Maxwell Field Quantization

The calculation of the field that is used here is based on the pure laws of Maxwell, without quantization. Indeed, using Quantum Field Theory is not a necessity in the domain studied here. It leads exactly to the same results (see [12]), with sometimes longer calculations. 

The authors of Ref. [12] reconsidered the problem of magnetic field in quark matter taking into account the rotated electromagnetism . They came to the conclusion that magnetic field can exist in superconducting quark matter in any case, although it does not form a quantized vortex lattice, because it obeys sourceless Maxwell equations and there is no Meissner effect. In our opinion this latter result is incorrect, since the equations for gauge fields were not taken into account and the boundary conditions were not posed correctly. 

The electromagnetic field is quantized as a set of harmonic oscillators. Maxwell s equations, and the resulting wave equations, are described by partial differential equations that formally have an infinite number of degrees of freedom. Physically this means that the electromagnetic held is described by an infinite number of harmonic oscillators, where one sits at every point in space. The modes of the electromagnetic held are then completely described by this ensemble of harmonic oscillators. 

Let us consider an isolated molecule perturbed by an electromagnetic field. According to the semiclassical approach, the external radiation is described as a plane monochromatic wave traveling with velocity c and obeying the Maxwell equations [21] (i.e., the fields are not quantized). 

Nevertheless, the introduction of the potentials (p and A reduces the number of fields, and they are to be considered as the dynamical variables of the theory. Owing to the gauge symmetry given by Eqs. (2.130) and (2.131), it is obvious that Maxwell s theory of electromagnetism features redundant degrees of freedom which will seriously hamper its quantization in chapter 7. The potentials cp and A themselves are denoted gauge potentials. 

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