How do I find the combined Hilbert space for this Hamiltonian?

Question

I knew that the operators in the folwing Hamiltonian act in different Hilbert spaces, so I cannot just multiply them.

$$\eqalign{ & H = g\left[ {\left( {a\sigma _1^ + + {a^\dagger }\sigma _1^ - } \right) + \left( {a\sigma _2^ + + {a^\dagger }\sigma _2^ - } \right)} \right] \cr & \,\,\,\,\,\, + J\left[ {\left( {S_1^ + \sigma _1^ - + S_1^ - \sigma _1^ + } \right) + \left( {S_2^ + \sigma _2^ - + S_2^ - \sigma _2^ + } \right)} \right] \cr} $$

How do I find a combined Hilbert space for this Hamiltonian?

Answer 1

I'm going to assume that the types of operators denoted by $a,\sigma$, and $S$ each act on separate Hilbert spaces $\mathcal H_a,\mathcal H_\sigma,\mathcal H_S$. Then your Hamiltonian should act on $\mathcal H:=\mathcal H_a\otimes \mathcal H_\sigma\otimes\mathcal H_S$, and the operators it's made up of should act like $$a(\vert\psi_a\rangle\otimes\vert\psi_\sigma\rangle\otimes\vert\psi_S\rangle)=(a\vert\psi_a\rangle)\otimes\vert\psi_\sigma\rangle\otimes\vert\psi_S\rangle,$$ and similarly for the other operators. Basically, each operator only acts on the corresponding part of the state.