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Feynman 24  The Weiner integral rediscovered by Feynman in imaginary time.
p.131 of James Gleick's GENIUS - a best seller in every book store
"..neither Dirac nor any other physicist had been able to follow up on the
notion of a Lagrangian in quantum mechanics - a way of expressing a
particle's history interms of a quantity of action.  Now Dirac's idea
served as an explosive release in Feynman's imagination.  The uneasy
elements of quantum mechanics broke loose and rearranged themselves into a
radically new formulation.  Where Dirac had pointed the way to calculating
how the wave function would evolve in an infinitesimal slice of time,
Feynman needed to carry the wave function farther, through finite time.  A
considerable barrier separated the infinitesimal from the finite.  Making
use of Dirac's infinitesimal slice required a piling up of many steps -
infinitely many of them.  Each step required an integration, a summing of
algebraic quantitites.  In Feynman's mind a sequence of multiplications and
compounded integrals took form.  He considered the coordinates that specify
a particle's position.  They churned through his compund integral.  The
quantity that emerged was, once again, a form of the action.  To produce
it, Feynman realized, he had to make a complex integral encompassing every
possible coordinate through which a particle could move.  The result was a
sum of .... a more abstract quantitiy called the probability amplitude.
Feynman summed the contributions of every conceivable path from the
starting position to the final position - though at first he saw more a
haystack of coordinate positions than a set of distinct paths.  Even so, he
realized that he had burrowed back to first pricniples and found a new
formulation of quantum mechanics ... his sense of paths in space-time
seemed somehow cleaner - more direct."

The simple non-relativistic quantum Schrodinger equation for a free
particle is formally a diffusion equation for Brownian motion in "imaginary
time".  Norbert Weiner at MIT in 20's (or earlier) had rigorously
formulated functional integrals of the kind Feynman rediscovered.
Feynman's jagged quantum worldlines are actually non-differentiable fractal
curves.  Non-Brownian "pink" and "black" noise characterize random time
series with hidden order.  This suggests new kinds of classical diffusion
partial differential equations with possibly new kinds of quantum analogs?