# Euclidean quantum mechanics

This is a general program of probabilistic reinterpretation of
Quantum Mechanics initiated by us in the mid eighties and founded on an
old suggestion of E. Schrödinger (1931). It is Euclidean and,
therefore, deals with heat equations instead of Schrödinger equations.
But, in contrast with Kac's approach, it preserves the time symmetry of
all well defined probability measures on path spaces involved in the
construction. Its key point is to use two heat equations, adjoint to
each other with respect to the time parameter. In the simplest one
dimensional free case, whose quantum Hamiltonian is
*H*_{0} = - 1/2
∂^{2}/∂*q*^{2},
this means that positive solutions of -∂η*/∂*t* =
*H*_{0}η* and ∂η/∂*t* =
*H*_{0}η are used to produce a real
valued diffusion *Z*_{t}, on
generally finite time intervals, such that ∫_{A}
ηη*(*q*,*t*)d*q* =
Pr{*Z*_{t} ∈ *A*} for
*A* a Borelian and Pr denoting a probability. This approach is
in no way limited to the position representation. Such processes are
often referred to as reciprocal, Bernstein, variational, or even local
Markov or two-sided Markov.

This Euclidean analogy has allowed, for instance, to discover a Quantum Noether Theorem in Hilbert space whose predictions are richer than the result available in textbooks.

The goal of Euclidean Quantum Mechanics is to construct the closest probabilistic analogy with Quantum Theory, preserving Feynman's intuition of "quantum paths" and allowing to discover qualitatively new quantum effects. This program also aims at introducing new structures in Stochastic Analysis,inspired by quantum theory.

In recent years, the above-mentioned suggestion of Schrödinger has been known as the "Schrödinger Problem" in the Mass Transportation community.