Projection hypothesis from the von Neumann-type interaction with a Bose–Einstein condensate

We derive the projection hypothesis in projective quantum measurement by restricting the set of observables. This projection hypothesis accompanies a bipartite system with the von Neumann-type interaction, which consists of a quantum mechanical system, with a meter variable to be measured, and a quantum field theoretically macroscopic extended object, that is, a spatiotemporally inhomogeneous Bose-Einstein condensate in quantum field theory with the quantum coordinate, that is, the zero-energy Goldstone mode(s) of the spontaneously broken global spatial translational symmetry. The key steps in the derivation are the return of the symmetry translation of this quantum coordinate to the inverse translation of the c-number spatial coordinate in quantum field theory and the reduction of quantum fluctuations to classical fluctuations with respect to the Goldstone mode(s) due to a superselection rule.