Suppose that some quantum wave function $\psi = \int a_i \,i\rangle \,\,di$ where pseudo-spanning ket "vectors" of $\psi$, $i \rangle$, are continuous. (thus the use of integral.)

By normalization, $a_i$ are assumed to satisfy $\int |a_i|^2 \,\,di = 1$.

Now to quantum decoherence. When spanning vectors are countable, it is easy to show how environment interaction leads to recovery of classical probability (Zurek: http://arxiv.org/ftp/quant-ph/papers/0306/0306072.pdf). However, when using pseudo-spanning vectors, I am not sure what will happen.

Environment Hilbert space spanning vectors are usually countable, and in such a case at least two pseudo-spanning vectors $i' \rangle$ and $i'' \rangle$ would have same environment-spanning vector "entangled". Thus standard method for recovering classical probability by quantum decoherence does not seem to work for continuous spectra.

In short words, in case an operator (let that be momentum operator or what) admits continuous spectra, can classical probability be recovered by quantum decoherence (for example, particle's momentum probability)? If so, what is derivation?

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    $\begingroup$ Permanent decoherence can only occur if 1) the Hilbert space of the environment is uncountably infinite, or 2) the system-environment Hamiltonian is time-dependent, which is just an approximation to scenario 1). Irreversible decoherence requires continuous spectra. $\endgroup$ – Mark Mitchison Feb 20 '16 at 18:41
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    $\begingroup$ Minor comment to the post (v2): In the future please link to abstract pages rather than pdf files, e.g., arxiv.org/abs/quant-ph/0306072 $\endgroup$ – Qmechanic Feb 20 '16 at 19:25

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