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Scattering requires a time-dependent Hilbert space

N. L. Chuprikov
Feb 2024
A new nonstationary theory of scattering a spinless particle on a $\delta(x)$ potential is presented. In contrast to the standard theory to be a perturbation theory our approach is based on an exact analysis of the asymptotes of nonstationary states, including the average values of observables on these states. The key role here is played by the unboundedness of the coordinate operator and the law of momentum conservation at $t\to\mp\infty$. In this scattering problem, the condition for the existence of asymptotically free dynamics actually coincides with the condition for the existence of the superselection rule in the theory. Von Neumann's postulate about the time independence of Hilbert space is not applicable here, since the configuration space $\mathbb{R}$, in which the particle moves at finite moments of time, is transformed at $t\to\mp\infty$ into the space $\mathbb{R}\setminus \{0\}$. This makes it incorrect to use perturbation theory to analyze the asymptotics of scattering states, and also requires a revision of the existing mathematical definition of the concept of the Schr\"{o}dinger representation, which underlies all versions of von Neumann's theorem on the irreducibility of the Schr\"{o}dinger representation.
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