Geometric Algebra and its Application to Mathematical Physics potx

[...]... of its utility and power 1.3 Linear Algebra We have illustrated a number of the properties of geometric algebra, and have given explicit constructions in two, three and four dimensions This introduction to the properties of geometric algebra is now concluded by developing an approach to the study of linear functions and non-orthonormal frames 1.3.1 Linear Functions and the Outermorphism Geometric algebra. .. representation theory of nite Clifford algebras 13, 48] It is also the usual route by which Cli ord algebras enter particle physics, though there the f ig are thought of as operators, and not as orthonormal vectors The geometric algebra we have de ned is associative and any associative algebra can be represented as a matrix algebra, so why not de ne a geometric algebra as a matrix algebra? There are a number... (volumes) with vectors perpendicular to it The trivector 1^ 2^ 3 commutes with all vectors, and hence with all multivectors The trivector (pseudoscalar) 1 2 3 also has the algebraic property of squaring to ;1 In fact, of the eight geometrical objects, four have negative square, f 1 2, 2 3, 3 1g and 1 2 3 Of these, the pseudoscalar 1 2 3 is distinguished by its commutation properties and in view of these... (1:63) 16 c α scalar a a vector line segment b bivector plane segment a b trivector volume segment Figure 1.1: Pictorial representation of the elements of the Pauli algebra The quaternion algebra sits neatly inside the geometric algebra of space and, seen in this way, the i, j and k do indeed generate 90 rotations in three orthogonal directions Unsurprisingly, this algebra proves to be ideal for representing... that geometric algebra, built on the framework of Cli ord algebra, provides a uni ed language for much of modern mathematics The work in this thesis is intended to o er support for Hestenes' ideas 1.2 Axioms and De nitions The remaining sections of this chapter form an introduction to geometric algebra and to the conventions adopted in this thesis Further details can be found in \Cli ord algebra to geometric. .. in two, three and four dimensions has shown how geometric algebra naturally encompasses the more restricted algebraic systems of complex and quaternionic numbers It should also be clear from the preceding section that geometric algebra encompasses both matrix and tensor algebra The following three chapters are investigations into how geometric algebra encompasses a number of further algebraic systems... Furthermore, exponentials of bivectors provide a very general method for handling rotations in geometric algebra, as is shown in Chapter 3 1.2.3 The Geometric Algebra of Space If we now add a third orthonormal vector geometric objects: 1 scalar f 1 2 3 g f 3 vectors 1 2 3 to our basis set, we generate the following 2 3 3 1 3 bivectors area elements g 1 2 3: trivector volume element (1:56) From these... this algebra are shared with the 2-dimensional case since the subsets f 1 2g, f 2 3g and f 3 1g generate 2-dimensional subalgebras The new geometric products to consider are ( ( 1 2) 3 1 2 3) k = = 1 2 3 k ( 1 2 3) (1.57) and 2 ( 1 2 3)2 = 1 2 3 1 2 3 = 1 2 1 2 3 = ;1: (1:58) These relations lead to new geometric insights: A simple bivector rotates vectors in its own plane by 90 , but forms trivectors... a b + a ^ b We call this the geometric product It has the following two properties: Parallel vectors (e.g a and a) commute, and the the geometric product of parallel vectors is a scalar Such a product is used, for example, when nding the length of a vector Perpendicular vectors (a, b where a b = 0) anticommute, and the geometric product of perpendicular vectors is a bivector This is a directed plane... sweeping round from the a direction to the b direction Examples of bivectors include angular momentum and any other object that is usually represented as an \axial" vector 4 trivectors have simply a handedness and a magnitude The handedness tells whether the vectors in the product a^b^c form a left-handed or right-handed set Examples include the scalar triple product and, more generally, alternating tensors