Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2005-07-11
Prog.Theor.Phys.Suppl.164:194-202,2007
Physics
High Energy Physics
High Energy Physics - Theory
10 pages with 2 figures. Talk given at the International Workshop "Frontiers in Quantum Physics", Yukawa Institute for Theoret
Scientific paper
10.1143/PTPS.164.194
The level crossing problem and associated geometric terms are neatly formulated by using the second quantization technique both in the operator and path integral formulations. The analysis of geometric phases is then reduced to the familiar diagonalization of the Hamiltonian. If one diagonalizes the Hamiltonian in one specific limit, one recovers the conventional formula for geometric phases. On the other hand, 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 topological proof of the Longuet-Higgins' phase-change rule, for example, thus 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.
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