9 - Some applications to physics

Summary

In this chapter, we briefly describe some applications of Clifford algebras to physics. The first of these concerns the spin of an elementary particle, when the spin is 1/2. Pauli introduced the Pauli spin matrices to describe this phenomenon. Physics is concerned with partial differential operators: building on Pauli's ideas, Dirac introduced a first order differential operator, now called the Dirac operator, in order to formulate the relativistic wave equation for an electron; this in turn led to the discovery of the positron. We shall study the Dirac operator more fully in the next chapter. Here we show how it can be used to formulate Maxwell's equation for an electromagnetic field in a particularly simple way, and will also describe the Dirac equation.

Particles with spin 1/2

The Pauli spin matrices were introduced by Pauli to represent the internal angular momentum of particles which have spin 1/2. Let us briefly describe how this can be interpreted in terms of the Clifford algebra A_0,3. In quantum mechanics, an observable corresponds to a Hermitian linear operator T on a Hilbert space H, and, when the possible values of the observable are discrete, these possible values are the eigenvalues of T.

The Stern-Gerlach experiment showed that elementary particles have an intrinsic angular momentum, or spin. If x is a unit vector in R³ then the component J_x of the spin in the direction x is an observable.