such that all terms in Û = ∞ n=0 1 n! (-i ℏ ) n ( P μ P) n Ân are properly confined within the subspace P, and upon gauge transformation, all light-matter interaction operators are now consistently confined in P for both gauges. Here, Û is defined analogously to the PZW gauge operator Û in the full space (Eq. 5), and P μ P ≡ μ(x, p) in principle is a function of both x and p, due to the finite level projection that ruins the locality of x31,62 . Further, Û is a unitary transformation operator in the P subspace and the identity operator in the subspace of 1r -P, such that we still have Û Û † = 1r ⊗ 1R ⊗ 1ph = Û Û † . Using Û, one can define the following Coulomb gauge Hamiltonian 36