Second-Quantized Field-Theoretical Formulation

Taking all this into account, consistent quantization of this gauge field theory may be achieved, e.g., by a constrained canonical procedure [138,139] or the manifestly covariant Gupta-Bleuler formalism, which employs an indefinite metric at the expense of an easy physical interpretation [140-142]. In the following the basic ideas of the canonical procedure will briefly be presented. 

By following the ideas of canonical quantization, both manifest Lorentz co-variance and gauge invariance of the theoretical description have been sacrificed. All physical quantities such as transition amplitudes and S-matrix elements, however, will be Lorentz covariant and independent of the chosen 

The last necessary ingredient toward a well-defined and consistent quantum field theory of radiation interacting with a fermionic matter field is the introduction of normal-ordered products of field operators. The necessity for this step is easily realized by consideration of the vacuum expectation value of Gauss law. 

Our discussion of the QED Hamiltonian is deliberately brief (and somewhat imprecise) because it would otherwise require a very extensive presentation of especially the interaction part of matter and photon fields. We come back to this issue in section 8.1.5 before we derive the electromagnetic interaction energies in semi-classical many-electron theory sufficient for molecular science. However, for more details on the QED Hamiltonian we refer to Ref. [65, p. 283 ff.]. 

Although the charge density. q x) is an operator not commuting with the Hamiltonian, the total charge 

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