the set of parameters by associating to a function that satisfies appropriate conditions the following operator in : The operator is symmetric if is real-valued, and it is self-adjoint (as a quadratic form) if is real and semi-bounded. The original is an upper symbol, usually non-unique, for the operator . It will be called a classical observable with respect to the family if the so-called lower symbol of , defined as has mild functional properties to be made precise according to further topological properties granted to the original set . A last point of this construction of the space of quantum states concerns its statistical aspects. There is indeed an interplay between two probability distributions: See also Coherent states Lieb conjecture Quantization References Mathematical physics