Returning to 2-dimensional space, we now use geometric algebra to reveal the structure of the Argand diagram. From any vector we can form an even multivector (a 2-dimensional spinor):
Using the vector to define the real axis, there is therefore a one-to-one correspondence between points in the Argand diagram and vectors in two dimensions. Complex conjugation,
now appears as a natural operation of reversion for the even multivector z, and (as shown above) it is needed when rotating vectors.
We now consider the fundamental derivative operator
and observe that
Generalising this behaviour, we find that
and define an analytic function as a function (or, equivalently, ) for which
Writing f = u + I v, this implies that
which are the Cauchy-Riemann conditions. It follows immediately that any non-negative, integer power series of z is analytic. The vector derivative is invertible so that, if
for some function s, we can find f as
Cauchy's integral formula for analytic functions is an example of this:
is simply Stokes's theorem for the plane. The bivector is necessary to rotate the line element into the direction of the outward normal.
This definition (4.8) of an analytic function generalises easily to higher dimensions, where these functions are called monogenic, although the simple link with power series disappears. Again, there are some surprises in three dimensions. We have all learned about the important class of harmonic functions, defined as those functions satisfying the scalar operator equation
Since monogenic functions satisfy
they must also be harmonic. However, this first-order equation is more restrictive, so that not all harmonic functions are monogenic. In two dimensions, the solutions of equation (4.13) are written in terms of polar coordinates as
Complex analysis tells us that there are special combinations (analytic functions) which have particular radial dependence:
In this way we can, in two dimensions, separate any given angular component into parts regular at the origin () and at infinity (). These parts are just the spinor solutions of the first-order equation (4.14).
The situation is exactly the same in three dimensions. The solutions of are
but we can find specific combinations of angular dependence which are associated with a radial dependence of or . We show this by example for the case l=1. Obviously, non-trivial solutions of must contain more than just a scalar part - they must be multivectors. For the position vector we find the following relations:
(Equation 4.20 can be derived from a more general formula given in Section 5.) We can assemble solutions proportional to r and :
where is the unit vector in the azimuthal direction.
Alternatively, we can generate a spherical monogenic from any spherical harmonic :
We have chosen to place the vector to the right of so as to keep within the even subalgebra of spinors. This practice is also consistent with the conventional Pauli matrix representation (2.20). As an example, we try this procedure on the l=0 harmonics:
For a selection of l=1 harmonics we obtain
Some readers may now recognise this process as similar to that in quantum mechanics when we add the spin contribution to the orbital angular momentum, making a total angular momentum . The combinations of angular dependence are the same as in stationary solutions of the Dirac equation. In particular, (4.25) indicates that only one monogenic arises from l=0. That is correct - only the state exists. Turning to (4.26) we see that there is one state with no angular dependence at all, and that the other has terms proportional to . These can also be interpreted in terms of and respectively.
The process by which we have generated these functions has, of course, nothing to do with quantum mechanics - another clue that many quantum-mechanical procedures are much more classical than they seem.