In quantum mechanics, the eigenstates of a self-adjoint operator correspond to the physical states with particular values of the corresponding observable physical quantity. A fundamental principle of quantum mechanics further states that after making a measurement the system is “projected” into the corresponding eigenstate. It is therefore only eigenstates which are unaffected by the measurement process; they are projected into themselves. In particular, in order to simultaneously make two measurements, the system must be in an eigenstate of each; otherwise, the order in which the measurements are made will matter.

As discussed in Section 14.1, the spin operators $\LL_m$ are imaginary multiples of the derivatives (“infinitesimal generators”) of the rotations $R_m$ introduced originally in § 7.3. Writing $\rrr_a$ for those derivatives, we have \begin{equation} \rrr_a = {dR_z\over d\alpha} \Bigg|_{\alpha=0} \end{equation} so that \begin{equation} \rrr_x = {1\over2} \begin{pmatrix}0& \ell\\\noalign{\smallskip} \ell& 0\\\end{pmatrix} \qquad \rrr_y = {1\over2} \begin{pmatrix}0& 1\\\noalign{\smallskip} -1& 0\\\end{pmatrix} \qquad \rrr_z = {1\over2} \begin{pmatrix}\ell& 0\\\noalign{\smallskip} 0& -\ell\\\end{pmatrix} \end{equation} where we have used $\ell$ rather than $i$ to denote the complex unit. One then normally multiplies by $-\ell$ (and adds a factor of $\hbar$, which we henceforth set to $1$) to obtain a description of the Lie algebra $\su(2)$ in terms of Hermitian matrices.

As discussed in [arXiv:hep-th/9807044,, arXiv:hep-th/9910010] , however, even in the quaternionic setting care must be taken with this last step; we must put the factor of $\ell$ in the right place! The right place turns out to be on the right; we define the operator \begin{equation} \LLL_m(\psi) = - (r_m \psi) \ell \end{equation} where $\psi$ is a 2-component octonionic column (representing a Majorana-Weyl spinor in 10 spacetime dimensions). The operators $\LLL_m$ are self-adjoint with respect to the inner product \begin{equation} \langle \psi,\chi \rangle = \pi \!\left( \psi^\dagger\chi \right) \end{equation} where the map \begin{equation} \pi(q) = {1\over2} (q + \ell q \bar\ell) \end{equation} projects $\OO$ to a preferred complex subalgebra $\CC\subset\OO$, namely the one containing $\ell$.

Spin eigenstates are obtained as usual as the eigenvectors of $L_z$ with eigenvalues $\pm{1\over2}$. What are the eigenvectors of $\LLL_z$? Unsurprisingly, we have \begin{equation} \LLL_z \begin{pmatrix}1\cr 0\cr\end{pmatrix} = \begin{pmatrix}1\cr 0\cr\end{pmatrix} \frac12 \end{equation} (We could of course have written the eigenvalue on the left since it is real.) Somewhat surprisingly, this is not the only eigenvector with eigenvalue $\hbar\over2$. For instance, we have \begin{equation} \LLL_z \begin{pmatrix}0\cr k\cr\end{pmatrix} = \begin{pmatrix}0\cr k\cr\end{pmatrix} \frac12 \end{equation} Note the crucial role played by the anticommutativity of the quaternions in this equation! Particular attention is paid in [arXiv:hep-th/9807044,, arXiv:hep-th/9910010] to the eigenstates \begin{equation} \eplus = \begin{pmatrix}1\\ k\\\end{pmatrix} \label{eplus} \qquad \eminus = \begin{pmatrix}-k\\   1\\\end{pmatrix} \end{equation} which satisfy \begin{equation} \LLL_z(\eplus) = \frac12\eplus \qquad \LLL_z(\eminus) = -\frac12\eminus \end{equation} and which were proposed as representing particles 1) at rest with spin $\pm{1\over2}$, respectively. The eigenstates $\eplus$ and $\eminus$ are orthogonal with respect to the above inner product, that is \begin{equation} \langle \eplus,\eminus \rangle = 0 \end{equation} We will therefore focus on these eigenstates, which have some extraordinary properties.

Consider now the remaining spin operators $\LLL_x$, $\LLL_y$ acting on these eigenstates. We have \begin{equation} \LLL_x(\eplus) = {1\over2} \begin{pmatrix}-k\\   1\\\end{pmatrix} = \eplus \left( -{k\over2} \right) \end{equation} and \begin{equation} \LLL_y(\eplus) = {1\over2} \begin{pmatrix}-k\ell\\ \ell\\\end{pmatrix} = \eplus \left( -{k\ell\over2} \right) \end{equation} with similar results holding for $\eminus$. This illustrates the fact that this quaternionic self-adjoint operator eigenvalue problem admits eigenvalues which are not real. More importantly, as claimed in [arXiv:hep-th/9807044,, arXiv:hep-th/9910010] , it shows that $\eplus$ is a simultaneous eigenvector of the three self-adjoint spin operators $\LLL_x$, $\LLL_y$, $\LLL_z$!

This result could have significant implications for quantum mechanics. In this formulation, the inability to completely measure the spin state of a particle, because the spin operators fail to commute, is thus ultimately due to the fact that the eigenvalues don't commute. Explicitly, we have \begin{align} 4\LLL_x \left( \LLL_y(\eplus) \right) &= 2\LLL_x (-\eplus \,k\ell) = 2\rrr_x (\eplus \,k\ell) \,\ell \nonumber\\ &= -2\rrr_x (\eplus \,\ell) \,k\ell = +2\LLL_x(\eplus) \,k\ell \nonumber\\ &= -\eplus \,k \, k\ell = + \eplus \,\ell \\ \noalign{\smallskip} 4\LLL_y \left( \LLL_x(\eplus) \right) &= -\eplus \,k\ell \, k = - \eplus \,\ell \end{align} which yields the usual commutation relation in the form \begin{equation} \left[\LLL_x,\LLL_y\right] \,(\eplus) = {1\over2} \, \eplus \,\ell = \LLL_z(\eplus) \,\ell \end{equation}

Furthermore, there is a phase freedom in (\ref{eplus}), since \begin{equation} \LLL_z \left( \eplus e^{\ell\theta} \right) = \left( \eplus e^{\ell\theta} \right) \left( {1\over2} \right) \end{equation} for any value of $\theta$. It is still true that $\eplus e^{\ell\theta}$ is a simultaneous eigenvector of all three spin operators, but the imaginary eigenvalues have changed. We have \begin{align} \LLL_x \left( \eplus e^{\ell\theta} \right) &= \left( \eplus e^{\ell\theta} \right) \left( -{k \, e^{2\ell\theta}\over2} \right) \\ \LLL_y \left( \eplus e^{\ell\theta} \right) &= \left( \eplus e^{\ell\theta} \right) \left( -{k\ell \, e^{2\ell\theta}\over2} \right) \end{align} so that the non-real eigenvalues depend on the phase. It is intriguing to speculate on whether the value of the non-real eigenvalues, which determine the phase, can be used to specify (but not measure) the actual direction of the spin, and whether this might shed some insight on basic properties of quantum mechanics such as Bell's inequality.

Finally, we point out that all eigenvectors of the complex operators $\LL_x$, $\LL_y$, $\LL_z$ turn out to be quaternionic; each eigenvector lies in some quaternionic subalgebra of $\OO$ which also contains $\ell$.