It is widespread common perception that Quantum Mechanics is about
{\it operators on abstract Hilbert spaces} $\mathcal{H}$, states which are normalized trace-class operator and observables which can be general linear operators on $\mathcal{H}$. The pairing comes from the Hilbert space of Hilbert-Schmidt operators on $\mathcal{H}$ via the trace
operation: $\langle T,S \rangle_{HS} = trace(T S^*)$. By complex interpolation one obtains the Schatten $S^p$-classes of compact operators, with $HS = S^2$.
This is quite analogous to scale of Lebesgue spaces $L^p$, say, with
$1 \leq p \leq \infty$, which play a significant role in functional analysis in general and Fourier Analysis in particular,
although sometimes wavelet theory appears to be more suitable for questions involving such spaces.

In this talk the author would like to challenge the audience with an alternative model, based on the Banach Gelfand
Triple $(S_0,L_2,S^*_0)$, based on the Feichtinger algebra and the dual space, called the space of mild distributions,
with the corresponding concept of ``mild convergence'' (namely the $w^*$-convergence). It also allows to give a proper
meaning to Dirac BRAs and KETs, which in its simple form allows to describe the basic operations of Linear Algebra
in a compact form, which is also taken as an inspiration for the general case (and in addition almost exactly what
can be done by using the MATLAB mathematical software).

{\bf We would like to announce a couple of bold claims}:
\begin{enumerate}
\item Typical signals is very well modelled as mild distributions;
\item Signals are measured by applying them on test functions;
\item Hence signals are very similar if they agree up to a small error for a large number of measurements;
\item Operators from $S_0(R^d)$ to $S^*_0(R^d)$ are characterized by their ``kernels'': signals
of $2d$ variables, spreading functions, or KNS-symbols;
\item They appear to be a suitable model for observables, larger than $\mathcal{L}(L^2)$;
\item The setting allows to describe Dirac-like bases in a clean way;
\item The invariance under the metaplectic group gives a lot of insight.
\end{enumerate}

As time permits we would like to point out that so-called {\it unitary Banach Gelfand Triple isomorphism}
are the correct analogue of unitary matrices in the finite dimensional case. The enrichment of
the structure from the abstract Hilbert space to the ambient triple (or rigged Hilbert space) provides
a lot of extra information. It can be used as an inspiration for the approximation of
continuous models by finite-dimensional ones of increasing size. Roughly speaking
it corresponds to learn a signal up to some {\it resolution}, like an audio-signal recorded
on a CD (providing the relevant information up to $20kHz$)