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Showing posts from December, 2021

Projective representations, central extensions, and covering groups

In my post on Wigner's theorem , I explained the famous result that any symmetry transformation on quantum states can be realized as a unitary or antiunitary operator on Hilbert space. But when we study symmetries of quantum systems, we usually have in mind not a single symmetry but a full group of symmetries; Wigner's theorem tells us nothing about how the operators corresponding to different symmetries in the same group should compose with one another. Suppose, for example, that a quantum system transforms under the symmetry group $G$, and that the unitary or antiunitary operator corresponding to the element $g \in G$ is denoted by $\hat{U}_g$. Because any two unitary operators related by a phase are physically equivalent, it may not be the case that $\hat{U}_{g_1}$ and $\hat{U}_{g_2}$ compose to $\hat{U}_{g_1 g_2}$; instead, we will have a relationship like $$\hat{U}_{g_1} \hat{U}_{g_2} = e^{i \phi(g_1, g_2)} \hat{U}_{g_1 g_2}.$$ At first glance, it seems like we just made

A quick note on infinite-dimensional Lie groups

I'm putting the finishing touches on a post about projective representations of Lie groups and Lie algebras — an essential topic for understanding how symmetries act on quantum systems — and discovered a gap in my own knowledge. See, in discussing projective representations of Lie algebras on a Hilbert space $\mathcal{H}$, it is very convenient to be able to talk about the Lie algebra of the unitary group $\operatorname{U}(\mathcal{H}),$ and even moreso about the Lie algebra of the projective unitary group $$\operatorname{PU}(\mathcal{H}) = \operatorname{U}(\mathcal{H})/\{\alpha I | \alpha \in \operatorname{U}(1)\}.$$ The trouble is that when $\mathcal{H}$ is infinite-dimensional, as is often the case in quantum systems of interest, $\operatorname{U}(\mathcal{H})$ is infinite-dimensional and thus not a manifold, at least not in the usual sense of a manifold being a space locally diffeomorphic to $\mathbb{R}^n.$ Everything I've ever learned about Lie groups and algebras has been