Representations in spacetime

The pinor and spinor reps of \({C\left(r,s\right)}\) and \({C_{0}\left(r,s\right)}\) turn out to still be irreducible when their actions are restricted to \({\textrm{Pin}\left(r,s\right)}\) and \({\textrm{Spin}\left(r,s\right)}\), so the representation theory of Clifford groups is identical to that of Clifford algebras. In this section we will list concrete representations for the various quantities we have defined in the two most common vector spaces in physics, corresponding to space and spacetime. We will choose bases that highlight the isomorphism \({C\left(3,0\right)\cong C_{0}\left(3,1\right)}\) and use the Pauli matrices \({\sigma_{i}}\).

Dealing first with space, we can view \({\textrm{Spin}\left(3,0\right)\cong SU\left(2\right)\cong Sp(1)}\) as either a group of quaternions sitting inside \({C_{0}\left(3,0\right)\cong\mathbb{H}}\), or as a group of complex matrices sitting inside \({C\left(3,0\right)\cong\mathbb{C}(2)}\). Both approaches can be fruitful, but we will focus on the latter, which since \({C^{\mathbb{C}}\left(3\right)\cong\mathbb{C}(2)\oplus\mathbb{C}(2)}\) can be viewed as the “odd Dirac” rep for signature \({(3,0)}\).

We can represent a chosen orthonormal basis \({\hat{e}_{i}}\) of \({\mathbb{R}^{3}}\) by the matrices \({\sigma_{i}}\), which as we saw generate the Clifford algebra \({C\left(3,0\right)\cong\mathbb{C}(2)}\). The bivectors are thus represented by \({\sigma_{i}\sigma_{j}=i\sigma_{k}}\), so that the elements of \({\textrm{Spin}\left(3,0\right)\cong SU\left(2\right)}\) are of the form

\(\displaystyle \textrm{exp}\left(-i\frac{\theta^{i}}{2}\sigma_{i}\right), \)

and \({\hat{u}=\hat{u}^{i}\hat{e}_{i}}\) is a unit length vector defined by \({\theta\hat{u}^{i}\equiv\theta^{i}}\). This corresponds to a rotation by \({\theta}\) in the plane orthogonal to \({\hat{u}}\).

In spacetime, \({\textrm{Spin}\left(3,1\right)^{e}\cong\textrm{Spin}\left(1,3\right)^{e}\cong SL\left(2,\mathbb{C}\right)}\) is a group of complex matrices sitting inside \({C_{0}\left(3,1\right)\cong C_{0}\left(1,3\right)\cong\mathbb{C}(2)}\), which can be viewed as the Weyl rep, the chiral decomposition of the Dirac rep \({C^{\mathbb{C}}\left(4\right)\cong\mathbb{C}(4)}\). A concrete representation is determined by the choice of signature and Dirac matrices; in the chiral basis for \({C(3,1)}\) we have

\(\displaystyle \gamma^{0}=\begin{pmatrix}0 & I\\ -I & 0 \end{pmatrix},\;\gamma^{i}=\begin{pmatrix}0 & \sigma_{i}\\ \sigma_{i} & 0 \end{pmatrix}\;\Rightarrow\gamma^{0}\gamma^{i}\psi=\begin{pmatrix}\sigma_{i} & 0\\ 0 & -\sigma_{i} \end{pmatrix}\begin{pmatrix}\psi_{L}\\ \psi_{R}\end{pmatrix}=\begin{pmatrix} \sigma_{i}\psi_{L}\\ -\sigma_{i}\psi_{R}\end{pmatrix}, \)

with the rotation component then \({\Omega\hat{e}_{0}\hat{e}_{i}=\hat{e}_{j}\hat{e}_{k}=\hat{e}_{0}\hat{e}_{j}\hat{e}_{0}\hat{e}_{k}\rightarrow\sigma_{j}\sigma_{k}=i\sigma_{i}}\). Every element of \({\textrm{Spin}\left(3,1\right)^{e}}\) can then be written as the two Weyl reps

\begin{aligned}U_{L} & =\pm\textrm{exp}\left(-i\frac{\theta^{i}}{2}\sigma_{i}-\frac{\phi^{i}}{2}\sigma_{i}\right),\\
U_{R} & =\pm\textrm{exp}\left(-i\frac{\theta^{i}}{2}\sigma_{i}+\frac{\phi^{i}}{2}\sigma_{i}\right),
\end{aligned}

where \({\theta\hat{u}^{i}\equiv\theta^{i}}\), \({\phi\hat{v}^{i}\equiv\phi^{i}}\), and the element corresponds to a rotation by \({\theta}\) in the space-like plane perpendicular to the unit vector \({\hat{u}=\hat{u}^{i}\hat{e}_{i}}\) and a Lorentz boost of rapidity \({\phi}\) in the direction of the unit vector \({\hat{v}=\hat{v}^{i}\hat{e}_{i}}\). These two representations can be seen to be inequivalent, i.e. there is no complex similarity transformation that goes between them. In the chiral basis for \({C(1,3)}\) we have

\begin{aligned}\gamma^{0}=\begin{pmatrix}0 & I\\
I & 0
\end{pmatrix},\;\gamma^{i}=\begin{pmatrix}0 & \sigma_{i}\\
-\sigma_{i} & 0
\end{pmatrix}\; & \Rightarrow\gamma^{0}\gamma^{i}\psi=\begin{pmatrix}-\sigma_{i} & 0\\
0 & \sigma
\end{pmatrix}\begin{pmatrix}\psi_{L}\\
\psi_{R}
\end{pmatrix}=\begin{pmatrix}-\sigma_{i}\psi_{L}\\
\sigma_{i}\psi_{R}
\end{pmatrix},\end{aligned}

with the rotation component then \({\Omega\hat{e}_{0}\hat{e}_{i}=\hat{e}_{j}\hat{e}_{k}=-\hat{e}_{0}\hat{e}_{j}\hat{e}_{0}\hat{e}_{k}\rightarrow-i\sigma_{i}}\). It turns out that with the reversed signature we have to reverse the signs in the exponential in order to keep the angles those of the corresponding rotation and boost, so the two Weyl reps end up keeping the same form as above.

Δ In mapping the above to other treatments, it is important to remember that the stacking order of the column vector depends on the choice of chiral Dirac matrices, but the form of the left and right Lorentz transformation reps remains the same; e.g. we can get alternative Dirac matrices by multiplying \({\gamma^{0}}\) or \({\gamma^{i}}\) by \({-1}\), but this does the same to \({\gamma_{5}}\), thus swapping the stacking order of the Weyl spinors in chiral decomposition and leaving the forms of \({U_{R}}\) and \({U_{L}}\) the same. Also again note that many treatments are for passive rotations, which would multiply all angles by \({-1}\).

85.spin-reps-3+1

The above depicts how An element \({U}\) of the group \({\textrm{Spin}(3,1)}\) acts as a Lorentz transformation on spacetime via the rep \({SO(3,1)}\), and as an exponential of bivectors expressed as multiplied pairs of Dirac matrices on spinors (in \({\mathbb{C}^{4}}\) for any basis, or in \({\mathbb{R}^{4}}\) for the Majorana basis). The rep acting on \({\mathbb{C}^{4}}\) is reducible to the two Weyl irreps, where \({U}\) then acts as an exponential of bivectors expressed as Pauli matrices \({\sigma_{i}}\) and \({i\sigma_{i}}\) on half-spinors. The bivector reps acting on spinors can be expressed in terms of the angles that generate Lorentz boosts and rotations on spacetime.

An Illustrated Handbook