Why does “real” physics need “imaginary” numbers?

William M. Pezzaglia Jr.

Department of PhysicsSanta Clara UniversitySanta Clara, CA 95053

mailto: wpezzag@clifford.org

This is a general overview talk that would be of general interest to faculty and jr/srstudents of either math or physics. It will help me gauge what you already know and don't. In particular it will give an overview of quaternions, octonions, grassmann and clifford algebras and applications compared to standard vectors, matrices and spinors, as well as introducing the new concepts of polydimensional physics.

Cal Poly Pomona, June 30-July 1, 2001, Lecture 1http://www.clifford.org/~wpezzag/talks.html

II.A. Complex Numbers, Algebra & Analysis Review the evolution from R -> C

1. An Imaginary Tale: The History of "i"

An evolution of the concept of "number"

a. Real Numbers Represent Things

b. Non-Physical Solutions to Real Equations

c. Geometric Interpretation of "i"

30,000 BC Cardinal Numbers Month has 29 or 30 days, Year 12 or 13 months3000 BC Rational Numbers Babylonians divide circle into parts of 60572 BC Irrational Numbers Pythagorean Triangle

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250 AD Diophantes rejects negative numbers as "less than nothing"

1545 Cardan's equation has "sophistic roots" even when cubic roots are real

1637 Descartes: "imaginary" number as impossibility of geometric construction

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1655? John Wallis gives a geometric construction for "i"

1793 Casper Wessel: Vectors in 2D, "i" rotates by 90 degrees

1806 Argand rediscovers the 2D concept, Modulus of complex number

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II.C. Octonions, rebels without a cause

Arthur Cayley

1821-1895

As for everything else, so for a mathematical theory: beauty can be perceived but not explained.

o 1845 Octonions o 1857 Matrix Algebra o 1858 Quaternions represented by Matrices

1. Cayley Numbers (Octonions)

(a) The Octaves

o 7 Imaginaries

o Commuting Identity

o Elements Anticommute

o ei ej + ej ei =-2δij

(b) Multiplication Table o Any two elements generate a quaternion group o 480 different permutations of multiplication tableo Heptagon Method: ea eb = ec

b = a+2n, c = a-2

n+1,

indices 1 to 7 mod 7

William Kingdom Clifford (1845-1879)

I hold in fact:

1) That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them.

2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave.

3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or etherial

4) That in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity.

“On the Space-Theory of Matter”Proceedings of the Cambridge Philosophical Society (1876)