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).