The level crossing problem and associated geometric terms are neatly formulated by the second quantized formulation. This formulation exhibits a hidden local gauge symmetry related to the arbitrariness of the phase choice of the complete orthonormal basis set. By using this second quantized formulation, which does not assume adiabatic approximation, a convenient exact formula for the geometric terms including off-diagonal geometric terms is derived. The analysis of geometric phases is then reduced to a simple diagonalization of the Hamiltonian, and it is analyzed both in the operator and path integral formulations. If one diagonalizes the geometric terms in the infinitesimal neighborhood of level crossing, the geometric phases become trivial (and thus no monopole singularity) for arbitrarily large but finite time interval T . The integrability of Schrödinger equation and the appearance of the seemingly non-integrable phases are thus consistent. The topological proof of the Longuet-Higgins' phase-change rule, for example, fails in the practical Born-Oppenheimer approximation where a large but finite ratio of two time scales is involved and T is identified with the period of the slower system. The difference and similarity between the geometric phases associated with level crossing and the exact topological object such as the Aharonov-Bohm phase become clear in the present formulation. A crucial difference between the quantum anomaly and the geometric phases is also noted.

We consider two quantization approaches to the Bateman oscillator model. One is Feshbach-Tikochinsky's quantization approach reformulated concisely without invoking the SU (1, 1) Lie algebra, and the other is the imaginaryscaling quantization approach developed originally for the Pais-Uhlenbeck oscillator model. The latter approach overcomes the problem of unboundedbelow energy spectrum that is encountered in the former approach. In both the approaches, the positive-definiteness of the squared-norms of the Hamiltonian eigenvectors is ensured. Unlike Feshbach-Tikochinsky's quantization approach, the imaginary-scaling quantization approach allows to have stable states in addition to decaying and growing states.

A gauge-fixing procedure for the Yang-Mills theory on an n-dimensional sphere (or a hypersphere) is discussed in a systematic manner. We claim that Adler's gauge-fixing condition used in massless Euclidean QED on a hypersphere is not conventional because of the presence of an extra free index, and hence is unfavorable for the gauge-fixing procedure based on the BRST invariance principle (or simply BRST gauge-fixing procedure). Choosing a suitable gauge condition, which is proved to be equivalent to a generalization of Adler's condition, we apply the BRST gauge-fixing procedure to the Yang-Mills theory on a hypersphere to obtain consistent results. Field equations for the Yang-Mills field and associated fields are derived in manifestly O(n + 1) covariant or invariant forms. In the large radius limit, these equations reproduce the corresponding field equations defined on the n-dimensional flat space.

The smooth topology change of Berry's phase from a Dirac monopole-like configuration to a dipole configuration, when one approaches the monopole position in the parameter space, is analyzed in an exactly solvable model. A novel aspect of Berry's connection A k is that the geometrical center of the monopole-like configuration and the origin of the Dirac string are displaced in the parameter space. Gauss' theorem S (∇ × A) · d S = V ∇ · (∇ × A)dV = 0 for a volume V which is free of singularities shows that a combination of the monopole-like configuration and the Dirac string is effectively a dipole. The smooth topology change from a dipole to a monopole with a quantized magnetic charge e M = 2π takes place when one regards the Dirac string as unobservable if it satisfies the Wu-Yang gauge invariance condition. In the transitional region from a dipole to a monopole, a half-monopole appears with an observable Dirac string, which is analogous to the Aharonov-Bohm phase of an electron for the magnetic flux generated by the Cooper pair condensation. The main topological features of an exactly solvable model are shown to be supported by a generic model of Berry's phase.

We discuss a quantum-theoretical aspect of the massive Abelian antisymmetric tensor gauge theory with antisymmetric tensor current. To this end, an