Pauli’s matrices – Geometric Algebra

In 1927 the German physicist Wolfgang Pauli developed a theory of the electron based upon the physical quantity spin, which has no equivalent in classical mechanics, although it is sometimes described as the rotation of the electron on itself.
The spin of the electron has a half integer value $| S | = 1/2$ and the spin axis in general is time-dependent, as it can also interact with the electromagnetic field.

Pauli, unable to use geometric algebra, exploited the mathematical arsenal of time, using vectors, matrices and complex numbers.
The theory is based on the following three matrices (plus the identity matrix, to be precise), associated but not identified with the three dimensions of 3D space:

$\sigma_x = \begin{pmatrix} 0&1 \\ 1&0 \end{pmatrix} \:\: \sigma_y = \begin{pmatrix} 0&-i \\ i&0 \end{pmatrix}\:\: \sigma_z = \begin{pmatrix} 1&0 \\ 0&-1 \end{pmatrix}$

which have the following properties:

$\sigma_x^2 = \sigma_y^2 = \sigma_z^2 = 1$
$\sigma_x \sigma_y = - \sigma_y \sigma_x = i \sigma_z$
$\sigma_y \sigma_z = - \sigma_z \sigma_y = i \sigma_x$
$\sigma_z \sigma_x = - \sigma_x \sigma_z = i \sigma_y$
$\sigma_x \sigma_y \sigma_z = i$
$\text{Tr}\:\sigma_x = \text{Tr}\:\sigma_y = \text{Tr}\:\sigma_z = 0$
$\det \sigma_x = \det \sigma_y = \det \sigma_z = -1$

It is shivering to verify that all the previous relations are satisfied by the basis of the geometric algebra $\mathbb{G}^3$ and therefore the Pauli matrices can be considered a matrix representation of the geometric algebra $\mathbb{G}^3$ .

Anyone wishing to deepen the relationship between geometric algebra and Pauli matrices will find an excellent reference in the recent book Electromagnetic fields with geometric algebra. For example, there is the formulation of the generic 3D multivector
$M = a_0 + \pmb a + \pmb{\hat B} + t$
in terms of Pauli matrices:

$\tilde{a_0} = \begin{pmatrix} a_0&0 \\ 0&a_0 \end{pmatrix}$

$\tilde a = \begin{pmatrix} a_3&a_1-i a_2 \\ i a_2+a_1&-a_3 \end{pmatrix}$

$\tilde{B} = \begin{pmatrix} i B_3&B_2+i B_1 \\ i B_1-B_2&-i B_3 \end{pmatrix}$

$\tilde{t} = \begin{pmatrix} i t&0 \\ 0&i t \end{pmatrix}$

which are added to give rise to a complex matrix with 8 parameters, precisely those necessary for geometric algebra to describe the elements of 3D space.

The fact that the Pauli matrix algebra, which is the foundation of quantum mechanics, finds a geometric representation, is encouraging in the perspective of a more intuitive interpretation than the cold “matrix mechanics”.

For example the canonical commutation relation
$\sigma_1 \sigma_2 - \sigma_2 \sigma_1 = 2 i \sigma_3$
can be matched with the expression of the GA
$e_1e_2 - e_2e_1 = 2i e_3$
in fact by reworking it, we will obtain the verification:
$e_1e_2 + e_1e_2 = 2 e_1e_2 = 2 (e_1e_2e_3) e_3$

Pauli’s formulation of the theory of electrons used a vector of matrices
$\pmb \sigma = (\sigma_1, \sigma_2, \sigma_3)$
to associate with the vector
$\pmb u = u_1 e_1 + u_2 e_2 + u_3 e_3$
by means of the matrix
$\pmb \sigma \cdot \pmb u = u_1 \sigma_1 + u_2 \sigma_2 + u_3 \sigma_3$

The following identity is of particular importance in Pauli’s algebra:

$(\pmb \sigma \cdot \pmb u)(\pmb \sigma \cdot \pmb v) = (\pmb u \cdot \pmb v) I + i\: \pmb \sigma \cdot (\pmb u \times \pmb v)$

but it must be admitted that it is a really obscure way of expressing a simple idea, if expressed in GA: its fundamental identity, the geometric product:

$\pmb u \pmb v = \pmb u \cdot \pmb v + \pmb u \wedge \pmb v$

The Pauli equation, here in the simplified version with a single stationary electron in a spatially uniform and time-varying magnetic field, can also be applied to other particles, such as atomic nuclei, and is the basis of nuclear magnetic resonance .

We compare the expressions of Pauli’s theory according to the classical formalism and the GA (the constant $\gamma$ is the gyromagnetic ratio of the electron).

\psi = \begin{bmatrix} a_0 + i a_3 \\ -a_2 + i a_1 \end{bmatrix}

	classical vector algebra	geometric algebra
Spin	$\psi = \begin{bmatrix} a_0 + i a_3 \\ -a_2 + i a_1 \end{bmatrix}$	$\Psi = a_0 + \sum a_k I e_k$ (*)
Magnetic field	$\pmb b$	$\pmb B = - \pmb b^*$
Pauli equation	$\begin{equation} \frac{d\psi(t)}{dt} = i \frac{1}{2} \gamma (\pmb \sigma \cdot \pmb b(t)) \psi(t)$	$\begin{equation} \frac{d\Psi(t)}{dt} = \frac{1}{2} \gamma \pmb B(t) \Psi(t)$

(*) since $a_0 ^ 2 + a_1 ^ 2 + a_2 ^ 2 + a_3 ^ 2 = 1$ it is a rotor and therefore $= e^{- i \theta / 2}$

Pauli’s theory, however correct, did not take into account the theory of relativity and was therefore extended by the English physicist Paul Dirac just a year later. This version also made use of matrices and employed the classical formalism. Dirac’s theory of the relativistic electron is translated into a geometric algebra in four dimensions: it is in short an algebra of spacetime $\mathbb{G}^{1,3}$