Transition probabilities between quantum states and their applications

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As a basic quantum algebra, we worked with the so-called CCR algebras and investigated their free states, which are known to be parametrized by covariance forms, from the view point of physical equivalence of the associated representations of quantum algebras.By fully utilizing our previous obtained determinant formula on the transition probability, we succeeded in establishing a quantum analogue of Kakutani dichotomy, which states that two representations arising from free states are physically equivalent unless they are disjoint in the sense that the transition probability vanishes between states. Moreover, the transition probability formula is further extended to coherent states, which are free states with non-trivial expectation values. The result is parallel to the case of free states with a term which evaluates expectation functionals by covariance forms being multiplied exponentially.