Next Monday David Hestenes gives the Oersted Medal Lecture at the AAPT
meeting in Philadelphia. His talk is on GEOMETRIC ALGEBRA, a unified
language for physics that he has developed during the past 40 years.
Geometric algebra could be taught in high schools; it would unify both
these subjects and clarify their use in physics.
Below are the first 2 pages of an article by Dr. Hestenes, on connections
among vectors, spinors, and complex numbers. He's going to talk about this
in Philly. If you must miss the meeting, you can vicariously learn from him
by studying this. If you go, you can get a head start by reading this. I
downloaded it from http://modelingnts.la.asu.edu/- Jane]
Vectors, Spinors, and Complex Numbers in Classical and Quantum Physics
by David Hestenes
(In the American Journal of Physics, Vol. 39/9, 1013-1027, September 1971)
Abstract:
Geometric calculus is shown to unite vectors, spinors, and complex numbers
into a single mathematical system with a comprehensive geometric
significance. The efficacy of this calculus in physical applications is
explicitly demonstrated.
[Note: The terms "multivector algebra" and "multivector calculus"
originally used in this paper have been replaced throughout by the terms
"geometric algebra" and "geometric calculus," which have since become
standard terms. DH]
INTRODUCTION
Among the many alternative mathematical systems in which the equations
of physics can be expressed, two of the most popular are the matrix form of
spinor calculus and the vector calculus formulated by Gibbs. These two
systems in some measure complement one another and are often used together
in problems concerning particles with spin. However, many theorems of
vector calculus are equivalent to theorems of matrix calculus. So when the
two systems are combined a great deal of redundancy occurs which makes
computations more difficult than necessary, much time being taken up with
mere translation between the two modes of expression. This paper shows how
the matrix and vector algebra can be replaced by a single mathematical
system, called geometric algebra, with which the tasks of theoretical
physics can be carried out more efficiently.
Geometric algebra derives its power from the fact that both the
elements and the operations of the algebra are subject to direct
geometrical interpretation. It can be argued, further, that geometric
algebra is the simplest system capable of providing a complete algebraic
expression of geometric concepts, though only some examples of the
effectiveness of the system are given in this paper. Additional perspective
on the cogency of geometric algebra can be gained by comparing it with
other approaches. For example, it is important to note that covariant
formulations of tensor and spinor algebra are by nature inefficient,
because at the very beginning they introduce irrelevant coordinates. The
coordinates function as a vehicle for the algebraic system. The actual
irrelevancy of the coordinates is finally admitted by requiring that
quantities which represent intrinsic geometric (and physical) entities be
covariant under coordinate transformations. In contrast, geometric algebra
is built out of objects with direct geometric interpretations; the
properties of these objects are specified by introducing algebraic
operations which directly determine their interrelations. Coordinates are
utilized only when they arise naturally in specific problems. Moreover,
geometric algebra retains all the advantages of tensor algebra, because
tensors can be introduced in a coordinate free fashion as multilinear
functions on geometric algebra; the operations of tensor algebra are then
already included in the operations of geometric algebra. However, a
systematic account of this approach to tensors has not yet been published.
[But see more recent references on the web site.]
Though the geometric algebra discussed here is isomorphic to the
so-called "Pauli (matrix) algebra," the interpretations of the two systems
differ considerably, and the practical consequences of this difference are
not trivial. Thus, questions of the representation of Pauli matrices and of
transformations among representations never arise in geometric algebra,
because they are irrelevant. Matrix algebra was invented to describe linear
transformations. So it should not be surprising to find that matrices have
irrelevant features when they are used to represent objects of a different
nature. From the geometric viewpoint of geometric algebra, matrices are
seen to arise in the theory of linear geometric functions; that is,
geometrics are taken to be more fundamental than matrices, rather than the
other way around. Simplifications which result from this reversal of
viewpoint are manifest in text and references of this paper.
Just as important as the mathematical simplifications that accrue is
the fact that the use of geometric algebra imbues many well known
mathematical expressions with new meaning. Thus, complex numbers arise
naturally with distinctive geometrical and physical interpretations
depending on the situation. Of particular significance is the realization
of a connection between spin and complex numbers in quantum theory and the
surprising conclusion that the Schrodinger theory already describes a
particle with spin. This will be discussed thoroughly in a subsequent paper
(hereinafter referred to as II[1]). [Also discussed in many other papers on
this web site.]
The purpose of this paper is to explain some of the advantages of
geometric calculus. The object is not to give a complete and balanced
account of the subject, but only to call attention to its major features
and some of the minor points that are often overlooked. ...
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Jane Jackson, Co-Director, Modeling Instruction Program
Box 871504, Dept.of Physics & Astronomy,ASU,Tempe,AZ 85287
480-965-8438/fax:965-7331 <http://modeling.asu.edu>
Genius must transform the world, that the world may produce more genius.