Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”)

1. Geometric algebra: A smallGeometric algebra: A small
introduction to a powerfulintroduction to a powerful
and general language ofand general language of
physicsphysics
Michael R.R. GoodMichael R.R. Good
Georgia Institute of TechnologyGeorgia Institute of Technology

2. Redundant LanguagesRedundant Languages
– Synthetic GeometrySynthetic Geometry
Coordinate GeometryCoordinate Geometry
Complex NumbersComplex Numbers
QuaternionsQuaternions
Vector AnalysisVector Analysis
Tensor AnalysisTensor Analysis
Matrix AlgebraMatrix Algebra
Grassmann AlgebraGrassmann Algebra
Clifford AlgebraClifford Algebra
Spinor AlgebraSpinor Algebra
etc…etc…
 There are unnecessary consequences of so many languages.There are unnecessary consequences of so many languages.
– Redundant learningRedundant learning
– Complicates access to knowledgeComplicates access to knowledge
– Frequent translationFrequent translation
– Lower concept density, i.e., theorems / definitionsLower concept density, i.e., theorems / definitions
Geometric
Concepts

3. Geometric algebraGeometric algebra
 A unifying language for mathematics.A unifying language for mathematics.
 A revealing language for large areas ofA revealing language for large areas of
theoretical and applied physics.theoretical and applied physics.
 Acts for both classical and quantumActs for both classical and quantum
physics.physics.
 Applications in robotics, computer vision,Applications in robotics, computer vision,
image processing, signal processing andimage processing, signal processing and
space dynamics.space dynamics.

4. What has geometric algebra doneWhat has geometric algebra done
for physics?for physics?
• Maxwell’s electrodynamics has been formulated in oneMaxwell’s electrodynamics has been formulated in one
equation revealing a more simple physical relationship.equation revealing a more simple physical relationship.
• Relativistic quantum mechanics has been reformulated,Relativistic quantum mechanics has been reformulated,
replacing abstract complex inner space Dirac matricesreplacing abstract complex inner space Dirac matrices
by real space-time basis vectors.by real space-time basis vectors.
• General relativity has been improved by the constructionGeneral relativity has been improved by the construction
of a new gauge theory using geometric calculusof a new gauge theory using geometric calculus
improving ease of calculations.improving ease of calculations.

5. ``Physicists quickly become impatient with any discussion ofPhysicists quickly become impatient with any discussion of
elementary conceptselementary concepts''
So why re-learn vectorsSo why re-learn vectors??
Geometric algebra:Geometric algebra:
 allows the division by vectors.allows the division by vectors.
 introduces a more general concept than the cross product, which isintroduces a more general concept than the cross product, which is
only defined in three dimensionsonly defined in three dimensions
– this is needed so that full information about relative directions can bethis is needed so that full information about relative directions can be
encoded in all dimensions.encoded in all dimensions.
 gives the imaginary unit concrete and natural geometricgives the imaginary unit concrete and natural geometric
interpretations.interpretations.
 is more intuitive than standard vector analysis.is more intuitive than standard vector analysis.
 is more efficient because it reduces the number of operations, andis more efficient because it reduces the number of operations, and
is coordinate free.is coordinate free.
 is well-defined for higher and lower dimensions.is well-defined for higher and lower dimensions.
 handles reflections and rotations with ease and power.handles reflections and rotations with ease and power.

6. So what is geometric algebra?So what is geometric algebra?
 A language for geometry.A language for geometry.
 The exploitation of the concept of a vector.The exploitation of the concept of a vector.
 The use of higher dimensional vectors, called k-The use of higher dimensional vectors, called k-
vectors.vectors.
 The combination of different dimensional concepts,The combination of different dimensional concepts,
scalars, vectors, bi-vectors, tri-vectors, and finally k-scalars, vectors, bi-vectors, tri-vectors, and finally k-
vectors to form multi-vectors.vectors to form multi-vectors.
Geometric
Concepts
Algebraic
Language

7. Geometric algebra makes use ofGeometric algebra makes use of
dimensions called gradesdimensions called grades
 PointPoint αα scalarscalar gradegrade
00
 VectorVector aa directed linedirected line grade 1grade 1
 Bi-vectorBi-vector BB directed plane grade 2directed plane grade 2
 Tri-vectorTri-vector TT directed volume grade 3directed volume grade 3
They are all called k-vectors:They are all called k-vectors:
 kk-vector-vector KK directed objectdirected object gradegrade kk

8. So what is a bi-vector?So what is a bi-vector?
 A bi-vector has the same magnitude as theA bi-vector has the same magnitude as the
familiar cross product.familiar cross product.
 The cross product is a vector, whereas a bi-The cross product is a vector, whereas a bi-
vector is an areavector is an area
 The bi-vector is a directed area and itsThe bi-vector is a directed area and its
orientation lies in the plane that it rests.orientation lies in the plane that it rests.
 The outer productThe outer product aa ∧∧ bb, or wedge product,, or wedge product,
defines a bi-vector and has magnitude:defines a bi-vector and has magnitude:
|a|a ∧∧ b| = |a| |b| sinb| = |a| |b| sin θθ

9. The outer product is the naturalThe outer product is the natural
partner of the inner product.partner of the inner product.
 The inner productThe inner product a · ba · b, or dot product, is a scalar and, or dot product, is a scalar and
has magnitude:has magnitude:
|a · b| = |a| |b| cos|a · b| = |a| |b| cos θθ
 The outer productThe outer product aa ∧∧ bb, or wedge product is a bi-vector, or wedge product is a bi-vector
and has magnitude:and has magnitude:
|a|a ∧∧ b| = |a| |b| sinb| = |a| |b| sin θθ
The outer product is more general than the cross product!The outer product is more general than the cross product!

10. Addition of different dimensions?Addition of different dimensions?
In complex analysis addition defines a relation:In complex analysis addition defines a relation:
z = x +z = x + i yi y
Clifford’s “geometric product” for vectors:Clifford’s “geometric product” for vectors:
abab == aa ⋅⋅ bb ++ aa ∧∧ bb
scalarscalar bi-vectorbi-vector
(inner product)(inner product) (outer product)(outer product)
additionaddition

11. How can you add a scalar to a bi-How can you add a scalar to a bi-
vector?vector?
 A scalar added to a bi-vector is the most basicA scalar added to a bi-vector is the most basic
axiom of geometric algebra, it is called theaxiom of geometric algebra, it is called the
geometric product, or Clifford product.geometric product, or Clifford product.
abab == aa ⋅⋅ bb ++ aa ∧∧ bb
 Adding different quantities is exactly what weAdding different quantities is exactly what we
want an addition to do! The product has scalarwant an addition to do! The product has scalar
and bi-vector parts, just like a complex numberand bi-vector parts, just like a complex number
has real and imaginary parts.has real and imaginary parts.

12. Sum of k-vectors are multi-vectorsSum of k-vectors are multi-vectors
 A multi-vectorA multi-vector MM is the sum ofis the sum of k-k-vectors.vectors.
( M =( M = αα ++ aa + B + T + … )+ B + T + … )
 A multi-vector has mixed grades. (gradesA multi-vector has mixed grades. (grades
are dimensions)are dimensions)

13. The geometric product is basicThe geometric product is basic
We can define our inner product and outer product in termsWe can define our inner product and outer product in terms
of the basic geometric product.of the basic geometric product.
Define dot product in terms of geometric product:Define dot product in terms of geometric product:
a · ba · b = 1/2 (= 1/2 (ab + baab + ba)) scalarscalar
Define wedge product in terms of geometric product:Define wedge product in terms of geometric product:
aa ∧∧ bb = 1/2 (= 1/2 (ab - baab - ba)) bi-vectorbi-vector
Using the geometric product:Using the geometric product: a · b + aa · b + a ∧∧ b = abb = ab

14. What happened to the crossWhat happened to the cross
product?product?
The cross product is translated from geometric algebra viaThe cross product is translated from geometric algebra via
a ∧ b = -i a × b
Or if you prefer viaOr if you prefer via
a × b = i a ∧ b
wherewhere i is a unit tri-vector, whose square is -1.is a unit tri-vector, whose square is -1. i = σσ11σσ22σσ33
The basis for geometric algebra in 3 dimensional space is:The basis for geometric algebra in 3 dimensional space is:
{1,{1, σσ1,1, σσ2,2, σσ3,3, σσ11σσ2,2, σσ11σσ3,3, σσ22σσ3,3, σσ11σσ22σσ3 }3 }
scalar vectors bi-vectors tri-vector

15. Why isn’t geometric algebra moreWhy isn’t geometric algebra more
widely known?widely known?
 Clifford was a student of Maxwell, butClifford was a student of Maxwell, but
died an early death allowing Gibbs’died an early death allowing Gibbs’
and Heaviside’s vector analysis toand Heaviside’s vector analysis to
dominate the 20dominate the 20thth
century.century.
 It is hard to learn a new language,It is hard to learn a new language,
especially one that asks you to revisitespecially one that asks you to revisit
elementary concepts.elementary concepts.
 Many physicists and teachers haveMany physicists and teachers have
not heard about geometric algebra’snot heard about geometric algebra’s
advantages.advantages.
W.K. Clifford 1845-1879

16. The Future of Geometric AlgebraThe Future of Geometric Algebra
Speculations and hope for:Speculations and hope for:
 An understanding of geometric algebra as a quantumAn understanding of geometric algebra as a quantum
algebra for a quantum theory of gravitation.algebra for a quantum theory of gravitation.
 Complex numbers, as mystical un-interpreted scalars, toComplex numbers, as mystical un-interpreted scalars, to
be proven unnecessary even in quantum mechanicsbe proven unnecessary even in quantum mechanics
 Unobserved higher dimensions to be provenUnobserved higher dimensions to be proven
unnecessary in the clarity created by geometric algebra.unnecessary in the clarity created by geometric algebra.
 Introducing geometric algebra:Introducing geometric algebra:
– High school: generalizing the cross product.High school: generalizing the cross product.
– Undergraduate: complimenting rotation matrices.Undergraduate: complimenting rotation matrices.
– Graduate: condensing the Maxwell equation’s into one equation.Graduate: condensing the Maxwell equation’s into one equation.

17. References and SourcesReferences and Sources
Online Presentations:Online Presentations:
 Introduction to Geometric Algebra, Course #53 by Alyn Rockwood, David Hestenes, Leo Dorst,Introduction to Geometric Algebra, Course #53 by Alyn Rockwood, David Hestenes, Leo Dorst,
Stephen Mann, Joan Lasenby, Chris Doran, and Ambjørn Naeve.Stephen Mann, Joan Lasenby, Chris Doran, and Ambjørn Naeve.
 GABLE: Geometric Algebra Learning Environment, Course #31 by Stephen Mann, and LeoGABLE: Geometric Algebra Learning Environment, Course #31 by Stephen Mann, and Leo
Dorst.Dorst.
 Gull, S. Lasenby A. & Doran,C. 1993 ‘Imaginary Numbers Are Not Real – The GeometricGull, S. Lasenby A. & Doran,C. 1993 ‘Imaginary Numbers Are Not Real – The Geometric
Algebra Of Space-Time’ http://www.mrao.cam.ac.uk/~clifford/introduction/intro/intro.htmlAlgebra Of Space-Time’ http://www.mrao.cam.ac.uk/~clifford/introduction/intro/intro.html
 Lasenby A.N Doran C.J lecture notes 2000-2001Lasenby A.N Doran C.J lecture notes 2000-2001
http://www.mrao.cam.ac.uk/~clifford/ptIIIcourse/http://www.mrao.cam.ac.uk/~clifford/ptIIIcourse/
 Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics by DavidOersted Medal Lecture 2002: Reforming the Mathematical Language of Physics by David
HestenesHestenes
Books:Books:
 Hestenes, D. 1966 ‘Spacetime algebra’ New York Gordon and BreachHestenes, D. 1966 ‘Spacetime algebra’ New York Gordon and Breach
 Jancewicz, B. 1988 ‘Multivectors and Clifford Algebra in Electrodynamics’ World ScientificJancewicz, B. 1988 ‘Multivectors and Clifford Algebra in Electrodynamics’ World Scientific
 Lounesto, P. 2001 ‘Clifford Algebras and Spinors’ Cambridge University PressLounesto, P. 2001 ‘Clifford Algebras and Spinors’ Cambridge University Press
Websites:Websites:
 Hestenes’s Site: http://modelingnts.la.asu.edu/Hestenes’s Site: http://modelingnts.la.asu.edu/
 Lounesto’s Site: http://www.helsinki.fi/~lounesto/Lounesto’s Site: http://www.helsinki.fi/~lounesto/
 Cambridge Group: http://www.mrao.cam.ac.uk/~clifford/Cambridge Group: http://www.mrao.cam.ac.uk/~clifford/