We begin with the simplest case of matter truncation. Consider a finite subset of electronic states {|α⟩}, where the projection operator P = α |α⟩⟨α| defines the truncation of the full electronic Hilbert space 1r = P + Q to the corresponding subspace P. This truncation reduces the size of the Hilbert space from originally 1r ⊗ 1R ⊗ 1ph to now P ⊗ 1R ⊗ 1ph , where 1R and 1ph represent the identity operators of the nuclear and photonic DOF, respectively. The truncated matter Hamiltonian is ĤM = P ĤM P = P T P + P V (x) P. (22) Throughout this review, we use calligraphic symbols (such as ĤM ) to indicate operators in the truncated Hilbert space. The simplest way to then write the QED Hamiltonians from Sects. II A and II B in this truncated subspace would then be to truncate each matter operator by P. Truncating the momentum operator and dipole operator as P pj P and P μ P, the QED Hamiltonians under the truncated subspace are commonly defined as