This concept of proper confinement is then used to resolve gauge ambiguities by ensuring that any two arbitrary gauges can be connected through unitary transformations within the P subspace. For either the dipole or Coulomb gauge in the full Hilbert space, the truncated analog can be formulated in four steps. First, represent the full space Hamiltonian in terms of ĤM , Ĥph and Û (as done in Eqs. 49 and 50). Second, truncate ĤM and Ĥph in their eigenbases. Third, redefine the PZW operator, Û , to be properly confined in the P subspace in terms of x and p. This can be done by applying the projection operator inside the exponential of Û as follows