Geometric phase in eigenspace evolution of invariant and adiabatic action operators

Area of Science:

• Quantum mechanics
• Differential geometry
• Topology

Background:

• Geometric phase, also known as Berry phase, describes phase changes in quantum systems undergoing cyclic evolution.
• Degenerate systems present unique challenges in defining and understanding geometric phases.
• The Stiefel and Grassmann manifolds are fundamental in understanding vector bundles and holonomy.

Purpose of the Study:

• To generalize the theory of geometric phase to N-fold degenerate eigenspaces.
• To interpret the geometric phase as a holonomy derived from a universal U(N) bundle.
• To formulate a rigorous theory for geometric phase in adiabatic evolution of eigenspaces.

Main Methods:

• Generalization of geometric phase theory for degenerate eigenspaces.
• Interpretation of geometric phase as holonomy over a Grassmann manifold.
• Formulation of a rigorous theory for adiabatic evolution using pullback U(N) bundles.

Main Results:

• The geometric phase for N-fold degenerate eigenspaces is identified as a holonomy from the universal Stiefel U(N) bundle.
• This holonomy accurately captures the geometric features of state evolution, irrespective of cyclicity.
• A theory for geometric phase in adiabatic evolution of eigenspaces is established, linked to a pullback U(N) bundle.

Conclusions:

• The generalized geometric phase provides a robust framework for understanding state evolution in degenerate quantum systems.
• Holonomy offers a powerful geometric interpretation of phase evolution, extending beyond cyclic processes.
• The developed theory is crucial for analyzing complex quantum dynamics and geometric properties.