The coherence properties of the classical waves are discussed in terms of the Cauchy problem for the wave equation, and of a discrete representation provided by a lattice of coupled Hamiltonian systems. Wave quanta are related to specific action–angle fields, and phase-space distributions of phonons and photons are obtained by Wigner transform. For photons in a thermal environment, the proposed Wigner function evolves towards the Planck equilibrium distribution. It is shown that the free electromagnetic field can also be found in states of definite helicity, described by a complex vector potential.

中文翻译：

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电磁场的量子分布

经典波的相干性质根据波动方程的柯西问题和由耦合哈密顿系统的晶格提供的离散表示进行讨论。波量子与特定的作用角场有关，声子和光子的相空间分布通过维格纳变换得到。对于热环境中的光子，建议的 Wigner 函数向普朗克平衡分布发展。结果表明，自由电磁场也可以在确定螺旋度的状态中找到，由复矢量势描述。