Transcript Geometric Algebra

Clifford Geometric Algebra (GA)
The natural algebra of 3D space
Complex numbers i
Quaternions q
j
Quaternions
• The generalization of complex numbers
to three dimensions
• i2 = j2 = k2 = -1, i j = k,
k
Non-commutative i j =-j i , try rotating a book
i
Quarternion rotations of vectors
• Bilinear transformation
• v’=R v R†
where R is a quaternion
v is a Cartesian vector
e3
Clifford’s Geometric Algebra
• Define algebraic elements e1, e2, e3
• With e12=e22=e32=1, and anticommuting
• ei ej= - ej ei
This algebraic structure unifies Cartesian coordinates, quaternions and
complex numbers into a single real framework.
e1e2
e2
e1 e2 e3
e1 e3
ei~σi
e1
e3
Geometric Algebra-Dual representation
e2 e3  e1 ,
e3e1  e2 ,
e1e2  e3
  e1e2 e3
e1e2
ι=e1e2e3
e2
e1 e3
e1
The product of two vectors….
To multiply 2 vectors we….just expand the brackets…distributive law of
multiplication over addition.
uv
 e1u1  e2u 2  e3u3 e1v1  e2 v2  e3v3 
 u1v1  u 2 v2  u3v3  u2 v3  v2u3 e2 e3  u1v3  v1u3 e1e3  u1v2  v1u 2 e1e2
 ui vi   u 2 v3  v2u3 e1  u1v3  v1u3 e2  u1v2  v1u 2 e3 
 u  v  u  v
  e1e2 e3
A complex-type number combining the dot and cross products!
Hence we now have an intuitive definition of multiplication and division of vectors,
subsuming the dot and cross products which also now has an inverse.
Spinor mapping
How can we map from complex spinors to 3D GA?
We see that spinors are rotation operators.
The geometric product is equivalent
to the tensor product
EPR setting for Quantum games
GHZ state
Ø=KLM
Probability distribution
Where
Doran C, Lasenby A (2003) Geometric algebra for physicists
<ρQ>0~Tr[ρQ]
References:
• Analyzing three-player quantum games in an
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•
EPR type setup
Analysis of two-player quantum games in an
EPR setting using Clifford's geometric algebra
N player games
Copies on arxiv, Chappell
A GAME THEORETIC APPROACH TO STUDY THE QUANTUM KEY
DISTRIBUTION BB84 PROTOCOL, IJQI 2011, HOUSHMAND
Google: Cambridge university geometric algebra
The geometric product magnitudes
In three dimensions we have:
Negative Numbers
• Interpreted financially as debts by Leonardo di
•
•
Pisa,(A.D. 1170-1250)
Recognised by Cardano in 1545 as valid
solutions to cubics and quartics, along with the
recognition of imaginary numbers as meaningful.
Vieta, uses vowels for unknowns and use
powers. Liebniz 1687 develops rules for symbolic
manipulation.
Diophantus 200AD
Modern
Precession in GA
Z=σ3
Spin-1/2
Re
Bz
ω
tIz
~
S  Rv0 R
v0  Sin 1  Cos  3
<Sx>=Sin θ Cos ω t
<Sy>=Sin θ Sin ω t
<Sz>=Cos θ
θ
x= σ1
ω = γ Bz
Y= σ2
Quotes
• “The reasonable man adapts himself to the world
•
•
around him. The unreasonable man persists in his
attempts to adapt the world to himself. Therefore,
all progress depends on the unreasonable man.”
George Bernard Shaw,
Murphy’s two laws of discovery:
“All great discoveries are made by mistake.”
“If you don't understand it, it's intuitively obvious.”
“It's easy to have a complicated idea. It's very hard
to have a simple idea.” Carver Mead.
Greek concept of the product
Euclid Book VII(B.C. 325-265)
“1. A unit is that by virtue of which each of
the things that exist is called one.”
“2. A number is a multitude composed of
units.”
….
“16. When two numbers having multiplied one
another make some number, the number so
produced is called plane, and its sides are the
numbers which have multiplied one another.”
e3
e1e2
ι=e1e2e3
e2
e1 e3
e1
Conventional Dirac Equation
“Dirac has redisovered Clifford algebra..”, Sommerfield
That is for Clifford basis vectors we have: {ei , e j }  ei e j  e j ei  2ei  e j  2 ij
isomorphic to the Dirac agebra.
Dirac equation in real space
F  a  E  B  b
  e1e2 e3
Same as the free Maxwell equation, except for the addition
of a mass term and expansion of the field to a full multivector.
Free Maxwell equation(J=0):
Special relativity
Its simpler to begin in 2D, which is sufficient to describe most phenomena.
We define a 2D spacetime event as

X  x   t
So that time is represented as the bivector of the plane and so an extra
Euclidean-type dimension is not required. This also implies 3D GA is sufficient to
describe 4D Minkowski spacetime.
We find:
2 2
X  x t
2
the correct spacetime distance.
We have the general Lorentz transformation given by:
X ' e
vˆ / 2  / 2
e
 / 2 vˆ / 2
Xe
e
Consisting of a rotation and a boost,
which applies uniformly to both
coordinate and field multivectors.
Compton scattering formula
C
Time after time
• “Of all obstacles to a thoroughly
penetrating account of existence, none
looms up more dismayingly than time.”
Wheeler 1986
• In GA time is a bivector, ie rotation.
• Clock time and Entropy time
The versatile multivector
(a generalized number)
M  a  v1e1  v2 e2  v3e3  w1e2 e3  w2 e3e1  w3e1e2  be1e2e3
 
 a  v  w  b
a+ιb
v
ιw
ιb
v+ιw
a+ιw
a+v
Complex numbers
Vectors
Pseudovectors
Pseudoscalars
Anti-symmetric EM field tensor E+iB
Quaternions or Pauli matrices
Four-vectors
Penny Flip game Qubit Solutions
Use of quaternions
Used in airplane
guidance systems
to avoid Gimbal lock
How many space dimensions do we
have?
• The existence of five regular solids implies
three dimensional space(6 in 4D, 3 > 4D)
• Gravity and EM follow inverse square laws
to very high precision. Orbits(Gravity and
Atomic) not stable with more than 3 D.
• Tests for extra dimensions failed, must be
sub-millimetre
Benefits of GA
• Maxwell’s equations reduce to a single
equation
• 4D spacetime embeds in 3D
• The Dirac equation in 4D spacetime
reduces to a real equation in 3D