: spin precession
• Field Theory: Dirac equation
• (General Relativity )
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 1: 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
Example 2: Foucault Pendulum
rωSgr mmm 2
ρωρρ zmmm 220
shz
zz
5
22
10724
2
sin 0
ml
S0
•Coriolis
S
mg
l
0)( ρρ ktti eet
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
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
Spinors 2-d C
3-d C x C
C
, ,'
'σ, Pauli matrices
Pauli G3
( ) p0 + i p = p
σk ( ) êk p ê3
j ( ) i p ê3
j σk ( ) i êk p
Interaction: spin with magnetic field B
• Pauli’s quantum Hamiltonian with
A = vector potential
p p – q A, minimal coupling,
AB
BAp
2
)(2
1 2 Vσm
mH I
magnetic moment : μ = S = ħ σ/2
Spin precession with uniform B :
In G3, spinor (quaternion) Ψ,
Be iiB kk 21
21 ˆ
)( 021
tiet BSolution in G3 :
Spin precesses on plane i B with angular freq.
B 21
0 independent of ħ !
Inverse of multivector in G3
defining Clifford conjugate:
• Generalizing:
AA
A
AA
AA
AAAA
iiAiiA
~
~
~
~
scalar ~~
~
1
baba
21
a
aa
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┴
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
')'(')'()(
BEfor
xnxxx
ic
daGjkS
F
F)F(
FJF jk~
1 1
0 1 2 3 4
0 1 1 2 2 3 0 2 0 3 1 3 6
0 1 2 1 2 3 2 3 0 0 1 3 4
0 1 2 3 1
IIII
gg
1
)1,1,1,1( diag 2,2
3210
6 bivectors split into two groups:
êi = i 0 and I êk = i j
Dirac Equation in G1,3
• Vector in Minkowski space :
• Quantization:
2200
0 , p withγp ppppp
)( 0 γtip
Propagator – first order Green function
• The propagator represents the conditional probability amplitude
• Fourier transform :
)'()'()( )4( xxxxSmp F
)0(lim ,1
)(
imp
pSF