PHYSICAL APPLICATIONS OF GEOMETRIC ALGEBRA Chris Doran and Anthony Lasenby COURSE AIMS To introduce Geometric Algebra as a new mathematical (cid:0) technique to add to your existing base as a theoretician or experimentalist. To develop applications of this new technique in the fields (cid:0) of classical mechanics, engineering, relativistic physics and gravitation. To introduce these new techniques through their (cid:0) applications, rather than as purely formal mathematics. To emphasise the generality and portability of geometric (cid:0) algebra through the diversity of applications. To promote a multi-disciplinary view of science. (cid:0) All material related to this course is available from http://www.mrao.cam.ac.uk/ clifford/ptIIIcourse (cid:2) or follow the link Cavendish Research Geometric (cid:3) (cid:3) Algebra Physical Applications of Geometric Algebra 2001. (cid:3) 1 A Q T UICK OUR In the following weeks we will Discover a new, powerful technique for handling rotations (cid:0) in arbitrary dimensions, and analyse the insights this brings to the mathematics of Lorentz transformations. Uncover the links between rotations, bivectors and the (cid:0) structure of the Lie groups which underpin much of modern physics. Learn how to extend the concept of a complex analytic (cid:0) function in 2-d (i.e. a function satisfying the Cauchy-Riemann equations) to arbitrary dimensions, and how this is applied in quantum theory and electromagnetism. Unite all four Maxwell equations into a single equation (cid:0) ( ), and develop new techniques for solving it. (cid:4)(cid:0) (cid:0) (cid:2) Combine many of the preceding ideas to construct a (cid:0) gauge theory of gravitation in (flat) Minkowski spacetime, which is still consistent with General Relativity. Use our new understanding of gravitation to quickly reach (cid:0) advanced applications such as black holes and cosmology. 2 S H OME ISTORY A central problem being tackled in the first part of the 19th Century was how to represent 3-d rotations. 1844 Hamilton introduces his quaternions, which generalize complex numbers. But confusion persists over the status of vectors in his algebra — do constitute the (cid:2)(cid:0)(cid:3) (cid:2)(cid:3) (cid:3)(cid:3) components of a vector? 1844 In a separate development, Grassmann. introduces the exterior product. (See later this lecture.) Largely ignored in his lifetime, his work later gave rise to differential forms and Grassmann (an- ticommuting) variables (used in super- symmetry and superstring theory) 1878 Clifford invents Geometric Algebra by uniting the scalar and exterior products into a single geometric product. This is invertible, so an equation such as has the solution (cid:4)(cid:5) (cid:0) (cid:6) . This is not possible with the separate scalar or (cid:0) (cid:0) (cid:5) (cid:0) (cid:4) (cid:6) exterior products. 3 Clifford could relate his product to the quaternions, and his system should have gone on to dominate mathematical physics. But (cid:7) (cid:7) (cid:7) Clifford died young, at the age of (cid:0) just 33 Vector calculus was heavily pro- (cid:0) moted by Gibbs and rapidly be- came popular, eclipsing Clifford and Grassmann’s work. 1920’s Clifford algebra resurfaces in the theory of quantum spin. In particular the algebra of the Pauli and Dirac matrices became indispensable in quantum theory. But these were treated just as algebras — the geometrical meaning was lost. 1966 David Hestenes recovers the geomet- rical meaning (in 3-d and 4-d respect- ively) underlying the Pauli and Dirac al- gebras. Publishes his results in the book Spacetime Algebra. Hestenes goes on to produce a fully developed geometric calculus. 4 In 1984, Hestenes and Sobczyk publish Clifford Algebra to Geometric Calculus This book describes a unified language for much for mathematics, physics and engineering. This was followed in 1986 by the (much easier!) New Foundations for Classical Mechanics 1990’s Hestenes’ ideas have been slow to catch on, but in Cambridge we now routinely apply geometric algebra to topics as diverse as black holes and cosmology (Astrophysics, Cavendish) (cid:0) quantum tunnelling and quantum field theory (cid:0) (Astrophysics, Cavendish) beam dynamics and buckling (Structures Group, CUED) (cid:0) computer vision (Signal Processing Group, CUED) (cid:0) Exactly the same algebraic system is used throughout. 5 PART 1 GEOMETRIC ALGEBRA IN TWO AND THREE DIMENSIONS LECTURE 1 In this lecture we will introduce the basic ideas behind the mathematics of geometric algebra (abbreviated to GA). The geometric product is motivated by a direct analogy with complex arithmetic, and we will understand the imaginary unit as a geometric entity. Multiplying Vectors - The scalar, complex and quaternion (cid:0) products. The Exterior Product - Encoding the geometry of planes (cid:0) and higher dimensional objects. The Geometric Product - Axioms and basic properties (cid:0) The Geometric Algebra of 2-dimensional space. (cid:0) Complex numbers rediscovered. The algebra of rotations (cid:0) has a particularly simple expression in 2-d, and leads to the identification of complex numbers with GA. 6 VECTOR SPACES Consist of vectors , , with an addition law which is (cid:4) (cid:5) commutative: (cid:4) (cid:4) (cid:5) (cid:0) (cid:5) (cid:4) (cid:4) associative: (cid:4) (cid:4) (cid:2)(cid:5) (cid:4) (cid:8)(cid:3) (cid:0) (cid:2)(cid:4) (cid:4) (cid:5)(cid:3) (cid:4) (cid:8)(cid:7) (cid:5) (cid:5) (cid:8) (cid:4) (cid:4) (cid:5) (cid:5) (cid:4) (cid:8) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:5) (cid:4) (cid:4) (cid:5) (cid:4) (cid:8) (cid:5) For real scalars and vectors and : (cid:9)(cid:3) (cid:10) (cid:4) (cid:5) 1. ; (cid:9)(cid:2)(cid:4) (cid:4) (cid:5)(cid:3) (cid:0) (cid:9)(cid:4) (cid:4) (cid:9)(cid:5) 2. ; (cid:2)(cid:9) (cid:4) (cid:10)(cid:3)(cid:4) (cid:0) (cid:9)(cid:4) (cid:4) (cid:10)(cid:4) 3. ; (cid:2)(cid:9)(cid:10)(cid:3)(cid:4) (cid:0) (cid:9)(cid:2)(cid:10)(cid:4)(cid:3) 4. If for all scalars then for all vectors . (cid:5)(cid:9) (cid:0) (cid:9) (cid:9) (cid:5)(cid:4) (cid:0) (cid:4) (cid:4) NB Two different addition operations. Get familiar concepts of dimension, linearly independent vectors, and basis. Have no rule for multiplying vectors. 7 MULTIPLYING VECTORS In your mathematical training so far, you will have various products for vectors: The Scalar Product The scalar, (or inner or dot) product, returns a scalar from (cid:4)(cid:5)(cid:5) two vectors. In Euclidean space the inner product is positive definite, (cid:2) (cid:6)(cid:4)(cid:6) (cid:0) (cid:4)(cid:5)(cid:4) (cid:11) (cid:6) (cid:7)(cid:4) (cid:8)(cid:0) (cid:6) From this we recover Schwarz inequality (cid:2) (cid:6)(cid:4) (cid:4) (cid:9)(cid:5)(cid:6) (cid:9) (cid:6) (cid:7)(cid:9) (cid:2) (cid:2) (cid:2) (cid:0)(cid:10) (cid:6)(cid:4)(cid:6) (cid:4) (cid:7)(cid:9)(cid:4)(cid:5)(cid:5) (cid:4) (cid:9) (cid:6)(cid:5)(cid:6) (cid:9) (cid:6) (cid:7)(cid:9) (cid:2) (cid:2) (cid:2) (cid:0)(cid:10) (cid:2)(cid:4)(cid:5)(cid:5)(cid:3) (cid:11) (cid:6)(cid:4)(cid:6) (cid:6)(cid:5)(cid:6) We use this to define the cosine of the angle between and (cid:4) (cid:5) via (cid:4)(cid:5)(cid:5) (cid:0) (cid:6)(cid:4)(cid:6)(cid:6)(cid:5)(cid:6) (cid:8)(cid:9)(cid:10)(cid:2)(cid:12)(cid:3) Can now do Euclidean geometry. In non-Euclidean spaces, such as Minkowski spacetime, Schwarz inequality does not hold. Can still introduce an orthonormal frame. Some vectors have squavre and some . (cid:4)(cid:5) (cid:12)(cid:5) 8 C N OMPLEX UMBERS A complex number defines a point on an (cid:13) Argand diagram. Com- (cid:15) plex arithmetic is a way (cid:14) of multiplying together (cid:14) vectors in 2-d. (cid:13) If then get length from (cid:15) (cid:0) (cid:13) (cid:4) (cid:16)(cid:14) (cid:2) (cid:2) (cid:2) (cid:15)(cid:15) (cid:0) (cid:2)(cid:13) (cid:4) (cid:16)(cid:14)(cid:3)(cid:2)(cid:13) (cid:12) (cid:16)(cid:14)(cid:3) (cid:0) (cid:13) (cid:4) (cid:14) Include a second , and form (cid:17) (cid:0) (cid:18) (cid:4) (cid:19)(cid:16) (cid:2) (cid:15)(cid:17) (cid:0) (cid:2)(cid:13) (cid:4) (cid:16)(cid:14)(cid:3)(cid:2)(cid:18) (cid:12) (cid:16)(cid:19)(cid:3) (cid:0) (cid:13)(cid:18) (cid:4) (cid:19)(cid:14) (cid:4) (cid:16)(cid:2)(cid:18)(cid:14) (cid:12) (cid:19)(cid:13)(cid:3)(cid:7) The real part is the scalar product. For imaginary term use polar representation (cid:0)(cid:2) (cid:0)(cid:3) (cid:15) (cid:0) (cid:6)(cid:15)(cid:6) (cid:11) (cid:3) (cid:17) (cid:0) (cid:6)(cid:17)(cid:6) (cid:11) (cid:0) (cid:2) (cid:0)(cid:0)(cid:2) (cid:3)(cid:2) (cid:15)(cid:17) (cid:0) (cid:6)(cid:15)(cid:6)(cid:6)(cid:17)(cid:6) (cid:11) (cid:7) Imaginary part is . The area of the (cid:16)(cid:6)(cid:15)(cid:6)(cid:6)(cid:17)(cid:6) (cid:10)(cid:12)(cid:13)(cid:2)(cid:12) (cid:12) (cid:20)(cid:3) parallelogram with sides and . Sign is related to (cid:15) (cid:17) handedness. Second interpretation for complex addition: a sum between scalars and plane segments. 9 Q UATERNIONS Quaternion algebra contains 4 objects, , (instead (cid:15)(cid:5)(cid:3) (cid:0)(cid:3) (cid:2)(cid:3) (cid:3)(cid:16) of 3). Algebra defined by (cid:2) (cid:2) (cid:2) (cid:0) (cid:0) (cid:2) (cid:0) (cid:3) (cid:0) (cid:0)(cid:2)(cid:3) (cid:0) (cid:12)(cid:5) Define a closed algebra. (Also a division algebra — not so important). Revolutionary idea: elements anticommute (cid:0)(cid:2) (cid:0) (cid:12)(cid:0)(cid:2)(cid:3)(cid:3) (cid:0) (cid:3) (cid:2)(cid:0) (cid:0) (cid:12)(cid:2)(cid:0)(cid:0)(cid:2)(cid:3) (cid:0) (cid:12)(cid:3) (cid:0) (cid:12)(cid:0)(cid:2) Problem: Where are the vectors? Hamilton used ‘pure’ quaternions — no real part. Gives us a new product: (cid:4) (cid:0) (cid:4)(cid:0)(cid:0) (cid:4) (cid:4)(cid:2)(cid:2) (cid:4) (cid:4)(cid:3)(cid:3) (cid:5) (cid:0) (cid:5)(cid:0)(cid:0) (cid:4) (cid:5)(cid:2)(cid:2) (cid:4) (cid:5)(cid:3)(cid:3) Result of product is (cid:4)(cid:5) (cid:0) (cid:8)(cid:4) (cid:4) (cid:6) is (minus) the scalar product. Vector term is (cid:8)(cid:4) (cid:6) (cid:0) (cid:2)(cid:4)(cid:2)(cid:5)(cid:3) (cid:12) (cid:5)(cid:2)(cid:4)(cid:3)(cid:3)(cid:0) (cid:4) (cid:2)(cid:4)(cid:3)(cid:5)(cid:0) (cid:12) (cid:5)(cid:3)(cid:4)(cid:0)(cid:3)(cid:2) (cid:4) (cid:2)(cid:4)(cid:0)(cid:5)(cid:2) (cid:12) (cid:5)(cid:0)(cid:4)(cid:2)(cid:3)(cid:3) Defines the cross product . Perpendicular to the plane (cid:4) (cid:17) (cid:5) of and , magnitude , and , and form a (cid:4) (cid:5) (cid:4)(cid:5) (cid:10)(cid:12)(cid:13)(cid:2)(cid:12)(cid:3) (cid:4) (cid:5) (cid:4) (cid:17) (cid:5) right-handed set. The cross product was widely adopted. 10