Quantum mechanics in magnetic backgrounds with manifest symmetry and locality
Quantum mechanics in magnetic backgrounds with manifest symmetry and locality
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具有明显对称性和局域性的磁背景中的量子力学
DOI：
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发表时间：
2019
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通讯作者：
Joseph Tooby
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文献类型：
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作者：
Joe Davighi;B. Gripaios;Joseph Tooby
通常用来表述和求解在磁场中运动的粒子的量子力学的方法，既不考虑局部性，也不考虑碰巧存在的任何整体对称性。例如，朗道的解决方案，一个粒子运动在一个均匀的磁场中的平面涉及到选择一个规范，其中既没有平移也没有旋转不变性是明显的。我们表明，局域性可以通过传递到一个冗余的描述，其中粒子在原始配置空间上的U（1）-主丛上移动，并且对称性可以通过传递到原始对称群的相应中心扩展U（1）来表现。有了对称性的证明，人们可以尝试用调和分析来解决这个问题，我们提供了一些成功的例子。一个是在任意规范（具有平移不变性或完全欧几里德群证明）中解决朗道问题。另一个例子是费米子刚体的运动，它可以通过由U（2）给出的位形空间SO（3）的非平凡U（1）-中心扩展上的平坦连接以明显局部和对称的方式来公式化和求解。
The usual methods for formulating and solving the quantum mechanics of a particle moving in a magnetic field respect neither locality nor any global symmetries which happen to be present. For example, Landau’s solution for a particle moving in a uniform magnetic field in the plane involves choosing a gauge in which neither translation nor rotation invariance are manifest. We show that locality can be made manifest by passing to a redundant description in which the particle moves on a U(1)-principal bundle over the original configuration space and that symmetry can be made manifest by passing to a corresponding central extension of the original symmetry group by U(1). With the symmetry manifest, one can attempt to solve the problem by using harmonic analysis and we provide a number of examples where this succeeds. One is a solution of the Landau problem in an arbitrary gauge (with either translation invariance or the full Euclidean group manifest). Another example is the motion of a fermionic rigid body, which can be formulated and solved in a manifestly local and symmetric way via a flat connection on the non-trivial U(1)-central extension of the configuration space SO(3) given by U(2).