In many-body theory (and quantum field theory I suppose) we often work in the grand canonical ensemble, where the number of particles in the system is only fixed on average. The density operator used to compute expectation values is
$$ \rho = \frac{e^{-\beta (H - \mu N)}}{Z_{G}} = \frac{e^{-\beta K}}{Z_{G}} $$
where $K$ is the so-called grand canonical hamiltonian.
My problem is when we substitute $H$ for $K$ in the expression of the evolution operator $ U(t) = e^{-iHt} \rightarrow e^{-iKt} $, which is done most of the time because it simplifies calculations. It seems equivalent to saying that Schrodinger equation is invariant under the change $H \rightarrow K$.
The justifications I've seen so far, are based on the fact that since the original hamiltonian conserves the number of particles and thus commutes with the operator $N$, this replacement is just a displacement in energy and does not essentially changes the dynamics of the system. I'm not really convinced by this because the replacement with $K$ is not equivalent with adding a simple constant to the hamiltonian.
Additionally, it seems that if this argument is true, then in general one would be justified to construct a new hamiltonian $H' = H + \mathcal{O}$ to describe the dynamics of a system, as long as $[H, \mathcal{O}]=0$.
Given that, my questions are as follow :
- Is the above statement that you can replace a hamiltonian $H' = H + \mathcal{O}$ when $[H, \mathcal{O}]=0$ true? If it'sn't, is there anything special with the case $\mathcal{O} = \mu N$, or are there some caveats?
- Thermodynamics ensembles are often defined by density operators on the form $\rho=e^{-\beta S}$ with $S$ that is a linear combination of $H$ and various conserved operators $c_j \mathcal{O}_j$. Could we get rid of those ensembles by just including the operators $c_j \mathcal{O}_j$ in the hamiltonian in the first place and finding $c_j$ such that $\langle \mathcal{O_j} \rangle$ is the value we want? $c_j$ would have the physical interpretation of a classical field that couples with $\mathcal{O}_j$.