正则对易关系的表示--代数与随机分析的算子
Representations of the canonical commutation relations--algebra and the operators of stochastic calculus
摘要：
我们研究了一个家庭的典型的交换关系（CCR）-代数（无限数量的自由度），我们称之为容许的表示。容许表示族包括福克真空表示。我们表明，每一个容许的表示，有一个相关的高斯随机分析，我们指出，福克真空CCR-表示的情况下，在一个自然的方式产生的Malliavin演算的运营商。因此，我们得到的算子Malliavin的变分法从一个更代数的方法比常见的。我们得到明确的和自然的公式和规则，随机分析的运营商。我们的方法利用对称（可关闭）对运营商的概念。福克真空表示产生一个极大对称对。这种对偶性的观点有进一步的优势，无界算子和稠密域的问题可以比其他工具更容易解决。随着CCR表示论的使用，我们也得到了，作为副产品，在多变量算子理论，我们觉得是独立的利益的一些新的结果。
英文摘要：
We study a family of representations of the canonical commutation relations (CCR)-algebra (an infinite number of degrees of freedom), which we call admissible. The family of admissible representations includes the Fock-vacuum representation. We show that, to every admissible representation, there is an associated Gaussian stochastic calculus, and we point out that the case of the Fock-vacuum CCR-representation in a natural way yields the operators of Malliavin calculus. And we thus get the operators of Malliavin's calculus of variation from a more algebraic approach than is common. And we obtain explicit and natural formulas, and rules, for the operators of stochastic calculus. Our approach makes use of a notion of symmetric (closable) pairs of operators. The Fock-vacuum representation yields a maximal symmetric pair. This duality viewpoint has the further advantage that issues with unbounded operators and dense domains can be resolved much easier than what is possible with alternative tools. With the use of CCR representation theory, we also obtain, as a byproduct, a number of new results in multi-variable operator theory which we feel are of independent interest.
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