Quantum Mechanics and Electromagnetism. Non-Relativistic Theory.

Abstract

We describe here the coherent formulation of electromagnetism in the nonrelativistic quantummechanical many-body theory. We use the mathematical frame of the field theory and its quantization in the spirit of the QED. This is necessary because of the manifold of misinterpretations emerging from the hystorical development of quantum mechanics, starting from the Schrödinger equation of a single particle in the presence of given electromagnetic fields, followed by the many-body theories of many charged identical particles having just Coulomb interactions inspired from the classical electromagnetic theory of point-like charges. However, this later is known to be inconsistent due to the self-interaction. This way could not be continued further to include properly the magnetic forces between the charged particles and lead to a lot of confusion about the interpretation of the magnetic field in the Hamiltonian, as well as about the gauge invariance. We emphasize the importance of the distinction between the applied (external fields) and the field in the matter. All these problems are length properly solved within the non-relativistic QED, nevertheless the confusion dominates in all the problems related to the magnetic properties of the solid state.

The Non-Relativistic Many-Body Quantum-Mechanical Hamiltonian with Diamagnetic Current-Current Interaction

“…Under this form it looks similar to the result we obtained before (in the absence of other external fields) and a quantization of the classical Hamiltonian of the point-like charged particles ( 14) is then immediate by second quantization of (16) as it was done in Ref. [5] and one may omit the configuration space quantization either of ( 14) or (13). This leads to the same Hamiltonian (6) as we deduced from the non-relativistic QED.…”

“…Such a theory may be constructed following the scheme of the relativistic QED, starting form the classical field theory based on the coupled Maxwell and the Schrödinger equations (instead Dirac's) followed by the quantization of the fields. Such an approach was described recently [13,14]. The standard presentation however starts from the Hamiltonian of a quantum mechanical charged particle in external electromagnetic field, quantizes both the wave function and the vector potential in the Coulomb gauge, while replacing the scalar potential with the standard quantized Coulomb interaction.…”

The Nonrelativistic Quantum-Mechanical Hamiltonian with Diamagnetic Current-Current Interaction

“…Such a theory may be constructed following the scheme of the relativistic QED, starting form the classical field theory based on the coupled Maxwell and the Schrödinger equations (instead Dirac's) followed by the quantization of the fields. Such an approach was described recently [9], [10]. The standard presentation however starts from the Hamiltonian of a quantum mechanical charged particle in external electromagnetic field, quantizes both the wave function and the vector potential in the Coulomb gauge, while replacing the scalar potential with the standard quantized Coulomb interaction.…”

Mean-Field Theory of the Meissner effect in Bulk Revisited

“…See in this context Dirac's theory of canonical formalism with constraints [16,17]. Actually one needs a new analysis of the problem in the frame of the non-relativistic QED (see [18,19] for a modern presentation). A natural choice of the gauge is here the Coulomb one.…”