We have now reached the point which is liable to cause the greatest intellectual shock. We have played an apparently harmless game with the algebra of 3-dimensional vectors and found a geometric quantity which has negative square and commutes with all multivectors. Multiplying this by , and in turn we get
which is exactly the algebra of the Pauli spin matrices used in the quantum mechanics of spin- particles! The familiar Pauli matrix relation,
is now nothing more than an expression of the geometric product of orthonormal vectors. We shall demonstrate the equivalence with the Pauli matrix algebra explicitly in a companion paper, but here it suffices to note that the matrices
comprise a matrix representation of our 3-dimensional geometric algebra. Indeed, since we can represent our algebra by these matrices, it should now be obvious that we can indeed add together the various different geometric objects in the algebra - we just add the corresponding matrices. These matrices have four complex components (eight degrees of freedom), so we could always disentangle them again.
Now it is clearly true that any associative algebra can be represented by a matrix algebra; but that matrix representation may not be the best interpretation of what is going on. In the quantum mechanics of spin- particles we have a case where generations of physicists have been taught nothing but matrices, when there is a perfectly good geometrical interpretation of those same equations! And it gets worse. We were taught that the were the components of a vector , and how to write things like and . But, geometrically, are three orthonormal vectors comprising the basis of space, so that in the are the components of a vector along directions and the result is a vector, not a scalar. With regard to , if you want to find the length of a vector, you must square and add the components of the vector along the unit basis vectors - not the basis vectors themselves. So the result is certainly true, but does not have the interpretation usually given to it.
These considerations all indicate that our present thinking about quantum mechanics is infested with the deepest misconceptions. We believe, with David Hestenes, that geometric algebra is an essential ingredient in unravelling these misconceptions.
On the constructive side, the geometric algebra is easy to use, and allows us to manipulate geometric quantities in a coordinate-free way. The -vectors, which play an essential role, are thereby removed from the mysteries of quantum mechanics, and used to advantage in physics and engineering. We shall see that a similar fate awaits Dirac's -matrices.
The algebra of 3-dimensional space, the Pauli algebra, is central to physics, and deserves further emphasis. It is an 8-dimensional linear space of multivectors, which we write as
where , , and we have reverted to bold-face type for 3-dimensional vectors. This is the exception referred to earlier; we use this convention to maintain a visible difference between spacetime 4-vectors and vectors of 3-dimensional space. There is never any ambiguity concerning the basis vectors , however, and these will continue to be written unbold.
The space of even-grade elements of this algebra,
is closed under multiplication and forms a representation of the quarternion algebra. Explicitly, identifying , , with , , , respectively, we have the usual quarternion relations, including the famous formula
Finally in this section, we relearn the cross product in terms of the outer product and duality operation (multiplication by the pseudoscalar):
Here we have introduced an operator precedence convention in which an outer or inner product always takes precedence over a geometric product. Thus is taken before the multiplication by i.
The duality operation in three dimensions interchanges a plane with a vector orthogonal to it (in a right-handed sense). In the mathematical literature this operation goes under the name of the `Hodge dual'. Quantities like or would conventionally be called `polar vectors', while the `axial vectors' which result from cross-products can now be seen to be disguised versions of bivectors.