共形群与光子量子态的关系
The connection between conformal group and quantum states of photons
摘要：
本文讨论共形群与光子量子态之间的关系。结果表明，光子量子态与共形群存在相似性，即真空态的时间发展算符（含自由哈密顿量）、位移算符和压缩算符分别对应于伸缩、平移和本征洛伦兹变换;光子的三个量子态（{ \it，即}，因此，量子光学中的三种变换（Fock态、相干态和压缩态）与共形群中的三种变换具有某种相似性。在此基础上，我们通过类比论证了，如果作用在真空态上的相变算符（因此产生了Fock态、相干态和压缩态）与保角变换完全相似，那么光子的第四个量子态（称为{ \it共形状态}），其对应于{ \it特殊保角}变换（加速度变换），因此具有特殊的物理意义。
英文摘要：
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.
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