量化约束系统：新观点
Quantizing Constrained Systems: New Perspectives
摘要：
本文研究具有非欧度量和可变高斯曲率的约束曲面上的量子力学。关于hbar ^2阶哈密顿量乘以高斯曲率的项的模糊性的老争论得到了解决。我们着手澄清这一问题，考虑约束是大的回复力的限制，因为约束坐标偏离其约束值。我们还发现了额外的hbar ^2级的二义性项，这些二义性项涉及约束势的自由度，这表明经典的约束哈密顿量或拉格朗日量不能唯一地确定量子化：直接量子化约束系统的二义性本质上是不可解的。然而，物理量子系统从来没有任何问题，它不可能有无限的约束力，并且总是围绕平均约束值波动。这个问题是从绝热近似的角度在量子力学，费曼路径积分，和半经典的绝热行动。
英文摘要：
We consider quantum mechanics on constrained surfaces which have non-Euclidean metrics and variable Gaussian curvature. The old controversy about the ambiguities involving terms in the Hamiltonian of order hbar^2 multiplying the Gaussian curvature is addressed. We set out to clarify the matter by considering constraints to be the limits of large restoring forces as the constraint coordinates deviate from their constrained values. We find additional ambiguous terms of order hbar^2 involving freedom in the constraining potentials, demonstrating that the classical constrained Hamiltonian or Lagrangian cannot uniquely specify the quantization: the ambiguity of directly quantizing a constrained system is inherently unresolvable. However, there is never any problem with a physical quantum system, which cannot have infinite constraint forces and always fluctuates around the mean constraint values. The issue is addressed from the perspectives of adiabatic approximations in quantum mechanics, Feynman path integrals, and semiclassically in terms of adiabatic actions.
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