The largest possible global symmetry of a 2-dimensional Hilbert space?

Question

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

Comments

I would say any group that admits a homomorphism to the general linear group. (In fact you should generalize to projective representations due to the structure of QM. This is related to Wigner's theorem.)

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