Bundle Description Research Articles

On quantum and parallel transport in a Hilbert bundle over spacetime

We study the Hilbert bundle description of stochastic quantum mechanics in curved spacetime developed by Prugovecki, which gives a powerful new framework for exploring the quantum mechanical propagation of states in curved spacetime. We concentrate on the quantum transport law in the bundle, specifically on the information which can be obtained from the flat space limit. We give a detailed proof that quantum transport coincides with parallel transport in the bundle in this limit, confirming statements of Prugovecki. Furthermore, we show that the quantum-geometric propagator in curved spacetime proposed by Prugovecki, yielding a Feynman path integral-like formula involving integrations over intermediate phase-space variables, is Poincaré gauge-covariant (i.e. is gauge-invariant except for transformations at the endpoints of the path) provided the integration measure is interpreted as a `contact point measure' in the soldered stochastic phase-space bundle raised over curved spacetime.

Torsion in fiber bundle models of gauge theories

We investigate the possibility of having non-zero torsion in the geometrical fiber bundle models of gauge theories. We specifically look at torsion in the principal bundle space itself rather than in the base space (space-time). The formalism is developed for a general non-Abelian gauge group. We then look in more detail at the example of torsion in the fiber bundle description of electromagnetism. In this case, we show that torsion is related to magnetic monopoles.

Gauge theory of a group of diffeomorphisms. III. The fiber bundle description

A new fiber bundle approach to the gauge theory of a group G that involves space-time symmetries as well as internal symmetries is presented. The ungauged group G is regarded as the group of left translations on a fiber bundle G(G/H,H), where H is a closed subgroup and G/H is space-time. The Yang–Mills potential is the pullback of the Maurer–Cartan form and the Yang–Mills fields are zero. More general diffeomorphisms on the bundle space are then identified as the appropriate gauged generalizations of the left translations, and the Yang–Mills potential is identified as the pullback of the dual of a certain kind of vielbein on the group manifold. The Yang–Mills fields include a torsion on space-time.

Higgs fields from symmetric connections—The bundle picture

The derivation (Forgács and Manton) of a classical gauge plus Higgs field theory from a purely symmetric gauge theory by dimensional reduction is translated into the pictorial language of symmetric connections on principal bundles. The bundle description provides a better understanding of symmetric gauge theories; its global nature excludes some gauge group-symmetry group combinations, and shows that the reduced gauge group is larger than the centralizer CH as previously thought. In general it is a certain factor group, NK/K, involving both the gauge and symmetry groups, and contains CH as a normal subgroup. The commonly used F2 Lagrangian of symmetric gauge theories will not prescribe the dynamics of fields associated with this larger group, unless these fields are suitably constrained. It is shown that a constrained Kaluza–Klein metric is required to construct the F2 Lagrangian, thus pointing the way to include dynamics for the NK/K group.

Another geometrical representation of supergravity

An alternative approach to a geometrical description of the contents of supergravity is presented. This description is developed in close analogy to the fibre bundle description of gauge fields. A curious feature is that the torsion tensor is not needed and is assumed to be zero throughout.

Internal geometrical structures on Minkowski space-time and Yang-Mills symmetries

Yang-Mills potentials and fields on Minkowski space-time are equivalent to connections and curvatures for corresponding principle fibre bundles over Minkowski space-time. Therefore, the usual Yang-Mills field equations, together with Bianchi's identities, define transport equations for the connection and curvature with respect to a given observer field (inertial system). This equivalence is worked out; furthermore, the notion of symmetric, particularly spherically symmetric Yang-Mills connections is transposed to the bundle description. We show that the restrictions on the Yang-Mills connection imposed by spherical symmetry are by no means as severe as assumed by Ikeda and Miyachi.