is ZPE of the other projected modes coming from P(m) (â † k + âk ) 2 P(m) = P(m) for k ≥ m (note that the only surviving term is P(m) âk â † k P(m) = P(m) ). This is very different from what happens in the case of a matter truncation. Now, the Hamiltonian that fails under a simple mode truncation is the dipole gauge Hamiltonian.