ABSTRACT
In this chapter, the authors prove that the free electromagnetic field is equivalent to an infinite set of uncoupled one-dimensional harmonic oscillators, one oscillator being associated with each value of the propagation vector k and polarization mode α. In order to quantize the electromagnetic field the authors shall have to express the field equations in Hamiltonian form and impose the canonical commutation rules on the canonically conjugate variables. Because the electromagnetic field possesses an infinite number of degrees of freedom it is also possible to formulate the theory in terms of a very different set of dynamical variables, the field operators themselves. To be more precise, the field operators only have off-diagonal matrix elements in the representation in which the photon occupation numbers are diagonal. In a state with a definite number of photons the expectation value of any field strength vanishes, and it is only meaningful to discuss fluctuations in the field strengths.
