Geometric phases, gauge symmetries and ray representation

Abstract

The conventional formulation of the non-adiabatic (Aharonov-Anandan) phase is based on the equivalence class $\{e^{i\alpha (t)}\psi (t,\vec{x})\}$ which is not a symmetry of the Schrödinger equation. This equivalence class when understood as defining generalized rays in the Hilbert space is not generally consistent with the superposition principle in interference and polarization phenomena. The hidden local gauge symmetry, which arises from the arbitrariness of the choice of coordinates in the functional space, is then proposed as a basic gauge symmetry in the non-adiabatic phase. This re-formulation reproduces all the successful aspects of the non-adiabatic phase in a manner manifestly consistent with the conventional notion of rays and the superposition principle. The hidden local symmetry is thus identified as the natural origin of the gauge symmetry in both of the adiabatic and non-adiabatic phases in the absence of gauge fields, and it allows a unified treatment of all the geometric phases. The non-adiabatic phase may well be regarded as a special case of the adiabatic phase in this re-formulation, contrary to the customary understanding of the adiabatic phase as a special case of the non-adiabatic phase. Some explicit examples of geometric phases are discussed to illustrate this re-formulation.

30 pages. Some clarifying sentences have been added in abstract and in the body of the paper. A new additional reference and some typos have been corrected. To appear in Int. J. Mod. Phys. A

References in corpus (7)

• Extending the quantal adiabatic theorem: Geometry of noncyclic motion
• Noncyclic geometric phase and its non-Abelian generalization
• Second quantized formulation of geometric phases
• Geometric phases and hidden local gauge symmetry
• Topological properties of Berry's phase
• Quantum anomaly and geometric phase; their basic differences
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Cited by in corpus (7)

• Geometric phases for mixed states and decoherence
• Characteristics of Chiral Anomaly in View of Various Applications
• Adiabatic approximation in the second quantized formulation
• Berry's phase and chiral anomalies
• A path integral derivation of the equations of anomalous Hall effect
• Lensing of Dirac monopole in Berry's phase
• Geometric phase and chiral anomaly; their basic differences