Second quantized formulation of geometric phases
Second quantized formulation of geometric phases
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DOI：
10.1103/physreva.72.012111
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发表时间：
2005-01
影响因子：
2.9
通讯作者：
S. Deguchi;K. Fujikawa
中科院分区：
文献类型：
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作者：
S. Deguchi;K. Fujikawa
水平交叉问题和相关的几何条款整齐地制定了第二个量化的配方。这个公式表现出一个隐藏的局部规范对称性有关的任意性的相位选择的完整的正交基组。通过使用这种二次量子化的公式，它不假设绝热近似，一个方便的精确公式的几何项，包括非对角几何项推导。几何相位的分析，然后减少到一个简单的对角化的哈密顿量，它是在运营商和路径积分公式进行分析。如果对角化水平交叉的无穷小邻域中的几何项，则对于任意大但有限的时间间隔T，几何相位变得平凡（因此没有奇异性）。薛定谔方程的可积性与不可积相的出现是一致的。例如，隆盖-希金斯相变规则的拓扑证明在实际的玻恩-奥本海默近似中失败了，其中涉及两个时间尺度的大而有限的比率，并且T与较慢系统的周期相同。与水平交叉相关联的几何相位和精确的拓扑对象（如Aharonov-Bohm相位）之间的差异和相似性在更现有的公式中变得清晰。量子反常和几何相位之间的一个重要区别也被指出。«少
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 Schroedinger equation and the appearance of the seemingly nonintegrable 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 themore » present formulation. A crucial difference between the quantum anomaly and the geometric phases is also noted.« less