  Coherent states in a projected Hilbert space have many useful properties.
When there are conserved quantities, a representation of the entire Hilbert
space is not necessary. The same issue arises when conditional observations are
made with post-selected measurement results. In these cases, only a part of the
Hilbert space needs to be represented, and one can define this restriction by
way of a projection operator. Here coherent state bases and normally-ordered
phase-space representations are introduced for treating such projected Hilbert
spaces, including existence theorems and dynamical equations. These techniques
are very useful in studying novel optical or microwave integrated photonic
quantum technologies, such as boson sampling or Josephson quantum computers. In
these cases states become strongly restricted due to inputs, nonlinearities or
conditional measurements. This paper focuses on coherent phase states, which
have especially simple properties. Practical applications are reported on
calculating recurrences in anharmonic oscillators, the effects of arbitrary
phase-noise on Schrodinger cat fringe visibility, and on boson sampling
interferometry for large numbers of modes.
