Disclaimer: Nothing in this note is original.

Consider the three Lie algebra generators of $SU(2)$ as three basis, then ${\frak su}(2)$ is a rank three vector space with real coefficients, thus an element of $SU(2)$ can be regarded as a rotation in ${\frak su}(2)$, namely a $SO(3)$ matrix. The adjoint representation of $g\in SU(2)$ acts on ${\frak su}(2)$,

\[\text{Ad}: SU(2)\to SO(3)\]

which is double valued. Inversely, each $SO(3)$ has a double valued representation by $2\times 2$ complex matrices acting on $\mathbb{C}^{2}$. The complex vector $(\psi_ {1},\psi_ {2})^{T}$ on which this $2\times 2$ matrix acts on are called spinors.

Mathematicians don’t like double-valued anythings since you can’t treat them as regular functions. So, we can say that $SU(2)$ matrices naturally furnish a spinor representation of the 2-fold cover of $SO(3)$. When $SU(2)$ is thought of as the 2-fold cover of $SO(3)$, it’s called the spinor group $\text{Spin}(3)$.

The topological reason that $SO(3)$ can admit a nontrivial double-cover is that $SO(3)$ is not simply connected, it is instead a projective manifold,

\[SO(2) \cong \mathbb{R}P^{3}.\]

An mysterious consequence of the projective nature of $SO(3)$ is that, one full rotation is different from two full rotations! A full rotation is by $2\pi$. The 1-parameter subgroup of $SO(3)$ defined by rotation along a certain axis (say $\hat{z}$) is given matrix

\[\theta \mapsto \begin{pmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix}\]

and for $\theta \in (0,2\pi)$ the trajectory is a closed curve in $\mathbb{R} P^{3}$, which can not be shrunk to a point. (This statement is a little abstract) In the mean while, for $\theta \in(0,4\pi)$ the trajectory can be shrunk to a point.

There are three ways to understand it more intuitively. The first example is the waiter with a platter picture. The second is Wyle’s double cone interpretation, which shows that a rotation by $4\pi$ is indeed continuously connected to doing nothing. A third interpretation will be shown later, until we have introduced the Dirac equation.

Hamilton on composing two rotations

The $SU(2)$ double cover is a power tool for investigating the product of two rotations, thanks to the nice algebraic properties of Pauli matrices. For example, given two rotations

\[R_ {1} = \exp \left\{ -i T^{a} \theta^{a} \right\} ,\quad R_ {2} = \exp \left\{ -i T^{a}\theta'^{a} \right\}\]

where

\[T^{a} := \frac{\sigma^{a}}{2}\]

is the generator for rotation, then we can write down the explicit expression of $R_ {1}R_ {2}$ in terms of $\theta$ and $\theta’$.

Note that $\sigma^{1,2,3}$ span a real vector space $V^{3}$. It is the space of traceless hermitian matrices. In this space there is a quadratic form given by

\[\left\langle h,h' \right\rangle := \frac{1}{2} \mathrm{Tr}\,(h h'), \quad h,h'\in V^{3}.\]

Then the basis satisfy a very nice relation

\[\left\langle \sigma_ {i},\sigma _ {j} \right\rangle = \delta_ {ij}.\]

Next we will expand $V^{3}$ by imposing on it further algebraic structure. We can introduce multiplication of the basis $\sigma_ {i}$. This can be easily done by adopting the multiplication of matrices. However, $\sigma_ {i}\sigma_ {2}=i\sigma_ {3}$ is not in $V^{3}$. The easy remedy is to expand $V^{3}$ simply to include it, say define $e_ {4}:=i\sigma_ {3}$. Let’s define

\[e_ {1}:=\sigma_ {1},\; e_ {2}:= \sigma_ {2}, \; e_ {3}:=\sigma_ {3},\; e_ {4}:=i\sigma_ {3}=\sigma_ {1}\sigma_ {2}.\]

But that wouldn’t be enough, to make the algebra closed under multiplication we also need to include

\[e_ {5}:= e_ {2}e_ {3}, \; e_ {6}:= e_ {1}e_ {3},\; e_ {7}:=e_ {1}e_ {2}e_ {3}=i I\]

where $I=\mathbb{1}_ {2}$ is the two dimensional identity matrix, and

\[e_ {8}:= I.\]

The algebra (over $\mathbb{R}$) given by all the eight basis is called the Pauli algebra, it is said to be generated by the Pauli matrices. It is an associative algebra, that is, a vector space with a composition (called product) that is associative and is distribution with respect to addition. In fact, in this case, this $8$-dimensional vector space is nothing but the space of all $2\times 2$ complex matrices!

The way we constructed the Pauli algebra can be generalized to Clifford algebra.

Clifford algebra. Let $C_ {n}$ be an associative algebra (over $\mathbb{R}$) with identity $I$, generated by an $n$-dimensional vector subspace $V^{n}$. Define a real quadratic form $\left\langle -,- \right\rangle$ over $V^{n}$. Let $e_ {1},\dots,e_ {n}$ be the basis of $V^{n}$, satisfying certain anti-commutation relation

\[\left\{ e_ {i} ,e_ {j} \right\} =2 g_ {ij} \,I\]

where $g_ {ij}$ is given by the quadratic form

\[g_ {ij} := \left\langle e_ {i},e_ {j} \right\rangle .\]

Then $C_ {n}$ is called the Clifford algebra generated by $V^{n}$.

Note that the quadratic form could be anything, and it is part of the definition of Clifford algebra. Pauli algebra is a Clifford algebra. If we set $g_ {ij}$ to be identically zero, then we recover the exterior algebra.

An important example of Clifford algebra are quaternions. Let $C_ {2}$ be the Clifford algebra generated by two basis, $e_ {1}$ and $e_ {2}$. Let the quadratic form be

\[\left\langle e_ {i} ,e_ {j} \right\rangle =-\delta_ {ij},\]

then we have

\[\left\langle e_ {1},e_ {1} \right\rangle =\left\langle e_ {2},e_ {2} \right\rangle =-1.\]

The product is also given by the anti-commutation relations, for example

\[\left\{ e_ {1},e_ {1} \right\} = 2 e_ {1}e_ {1}=2g_ {11}I = -2\implies e_ {1}e_ {1}=-1.\]

The same goes for $e_ {2}$. The product $e_ {1}e_ {2}$ defines a new basis of the algebra,

\[e_ {3}:=e_ {1}e_ {2}, \quad e_ {3}e_ {3}=e_ {1}e_ {2}e_ {1}e_ {2}=-1.\]

Rewrite the basis as $j=e_ {1},k=e_ {2}$ and $i=e_ {3}$, the resulting algebra is the quaternion, a quaternion in general can be expanded (over $\mathbb{R}$) in terms of these basis,

\[a+bi+cj+dk,\quad a,b,c,d \in \mathbb{R}.\]

The Dirac algebra

The Lorentz group and its geometry

Given the Minkowski metric $g=\text{diag}(-1,1,1,1)$, the Lorentz group $L$ is by definition the isometry group,

\[L := \left\{ 4\times 4 \text{ real matrices }B \,\middle\vert\, \left\langle Bx,By \right\rangle=\left\langle x,y \right\rangle \right\}\]

acting on the 4-dimensional Minkowski space $M$. In metric notation,

\[\left\langle x,y \right\rangle := g_ {\mu \nu} \, x^{\mu}y^{\nu}.\]

It can be shown that $L$ breaks down to four connected components, depending on the sign of the determinant and weather $Bx$ changes the sign of the time component of $x$. The so-called proper Lorentz group, whose elements has determinant $1$ and doesn’t change the sign in the time-component, is denoted as $L_ {0}$, this corresponds to the physical Lorentz transforms.

Similar to $SO(3)$ is double-covered by $SU(2)$, $L_ {0}$ turns out to be double covered by $SL(2,\mathbb{C})$, the special linear $2\times 2$ complex matrix.