A truly exciting exercise in complex analysis, asymptotic analysis
and innovative thinking. The basic idea is inspired by quantum
mechanics. Starting from this background the usual
probability-density interpretation of wave functions $\psi(x)$ is
assumed transferred to what mathematicians would call the
Krein-space context. Briefly, the Krein-space-like inner
self-product is defined in the form $\varrho_\psi=\langle
\psi|{P|\psi \rangle$  evaluated as integral $\varrho_\psi= \int_{C}
\psi*(-x) \psi(x) dx$ not necessarily over the real line
$C=C_{real}\equiv \mathbb{R}$) but over certain {\it ad hoc} complex
curve $C=C_\psi\neq \mathbb{R}$ called ``eigenpath".

For a few concrete input wave functions the idea is illustrated, in
more detail, Numerically. With the latter curves constructed and
discussed. It is shown that the corresponding ``generalized
probability density" may really be kept locally real and positive
(this is assumed achieved via the {\em local} choice of direction of
$C$) plus integrable (so that one can normalize the given $\psi(x)$
{\em globally} to one).

Needless to add: the reader of this text is rewarded by being shown
a number of amazing technical tricks (too many for being even listed
here) accompanied by a number of some even more amazing speculations
about the possible structure of some future ``final" physical
theory, to be based on the use of similar concepts.



MR2718550 Bender, Carl M.; Hook, Daniel W.; Meisinger, Peter N.;
Wang, Qing-hai Probability density in the complex plane. Ann.
Physics 325 (2010), no. 11, 2332--2362. 81P05