Time evolution generated by grand canonical ensemble Our professor in condensed matter physics derived a differential equation for the zero temperature Greens functions. In the derivation he makes use of the Heisenberg equation of motion to find the derivative of the particle field operator, i.e.
$i\partial _t \Psi(x, t) = [\Psi, H].$
Here $H = H_{s} - \mu N$, with the "normal" Hamiltonian describing the system $H_s$, chemical potential $\mu$ and the number operator $N$.
Why is this reasonable? In some situations it might come handy to make the above redefinition but why would the the chemical potential part impact the time evolution of the system on the microscopic level?
The greens function are introduced as $G = -i<T\{\Psi(x,t)\Psi^\dagger(x',t')\}>$.
 A: In most cases of interest adding term $\mu N = \mu\sum_k c_k^\dagger c_k$ merely shifts the energy of the particles:
$$\sum_k E_kc_k^\dagger c_k \rightarrow \sum_k (E_k-\mu)c_k^\dagger c_k.$$
Thus, nothing changes from the point of view of the dynamics. On the other hand, having the evolution operator to coincide with the one in the density matrix used for averaging,
$$\rho = e^{-\beta(H_s-\mu N)},$$
simplifies a lot of math, e.g. when deriving the Lehmann representation, Matsubara expansion, etc.
Update
Following the discussion, it is necessary to add a correction: the expositions of the Green's function formalism usually deal with the situations where the number of particles is conserved ($[N, H_s]=0$) or where the particles are created/annihilated in pairs. If the Hamiltonian contains source terms adding odd numbers of particles, it usually requires extension of the formalism, as, e.g., in the case of superfluidity/superconductivity.
A: It's because $\mu$ determines the equilibrium particle density and having a lot of particles about  definitely affects the behaviour of an interacting  system.
