# Superposition, transition probabilities and primitive observables in infinite quantum systems

November 08, 2014

The concepts of superposition and of transition probability, familiar from
pure states in quantum physics, are extended to locally normal states on
funnels of type I factors. Such funnels are used in the description of
infinite systems, appearing for example in quantum field theory or in quantum
statistical mechanics; their respective constituents are interpreted as
algebras of observables localized in an increasing family of nested spacetime
regions. Given a generic reference state (expectation functional) on a funnel,
e.g, a ground state or a thermal equilibrium state, it is shown that
irrespective of the global type of this state all of its excitations, generated
by the adjoint action of elements of the funnel, can coherently be superimposed
in a meaningful manner. Moreover, these states are the extreme points of their
convex hull and as such are analogues of pure states. As further support of
this analogy, transition probabilities are defined, complete families of
orthogonal states are exhibited and a one-to-one correspondence between the
states and families of minimal projections on a Hilbert space is established.
The physical interpretation of these quantities relies on a concept of
primitive observables. It extends the familiar framework of observable algebras
and avoids some counter intuitive features of that setting. Primitive
observables admit a consistent statistical interpretation of corresponding
measurements and their impact on states is described by a variant of the von
Neumann-Lüders projection postulate.

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