The connection between conformal group and quantum states of photons

Abstract

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.