论文标题
量子力学中的自洽方程的经典动力学 - 扩展版本
Classical Dynamics from Self-Consistency Equations in Quantum Mechanics -- Extended Version
论文作者
论文摘要
在过去的三十年中，P.Bóna基于正常状态的象征性结构，并提供了一种通用环境，并提供了一种很方便地研究微观量子过程的宏观经典动力学的出现。我们在这里提出了一种新的数学方法，对Bona的一种，具有许多适用性的兄弟领域。它突出了自遇到的核心作用。这导致了一个数学框架，在该框架中，经典和量子世界自然纠缠在一起。我们在任何$ c^{\ ast} $ -Elgebra上构建了在Hermitian弱$^{\ ast} $连续函数上的多项式函数的泊松支架。这让人想起有限维谎言代数的众所周知的结构。然后，我们将此泊松支架限制在此$ c^{\ ast} $ - 代数的状态下，以占用泊松理想的商。这导致在换向的$ c^{\ ast} $上的密集定义的对称派生 - 在一组状态下实现函数的代数。直到关闭，这些事实证明可以生成$ c_ {0} $ - 收缩组。事实上，总的来说，交换性$ c^{\ ast} $ - 代数，即使是无限的对称派生的封闭性也是一个非平凡的问题。引入了一些新的数学概念，它们本身可能很有趣：convex弱$^{\ ast} $gâteaux导数，与状态相关的$ c^{\ ast} $ - 动力学系统和弱$ $^{\ ast} $ - hausdorff hypertopology，一种新的$ hypertopology，又是$ hypertopology，又是其他损失的事物。 } $ - 紧凑型集合通常具有弱$^{\ ast} $ - 无限尺寸的密集极端边界。我们最近在具有远距离或均值场相互作用的晶格 - 纤维和量子旋转系统的宏观动力学特性的结果证实了我们此处介绍的一般方法的相关性。
During the last three decades, P. Bóna has developed a non-linear generalization of quantum mechanics, based on symplectic structures for normal states and offering a general setting which is convenient to study the emergence of macroscopic classical dynamics from microscopic quantum processes. We propose here a new mathematical approach to Bona's one, with much brother domain of applicability. It highlights the central role of self-consistency. This leads to a mathematical framework in which the classical and quantum worlds are naturally entangled. We build a Poisson bracket for the polynomial functions on the hermitian weak$^{\ast }$ continuous functionals on any $C^{\ast }$-algebra. This is reminiscent of a well-known construction for finite-dimensional Lie algebras. We then restrict this Poisson bracket to states of this $C^{\ast }$-algebra, by taking quotients with respect to Poisson ideals. This leads to densely defined symmetric derivations on the commutative $C^{\ast }$-algebras of real-valued functions on the set of states. Up to a closure, these are proven to generate $C_{0}$-groups of contractions. As a matter of fact, in general commutative $C^{\ast }$-algebras, even the closableness of unbounded symmetric derivations is a non-trivial issue. Some new mathematical concepts are introduced, which are possibly interesting by themselves: the convex weak $^{\ast }$ Gâteaux derivative, state-dependent $C^{\ast }$-dynamical systems and the weak$^{\ast }$-Hausdorff hypertopology, a new hypertopology used to prove, among other things, that convex weak$^{\ast }$-compact sets generically have weak$^{\ast }$-dense extreme boundary in infinite dimension. Our recent results on macroscopic dynamical properties of lattice-fermion and quantum-spin systems with long-range, or mean-field, interactions corroborate the relevance of the general approach we present here.
