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 have the correct properties of squaring to 1 and anti-commuting under Clifford (matrix) multiplication. These basis vectors then generate the Clifford algebra \({C\left(3,0\right)\cong\mathbb{C}(2)}\). The bivectors are thus naturally 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 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 \({C^{\mathbb{C}}\left(4\right)\cong\mathbb{C}(4)}\). We choose to represent the bivectors \({\hat{e}_{0}\hat{e}_{i}}\) by the matrices \({\sigma_{i}}\) (which again have the correct properties of squaring to 1 and anti-commuting), the remaining bivectors \({\hat{e}_{i}\hat{e}_{j}=\hat{e}_{0}\hat{e}_{i}\hat{e}_{0}\hat{e}_{j}}\) therefore being represented by the matrices \({\sigma_{i}\sigma_{j}=i\sigma_{k}}\). Since every element of \({\textrm{Spin}\left(3,1\right)^{e}\cong\textrm{Spin}\left(1,3\right)^{e}}\) is of the form of plus or minus the exponential of a bivector, every element can then be written as

\(\displaystyle U_{L}=\pm\textrm{exp}\left(-i\frac{\theta^{i}}{2}\sigma_{i}-\frac{\phi^{i}}{2}\sigma_{i}\right), \)

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 \({\hat{v}=\hat{v}^{i}\hat{e}_{i}}\).

Note that we also have the choice to represent the bivectors \({\hat{e}_{0}\hat{e}_{i}}\) by the matrices \({-\sigma_{i}}\) (which also square to 1 and anti-commute), which leaves the remaining bivectors \({\hat{e}_{i}\hat{e}_{j}}\) still represented by \({\sigma_{i}\sigma_{j}=i\sigma_{k}}\), so that the alternative general bivector form is

\(\displaystyle U_{R}=\pm\textrm{exp}\left(-i\frac{\theta^{i}}{2}\sigma_{i}+\frac{\phi^{i}}{2}\sigma_{i}\right). \)

These representations can be seen to be inequivalent, i.e. there is no similarity transformation that goes between them. Referring back to the Dirac matrices in the chiral basis for \({C(3,1)}\)

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

we can see that the first representation of the bivectors \({\hat{e}_{0}\hat{e}_{i}}\) by the matrices \({\sigma_{i}}\) corresponds to the left-handed Weyl rep, explaining the subscripts above.

Δ Note that with a \({(1,3)}\) signature, the bivector reps would remain the same, but referring back to the Dirac matrices in the chiral basis for \({C(1,3)}\)

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

we would swap the definitions \({U_{R}}\) and \({U_{L}}\) above. 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 Dirac matrices, but the form of the left and right Lorentz transformation reps depend only on the signature; e.g. we can get alternative Dirac matrices by multiplying \({\gamma^{0}}\) 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