深入探究诺特定理
Getting to the Bottom of Noether's Theorem
AI总结：
本文采用代数方法比较经典与量子诺特定理，揭示 Alfsen 和 Shultz 提出的动力学对应第二条件表达了量子与统计力学间的关系，与“逆温度是虚时间”原理密切相关。
摘要：
我们考察了诺特定理背后的假设，该定理将对称性与守恒定律联系起来。为了比较该定理的经典版本与量子版本，我们采用了代数方法。在经典力学与量子力学中，可观测量自然是 Jordan 代数的元素，而单参数变换群的生成元自然是 Lie 代数的元素。只要我们能将可观测量映射为生成元，使得每个可观测量生成一个保持自身的单参数群，诺特定理便成立。在普通的复量子力学中，这种映射是乘以 $\sqrt{-1}$。在 unital JB-algebras 的更一般框架中，Alfsen 和 Shultz 将这种映射称为“动力学对应”，并证明其存在使我们能够将该 unital JB-algebra 等同于复 C*-algebra 的自伴部分。然而，为了证明其结果，他们对动力学对应施加了第二个更隐蔽的条件。我们证明这表达了量子力学与统计力学之间的关系，与“逆温度是虚时间”这一原理密切相关。
英文摘要：
We examine the assumptions behind Noether's theorem connecting symmetries and conservation laws. To compare classical and quantum versions of this theorem, we take an algebraic approach. In both classical and quantum mechanics, observables are naturally elements of a Jordan algebra, while generators of one-parameter groups of transformations are naturally elements of a Lie algebra. Noether's theorem holds whenever we can map observables to generators in such a way that each observable generates a one-parameter group that preserves itself. In ordinary complex quantum mechanics this mapping is multiplication by $\sqrt{-1}$. In the more general framework of unital JB-algebras, Alfsen and Shultz call such a mapping a "dynamical correspondence", and show its presence allows us to identify the unital JB-algebra with the self-adjoint part of a complex C*-algebra. However, to prove their result, they impose a second, more obscure, condition on the dynamical correspondence. We show this expresses a relation between quantum and statistical mechanics, closely connected to the principle that "inverse temperature is imaginary time".
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