We prove a theorem showing that quantum mechanics is not directly a stochastic process characterizing Brownian motion but rather its square root. This implies that a complex-valued stochastic process is involved. Schrödinger equation is immediately derived without further assumptions using Itō integrals that are properly generalized. Fluctuations in space arise from a Brownian motion and the combined effect of a stochastic process with a symmetric Bernoulli distribution typical of tossing a coin.
Abstract Feynman developed the path integral approach to quantum mechanics as an alternative to wave or matrix mechanics that has found fruitful implementation in quantum mechanics, quantum electrodynamics, and statistical physics. Despite its many successes, the usual Feynman path integral formalism suffers from a lack of mathematical rigor. Questions of mathematical rigor still exist, but Mark Kac provided a rigorous path integral approach to quantum mechanics. The Kac formulation for quantum systems is given in terms of Wiener measure for a path integral and employs the Brownian motion distribution. In this paper, we review the Feynman–Kac approach and other stochastic processes related to Brownian motion. We also give an account of the successes and difficulties we faced during our journey through numerical path integral problems involved in physics.
We propose a stochastic Schrödinger equation in which the momentum is coupled to a white Gaussian noise. In the stochastic representation, the kinetic energy representing the self-interaction of momentum is reduced to a linear term of momentum. As such, the quantum evolution operator factorizes into two contributions due to the momentum and the potential, respectively. The exact quantum propagator thereby becomes an expectation of the stochastic one in which the amplitude results from the potential with a fluctuating position: The particle moves with a constant velocity, subjected to a complex Brownian motion. We demonstrate that the stochastic Schrödinger equation can be feasibly used to derive the quantum propagators for the linear potential and the harmonic oscillator system. Novel semiclassical and other approximations may be developed from the new representation of quantum mechanics.
We present a detailed study of a simple quantum stochastic process, the quantum phase space Brownian motion, which we obtain as the Markovian limit of a simple model of an open quantum system. This physical description of the process allows us to specify and to construct the dilation of the quantum dynamical maps, including conditional quantum expectations. The quantum phase space Brownian motion possesses many properties similar to those of the classical Brownian motion, notably its increments are independent and identically distributed. Possible applications to dissipative phenomena in the quantum Hall effect are suggested.
Building upon the work of Hu, Paz, and Zhang [1,2] on open quantum systems we consider the quantum Brownian motion (QBM) model with one oscillator (position variable
x) as the system, {\it nonlinearly} coupled to an environment of
N harmonic oscillators (with mass
mn, natural frequency
ωn, position
qn and momentum
pn variables) in the form
∑n(vn1(x)qnk+vn2(x)pnl) where
k,l are integers (the present work only considers the
k=l=2 cases). The vertex functions $v_{n1}, v_{n2} $ are of the form
vn1=λCn1f(x),vn2(x)=−λCn2mn−2ωn−2f(x) where
Cn1,2 are the coupling constants with the
nth oscillator,
f(x) is any arbitrary function of
x, and
λ is a dimensionless constant. Employing the closed-time-path formalism the influence action
SIF is calculated using a perturbative expansion in
λ. It is possible to identify the terms in
SIF quadratic or higher in
Δ(s)≡f(x+(s))−f(x−(s)) to constitute the noise kernel, while terms linear in
Δ to that of the dissipation kernel. The non-Gaussian noise kernel gives rise to non-zero three-point correlation function of the corresponding stochastic force. The pathway presented here should be useful for the exploration of \textit{non-Gaussian properties of systems nonlinearly coupled with their environments}; examples in early universe cosmology and in quantum optomechanics (QOM) are mentioned. A modified fluctuation-dissipation relation (FDR) is also established, which ensures the consistency of the model and the accuracy of results even at higher perturbative orders. Another result of significance is the derivation of a nonlinear Langevin equation which is expected to be useful for many open quantum system applications.
In the frames of classical mechanics the generalized Langevin equation is derived for an arbitrary mechanical subsystem coupled to the harmonic bath of a solid. A time-acting temperature operator is introduced for the quantum Klein-Kramers and Smoluchowski equations, accounting for the effect of the quantum thermal bath oscillators. The model of Brownian emitters is theoretically studied and the relevant evolutionary equations for the probability density are derived. The Schrodinger equation is explained via collisions of the target point particles with the quantum force carriers, transmitting the fundamental interactions between the point particles. Thus, electrons and other point particles are no waves and the wavy chapter of quantum mechanics originated for the force carriers. A stochastic Lorentz-Langevin equation is proposed to describe the underlaying Brownian-like motion of the point particles in quantum mechanics. Considering the Brownian dynamics in the frames of the Bohmian mechanics, the density functional Bohm-Langevin equation is proposed, and the relevant Smoluchowski-Bohm equation is derived. A nonlinear master equation is proposed by proper quantization of the classical Klein-Kramers equation. Its equilibrium solution in the exact canonical Gibbs density operator, while the well-known Caldeira-Leggett equation is simply a linearization at high temperature. In the case of a free quantum Brownian particles, a new law for the spreading of the wave packet it discovered, which represents the quantum generalization of the classical Einstein law of Brownian motion. A new projector operator is proposed for the collapse of the wave function of a quantum particle moving in a classical environment. Its application results in dissipative Schrodinger equations, as well as in a new form of dissipative Liouville equation in classical mechanics.