May 29, 2012 Geometric Algebra Eric Chisolm Abstract This is an introduction to geometric algebra, an alternative to traditional vector algebra that expands on it in two ways: 1. In addition to scalars and vectors, it defines new objects representing subspaces of any dimension. 2. It defines a product that’s strongly motivated by geometry and can be taken between any two objects. For example, the product of two vectors taken in a certain way represents their common plane. This system was invented by William Clifford and is more commonly known as Clifford algebra. It’s actually older than the vector algebra that we use today (due to Gibbs) and includes it as a subset. Over the years, various parts of Clifford algebra have been reinvented independently by many people who found they needed it, often not realizing that all those parts belonged in one system. This suggests that Clifford had the right idea, and that geometric algebra, not the reduced version we use today, deserves to be the standard “vector algebra.” My goal in these notes is to describe geometric algebra from that standpoint and illustrate its usefulness. The notes are work in progress; I’ll keep adding new topics as I learn them myself. Contents 1 Introduction 3 1.1 Motivation ......................................... ..... 3 1.2 Simple applications . ...... 5 1.3 Wherenow?........................................ ...... 8 1.4 Referencesandcomments .. .. .. .. .. .. .. .. .. .. .. .. .......... 9 2 Definitions and axioms 10 3 The contents of a geometric algebra 13 arXiv:1205.5935v1 [math-ph] 27 May 2012 4 The inner, outer, and geometric products 16 4.1 The inner, outer, and geometric products of a vector with anything............... 17 4.2 The general inner product, outer product, and geometric product................ 20 4.3 The geometric meaning of the inner and outer products . .............. 27 5 Other operations 31 5.1 Gradeinvolution ..................................... ...... 31 5.2 Reversion ......................................... ...... 32 5.3 Cliffordconjugation .................................. ....... 34 5.4 Thescalarproduct .................................. ........ 35 5.5 Thedual.......................................... ...... 39 5.6 Thecommutator .................................... ....... 42 1 6 Geometric algebra in Euclidean space 44 6.1 Twodimensionsandcomplexnumbers . ......... 44 6.2 Three dimensions, Pauli matrices, and quaternions . .............. 45 7 More on projections, reflections, and rotations 48 7.1 Orthogonalprojectionsandrejections . .............. 48 7.1.1 Projectingavectorintoasubspace. ......... 48 7.1.2 Projecting a multivector into a subspace . ......... 50 7.2 Reflections........................................ ....... 51 7.2.1 Reflectingavectorinasubspace . ....... 51 7.2.2 Reflecting a multivector in a subspace . ....... 52 7.3 Rotations ......................................... ...... 53 7.3.1 Rotating a multivector in a plane . ..... 54 7.3.2 Rotations in three dimensions . ...... 54 8 Frames and bases 54 8.1 Reciprocalframes.................................. ......... 55 8.2 Multivectorbases................................... ........ 56 8.3 Orthogonalprojectionsusingframes . ............. 60 9 Linear algebra 61 9.1 Preliminaries . ..... 61 9.2 Theadjoint ........................................ ...... 62 9.3 Normaloperators.................................. ......... 64 9.4 Symmetricandskewsymmetricoperators . ............ 65 9.5 Isometries andorthogonaltransformations . ................ 66 9.5.1 Isometriesandversors ............................. ....... 66 9.5.2 Rotorsandbiversors............................... ...... 68 9.6 Extending linear functions to the whole algebra . ............ 70 9.6.1 Outermorphisms ................................... .... 70 9.6.2 Outermorphismextensions . ...... 72 9.7 Eigenblades and invariant subspaces . ........... 75 9.8 Thedeterminant .................................... ....... 76 10 Applications 78 10.1 Classical particle mechanics . .......... 78 10.1.1 Angularmomentumasabivector. ....... 78 10.1.2 TheKeplerproblem ................................. .... 79 A Summary of definitions and formulas 81 A.1 Notation.......................................... ...... 81 A.2 Axioms ............................................ .... 82 A.3 Contentsofageometricalgebra. ........... 83 A.4 Theinner,outer,andgeometricproducts . ............. 83 A.5 The geometric meaning of the inner and outer products . .............. 85 A.6 Gradeinvolution ..................................... ...... 85 A.7 Reversion ......................................... ...... 86 A.8 Cliffordconjugation .................................. ....... 86 A.9 Thescalarproductandnorm . .. .. .. .. .. .. .. .. .. .. .. ......... 87 A.10Thedual......................................... ....... 88 A.11Thecommutator ................................... ........ 88 A.12Framesandbases.................................. ......... 89 A.13Theadjointofalinearoperator. ........... 90 A.14Symmetricandskewsymmetricoperators . ............. 90 2 A.15Isometries and orthogonal transformations . ................. 90 A.16 Eigenvalues, invariant subspaces, and determinants . ................. 91 B Topics for future versions 91 References 92 1. Introduction 1.1. Motivation I’d say the best intuitive definition of a vector is “anything that can be represented by arrows that add head-to-tail.” Such objects have magnitude (how long is the arrow) and direction (which way does it point). Real numbers have two analogous properties: a magnitude (absolute value) and a sign (plus or minus). Higher-dimensional objects in real vector spaces also have these properties: for example, a surface element is a plane with a magnitude (area) and an orientation (clockwise or counterclockwise). If we associate real scalars with zero-dimensional spaces, then we can say that scalars, vectors, planes, etc. have three features in common: 1. An attitude: exactly which subspace is represented. 2. A weight: an amount, or a length, area, volume, etc. 3. An orientation: positive or negative, forward or backward, clockwise or counterclockwise. No matter what the dimension of the space, there are always only two orientations. If spaces of any dimension have these features, and we have algebraic objects representing the zero- and one-dimensional cases, then maybe we could make objects representing the other cases too. This is exactly what geometric algebra gives us; in fact, it goes farther by including all of these objects on equal footing in a single system, in which anything can be added to or multiplied by anything else. I’ll illustrate by starting in three-dimensional Euclidean space. My goal is to create a product of vectors, called the geometric product, which will allow me to build up objects that represent all the higher-dimensional subspaces. Given two vectors u and v, traditional vector algebra lets us perform two operations on them: the dot product (or inner product) and the cross product. The dot product is used to project one vector along another; the projection of v along u is u v P (v)= · u (1) u u 2 | | where u v is the inner product and u 2 = u u is the square of the length of u. The cross product represents the oriented· plane defined by u and| v| ; it points· along the normal to the plane and its direction indicates orientation. This has two limitations: 1. It works only in three dimensions, because only there does every plane have a unique normal. 2. Even where it works, it depends on an arbitrarily chosen convention: whether to use the right or left hand to convert orientations to directions. So the resulting vector does not simply represent the plane itself. Because of this, I’ll replace the cross product with a new object that represents the plane directly, and it will generalize beyond three dimensions as easily as vectors themselves do. I begin with a formal product of vectors uv that obeys the usual rules for multiplication; for example, it’s associative and distributive over addition. Given these rules I can write 1 1 uv = (uv + vu)+ (uv vu). (2) 2 2 − 3 The first term is symmetric and bilinear, just like a generic inner product; therefore I set it equal to the Euclidean inner product, or 1 (uv + vu) := u v. (3) 2 · I can immediately do something interesting with this: notice that u2 = u u = u 2, so the square of any vector is just its squared length. Therefore, the vector · | | u u−1 := (4) u2 is the multiplicative inverse of u, since obviously uu−1 = u2/u2 = 1. So in a certain sense we can divide by vectors. That’s neat. By the way, the projection of v along u from Eq. (1) can now be written P (v) = (v u)u−1. (5) u · In non-Euclidean spaces, some vectors are null, so they aren’t invertible. That means that this projection operator won’t be defined. As it turns out, projection along noninvertible vectors doesn’t make sense geometrically; I’ll explain why in Section 7.1. Thus we come for the first time to a consistent theme in geometric algebra: algebraic properties of objects frequently have direct geometric meaning. What about the second term in Eq. (2)? I call it the outer product or wedge product and represent it with the symbol , so now the geometric product can be decomposed as ∧ uv = u v + u v. (6) · ∧ To get some idea of what u v is, I’ll use the fact that it’s antisymmetric in u and v, while u v is symmetric, to modify Eq. (6) and get ∧ · vu = u v u v. (7) · − ∧ Multiplying these equations together I find uvvu = (u v)2 (u v)2. (8) · − ∧ Now vv = v 2, and the same