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I've been working with quantum field theory on the basis of it being a stochastic signal processing formalism for a number of years. The projection to fourier components within the forward light-cone in 1+3 dimensions is comparable to projection to positive frequency components in 1+0 dimensions (and hence comparable to projection to the analytic signal).

Modern physics is largely about storing and processing digital records of multiple (often multiple thousands of) electronic signals. The digital records are most often somewhat ad-hoc, very sparse, and very nonlinear in their relation to the original signals, typically as times at which an analogue voltage makes a transition between distinct metastable levels ("a particle was detected at time ... and position ...", but in fact, for example, a micron-sized device transitioned to a current-flowing state for a few nanoseconds before being reset to a zero current state), nonetheless this is signal processing.

As a partial motivation for thinking this way, we can construe quantum fluctuations, in a signal processing PoV, as a Poincaré invariant noise, very comparably to the translation and euclidean invariance of thermal noise (see my Phys. Lett. A 338, 8-12(2005); non-paywalled arXiv preprint).

I'm very much aware of Leon Cohen's work on relationships between deterministic SP and quantum theory (but very happy to hear of anything recent in the SP literature), and Hilbert spaces are routinely used for deterministic SP, but I'm interested to hear of any uses, in the literature or otherwise, of Hilbert space mathematics for stochastic SP (particularly any that take an SP approach, but stochastic fields are also called "random fields" in the mathematical literature).

I'm also interested in Hilbert space methods for nonlinear control theory that take a more-or-less SP approach, if anyone can point me to such (particularly stochastic nonlinear control theory, but deterministic if needs must).

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  • $\begingroup$ You are describing several physics & maths concepts and theories, and asking for SP works related with quantum theory is a broad and unanswerable question. SP is a tool, and as such, this can be used in thousand of ways for dealing with records. With or without stochastic components. By the way, control theory is just another game, which i think is totally unrelated to the previous question, again, depending on which will be the control system defined to be "controlled", and also unanswerable if left to the reader. I suggest you to clarify your question, or elaborate your specific framework. $\endgroup$ – Brethlosze Jan 16 '17 at 4:45
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    $\begingroup$ So what do you think about this Quantum Signal Processing reference? webee.technion.ac.il/people/YoninaEldar/Download/EO02m.pdf $\endgroup$ – Brethlosze Jan 16 '17 at 5:13
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    $\begingroup$ There is even an Amazon book on Quantum Mechanics Signal Processing and Spectral Analysis amazon.com/… $\endgroup$ – Brethlosze Jan 16 '17 at 5:16
  • $\begingroup$ Thanks for the Eldar reference, which was interesting. I don't see a necessity for the move to nonlinearity in the subsection "Rank-One QSP Measurements", however. For stochastic signals, at least, I think complex Hilbert spaces are sufficient (because the axioms of probability are linear; see Leon Cohen, "Rules of Probability in Quantum Mechanics", Found. Phys. 18, 983(1988), for an elementary approach), however I take it that introducing nonlinearity might simplify problem solving in some SP cases. There are theorems about the inadvisability of introducing nonlinear operators in QM. $\endgroup$ – Peter Morgan Jan 17 '17 at 17:30
  • $\begingroup$ I definitely see your point on control theory, however perhaps someone else will think of something that seems to them marginally relevant. $\endgroup$ – Peter Morgan Jan 17 '17 at 17:31

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