Using the projection operator, one can define the pro-jected matter Hamiltonian (or the truncated matter Hamiltonian) as follows ĤM = P ĤM P = P T P + P V (x) P. (27) Throughout this review, we use calligraphic symbols (such as ĤM ) to indicate operators in the truncated Hilbert space, which we have already started in Eq. 1 of the Introduction.