Holonomy of a principal composite bundle connection, non-abelian geometric phases and gauge theory of gravity

Abstract

We show that the holonomy of a connection defined on a principal composite bundle is related by a non-abelian Stokes theorem to the composition of the holonomies associated with the connections of the component bundles of the composite. We apply this formalism to describe the non-abelian geometric phase (when the geometric phase generator does not commute with the dynamical phase generator). We find then an assumption to obtain a new kind of separation between the dynamical and the geometric phases. We also apply this formalism to the gauge theory of gravity in the presence of a Dirac spinor field in order to decompose the holonomy of the Lorentz connection into holonomies of the linear connection and of the Cartan connection.

15 pages, 1 figure

References in corpus (6)

• Nonabelian Bundle Gerbes, their Differential Geometry and Gauge Theory
• Product Integral Formalism and Non-Abelian Stokes Theorem
• Quantum mechanics with time-dependent parameters
• Covariant Spin Structure
• Control aspects of holonomic quantum computation
• Topological Quantum Field Theory on Non-Abelian Gerbes

Cited by in corpus (5)

• Non-abelian higher gauge theory and categorical bundle
• A new kind of geometric phases in open quantum systems and higher gauge theory
• Adiabatic theorem for bipartite quantum systems in weak coupling limit
• Adiabatic transport of qubits around a black hole
• Geometric phases in quantum control disturbed by classical stochastic processes