Formal GNS Construction and States in Deformation Quantization
Formal GNS Construction and States in Deformation Quantization
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形式 GNS 构造和变形量化状态
DOI：
10.1007/s002200050402
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发表时间：
1996
影响因子：
2.4
通讯作者：
S. Waldmann
中科院分区：
文献类型：
--
作者：
M. Bordemann;S. Waldmann
摘要：在本文中，我们开发了一种构造希尔伯特空间的方法，以及变形量子化中量子可观测量的形式代数的表示，这是复C*-代数的著名GNS构造的类似物：在这种方法中，相应的正线性泛函（“状态”）不取它们在复数域中的值，而是取它们在形式参数中的形式复洛朗级数域（的适当扩展域）中的值。通过使用这些领域的代数和拓扑性质，我们证明，这种结构是有意义的，并显示在物理的例子，标准的表示，如巴格曼和薛定谔表示出来正确的，无论是正式的和在一个适当的收敛计划。对于某些哈密顿函数（包含在正泛函的Gel'fand理想中），含时薛定谔方程的形式解是存在的。此外，我们证明了对于任何一个具有Wick型Fedosov星星积的Kähler流形，所有的经典δ泛函都是正的，并给出了变形代数的某些形式的Bargmann表示.
Abstract: In this paper we develop a method of constructing Hilbert spaces and the representation of the formal algebra of quantum observables in deformation quantization which is an analog of the well-known GNS construction for complex C*-algebras: in this approach the corresponding positive linear functionals (“states”) take their values not in the field of complex numbers, but in (a suitable extension field of) the field of formal complex Laurent series in the formal parameter. By using the algebraic and topological properties of these fields we prove that this construction makes sense and show in physical examples that standard representations such as the Bargmann and Schrödinger representation come out correctly, both formally and in a suitable convergence scheme. For certain Hamiltonian functions (contained in the Gel'fand ideal of the positive functional) a formal solution to the time-dependent Schrödinger equation is shown to exist. Moreover, we show that for every Kähler manifold equipped with the Fedosov star product of Wick type all the classical delta functionals are positive and give rise to some formal Bargmann representation of the deformed algebra.