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Rotations and Geometric algebra

Geometric algebra is useful to a physicist because it automatically incorporates the structure of the world we inhabit, and accordingly provides a natural language for physics. One of the clearest illustrations of its power is the way in which it deals with reflections and rotations. The key to this approach is a theorem due to Hamilton[25]: given any unit vector n (, we can resolve an arbitrary vector x into parts parallel and perpendicular to n: . These components are identified algebraically through their commutation properties:

The vector can therefore be written -nxn. Geometrically, the transformation represents a reflection in a plane perpendicular to n. To make a rotation we need two of these reflections:

where is called a `rotor'. We call the `reverse' of R, because it is obtained by reversing the order of all geometric products. The rotor is even (i.e. viewed as a multivector it contains only even-grade elements), and is unimodular, satisfying .

Figure 1: A rotation composed of two reflections

As an example, let us rotate the unit vector a into another unit vector b, leaving all vectors perpendicular to a and b unchanged (a simple rotation). We can accomplish this by a reflection perpendicular to the unit vector which is half-way between a and b (see Figure 1):

This reflects a into -b, which we correct by a second reflection perpendicular to b. Algebraically,

which represents the simple rotation in the plane. Since , we define

so that the rotation is written

which is a `bilinear' transformation of a. The inverse transformation is

The bilinear transformation of vectors is a very general way of handling rotations. In deriving this transformation the dimensionality of the space of vectors was at no point specified. As a result, the transformation law works for all spaces, whatever dimension. Furthermore, it works for all types of geometric object, whatever grade. We can see this by considering the product of vectors

which holds because .

As an example, consider a 2-dimensional rotation:

A rotation by angle is performed by the even element (the equivalent of a complex number)

As a check:


The bilinear transformation is much easier to use than the one-sided rotation matrix, because the latter becomes more complicated as the number of dimensions increases. Although this is less evident in two dimensions, in three dimensions it is obvious: the rotor


represents a rotation of radians about the axis along the direction of . If required, we can decompose rotations into Euler angles , the explicit form being

We now examine the composition of rotors in more detail. In three dimensions, let the rotor R transform the unit vector along the z-axis into a vector s:


Now rotate the s vector into another vector , using a rotor . This requires

so that the transformation is characterised by

which is the (left-sided) group combination rule for rotors. Now suppose that we start with s and make a rotation of about the z-axis, so that returns to s. What happens to R is surprising; using (3.13) above, we see that

This is the behaviour of spin- particles in quantum mechanics, yet we have done nothing quantum-mechanical; we have merely built up rotations from reflections.

How can this be? It turns out[24] that it is possible to represent a Pauli spinor (a 2-component complex spinor) as an arbitrary even element (four real components) in the geometric algebra of 3-space pauli1. Since is a positive-definite scalar in the Pauli algebra we can write

Thus, the Pauli spinor can be seen as a (heavily disguised) instruction to rotate and dilate. The identification of a rotor component in then explains the double-sided action of spinors on observables. The spin-vector observable, for example, can be written in geometric algebra as

which has the same form as equation (3.15). This identification of quantum spin with rotations is very satisfying, and provides much of the impetus for David Hestenes' work on Dirac theory.

A problem remaining is what to call an arbitrary even element . We shall call it a spinor, because the space of even elements forms a closed algebra under the left-sided action of the rotation group: gives another even element. This accords with the usual abstract definition of spinors from group representation theory, but differs from the column vector definition favoured by some authors[26].

next up previous
Next: Analytic and Monogenic Up: Imaginary Numbers are not Previous: Interlude

Mark Ashdown