As discussed in Ref. 81, this idea can be generalized to any kind of truncation of a Hilbert space, even for those going beyond just material truncation. For example, the gauge-transformation operator can also be constructed for cavity photonic mode truncation, where the projection operator P will also include the cavity mode truncation (by projecting out the corresponding Fock states of those truncated modes, except for the group Fock state). For that case, the most general expression of Û becomes 81