New Foundations in Mathematics: The Geometric Concept of Number

Transcription

1 New Foundations in Mathematics: The Geometric Concept of Number by Garret Sobczyk Universidad de Las Americas-P Cholula, Mexico November 2012

2 What is Geometric Algebra? Geometric algebra is the completion of the real number system to include new anticommuting square roots of plus and minus one, each such root representing an orthogonal direction in successively higher dimensions.

3 Contents I. Beyond the Real Numbers. a) Clock arithmetic. b) Modular polynomials and approximation. b) Complex numbers. c) Hyperbolic numbers. II. The Geometric Concept of Number. a) Geometric numbers of the plane. b) Geometric numbers of 3-space. c) Reflections and rotations. d) Geometric numbers of Euclidean Space R n.

4 Contents III. IV. Linear Algebra and Matrices. a) Matrices of geometric numbers. b) Geometric numbers and determinants. c) The spectral decomposition. Splitting Space and Time. a) Minkowski spacetime. b) Spacetime algebra

5 Contents V. Geometric Calculus. a) The vector derivative. VI. VII. b) Fundamental theorem of calculus. Differential Geometry. a) The shape operator. b) The Riemann curvature bivector c) Conformal mappings Non-Euclidean and Projective Geometries a) The affine plane b) Projective geometry c) Conics d) The horosphere

6 CONTENTS Lie groups and Lie algebras. a) Bivector representation b) The general linear group c) Orthogonal Lie groups and algebras d) Semisimple Lie Algebras IX. Conclusions X. Selected References

7 Clock Arithmetic 12 = 3x2 2 Spectral equation: s 1 + s 2 = 1 or 3(s 1 + s 2 ) = 3 s 2 = 3. This implies that 9 s 2 = s 2 = 9, and s 1 = 4. Now define q 2 = 2 s 2 = 6. Spectral basis: { s 1, s 2, q 2 } idempotents: s 1 2 = 16 = 4 mod 12 = s 1 s 2 2 = 81 = 9 mod 12 = s 2 nilpotent: q 2 2 = 36 = 0 mod 12, s 1 s 2 = 0 mod 12

8 Clock Arithmetic: 12 = 3x2 2 A calculation: 5s 1 + 5s 2 = 5mod(12) or 2s 1 + 1s 2 = 5 mod(12). It follows that 2 n s n s 2 = 5 n mod(12) for all integers n. n=-1 gives 1/5 = 2s 1 + 1s 2 = 5 mod(12) and n=100 gives = s 1 + s 2 = 1 mod(12).

9 Modular Polynomials and Interpolation mod(h(x))

10

11 Complex and Hyperpolic Numbers u 2 =1

12 Hyperbolic Numbers

13 : Geometric Numbers G 2 of the Plane Standard Basis of G 2 ={1, e 1, e 2, e 12 }. where i=e 12 is a unit bivector.

14 ab =a. b+a^b a. b=½(ab+ba) a^b=½(ab-ba) a 2 =a. a= a 2 Basic Identities

15 Geometric Numbers of 3-Space a^b=i axb a^b^c=[a. (bxc)]i where i=e 1 e 2 e 3 =e 123 a. (b^c)=(a. b)c-(a. c)b = - ax(bxc).

16 Reflections L(x) and Rotations R(x) where a = b =1 and

17 Geometric Numbers G n of R n Standard basis of the geometric algebra G n of the Euclidean space R n. There are (n:k) basis k-vectors in G n. It follows that the dimension of G n is

18 Matrices of the Geometric Algebra G 2 Recall that G 2 =span{1, e 1, e 2, e 12 }. By the spectral basis of G 2 we mean where are mutually annihiliating idempotents. Note that e 1 u + = u - e 1.

19 For example, if then the element g Ɛ G 2 is We find that

20 Matrices of the Geometric Algebra G 3 We can get the complex Pauli matrices from the matrices of G 2 by noting that e 1 e 2 = i e 3 or e 3 = -i e 1 e 2, where i = e 123 is the unit element of volume of G 3. We get

21 Geometric numbers and determinants Let a 1, a 2,..., a n be vectors in R n, where Then

22 Spectral Decomposition Let with the characteristic polynomial φ(x)=(x-1)x 2. Recall that the spectral basis for this polynomial was Replacing x by the matrix X, and 1 by the identity 3x3 matrix gives

23 It follows that the spectral equation for X is X=1 S S 2 + Q 2, with the eigenvectors We now obtain the Jordan Normal Form for X

24 The ordinary rotation Splitting Space and Time is in the blue plane of the bivector i=e 12. The blue plane is boosted into the yellow plane by with the velocity v/c = Tanh ɸ. The light cone is shown in red.

25 Minkowski Space R 1,3 g 0 is timelike, g 1 g 2 g 3 spacelike

26 Spacetime Algebra G 1,3 We start with We factor e 1, e 2, e 3 into Dirac bivectors, where

27 Geometric Calculus The vector derivative at a point x in R n. Definition: Formulas:

28 Fundamental theorem of calculus. Let M be a k-surface in R n. A point x Ɛ M is given by x=x(s 1,s 2,,s k ) for the coordinates s i Ɛ R. The tangent vectors x i at the point x Ɛ M are defined by and generate the tangent geometric algebra T x at the point x Ɛ M.

29 Classical Integral Theorems A function is monogenic if for all x Ɛ M.

30 Differential Geometry Let M be a k-surface in R n. Define the tangent pseudoscalar I x at x Ɛ R n by the projection operator P x at x Ɛ R n by and the shape operator S(A r ) by

31 The Riemann Curvature Bivector The Riemann curvature bivector R(a^b) is defined by We have the basic relationship is the induced connection on the k-surface M. The classical Riemann curvature tensor is

32 Conformal Mappings Conformal mapping the unit cylinder onto the figure shown.

33 A more exotic conformal mapping of the hyperboloid like figure into the figure surrounding it.

34 Non-Euclidean and Projective The affine plane. Each point x in R n determines a unique point x h in the affine plane. Geometries

35 Desargue s Configuration Thm: Two triangles are in perspective axially if and only if they are in perspective centrally.

36 Any conformal transformation can be represented by an orthogonal transformation on the horosphere. The Horosphere

37

38 Lie Algebras and Lie Groups Let G n,n be the 2 2n -dimensional geometric algebra with neutral signature. The Witte basis consists of two dual null cones: We now construct the matrix of bivectors

39 These bivectors are the generators of the general linear Lie algebra gl n, with the Lie bracket product Each bivector F generates a linear transformation f, defined by

40 General Linear Group The general linear group GL n is obtained from the Lie algebra gl n by exponentiation. We have GL n = { G=e F F Ɛ gl n }. Consider now the one parameter subgroups defined for each G Ɛ gl n by g t (x)=e ½tF x e -½tF where x= x i a i and t Ɛ R. Differentiating gives It follows that

41 Conclusions Since every (finite dimensional) Lie algebra can be embedded in gl n, it follows that every Lie algebra can be represented as a Lie algebra of bivectors. Complex semi-simple Lie algebras are classified by their Dynkin diagrams. Geometric algebra offers new geometric tools for the study of representation theory, differential geometry, and provides a unified algebraic approach to many areas of mathematics. I hope my selection of topics has been sufficiently broad to support my contention that geometric algebra and the Geometric Concept of Number should be viewed as a New Foundation for Mathematics.

42 Selected References R. Ablamowicz, G. Sobczyk, Lectures on Clifford (Geometric) Algebras and Application, Birkhauser, Boston W.K. Clifford, Applications of Grassmann's extensive algebra, Amer. J. of Math. 1 (1878), T. Dantzig, NUMBER: The Language of Science, Fourth Edition, Free Press, P. J. Davis, Interpolation and Approximation, Dover Publications, New York, F.R. Gantmacher, Theory of Matrices, translated by K. A. Hirsch, Chelsea Publishing Co., New York (1959). T.F. Havel, GEOMETRIC ALGEBRA: Parallel Processing for the Mind (Nuclear Engineering) D. Hestenes, New Foundations for Classical Mechanics, 2nd Ed., Kluwer D. Hestenes, Point Groups and Space Groups in Geometric Algebra, In: L. Doerst, C. Doran, J. Lasenby (Eds), Applications of Geometric Algebra with Applications in Computer Science and Engineering, Birkhauser, Boston (2002). p D. Hestenes, Space Time Algebra, Gordon and Breach, 1966.

43 D. Hestenes and G. Sobczyk. Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, 2nd edition, Kluwer P. Lounesto, Clifford Algebras and Spinors, 2nd Edition. Cambridge University Press, Cambridge, P. Lounesto, CLICAL software packet and user manual. Helsinki University of Technology of Mathematics, Research, Report A248, G. Sobczyk, The missing spectral basis in algebra and number theory, The American Mathematical Monthly 108 April 2001, pp G. Sobczyk, Geometric Matrix Algebra, Linear Algebra and its Applications, 429 (2008) G. Sobczyk, Hyperbolic Number Plane, The College Mathematics Journal, Vol. 26, No. 4, pp , September G. Sobczyk, A Complex Gibbs-Heaviside Vector Algebra for Space-time, Acta Physica Polonica, Vol. B12, No.5, , G. Sobczyk, Spacetime Vector Analysis, Physics Letters, 84A, 45-49, G. Sobczyk, Noncommutative extensions of Number: An Introduction to Clifford's Geometric Algebra, Aportaciones Matematicas Comunicaciones}, 11 (1992)

44 G. Sobczyk, Hyperbolic Number Plane, The College Mathematics Journal, 26:4 (1995) G. Sobczyk, The Generalized Spectral Decomposition of a Linear Operator, The College Mathematics Journal, 28:1 (1997) G. Sobczyk, Spectral integral domains in the classroom, APORTACIONES MATEMATICAS, Serie Comunicaciones Vol. 20, (1997) G. Sobczyk, Spacetime vector analysis, Physics Letters, 84A, 45 (1981). G. Sobczyk, New Foundations in Mathematics: The Geometric Concept of Number, San Luis Tehuiloyocan, Mexico J. Pozo and G. Sobczyk. Geometric Algebra in Linear Algebra and Geometry}. Acta Applicandae Mathematicae, 71: , Note: Copies of many of my papers and talks can be found on my website: