Geometric Algebra and Applications to Physics doc

[...]... interpretation to all mathematical elements of the algebra This intrusion of geometric consideration into the abstract system of Clifford algebra has enriched geometric algebra as a powerful mathematical theory Geometric algebra is, in fact, the largest possible associative division algebra that integrates all algebraic systems (viz., algebra of complex numbers, vector algebra, matrix algebra, quaternion algebra, ... we call the multivector of any grade a simple multivector to distinguish it from the generic multivector consisting of four parts: scalar, vector, bivector, and trivector 1.4 Geometric Algebra as a Symbolic System Mathematical objects of geometric algebra have one kind of addition rule, different from Gibbs’ vector algebra, and one general kind of multiplicative rule, known as the geometric product The... in an abstract manner, and Descartes accepted the existence of similar rules for manipulating line segments and greatly improved symbolism and algebraic P1: Binaya Dash October 24, 2006 8 14:30 C7729 C7729˙C001 Geometric Algebra and Applications to Physics technique Thus, it seemed that numbers might be put into one -to- one correspondence with points on a geometric line, leading to a significant step in... Grassmann’s algebra of extension and Hamilton’s quaternion algebra by introducing a new type of product ab of two proper (non-zero) vectors, called geometric product He constructed a powerful algebraic system, now popularly known 3 P1: Binaya Dash October 24, 2006 4 14:30 C7729 C7729˙C001 Geometric Algebra and Applications to Physics as Clifford algebra, in which vectors are equipped with a single associative... we are in a position to extract the inner and outer product of a vector a and a bivector A = b ∧ c from the geometric product a A by using the associative rule (1.15) and noting that a · A and a ∧ A must have opposite symmetries, i.e., a ∧ A = A∧ a, a · A = −A · a (1.16a) (1.16b) P1: Binaya Dash October 24, 2006 14:30 C7729 C7729˙C001 16 Geometric Algebra and Applications to Physics The anticommutability... 72 6 Spinor and Quaternion Algebra 75 Spinor Algebra: Quaternion Algebra 75 Vector Algebra 77 Clifford Algebra: Grand Synthesis of Algebra of Grassmann and Hamilton and the Geometric Algebra of Hestenes ... multivectors Thus scalars are termed as multivectors of grade 0, vectors as multivectors of grade 1, bivectors as multivectors of grade 2, trivectors as multivectors of grade 3, etc A volume-like object having magnitude as well as a choice of handedness is graphically represented by an oriented parallelepiped with handedness defined by three vectors a, b, and c, and mathematically represented by trivector... three vectors a , b, and c with the head of a attached to ¯ ¯ ¯ ¯ and with the head of b attached to the tail of c , and math¯ the tail of b ¯ ematically represented by trivectors a ∧ b ∧ c The order of vectors ¯ ¯ ¯ in a ∧ b ∧ c determines the handedness and the sign of the oriented ¯ ¯ ¯ parallelopiped A trivector represents the essential abstraction of volume orientation with handedness and magnitude... of M In geometric algebra for three-dimensional space, unit multivector may be scalar, vector, bivector, or trivector We take any set of three orthonormal vectors as a basis for vectors The three mutually orthogonal unit bivectors constructed out of three orthonormal basis vectors are taken as a basis for bivectors There is only one unit scalar 1 Also, there is only one unit trivector, equal to the... equal to the product P1: Binaya Dash October 24, 2006 14:30 C7729 C7729˙C001 12 Geometric Algebra and Applications to Physics a a scalar grade = 0 vector grade = 1 a b a b bivector grade = 2 c _ a b a b c trivector grade = 3 FIGURE 1.1 Four mathematical elements of the geometric algebra for three-dimensional space are represented graphically of the three orthonormal vectors considered because there is only . textbook on geometric algebra with applications to physics and serves also as an introduction to geometric algebra intended for research workers in physics. of geometric algebra to problems central to the quantization of gravity. Spin and torsion play key roles here, and the thought emerges that geometric algebra