Group Transformations of Semiclassical Gauge Systems

Semiclassical systems being symmetric under Lie group are studied. A state of a semiclassical system may be viewed as a set (X,f) of a classical state X and a quantum state f in the external classical background X. Therefore, the set of all semiclassical states may be considered as a bundle ("semiclassical bundle"). Its base {X} is the set of all classical states, while a fibre is a Hilbert space of quantum states in the external background X. Symmetry transformation of a semiclassical system may be viewed as an automorphism of the semiclassical bundle. Automorphism groups can be investigated with the help of sections of the bundle: to any automorphism of the bundle one assigns a transformation of section of the bundle. Infinitesimal properties of transformations of sections are investigated; correspondence between Lie groups and Lie algebras is discussed. For gauge theories, some points of the semiclassical bundle are identified: a gauge group acts on the bundle. For this case, only gauge-invariant sections of the semiclassical bundle are taken into account.