Quantum Gauge Transformation, Gauge-Invariant Extension and Angular Momentum Decomposition in Abelian Higgs Model
I discuss the momentum and angular momentum decomposition problem in the Abelian Higgs model. The usual gauge-invariant extension (GIE) construction is incorporated naturally into the framework of quantum gauge transformation $\grave{a}$ ${\it la}$ Strocchi and Wightman and with this I investigate the momentum and angular momentum separation in a class of GIE conditions which correspond to the so-called "static gauges". Using this language I find that the so-called "generator criterion" does not generally hold even for the dressed physical field. In the case of $U(1)$ symmetry breaking, I generalize the standard GIE construction to include the matter field degrees of freedom so that the usual separation pattern of momentum and angular momentum in the unitarity gauge can be incorporated into the same universal framework. When the static gauge condition could not uniquely fix the gauge, I show that this GIE construction should be expanded to take into account the residual gauge symmetry. In some cases I reveal that the usual momentum or angular momentum separation pattern in terms of the physical dressed field variables needs some type of modification due to the nontrivial commutator structure of the underlying quantum gauge choice. Finally, I give some remarks on the general GIE constructions in connection with the possible commutator issues and modification of momentum and angular momentum separation patterns.