论文标题
到达Noether定理的底部
Getting to the Bottom of Noether's Theorem
论文作者
论文摘要
我们检查了连接对称和保护法的Noether定理背后的假设。为了比较该定理的经典和量子版本，我们采用代数方法。在经典和量子力学中，可观察物是约旦代数的自然要素，而单参数转换组的发电机自然是Lie代数的要素。每当我们可以将可观察到的发电机映射到以使每个可观察到的一个可观察到的单参数群中，Noether的定理都会保持。在普通复杂的量子力学中，此映射是乘以$ \ sqrt {-1} $的乘法。在Unital JB-Elgebras的更一般框架中，Alfsen和Shultz称其为“动态对应关系”，并显示其存在使我们能够识别具有复杂C*-Algebra的自相关部分的Unital JB-Elgebra。但是，为了证明他们的结果，他们对动态对应关系施加了第二，更晦涩的条件。我们表明，这表达了量子和统计力学之间的关系，与“逆温度为假想时间”的原则紧密相关。
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".
