培养量子宝宝
Bringing Up a Quantum Baby
摘要：
任何两个无限维（可分）希尔伯特空间是酉同构的。因此，它们的所有自伴算子的集合也酉等价。因此，如果所有的自伴算符都能被观测到，并且如果量子物理学中没有比狄拉克的“量子力学”更重要的公理，那么量子物理学家将无法区分环面和地面上的洞。我们认为，确实存在这样的公理，涉及到哈密尔顿域中的向量：在这个域中的“概率密度”（厄米形式）生成一个代数，从这个代数中，经典的位形空间及其拓扑（以及公理的进一步改进，其C^K和C^无穷结构）可以使用Gel'fand - Naimark理论重建。对于这些公理来说，经典拓扑只是某些量子态的一个属性，从量子物理学中出现的位形空间随着能量激发的增加而逐渐变得不可微，最终完全不复存在。在建立了这些公理之后，我们应用它们来证明拓扑变化的可能性和讨论量化的模糊拓扑。关于时间在量子物理中的作用的基本问题也得到了解决。
英文摘要：
Any two infinite-dimensional (separable) Hilbert spaces are unitarily isomorphic. The sets of all their self-adjoint operators are also therefore unitarily equivalent. Thus if all self-adjoint operators can be observed, and if there is no further major axiom in quantum physics than those formulated for example in Dirac's `Quantum Mechanics', then a quantum physicist would not be able to tell a torus from a hole in the ground. We argue that there are indeed such axioms involving vectors in the domain of the Hamiltonian: The ``probability densities'' (hermitean forms) ψ^\dagger χfor ψ,χin this domain generate an algebra from which the classical configuration space with its topology (and with further refinements of the axiom, its C^K and C^infinity structures) can be reconstructed using Gel'fand - Naimark theory. Classical topology is an attribute of only certain quantum states for these axioms, the configuration space emergent from quantum physics getting progressively less differentiable with increasingly higher excitations of energy and eventually altogether ceasing to exist. After formulating these axioms, we apply them to show the possibility of topology change and to discuss quantized fuzzy topologies. Fundamental issues concerning the role of time in quantum physics are also addressed.
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