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Geometric Algebra 1
Chris Doran
Astrophysics Group Cavendish Laboratory Cambridge, UK


Resources
· A complete lecture course, including handouts, overheads and papers available from www.mrao.cam.ac.uk/~Clifford · Geometric Algebra for Physicists out in March (C.U.P.) · David Hestenes' website modelingnts.la.asu.edu

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What is Geometric Algebra?
· Geometric Algebra is a universal Language for physics based on the mathematics of Clifford Algebra · Provides a new product for vectors · Generalizes complex numbers to arbitrary dimensions · Treats points, lines, planes, etc. in a single algebra · Simplifies the treatment of rotations · Unites Euclidean, affine, projective, spherical, hyperbolic and conformal geometry

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Grassmann
German schoolteacher 1809-1877 Published the Lineale Ausdehnungslehre in 1844 Introduced the outer product

a b b a
Encodes a plane segment
b a
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2D Outer Product
· Antisymmetry implies · Introduce basis vectors · Form product

· Returns area of the plane + orientation. · Result is a bivector · Extends (antisymmetry) to arbitrary vectors
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Complex Numbers
· Already have a product for vectors in 2D · Length given by aa* · Suggests forming

· Complex multiplication forms the inner and outer products of the underlying vectors! · Clifford generalised this idea
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Hamilton
Introduced his quaternion algebra in 1844

i 2 j2 k 2 ijk 1
Generalises complex arithmetic to 3 (4?) dimensions Very useful for rotations, but confusion over the status of vectors
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Quaternions
· Introduce the two quaternion `vectors'

· Product of these is · where c0 is minus the scalar product and

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W.K. Clifford 1845-1879
Introduced the geometric product

Product of two vectors returns the sum of a scalar and a bivector Think of this sum as like the real and imaginary parts of a complex number
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History
· Foundations of geometric algebra (GA) were laid in the 19th Century · Key figures: Hamilton, Grassmann, Clifford and Gibbs · Underused (associated with quaternions) · Rediscovered by Pauli and Dirac for quantum theory of spin · Developed by mathematicians (Atiyah etc.) in the 50s and 60s · Reintroduced to physics in the 70s by David Hestenes
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Properties
· Geometric product is associative and distributive

· Square of any vector is a scalar · Define the inner (scalar) and outer (exterior) products

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2D Algebra
· Orthonormal basis is 2D · Parallel vectors commute · Orthogonal vectors anticommute since · Unit bivector has negative square

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2D Basis
· Build into a basis for the algebra 1 scalar 2 vectors 1 bivector

· Even grade objects form complex numbers · Map between vectors and complex numbers y x,z x
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2D Rotations
· In 2D vectors can be rotated using complex phase rotations v, y



u, x

· But · Rotation
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3 Dimensions
· Now introduce a third vector e 1 , e 2 , e 3 · These all anticommute e 1 e2 e 2 e1 etc.

e

3

e

2

e1 · Have 3 bivectors now: e 1 e2 , e 2 e3 , e 3 e1 e1e

2

e2e
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e3e

1
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Bivector Products
· Various new products to form in 3D · Product of a vector and a bivector e 1 e1 e 2 e 2 e 1 e 2 e 3 e 1 e 2e 3 I · Product of two perpendicular bivectors: e 2 e3 e 3 e1 e 2 e3 e 3 e 1 e 2 e 1 e 1 e · Set
2

i e 2 e3 ,

j e 3 e 1 ,

k e1e

2

· Recover quaternion relations i 2 j2 k 2 ijk 1
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3D Pseudoscalar
· 3D Pseudoscalar defined by · Represents a directed volume · Has negative square I 2 e1 e 2 e3 e 1 e2 e 3 e 2 e 3 e 2 · Commutes with all vectors e 1 I e 1 e 1 e 2 e 3 e 1 e 2 e 1 e 3 e

I e 1 e2 e
element

3

e 3 1
1e 2 e 3e 1

Ie

1

· Interchanges vectors and planes

e 1 I e2 e 3 Ie 2 e 3 e

e2e
1

3

e

1
17

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3D Basis
Different grades correspond to different geometric objects

Grade 0 Scalar

Grade 1 Vector
3

Grade 2 Bivector

Grade 3 Trivector
1

1

e 1 , e2 , e

e1 e 2 , e2 e 3 , e 3e

I
k

Generators satisfy Pauli relations e i ej ij ijk Ie Recover vector cross product
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a b I a b
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Reflections
· Build rotations from reflections · Good example of geometric product ­ arises in operations

a a nn a a a nn
Image of reflection is

b

a n

b a a a 2a nn a an nan nan
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Rotations
· Two rotations form a reflection a mnanm mnanm · Define the rotor R mn · This is a geometric product! Rotations given by a RaR R nm · Works in spaces of any dimension or signature · Works for all grades of multivectors A RAR · More efficient than matrix multiplication
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3D Rotations
· Rotors even grade (scalar + bivector in 3D) · Normalised: RR mnnm 1 · Reduces d.o.f. from 4 to 3 ­ enough for a rotation · In 3D a rotor is a normalised, even element R B RR 2 B 2 1 · Can also write R expB/2 · Rotation in plane B with orientation of B · In terms of an axis R expIn/2
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Group Manifold
· Rotors are elements of a 4D space, normalised to 1 · They lie on a 3-sphere · This is the group manifold · Tangent space is 3D · Can use Euler angles R expe 1 e2 /2 expe 2e 3 /2 expe1 e 2 /2 · Rotors R and ­R define the same rotation · Rotation group manifold is more complicated
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Lie Groups
· Every rotor can be written as R expB/2 · Rotors form a continuous Lie group · Bivectors form a Lie algebra under the commutator product · All finite Lie groups are rotor groups · All finite Lie algebras are bivector algebras · (Infinite case not fully clear, yet) · In conformal case starting point of screw theory (Clifford, 1870s)!
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Rotor Interpolation
· How do we interpolate between 2 rotations? · Form path between rotors R0 R 0 R R 0 expB R1 R 1 · Find B from expB R R 1 · This path is invariant. If points transformed, path transforms the same way · Midpoint simply R1/2 R 0 expB/2 · Works for all Lie groups
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Interpolation 2
· · · · For rotors in 3D can do even better! View rotors as unit vectors in 4D Path is a circle in a plane Use simple trig' to get SLERP

R

1

R

0

1 sin1 R 0 sin R 1 sin · For midpoint add the rotors and normalise! R

sin/2 R1/2 R 0 R 1 sin
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Exercises
Verify the following

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