For the Lorentz group, the representations and of the generators and fulfill the following commutation rules.
2.
The latter expression has further problems, as it can be shown that the two terms are not true angular momenta as they do not obey the correct quantum commutation rules.
3.
To this end, we shall endow it with ladder operators for the energy-momentum eigenconfigurations ( particles ) of each field, complete with appropriate ( anti-) commutation rules, as well as a singular exclusively along directions that correspond to BRST exact eigenstates of the unperturbed Hamiltonian.
4.
In the current example the coupling of a dipole oscillator to the electromagnetic field has a dissipative component, in the form of the zero-point ( vacuum ) field; given the existence of radiation reaction, the vacuum field must also exist in order to preserve the canonical commutation rule and all it entails.
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