Anonym said:
First of all: thank you. But what you mean "these are not the usual basis vectors in Fock space, which is usually spanned with the eigenvectors of the (E,P) observables, and not the EM configuration states"?
For fields, this is a kind of tricky issue. Compare it to the NR single-particle quantum system. The configuration space of a featureless point particle is simply R^3 (generalized coordinates x,y,z). The hilbert space of this single-particle quantum system is then, by definition, spanned by basis vectors which are indexed upon the configuration space: each |x,y,z> is a (generalized) basis vector spanning the Hilbert space. But the "fock space basis" equivalent of this Hilbert space is spanned by the common eigenstates of the {E,Px,Py,Pz} set of operators, which turn out to be non-degenerate in this case and correspond to the |E,px,py,pz> vectors, with E = 1/(2m) (px^2 + py^2 + pz^2).
Of course, in this example, they are simply related by a Fourier transform.
For the EM case it is more tricky. The Fock space basis is again given by the eigenvectors of {E,Px,Py,Pz}, but this time, there is a lot of degeneracy, and it turns out that a tidy way of writing down all these eigenvectors is by using finite n-tuples of natural numbers, which correspond to "photon states" with n different modes. The relationship to a classical EM *configuration* is not so evident, and I'm kinda struggling myself to get a clear view on this. Mind you that a classical *mode* is not a configuration, but an entire solution to the classical EM field equation ; one should picture a classical configuration rather as a totally arbitrary vector potential distribution *at a single instant*. (if I understand this well, one shouldn't even take A but a related quantity). Note that a configuration doesn't determine the energy (we also need the derivatives of the potential for that) for instance. To each such configuration must correspond then also a basis vector, which should be able to be expressed as a function of the Fock elements.
This is (partially) treated in Mandel and Wolf, section 10.4 - but I have to say that this is some time ago that I studied this, and I remember vaguely not having understood everything clearly myself. But the idea goes more or less as follows: to each classical EM mode is assigned a harmonic oscillator, which has a single q and a single p variable assigned to it. The independent quantization of each of these harmonic oscillators corresponds to the canonical quantization procedure for the EM field and leads to the Fock basis. But the set of all these q variables of all these harmonic oscillators is nothing else but a set of generalized coordinates of the configuration space of the EM field. We can write the inproduct between a specific Fock basis state and a configuration eigenstate, using the position representation for the energy eigenstates of harmonic oscillators.
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