All the Hamiltonians previously discussed in this review are under the single mode approximation; however, many types of cavities such as the Fabry-Pérot cavity has a infinite number of quantized modes (see Sect. VI). It is then pertinent to consider how to truncate this manymode Hilbert space to a more computable space with a small finite number of modes. Whenever a truncation occurs, the gauge invariance condition may become ambiguous.