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Concluding Remarks

Most of the above is well known - the vast majority of the theorems presented date back at least a hundred years. The trouble, of course, is that these facts, whilst `known', were not known by the right people at the right time, so that an appalling amount of reinvention and duplication took place as physics and mathematics advanced. Spinors have a central role in our understanding of the algebra of space, and they have accordingly been reinvented more often than anything else. Each reincarnation emphasises different aspects and uses different notation, adding new storeys to our mathematical Tower of Babel. What we have tried to show in this introductory paper is that the geometric algebra of David Hestenes provides the best framework by which to unify these disparate approaches. It is our earnest hope that more physicists will come to use it as the main language for expressing their work.

In the three following papers, we explore different aspects of this unification, and some of the new physics and new insights which geometric algebra brings. Paper II[24] discusses the translation into geometric algebra of other languages for describing spinors and quantum-mechanical states and operators, especially in the context of the Dirac theory. It will be seen that Hestenes' form of the Dirac equation genuinely liberates it from any dependence upon specific matrix representations, making its intrinsic geometric content much clearer.

Paper III[27] uses the concept of multivector differentiation[6] to make many unifications and improvements in the area of Lagrangian field theory. The use of a consistent and mathematically rigorous set of tools for spinor, vector and tensor fields enables us to clarify the role of antisymmetric terms in stress-energy tensors, about which there has been some confusion. A highlight is the inclusion of functional differentiation within the framework of geometric algebra, enabling us to treat `differentiation with respect to the metric' in a new way. This technique is commonly used in field theories as one means of deriving the stress-energy tensor, and our approach again clarifies the role of antisymmetric terms.

Paper IV[16] examines in detail the physical implications of Hestenes' formulation and interpretation of the Dirac theory. New results include predictions for the time taken for an electron to traverse the classically-forbidden region of a potential barrier. This is a problem of considerable interest in the area of semiconductor technology.

We have shown elsewhere how to translate Grassmann calculus[30,31], and some aspects of twistor theory[32] into geometric algebra, with many simplifications and fresh insights. Thus, geometric algebra spans very large areas of both theoretical and applied physics.

There is another language which has some claim to achieve useful unifications. The use of `differential forms' became popular with physicists, particularly as a result of its use in the excellent, and deservedly influential, `Big Black Book' by Misner, Thorne & Wheeler[33]. Differential forms are skew multilinear functions, so that, like multivectors of grade k, they achieve the aim of coordinate independence. By being scalar-valued, however, differential forms of different grades cannot be combined in the way multivectors can in geometric algebra. Consequently, rotors and spinors cannot be so easily expressed in the language of differential forms. In addition, the `inner product', which is necessary to a great deal of physics, has to be grafted into this approach through the use of the duality operation, and so the language of differential forms never unifies the inner and outer products in the manner achieved by geometric algebra.

This leads us to say a few words about the widely-held opinion that, because complex numbers are fundamental to quantum mechanics, it is desirable to `complexify' every bit of physics, including spacetime itself. It will be apparent that we disagree with this view, and hope earnestly that it is quite wrong, and that complex numbers (as mystical uninterpreted scalars) will prove to be unnecessary even in quantum mechanics.

The same sentiments apply to theories involving spaces with large numbers of dimensions that we do not observe. We have no objection to the use of higher dimensions as such; it just seems to us to be unnecessary at present, when the algebra of the space that we do observe contains so many wonders that are not yet generally appreciated.

We leave the last words to David Hestenes and Garret Sobczyk[6]:

Geometry without algebra is dumb! - Algebra without geometry is blind!


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Mark Ashdown