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Suppose we have an orthonormal basis of states $|j,m,p\rangle$ where

  • $j=\frac{1}{2},\frac{3}{2},\ldots$ is the angular momentum quantum number associated with some angular momentum operator $\mathbf{J}=(J_x,J_y,J_z)$,
  • $m=-j,\ldots,j$ is the projection of the angular momentum $J_z$, and
  • $p=\pm1$ is the parity of the state under inversion operator $i$, where $i^2=1$.

Now consider the family operators formed by arbitrary rotations $R^z_\phi := e^{-i J_z \phi}$ by an angle $\phi \in [0,2 \phi)$ about the $z$-axis and reflections $\Sigma^z_\phi = R^z_{\phi/2} \left( P R^y_\pi \right) R^z_{-\phi/2} = R^y_\pi R^z_{-\phi} P$ about a plane containing the $z$-axis and making an angle $\phi/2 \in [0,\pi)$ with the $y$-axis, where $P$ is the inversion operator.

The geometric description of these operators suggests that together they form a representation of the $O(2) \cong C_{\infty v}$ group of the symmetries of the circle, but we can quickly check that for half-integer angular momenta $j$ we have $R^z_{2\pi} = -1 \neq +1 = R^z_0$ so that the operators do not satisfy the defining relations for a group representation. By a similar token we would expect a composition $\Sigma^z_{\phi_1} \Sigma^z_{\phi_2}$ of reflections to yield a rotation $R^z_{\phi_1-\phi_2}$, but in fact we find $\Sigma^z_{\phi_1} \Sigma^z_{\phi_2} = -R^z_{\phi_1-\phi_2}$, with the factor $-1$ ultimately originating from $R^y_\pi R^y_\pi = R^y_{2 \pi} = -1$.

However, by fiddling around a bit I find that the association $$ r_\phi \to R^z_{2 \phi} $$ and $$ \sigma_\phi \to i \Sigma^z_{2\phi},\,\,\,(i \equiv \sqrt{-1}) $$ between the abstract rotations $r_\phi$ and reflections $\sigma_\phi$ of $O(2)$ and the quantum mechanical operators, then we obtain a true representation of $O(2)$. One then straightforwardly finds that the pairs $| j, m, P \rangle$ and $|j, -m, P \rangle$ together form 2-dimensional irreducible representations of symmetry $\Lambda = |m|$ under the group action.

Two questions:

  1. What is the physical meaning of this doubling of the angle $\phi \to 2\phi$ and the insertion of the factor $i$?

  2. As I understand it, the naive association $r_\phi \to R^z_\phi$ and $\sigma_\phi \to \Sigma^z_\phi$ constitutes a projective representation of $O(2)$, but what can I do with this information? I am interested in understanding the possible coupling that can occur between zeroth-order energy eigenstates $|j,m,p\rangle$ under the influence of a pertubation commuting with all the $R^z_\phi$ and $\Sigma^z_\phi$. From the true representation constructed above I can for instance infer that coupling only occurs between states of equal $m$ and that eigenstates occur in degenerate pairs $\pm m$. Can the same conclusions be drawn by analyzing the projective representation, or are there indeed additional constraints which can be inferred?

Qmechanic
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creillyucla
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1 Answers1

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I have a partial answer to some of the questions posed. First off, by analyzing the characters of the action of group $G \equiv \{R^z_\phi : \phi \in [0,4\pi)\} \cup \{\Sigma^z_\phi : \phi \in [0,4\pi)\} \ncong O(2)$ on the pairs $|j, \pm m, p\rangle$ one can establish that these pairs form irreducible representations of $G$ and that pairs are equivalent as representations of $G$ if and only if they have equal $|m|$ (but perhaps differing $j,p$). Therefore couplings between states induced by a pertubation commuting with $G$ is only allowed between states of equal $m$, and eigenstates of the perturbation necessarily occur in degenerate pairs. This is the same conclusion arrived at using the correspondance $r_\phi \to R^z_{2\phi}$ and $\sigma_\phi \to i \Sigma^z_{2 \phi}$ defining a non-projective representation of $O(2)$, so the symmetry decomposition of the $|j, m, p \rangle$ is the same here whether one uses the projective or non-projective representations.

The group $G$ can alternatively be characterized as the subset of unit quaternions $a + b i + c j + d k$ satisfying $(a^2 + b^2)(c^2 + d^2) = 0$. This group can be parameterized by a pair $(\phi,\pm 1)$ where $\phi \in [0,2 \pi)$ and $(\phi, +1) \cong \cos \phi + i \sin \phi$ and $(\phi, -1) \cong j \cos \phi + k \sin \phi$. We can parameterize $O(2)$ in the same way via $(\phi,+1) \cong r_\phi$ and $(\phi,-1) \cong \sigma_\phi$. I emphasize that though the two groups $G$ and $O(2)$ permit the same parameterization, they are nonequivalent as groups.

$G$ forms a double cover of $O(2)$ via the mapping $(\phi,\pm 1) \to (2 \phi, \pm1)$. $G$ must in fact be a covering group of $O(2)$ in light of this post.

For more on the mathematical details concerning $G$, $O(2)$ and their relationship, see Robert Taylor, Proc. Amer. Math. Soc. 5 (1954), 753-768. In the article Taylor refers to $O(2)$ and $G$ as $G^2$ and $G^3$ respectively.

I still don't have a compelling physical explanation for the angle-doubling $\phi \to 2 \phi$ and the phase factor $i$ for the reflections. The doubling seems as though it should be related to the double-covering mentioned earlier, and the phase factor should bear some connection to the fact that the multiplication $(\phi_1,-1) . (\phi_2,-1)$ is $(\phi_1 - \phi_2, +1)$ for $O(2)$ but $(\phi_1 - \phi_2 + \pi, +)$ for $G$. However these observations do not constitute a physical explanation in any meaningful sense.

creillyucla
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