H*-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics
H*-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics
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H*-代数和非单位 Frobenius 代数：无限维分类量子力学的第一步
DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
C. Heunen
中科院分区：
文献类型：
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作者：
S. Abramsky;C. Heunen
一类Frobenius代数被用来描述有限维Hilbert空间上的标准正交基和可观测值。这些代数中单位的存在意味着它们只能在有限维度上实现。我们寻求一个合适的推广，这将允许任意基础和可观察到的描述在量子力学的范畴公理化。我们发展了H*-代数的一个定义，它可以在任何对称的一元剑型范畴中解释，从希尔伯特空间（可能是无限维）范畴的泛函分析简化到经典概念，从而提供了一种讨论任意维标准正交基和量子可观测的分类方法。此外，这些代数在紧范畴中被简化为通常的Frobenius代数的概念。然后研究了非一元Frobenius代数与H*-代数之间的关系。我们给出了一些等价条件来刻画它们在希尔伯特空间范畴内重合的情况。我们还证明了它们在广义关系和正矩阵的范畴中总是重合的。
A certain class of Frobenius algebras has been used to characterize orthonormal bases and observables on finite-dimensional Hilbert spaces. The presence of units in these algebras means that they can only be realized finite-dimensionally. We seek a suitable generalization, which will allow arbitrary bases and observables to be described within categorical axiomatizations of quantum mechanics. We develop a definition of H*-algebra that can be interpreted in any symmetric monoidal dagger category, reduces to the classical notion from functional analysis in the category of (possibly infinite-dimensional) Hilbert spaces, and hence provides a categorical way to speak about orthonormal bases and quantum observables in arbitrary dimension. Moreover, these algebras reduce to the usual notion of Frobenius algebra in compact categories. We then investigate the relations between nonunital Frobenius algebras and H*-algebras. We give a number of equivalent conditions to characterize when they coincide in the category of Hilbert spaces. We also show that they always coincide in categories of generalized relations and positive matrices.