Given a Hilbert space $\mathcal{H}$, we pick up some state vector $| \psi\rangle$ which lives in the Hilbert space $\mathcal{H}$.

The $| \psi\rangle$ is a vector of the Hilbert space satisfies the rule like
$\langle\psi| \psi\rangle =1$.

My question is about when a Hilbert space's state vector $| \psi\rangle$ can become a spinor? Of course I need to clarify what I meant and possible interpretations.

can the state vector $| \psi\rangle$ be a spinor as a spinor representation of the spacetime symmetry? (the projective representation of the rotational group $SO(d,1)$)? If so, if this state vector $| \psi\rangle$ is a ground state of the Hamiltonian system, we can transform the state vector $| \psi\rangle$ as the spinor under the rotational group $SO(d,1)$ or more precisely the spin group $Spin(d,1)$. Does this imply that the system have finite or infinite many ground states? [Well, if the spinor representation is finite dimensional, it looks that we still have a finite dimensional ground state subspace of the Hilbert space.] In any case, is this possible?

can the state vector $| \psi\rangle$ be a spinor as a spinor representation of the flavor or internal symmetry, of say some $Spin$ group? (For example, we can have a solid state spin system with $Spin(3)=SU(2)$, symmetry, and one of the state vector $| \psi\rangle$ is a spinor representation (spin-1/2) of the $Spin(3)=SU(2)$.) If so, what are the implications of the properties and dynamics of the system?

This post imported from StackExchange Physics at 2020-12-03 17:31 (UTC), posted by SE-user annie marie heart