thesis Geometric intuition

Richard Feynman once made a statement represent the consequences of rotations in to the effect that the history of mathematics One day, perhaps, 3-space faithfully. is largely the history of improvements in Clifford’s algebra will It is also completely natural not only to add notation — the progressive invention of or subtract multi-vectors, but to multiply or ever more efficient means for describing be taught routinely to divide them — something not possible with logical relationships and making them students in place of ordinary vectors. The result is always another easier to grasp and manipulate. The Romans multi-vector. In the particular case of a were stymied in their efforts to advance vector analysis. multi-vector that is an ordinary vector V, the mathematics by the clumsiness of Roman inverse turns out to be V/v2, where v2 is the numerals for arithmetic calculations. After to j, that is, eiej = −ejei. Another way to put squared magnitude of V. It’s a vector in the Euclid, geometry stagnated for nearly it is that multiplication between parallel same direction but of reciprocal magnitude. 2,000 years until Descartes invented a new vectors is commutative, whereas it is anti- Write out the components for the notation with his coordinates, which made commutative for orthogonal vectors. product of two vectors U and V, and you it easy to represent points and lines in These rules are enough to define find the resultUV = U•V + ĬU × V, with • space algebraically. the algebra, and it’s then easy to work and × being the usual dot and cross product Feynman himself, of course, introduced out various implications. For example, of vector analysis. Hence, geometric algebra 2 2 2 into physics a profound change in notation (e1e2) = (e2e3) = (e3e1) = −1. Something blends both operations in a natural way. with his space–time diagrams for quantum like e1e2 is called a bi-vector, but isn’t a For nearly 40 years, physicist field theory. Previously, writing out the vector at all; rather it is a novel thing in its David Hestenes of Arizona State University terms in an infinite series for a probability own right. Similarly, e1e2e3 also isn’t a vector, has waged a one-man crusade to advertise amplitude involved a laborious algebraic or a bi-vector, but a tri-vector, another Clifford’s geometric algebra and to lift it procedure, which Feynman replaced totally new thing, the square of which also up to what he sees as its rightful place in with simple pictures and explicit rules comes to −1. Within this algebra, the most physics. It hasn’t worked yet. The standard to translate them into mathematical general object is a multi-vector — the sum techniques of vector analysis as originally expressions. This was an advance in of a scalar, vector, bi-vector and tri-vector. introduced by Gibbs remain dominant housekeeping, if you will, but also among In a sense, this is an advance over Descartes instead, which is too bad. the most important advances in twentieth- in that it provides a way to combine lines, Maxwell’s equations in vector notation century mathematical physics. areas and volumes within one formalism. are often cited as a prime example of the However, one of the most important The resulting algebra has remarkable beauty of physics, but the elegance is only and elegant advances in mathematical richness within it. The bi-vectorse 1e2, e2e3, enhanced in geometric algebra. It’s natural notation has perhaps not yet achieved the e3e1, for example, can be thought of as to combine the electric and magnetic fields wide recognition it deserves. In 1873, the oriented areas. They are linked to rotations into one field quantity: F = E + ĬcB. The full

English mathematician and philosopher respectively about the e3, e1, e2 axes, and act equations then take the simple form ∇F = J William Clifford invented a deceptively identically to the basis elements of William where ∇ is the four-gradient 1/c ∂t + ∂r and J simple algebraic system unifying Cartesian Rowan’s quaternions, which he introduced the four-current 1/ε0 ρ – cμ0J. By combining coordinates with complex numbers, and in 1843 in an attempt to generalize complex the dot and cross products, Maxwell’s four offering a compact representation of lines, numbers to three dimensions. The tri- equations collapse into one (of course, this areas and volumes, as well as rotations, vector — for simplicity, we can— denote it as can also be achieved in tensor notation). in 3-space. In more advanced physics, Ĭ — acts analogously to i = √−1; it’s square The improvement is even more startling for Clifford’s algebra — he called it ‘geometric is −1 and it commutes with all the basis the Dirac equation, which actually takes the algebra’ — is now well recognized as the vectors. Using this shorthand, the bi-vectors form of a simple generalization of Maxwell’s natural algebra for describing physics and tri-vectors together satisfy the Pauli equations in which the fieldF becomes a full in 3-space, but it hasn’t yet caught on in algebra eiej = Ĭεijkek central to the description multi-vector. This and other examples are engineering, or even in standard treatments of rotations in three dimensions (here k is explored in more detail in a short review of of electricity and magnetism or fluid summed over, and ε123 = 1 changes sign for geometric algebra (J. M. Chapell et al., http:// dynamics, where vector analysis with its any permutation of indices, and vanishes if arxiv.org/abs/1101.3619; 2011), and Hestenes ugly cross product still holds sway. any two are equal). has created a wide variety of introductory Clifford’s geometric algebra begins For example, the rotation of any materials (http://geocalc.clas.asu.edu). with the three coordinate vectors e1, e2, vector about the e3 axis is generated by One day, perhaps, Clifford’s geometric e3 inherited from Descartes for the three multiplying the vector from the left by algebra will be taught routinely to students independent directions in space. These e2e1; this bi-vector is a ‘rotor’ that acts as in place of vector analysis. It would probably satisfy the usual rules of orthonormality, an operator generating a rotation through eliminate a great deal of confusion, and ei • ej = δij,; they are mutually perpendicular the arc defined ase 1 sweeps through to improve the geometric intuition of many and of unit length. Clifford then introduced e2. For any two unit vectors ea and eb, ebea practising scientists. ❐ another kind of multiplication between generates a similar rotation in the plane vectors, denoted as eiej. His key point was defined by the two vectors. Of course, MARK BUCHANAN to assume that this kind of multiplication these rotations satisfy a non-commutative would be anti-commutative for i not equal algebra as must be true if they are to Corrected online: 28 October 2011

442 NATURE PHYSICS | VOL 7 | JUNE 2011 | www.nature.com/naturephysics

© 2011 Macmillan Publishers Limited. All rights reserved Correction In the Thesis article, ‘Geometric intuition’ (Nature Phys. 7, 442; 2011), in the final equation quoted in the article, the term 1/c should instead have been 1/ε0. This error has been rectified in the HTML and PDF versions.

© 2011 Macmillan Publishers Limited. All rights reserved.