5 Physically Relevant Representations

Franco Strocchi

Roberto Beneduci

We study two relevant characterizations of a commutative positive operator valued measure (POVM) F . The first one is a Choquet type of an integral representation. It introduces a measure ν on the space of the projection valued measures (PVMs) and describes F as an integral over this space. The second one represents a commutative POVM F as the randomization of a single PVM E by means of a Markov kernel μ . We show that one can be derived from the other. We also elaborate upon some previous results on Choquet’s representation of Markov kernels and find a functional relationship between ν and μ . Finally, we analyze some relevant particular cases and provide some physically relevant examples which include the unsharp position observables.

Emilio d’Emilio, Luigi E. Picasso

Edgard Elbaz

Henk Hoogland

The projective equivalence relation for representations of symmetry groups in quantum mechanics is refined to a so-called gauge equivalence relation. An example is given where this leads to a physically more relevant classification of elementary quantum mechanical systems. It is indicated how this idea might be applied to projective bundle .representations.

Franco Strocchi

From the examples and the discussion of the previous chapter, it appears that for infinite systems the choice of the representation for the algebra of canonical variables (a basic preliminary step for even defining the dynamical problem) is a highly non-trivial problem (unless the model is exactly soluble).