Extended statistical modeling under symmetry; The link toward quantum mechanics

被引:2
|
作者
Helland, Inge S. [1 ]
机构
[1] Univ Oslo, Dept Math, N-0316 Oslo, Norway
来源
ANNALS OF STATISTICS | 2006年 / 34卷 / 01期
关键词
Born's formula; complementarity; complete sufficient statistics; Gleason's theorem; group representation; Hilbert space; model reduction; quantum mechanics; quantum theory; symmetry; transition probability;
D O I
10.1214/009053605000000868
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
020208 ; 070103 ; 0714 ;
摘要
We derive essential elements of quantum mechanics from a parametric structure extending that of traditional mathematical statistics. The basic setting is a set A of incompatible experiments, and a transformation group G on the cartesian product Pi of the parameter spaces of these experiments. The set of possible parameters is constrained to lie in a subspace of Pi, an orbit or a set of orbits of G. Each possible model is then connected to a parametric Hilbert space. The spaces of different experiments are linked unitarily, thus defining a common Hilbert space H. A state is equivalent to a question together with an answer: the choice of an experiment a c A plus a value for the corresponding parameter. Finally, probabilities are introduced through Born's formula, which is derived from a recent version of Gleason's theorem. This then leads to the usual formalism of elementary quantum mechanics in important special cases. The theory is illustrated by the example of a quantum particle with spin.
引用
收藏
页码:42 / 77
页数:36
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