Our account begins in 1931/1932, when one of the fathers of quantum theory, Erwin Schrodinger, proposed a Gedankenexperiment to comprehend the propagation of a cloud of Brownian particles. He proposed [1], [2] the inference problem to identify the most likely evolution of a large collection of particles as they traverse between two endpoints in time- their distribution being measured and available at the two endpoints. This random evolution, now known as the Schrodinger bridge, entails a flow of one-time distributions that interpolate the known initial and final marginals.
This is an exploration with today’s tools of probability theory into the mystery of ‘Über die Umkehrung der Naturgesetze’ (On time-reversal of natural laws) written by Erwin Schrödinger in 1931. He propagates the idea that diffusions given their marginal distributions at finite initial and final time are time-reversible. We are going to meet clouds of identical and independent particles considering them as realizations of diffusions. The particle dynamics is assumed to be known. Given the observation of such a cloud in terms of distribution densities at finite initial and final time, the intention is to find the ‘most probable’ distribution density of the cloud at intermediate times.
In Nelson [10], it is shown that for each solution of a Schr\"odinger equation
ϕ(x,t), x∈Rd, t≧0 , with
∥ϕ∥=1 , there exists a diffusion process which has ... probability density at time
t given byMore precisely the following is shown in [10]: suppose that we are given a real valued function
V(x), x∈Rd , and a complex valued function
ϕ0(x), x∈Rd , with
∥ϕ0∥=1 , and consider the Schrodinger equation Then, under the assumption that
ϕ and
b~ are sufficiently regular, there exists a diffusion process
{Xt}, t≧0 , with initial density
∣ϕ0(x)∣2 and generator
(1/2)Δ+b~(x,t)⋅∇ such that the probability density of this diffusion at time
t≧0 is
∣ϕ(x,t)∣2 .In addition, in [10] it is also shown that the diffusion process
{Xt}, t≧0 , solves the second order stochastic differential equation
(1/2)(D∗D+DD∗)Xt=−∇V(Xt) , where
D and
D∗ are Nelson's forward and backward stochastic derivatives.Generally sPeaking, the functions
ϕ and
b~ may be singular, and the existence of the corresponding diffusion process must be studied carefully.Carlen
[1,2] , Meyer and Zheng [8] and Nagasawa [9] considered rigorously the construction of a diffusion process with a given initial density and a generator.
ln order to construct a Markovian propagator for the diffusion process, Carlen
[1,2] used partial differential equation methods.Meyer and Zheng [8] restricted themselves to the case when
ϕ(x,t)=ϕ0(x), ∀t≧0 and considered the construction of the diffusion process through the relationships between Markov processes and Dirichlet forms.We may consult Nagasawa [9] about recent developments of this area.In Carlen [1], it is assumed that
V(x), x∈Rd , is a Rellich class potential and
∥∇ϕ0∥2<∞ , and shown that there exists a diffusion process
{Xt}, t≧0 , which has the probability density
∣ϕ(x,t)∣2 at time
t≧0 and admits the representation
Our modern understanding of atoms, molecules, solids, atomic nuclei, and elementary particles is largely based on quantum mechanics. Quantum mechanics grew in the mid-1920s out of two independent developments: the matrix mechanics of Werner and the wave mechanics of Erwin Schrödinger. For the most part this chapter follows the path of wave mechanics, which is more convenient for all but the simplest calculations. The general principles of the wave mechanical formulation of quantum mechanics are laid out and provide a basis for the discussion of spin, identical particles. and scattering processes. The general principles are supplemented with the canonical formalism to work out the Schrödinger equation for charged particles in a general electromagnetic field. The chapter ends with the unification of the approaches of wave and matrix mechanics by Paul Dirac, and a modern approach, known as Hilbert space, is briefly described.