Hermitian切值形式的分次李代数
Graded Lie algebra of Hermitian tangent valued forms
摘要：
在所谓的“协变量子力学“理论中，伽利略和爱因斯坦时空框架中的复线丛上的厄米矢量场起着基本作用。事实上，已经证明了厄米特向量场的李代数自然同构于相空间的“特殊函数”的李代数。事实上，这是上述特殊函数的协变量化的来源。在理论的最初版本中，这个结果是以一种相当复杂的方式来表述和证明的;现在，我们已经实现了一种更直接和简单的方法来分类厄米矢量场，并通过特殊的相位函数来表示它们。鉴于一个可能的协变量化的一个更大的类的“可观”，这是很自然的考虑厄米切值形式。因此，本文对复线丛的厄米切值形式的分次李代数进行了完备的分析，并根据切值形式和基空间形式对它们进行了分类。在坐标系中获得局部分类。对于整体分类，我们需要一个厄米特联系：事实上，这正是规范理论所需要的联系。
英文摘要：
In the theory of so called "Covariant Quantum Mechanics" a basic role is played by Hermitian vector fields on a complex line bundle in the frameworks of Galilei and Einstein spacetimes. In fact, it has been proved that the Lie algebra of Hermitian vector fields is naturally isomorphic to a Lie algebra of "special functions" of the phase space. Indeed, this is the source of the covariant quantisation of the above special functions. In the original version of the theory, this result was formulated and proved in a rather involved way; now, we have achieved a more direct and simple approach to the classification of Hermitian vector fields and to their representation via special phase functions. In view of a possible covariant quantisation of a larger class of "observables" it is natural to consider the Hermitian tangent valued forms. Thus, this paper is devoted to a self--contained analysis of the graded Lie algebra of Hermitian tangent valued forms of a complex line bundle and to their classification in terms of tangent valued forms and forms of the base space. The local classification is obtained in coordinates. For the global classification we need a Hermitian connection: indeed, this is just the connection required in gauge theories.
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