Suppose we have a quantum system of a 2-dimensional Hilbert space $\mathcal{H}$ and a Hamiltonian $\hat H$.

My puzzle: *What is the largest possible global symmetry for the Hilbert space $\mathcal{H}$ and Hamiltonian $\hat H$?*

My attempt: By symmetry of a quantum system, in some sense, we meant to find the quantum symmetry transformation $\hat S$ as an operator in terms of matrx such that
$$
\hat S \hat H \hat S^{-1} =\hat H .
$$

Naively, if we have $\hat H $ is proportional to an identity matrix $\mathbb{I}$ acting on 2-dim state vector in $|v \rangle \in\mathcal{H}$ , we have at most a constraint for
$$
\hat S \hat S^{-1} =\mathbb{I}.
$$
This means that the symmetry forms a invertible matrices with complex entries known as the general linear matrix group
$$
GL(2,\mathbb{C}).
$$
However, this does only include the linear symmetry, but not the anti-linear symmetry such as complex conjugation $\hat K$. So again, **What is the largest possible global symmetry for the Hilbert space $\mathcal{H}$ and Hamiltonian $\hat H$?**

This post imported from StackExchange Physics at 2020-12-04 11:33 (UTC), posted by SE-user annie marie heart