希尔伯特第六问题与量子信息学原理
The sixth Hilbert's problem and the principles of quantum informatics
摘要：
以傅立叶变换为例，分析了希尔伯特空间几何在统计模型构造中的应用。根据玻尔的互补原理，提出了互补的坐标和动量表象。证明了坐标分布的特征函数可以看作是动量表象中ψ函数的卷积，反之亦然。论证了引入坐标和动量算符的自然性。给出了希尔伯特空间几何的概率解释。Cauchy-Bunyakowsky（Cauchy-Schwartz），Cramer-Rao和不确定性不等式被认为是在同一框架。量子信息学作为一门自然科学的主要假设。量子信息学是概率论和量子力学的理论基础。
英文摘要：
By the example of a Fourier transform, the possibilities of Hilbert space geometry applications for statistical model construction are analyzed. In accordance with Bohr's complementarity principle, mutually-complementary coordinate and momentum representations are presented. It was demonstrated that the characteristic function of coordinate distribution may be considered as a convolution of the psi-function in momentum representation and vice versa. The naturalness of coordinate and momentum operators introduction is demonstrated. A probabilistic interpretation of Hilbert space geometry is given. Cauchy-Bunyakowsky (Cauchy-Schwartz), Cramer-Rao and uncertainty inequalities are considered in the same framework. The principal postulates of quantum informatics as a natural science are presented. It is demonstrated that quantum informatics serves as a theoretic basis for both probability theory and quantum mechanics.
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