TRACING THE HISTORY OF "CLIFFORD ALGEBRA"

In his book, Eddington said, "I cannot believe that anything so ugly as multiplication of matrices is an essemtial part of the scheme of nature."

(Mathematical physicist, N. Salingaros, says: "Eddington apparently defined the real Majorana [spinor] matrices (actually an equivalent set) before Majorana; he introduced the now standard g⁴ notation, and also the concept of chirality; he investigated the group structure of the Dirac algebra; and he defined the complex Clifford algebra over an eight-dimensional space which appears today in formulations of supersymmetry and supergravity.")

However, it seemed that no Columbia math or physics prof could tell me anything about Clifford Algebra. But then I was told that the French-American mathematician, Claude Chevalley, was writing a paper on Clifford Algebra. However, when I innocently asked about "an elementary introduction to Clifford Algebra", Chevalley began spluttering and almost had an "attack". (Apparently there was no elementary introduction to Clifford Algebrta until I developed one.)

I searched for leads on this for 30 years, writing many many letters to mathematicians who should know.

I didn't realize, from reading Eddington, that Clifford united the work of W. R. Hamilton and Hermann Grassmann. At age 17, I had read about Hamilton in Men of Mathematics by Eric Temple Bell (1883-1960). Bell devotes a chapter to Hamilton and praises his early work, but makes him appear to be a drunken recluse who fudged something called "quaternions". This prejudiced me (and others) and diverted my search until the following occurred.

In 1981, I learned that a French mathematician, M. Riesz, (following the lead of physicist, Arnold Sommerfeld (1868-1951)) had lectured on Clifford Algebra in 1958 at the University of Maryland at College Park. Then I learned that an American physicist, David Hestenes (now at Arizona State University), had learned from Riesz and started writing about Clifford Algebra.

A second opinion on the importance of this subject was given by a Cambridge U. crystallographer, Simon L. Altman, published (in a Mathematical Association of America journal, Mathematics Magazine, December, 1989), "Hamilton, Grassmann, Rodrigues, and the Quaternion Scandal -- What went wrong with one of the major mathematical discoveries of the nineteenth century?" Unfortunately, Altman does not mention Clifford, but does describe the basic mathematics.

Before William Clifford (1845-1879), we have:

• Portuguese-French banker and "recreational" mathematician, Olinde Rodrigues (1794-1851).
• German Sanskrit scholar and "recreational" mathematician, Hermann Grassmann (1809-1877).
• Irish mathematician, Sir William Rowan Hamilton (1805-1865).

I found reference to Rodrigues in Rotations, Quaternions, and Double Groups, S. L. Altman, Clarendon Press, 1986. Altman is an Oxford University crystallographer, who taught rotational theory for a decade before writing this book. On p. vii, "... rotation operators are often obtained as by-products of the angular momentum operators in quantum mechanics. Partly as a result of this approach, rotations are then parametrized by means of the familiar Euler angles, which suffer from three defects: they are not always unique, they are very cumbersome to determine in the finite rotation groups (point groups), and they do not provide a scheme for the multiplication of rotations. An entirely different approach to rotations is possible, which was introduced by Olinde Rodrigues in 1840 but which has never been used. The rotation operators in this approach are obtained by an entirely geometric method, which ... leads most naturally to the parametrization of rotations by parameters that coincide with quaternions. These parameters are unique, exceedingly easy to determine, and -- because they are quaternions -- they provide an algebra that permits the multiplication of rotations in a simple way. At the same time, and most importantly, these parameters determine unambiguously the phase factors that appear in the angular momentum representations for half-integral quantum numbers. [Quaternions discovered independently, and spinors in 1840!] This result leads to a rigorous formulation of the representations of the rotation group, either as projective representations or by means of double groups."

There is so little about Rodrigues in The Literature that many misconceptions are said about him. Elie Cartan (1869-1951), who is credited with discovering the spinor, invented a nonexistent collaborator for Rodrigues by the name "Olinde" (Rogridgues' middle name), a mistake repeated by Temple. Others misspelled his name as "Rodrigue" and "Rodrigues".

Altman refers to the familiar "Euler parameters for rotation" as "the Euler-Rodrigues parameters".

Rodrigues became the patron and financial supporter of Count Louis Saint-Simon (1760-1825), founder of French Socialism. After the death of Saint-Simon, Rodrigues became head of The Socialist Party.

So religious and ethnic discrimination, political discrimination, and the discrimination of Mainstream mathematicians contributed to the present obscurity of this creative man.

Says Hestenes: "Hermann Grassmann completed the algebraic formulation of basic ideas in Greek Geometry begun by Descartes. The Greek theory of ratio and proportion is now incorporated in the properties of scalar and vector multiplication. The Greek idea of projection is incorporated in the inner product. And the Greek geometrical product is expressed by outer multiplication. ....Only in ... Grassman's outer product is it possible to understand that the careful Greek distinction between number and magnitude as real geometric signigicance ... correspond[ing] roughly to the distinction between a scalar and a vector....Only in the work of Grassmann are the notions of direction, dimension, orientation and scalar magnitude finally disentangled ... impossible without the earlier vague distinction of the Greeks and perhaps without its reformulation in quasi-aruthmetic terms by his father.... Grassmann was the first to define multiplication simply by specifying a set of algebraic rules."

In the almost unreadable Ausdehninglehre (Calculus of Extension), Grassmann built "a calculus of extensive magnitudes", a kind of vector algebraization of geometry, beginning with vector sum and difference. It's been said that Grassmann created many different kinds of vector products, but the notable ones are interior product (inner product), exterior product (related to outer product) and, late in life, a multiproduct, as sum of these two product to recapture Hamilton's quaternions. Today, in differential forms, interior product and exterior product reside in different systems, bridged by an recondite transformation -- in stark contrast to their complementary relation in Clifford Algebra, and my own derivation of outer product from inner product and derivation of multiproduct from both.

The difference between a Clifford Algebra and what is today "a Grassmann Algebra" is articulated by a multiproduct added to its reversion:

• Grassmann: d₁ Äd₂ + d₁ Äd₂ = 0. (This Grassmann product is used in quantum theory mathematics, for "fermionic integrals".)

Hestenes says: "It is fair to say, for example, that Grassmann laid the foundations for linear algebra .... Yet the standard linear algebra grew up without his contribution. The frequent duplication of Grassmann's discoveries is not a mark of limited originality but rather a sign that Grassmann was keenly attuned to a powerful thematic force driving mathematical development, namely, the subtle interplay between geometry and algebra. What sets Grassmann ahead of other creative mathematicians is his systemic vision of a universal geometric calculus. This marks him as one of the great conceptual synthesizers of all time."

The American combinatorist, Gian-Carlo Rota has also commented upon another misunderstanding of Grassmann's work: "With the rise of functional analysis, another dogma was making headway namely, the distinction between a vector spaceV and its dual, V^*, and the pairing of the two viewed as a bilinear form. .... Grassmann's idea was to develop a calculus for the join and meet of linear varieties in projective space, a calculus that is actually realized by the progressive [outer] and the regressive [inner] products....[T]he dual space V^* of a vector space V [is] a hyperplane ... living in V^*] with a linear functional is a step backwards in clarity." Grassmann's exterior product was also discovered by Saint-Venant in 1845; by ?. O'Brien in 1846; and by Augustin Cauchy in 1853.

When young, Hamilton (1805-1965) found an interpretation of mechanics in terms of optics, which prepared the way for modern quantum theory. His reformulation of Lagrangian theory in analytical mechanics became the favorite treatment in quantum mathematics, until Feynmann's Lagrangian approach. And, elsewhere, I've noted that Hamilton invented the word "vector", applying it to his reformulation of complex numbers as vectors of real numbers, and his extension of this to quaternions.

Hamilton knew that the complex number modeled rotation in 2-D, so he sought and extension of complex nunbers to model rotation in 3-D. Since the complex number adjoined a new unit, i = Ö ¯1, Hamilton considered adjoining another new unit, j = Ö¯1.

The story of how Hamilton struggled with this for years has been told over and over. Only recently, in A History of Algebra by B. L. van der Waerden, did I learn the difficulty. Hamilton knew that the complex number satisfied a "law of moduli", which van der Waerden states in what I consider a confusing way: "the length of the product equals the product of the lengths of the factors". I believe my definition is better: the binary product of sums of squares equal a sum of squares.

Consider the product coefficients in the product (a + bi)(c + di) = ac + adi + bci - bd = (ac - bd) + (ad + bc)i. Those coefficients are (ac - bd), (ad + bc). Now, consider the sum of their squares: (ac - bd)² + (ad + bc)² = a²c² - 2abcd + b² d² + a²d² + 2abcd + b²c² = a ²c² + b²d² + a²d² + b² c² = a²c² + a²d² + b²c² + b²d² = a²(c² + d²) + b² (c² + d²) = (a² + b²)(c² + d² ). That is, (ac - bd)² + (ad + bc)² = (a² + b²)(c² + d²). This last is "the law of the moduli". What does that mean?

The term (a² + b²)^1/2 is The Modulus of the complex number, a + bi. And (c² + d²)^1/2 is The Modulus of the complex number, c + di. So "the law" says "the product of thesquares of the moduli equals the sum of squares of the product terms.

This "law" failed for Hamilton in trying to use a hypercomplex number of the form, u + vi + wj. The problem in multiplying two such numbers involved the product term ij. If he set it to zero, the moduli law failed, and he tried various tricks regarding this product. Each morning, his little son would ask: "Well, Papa, can you multiply triplets?" And Hamilton had to reply, "No, I can only add and subtract them." Actually, he could multiply them, but the product did not fit "the law of the moduli" and, wihout this, Hamilton considered "the whole enterprise a failure".

The fundamental formula, i² = j² = k² = ijk = -1 of the theory of quaternions came to Hamilton as he was walking with his wife from Dunsink Observatory to Dublin along the Royal Canal on 16th October, 1843. He carved this formula on a stone of Broome Bridge, where it was seen for decades -- perhaps even today. (Quaternions and the Irish Revolution.)

However, any one reading Hamilton's papers on this might be confused with the fact that Hamilton spent much time -- apart from the algebra and geometry of the problem -- playing around with squares of numbers. Why?

In A History of Algebra, van der Waerden gives us some clues, but does not completely explicate the matter. It happened that the great Swiss mathematician, Leonhard Euler (1707-1783) had, in 1748, found a pattern similar to the Law of Moduli in finding that every integer can be written as the sum of four squares of integers. So, this motivated Hamilton to work with squares of integers. But the French mathematician, Adrien Marie Legendre (1752-1833) had showed, in 1830, that you cannot, in general, rite an integer as sum of three integral squares. Had Hamilton known this, he might not have wasted years on the "triplet" problem.

Withal, Hamilton developed inner product, outer product, multiproduct of multivectors, but did little with the multiproduct.

Educated at Trinity College of Cambridge University, Clifford, at age 26, was appointed Professor of Appled Mathematics and Mechanics at University College, London, on the recommendation of the great William Clerk Maxwell (1831-79). (University College was the first in Great Britain to admit women students.) The Chairman of the Mathematics Department, Joseph L. Sylvester (1814-97) (once math tutor of teen-age Florence Nightingale), had especial praise for a paper of Clifford which anticipated the modern field of geometrodynamics, notably beginning with the notion of Einstein of space curved by gravity, and the ideas of Wheeler and Misner about "worm-holes" in space. Clifford wrote: "I hold in fact:
"(1)That small portions of space are in fact of a nature analogous to litle hills on the surface which is on the average flat: namely, that the ordinary laws of geometry are not valid to them.
"(2)That this property of being curved or distorted is continually being passed on 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.
"(4)That in the physical world nothing else takes place but this variation, subject (possibly) to the laws of continuity."

Clifford gave a remarkable testimony to the notion ofstewardship in his essay, "The ethics of personal belief":
"No one man's belief is a private matter which concerns him alone....Our words, our phrases, our forms and processes and modes of thought, are common property, fashioned and perfected from age to age; an heirloom which every succeeding generation as a precious deposit and a sacred trust to be handed on to the next one, not changed but enlarged and purified. Into this, for good or ill, is woven every belief of every man who has speech of his fellows. An awful priviled\ge and an awful responsibility that we should help to create the world in which posterity will live."

Hestenes says: "Clifford may have been the first person to [realize] ... that two different interpretations of number can be distinguished, the quantitative and the operational ... number as a measure of 'how much' or 'how many' of something ... [contrasted with] a relation between different quantities.... Interpreted quantitatively, [the unit bivector] i is a measure of directed area. Operationally interpreted, i specifies a rotation in the i-plane. Clifford observed that Grassmann developed the idea of directed number from the quantitative point of view, while Hamilton emphasized the operational interpretation. The two approaches are brought together by the geometric product [multiproduct]....[V]ectors are usually interpreted quantitatively, while spinors are usually interpreted operationally."

Clifford wrote two papers setting forth his ideas, but died prematurely of tuberculosis, leaving a young wife and two tiny girls.

The development soon after of the Gibbs-Heaviside vector algebra and analysis diverted attention from his work until Riesz and Hestenes revived it.

Then, with the Polish mathematician, G. Sobczyk, Hestenes wrote, Clifford Algebra to Geometric Calculus, in which Linear Algebra and Set Theory and Analysis and Differential Forms are derived in Clifford Algebra.

In 1986 appeared a collection of articles by various mathematicians and physicists, Clifford Algebras and Their Applications in Mathematical Physics, Eds. J. S. R. Chisholm and R. K. Common, from a 1985 NATO and SERC Workshop at the University of Canterbury, Kent, England.

In 1985 appeared Hestenes' New Foundations for Classical Mechanics, showing how Rotational Mechanics is facilitated by Clifford Algebra and Planetary Mechanics by Spinors. (Personal note.)

At that time there was promise of an extension, New Foundations of Mathematical Physics, to articulate further what Hestenes first did in Space-Time Algebra, but it has never appeared.

If you search ONLINE, say via Google, you will find many sites devoted to "Clifford Algebra".

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