  This note is concerned with the connections between the conformal group and
the quantum states of photons. It is shown that there exist analogies between
the photonic quantum states and the conformal group, namely, the
time-development operator (with a free Hamiltonian), displacement and squeezing
operators of the vacuum state corresponds to the dilatation, translation,
proper Lorentz transformations, respectively, and that the three quantum states
of photons ({\it i.e.}, Fock state, coherent state and squeezed state) in
quantum optics thus bear some analogy to the above three transformations in the
conformal group. Based on this comparison, we argue by analogy that if the
phase transformation operators acting on a vacuum state (hence the Fock state,
coherent state and squeezed state are generated) are truly exactly analogous to
the conformal transformations, then a fourth quantum state of photons (referred
to as the {\it conformal state}), which corresponds to the {\it special
conformal} transformation (acceleration transformation) and will therefore be
of special physical interest, can be suggested.
