Geometric algebra: A Geometric algebra: A small introduction to a small introduction to a powerful and general powerful and general language of physicslanguage of physics

Michael R.R. GoodMichael R.R. GoodGeorgia Institute of Georgia Institute of

TechnologyTechnology

Redundant LanguagesRedundant Languages– Synthetic GeometrySynthetic Geometry

Coordinate GeometryCoordinate GeometryComplex NumbersComplex NumbersQuaternionsQuaternionsVector AnalysisVector AnalysisTensor AnalysisTensor AnalysisMatrix AlgebraMatrix AlgebraGrassmann AlgebraGrassmann AlgebraClifford AlgebraClifford AlgebraSpinor AlgebraSpinor Algebraetc…etc…

There are unnecessary consequences of so many There are unnecessary consequences of so many languages.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

GeometricConcepts

Geometric algebra Geometric algebra A unifying language for mathematics.A unifying language for mathematics. A revealing language for large areas A revealing language for large areas

of theoretical and applied physics.of theoretical and applied physics. Acts for both classical and quantum Acts for both classical and quantum

physics.physics. Applications in robotics, computer Applications in robotics, computer

vision, image processing, signal vision, image processing, signal processing and space dynamics. processing and space dynamics.

What has geometric algebra What has geometric algebra done for physics?done for physics?

• Maxwell’s electrodynamics has been formulated in Maxwell’s electrodynamics has been formulated in one equation revealing a more simple physical one equation revealing a more simple physical relationship.relationship.

• Relativistic quantum mechanics has been Relativistic quantum mechanics has been reformulated, replacing abstract complex inner space reformulated, replacing abstract complex inner space Dirac matrices by real space-time basis vectors.Dirac matrices by real space-time basis vectors.

• General relativity has been improved by the General relativity has been improved by the construction of a new gauge theory using geometric construction of a new gauge theory using geometric calculus improving ease of calculations.calculus improving ease of calculations.

`̀Physicists quickly become impatient with any Physicists quickly become impatient with any discussion of elementary conceptsdiscussion of elementary 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, introduces a more general concept than the cross product,

which is only defined in three dimensionswhich is only defined in three dimensions– this is needed so that full information about relative directions this is needed so that full information about relative directions

can be encoded in all dimensions.can be encoded in all dimensions. gives the imaginary unit concrete and natural geometric gives 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 is more efficient because it reduces the number of

operations, and is coordinate free.operations, and 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.

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 The combination of different dimensional

concepts, scalars, vectors, bi-vectors, tri-vectors, concepts, scalars, vectors, bi-vectors, tri-vectors, and finally k-vectors to form multi-vectors.and finally k-vectors to form multi-vectors.

GeometricConcepts

AlgebraicLanguage

Geometric algebra makes use Geometric algebra makes use of dimensions called gradesof dimensions called grades

PointPoint scalarscalar grade 0 grade 0 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 grade grade kk

So what is a bi-vector?So what is a bi-vector? A bi-vector has the same magnitude as the A bi-vector has the same magnitude as the

familiar cross product.familiar cross product. The cross product is a vector, whereas a The cross product is a vector, whereas a

bi-vector is an areabi-vector is an area The bi-vector is a directed area and its The 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 product The outer product a a b b, 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| sin b| = |a| |b| sin

The outer product is the The outer product is the natural partner of the inner natural partner of the inner

product.product. The inner product The inner product a · ba · b, or dot product, is a scalar , or dot product, is a scalar

and has magnitude:and has magnitude:

|a · b| = |a| |b| cos |a · b| = |a| |b| cos

The outer product The outer product aa bb, or wedge product is a bi-, or wedge product is a bi-vector and has magnitude:vector and has magnitude:

|a |a b| = |a| |b| sin b| = |a| |b| sin

The outer product is more general than the cross The outer product is more general than the cross product!product!

Addition of different Addition of different dimensions?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 = = a a bb + + aa bb

scalarscalar bi-vector bi-vector (inner product)(inner product) (outer product) (outer product)

additionaddition

How can you add a scalar to a How can you add a scalar to a bi-vector?bi-vector?

A scalar added to a bi-vector is the most A scalar added to a bi-vector is the most basic axiom of geometric algebra, it is called basic axiom of geometric algebra, it is called the geometric product, or Clifford product.the geometric product, or Clifford product.

abab = = a a bb + + aa bb Adding different quantities is exactly what we Adding different quantities is exactly what we

want an addition to do! The product has want an addition to do! The product has scalar and bi-vector parts, just like a complex scalar and bi-vector parts, just like a complex number has real and imaginary parts. number has real and imaginary parts.

Sum of k-vectors are multi-Sum of k-vectors are multi-vectorsvectors

A multi-vector A multi-vector MM is the sum of is the sum of k-k-vectors.vectors.

( M = ( M = + + aa + B + T + … ) + B + T + … )

A multi-vector has mixed grades. A multi-vector has mixed grades. (grades are dimensions)(grades are dimensions)

The geometric product is The geometric product is basicbasic

We can define our inner product and outer product in We can define our inner product and outer product in terms of the basic geometric product.terms of the basic geometric product.

Define dot product in terms of geometric product:Define dot product in terms of geometric product: a · b a · b = 1/2 (= 1/2 (ab + baab + ba)) scalarscalarDefine wedge product in terms of geometric product:Define wedge product in terms of geometric product: a a b b = 1/2 (= 1/2 (ab - baab - ba)) bi-vectorbi-vector

Using the geometric product:Using the geometric product: a · b + a a · b + a b = ab b = ab

What happened to the cross What happened to the cross product?product?

The cross product is translated from geometric algebra The cross product is translated from geometric algebra viavia

a b = -i a bOr if you prefer viaOr if you prefer via

a b = i a b

where where i is a unit tri-vector, whose square is -1. is a unit tri-vector, whose square is -1. i = 11223 3

The basis for geometric algebra in 3 dimensional space is: The basis for geometric algebra in 3 dimensional space is:

{1, {1, 1, 1, 2, 2, 3, 3, 112, 2, 113, 3, 223, 3, 11223 }3 } scalar vectors bi-vectors tri-vector

Why isn’t geometric algebra Why isn’t geometric algebra more widely known?more widely known?

Clifford was a student of Maxwell, Clifford was a student of Maxwell, but died an early death allowing but died an early death allowing Gibbs’ and Heaviside’s vector Gibbs’ and Heaviside’s vector analysis to dominate the 20analysis to dominate 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 especially one that asks you to revisit elementary concepts.revisit elementary concepts.

Many physicists and teachers Many physicists and teachers have not heard about geometric have not heard about geometric algebra’s advantages.algebra’s advantages.

W.K. Clifford 1845-1879

The Future of Geometric The Future of Geometric AlgebraAlgebra

Speculations and hope for:Speculations and hope for: An understanding of geometric algebra as a quantum An 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, Complex numbers, as mystical un-interpreted scalars,

to be proven unnecessary even in quantum mechanicsto be proven unnecessary even in quantum mechanics Unobserved higher dimensions to be proven Unobserved higher dimensions to be proven

unnecessary in the clarity created by geometric unnecessary in the clarity created by geometric algebra.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 Graduate: condensing the Maxwell equation’s into one

equation.equation.

References and SourcesReferences and SourcesOnline 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 Leo GABLE: 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 Geometric Gull, S. Lasenby A. & Doran,C. 1993 ‘Imaginary Numbers Are Not Real – The Geometric Algebra Of Space-Time’ Algebra Of Space-Time’ http://www.mrao.cam.ac.uk/~clifford/introduction/intro/intro.html

Lasenby A.N Doran C.J lecture notes 2000-2001 Lasenby A.N Doran C.J lecture notes 2000-2001 http://www.mrao.cam.ac.uk/~clifford/ptIIIcourse/

Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics by David Oersted 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: Hestenes’s Site: http://modelingnts.la.asu.edu/ Lounesto’s Site: Lounesto’s Site: http://www.helsinki.fi/~lounesto/ Cambridge Group: Cambridge Group: http://www.mrao.cam.ac.uk/~clifford/