Jared R. Stenson and Dr. Jean-Francois S. Van Huele, Physics and Astronomy
In order to progress we are often forced to adopt certain conventions. Just over a century ago, several new scientific ideas were emerging which increased the need to revise our mathematical methods and adopt new standard approaches. The Clifford Algebra was one proposed formalism that met the needs of the day yet, for whatever reason, was left behind in favor of the conventional math we are familiar with. However, as we’ve delved deeper into modern science many are now looking back to this forgotten approach because of its conceptual ease and geometrical structure . Understanding the nature, benefits, and limitations of one of these approaches, the 3-dimensional Clifford Algebra ( 3 Cl ), was the focus of this project. This was done by developing a computational package, CliffAlg3, using MAPLE VII.
Procedures and Problems
At first it was not clear how a new algebra could be constructed. This is achieved through redefining a product. In Cl3 three objects exist (e1,e2,e2) along with a mathematical relation defining the way they multiply. From every possible product of any two of these three “cliffors” a basis set of eight ( 23 ) elements can be formed (written as i e where i = 0,1,..7 and 1 0 e º and e7 º e1 e2 e3) . We then constructed an 8 x 8 table consisting of every possible product of the 8 basis elements with each other. This revealed the structure and symmetry of the 3- dimensional Clifford space. For example, we learned of several interesting parallels and differences between the cliffor 7 e and the imaginary number i that may become important later.
After a short time studying the structure of 3 Cl coding began. We found it useful to define 3 notations for displaying a general cliffor (i.e. a general linear combination of the 8 basis elements). (1) The array notation treats a cliffor as a complex scalar plus a complex 3- dimensional array. Since several of the procedures we have coded convert the cliffor to this notation in order to perform the requested operation it is perhaps most fundamental. (2) The 4- basis notation represents a cliffor as an explicit linear combination of the four basis elements 0 1 2 3 e ,e ,e ,e with complex coefficients. (3) The 8-basis notation displays a cliffor explicitly as a real linear combination of all 8 basis elements. This would keep the imaginary number i and the cliffor 7 e separate should we ever extend our research to Clifford Algebras of other dimensions, such as 4 Cl . MAPLE procedures for converting between these notations were also created. Since the Clifford Algebra is entirely built on the special properties of the Clifford product, as opposed to the conventional vector product, a corresponding procedure was first completed. A Clifford sum for MAPLE was also defined, though it is much less vital. It is useful in consolidating expressions and simplifying some computations. However, in other cases it may only serve to lengthen coding time. The standard addition command in MAPLE, “+”, works for most applications though in some it proves counterintuitive and can lengthen coding time as well.
Combining the Clifford sum with a procedure that takes any positive integer power of a cliffor can be used to approximate any function that is expandable using a power series approximation. In this way we first approximated the exponential function. However, in the literature a more general and exact algorithm for converting a real-valued function to a Clifford-valued function was found . After coding this procedure we corroborated our exponential approximations with exact solutions, and also extended our capabilities to include exact, cliffor-valued functions such as the trigonometric, logarithmic, Bessel, and Legendre functions. The procedure taking only positive integer powers mentioned above was also extended to include all rational powers. Additionally, this algorithm adapts several specific MAPLE commands such as simplify, convert, eval, and expand to operate on cliffors. Many other functions and commands either were or can be generated with only a few variations of this algorithm.
Because of the lack of consistency that comes with young ideas we met some challenges as we checked our results against accepted identities. For example, we found at least 2 separate definitions of the norm in 3 Cl . Also, we noticed that various authors use the term “vector” differently in their approach to 3 Cl . Most problems though were conceptual in nature and did not affect our code. We realize that in many cases no standard has been set and so have attempted to keep the code general, including as many options as possible.
Conclusions and Further Research
As a whole, the MAPLE package CliffAlg3 can accurately perform several useful mathematical operations although progress must still be made. For example, we would like to enable CliffAlg3 to perform symbolic manipulations in a more compact way. As it is now it is most useful as a numerical device. Also, several changes that need to be made are of an aesthetic nature. Once more standards are set the package may be further consolidated. We would also like to extend it to include other functions such as calculus operations and further plotting capabilities. Since the Clifford Algebra is both a natural basis for studying spin and is proposed as the most complete formalism in which to develop certain other interpretations of quantum mechanics further work is anticipated applying CliffAlg3 to study Stern-Gerlach type experiments under the ontological interpretation proposed by David Bohm .
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