The structure of the projection-operator theory of Mori for quantum Brownian motion is reconsidered, emphasizing the nature of underlying quantum processes. Quantum-mechanical assumptions of the theory are extracted to sets of axioms. The Hilbert space scrV formed with relevant observables for quantum generalized Langevin equations (QGLE's) includes, in general, elements that cannot be operators on state vectors of the original system. The definition of Liouville operator L, as a mapping from one operator to another, thus requires its reconstruction in scrV. The work shows that the mentioned axioms not only ensure a unified reconstruction of L (hence of QGLE's) in systems in stationary states that may be nonequilibrium as well as equilibrium, but also extend intrinsically the domain of applicability of QGLE's. Illustrations of the setting for these extended QGLE's are presented by concrete examples of Hilbert spaces formed with quantum-mechanical Gaussian and non-Gaussian random variables, showing specifically the significance of a response-function-type inner product that assigns \ensuremath{\delta} correlation to the quantized Gaussian white noise which is an operator-valued distributional process with Kubo-Martin-Schwinger property.
The structure of the projection-operator theory of Mori for quantum Brownian motion is reconsidered, emphasizing the nature of underlying quantum processes. Quantum-mechanical assumptions of the theory are extracted to sets of axioms. The Hilbert space scrV formed with relevant observables for quantum generalized Langevin equations (QGLE's) includes, in general, elements that cannot be operators on state vectors of the original system. The definition of Liouville operator L, as a mapping from one operator to another, thus requires its reconstruction in scrV. The work shows that the mentioned axioms not only ensure a unified reconstruction of L (hence of QGLE's) in systems in stationary states that may be nonequilibrium as well as equilibrium, but also extend intrinsically the domain of applicability of QGLE's. Illustrations of the setting for these extended QGLE's are presented by concrete examples of Hilbert spaces formed with quantum-mechanical Gaussian and non-Gaussian random variables, showing specifically the significance of a response-function-type inner product that assigns \ensuremath{\delta} correlation to the quantized Gaussian white noise which is an operator-valued distributional process with Kubo-Martin-Schwinger property.
We systematically derive the quantum generalized nonlinear Langevin equation using Morozov's projection operator method. This approach extends the linear Mori-Zwanzig projection operator technique, allowing for the inclusion of nonlinear interactions among macroscopic modes. Additionally, we obtain the quantum generalized Fokker-Planck equation within the Heisenberg picture, which is consistent with Morozov's original formulation. These equations are fundamentally significant in nonequilibrium statistical physics, particularly in scenarios characterized by enhanced fluctuations, such as anomalous transport phenomena near critical points. The quantum nature of the derived generalized Langevin and Fokker-Planck equations is anticipated to provide a more detailed description than their classical equivalents. Specifically, the noise kernel in the quantum generalized Langevin equation is multiplicative, which broadens the applicability beyond Gaussian approximations. Given specific interactions, these equations are expected to be instrumental in investigating critical transport phenomena.
We first point out that nonrelativistic quantum mechanics formulated at imaginary time s = it can formally be recognized as the Fokker-Planck description of a classical Brownian motion, which occurs (in general) in a medium where particles are absorbed and emitted. We next offer a new formulation of quantum dynamics, which is basically the Langevin treatment of that Brownian motion. More explicitly, we derive a noise average representation of the transition probability W ( X", s"|X', s' ) in terms of the solutions to a Langevin equation with a Gaussian white noise. Upon analytic continuation back to real time W ( X", s"|X', s' ) becomes the propagator of the original Schrödinger equation, namely we have ⟨ X", t"|X', t' ⟩ = W ( X", it"|X', it' ). The main advantage of this approach is to permit a systematic application to quantum-dynamical problems of the mathematical techniques developed in the theory of classical stochastic processes, even when vector potentials and/or time-dependent potentials are present.
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.
A model independent generalization of quantum mechanics, including the usual as well as the dissipative quantum systems, is proposed. The theory is developed deductively from the basic principles of the standard quantum theory, the only new qualitative assumption being that we allow the wave operator at time t of a quantum system to be non-differentiable (in t) in the usual sense, but only in an appropriately defined (Sec. 5) stochastic sense. The resulting theory is shown to lead to a natural generalization of the usual quantum equations of motion, both in the form of the Schrödinger equation in interaction representation (Sec. 6) and of the Heisenberg equation (Sec. 8). The former equation leads in particular to a quantum fluctuation-dissipation relation of Einstein’s type. The latter equation is a generalized Langevin equation, from which the known form of the generalized master equation can be deduced via the quantum Feynmann-Kac technique (Secs. 9 and 10). For quantum noises with increments commuting with the past the quantum Langevin equation defines a closed system of (usually nonlinear) stochastic differential equations for the observables defining the coefficients of the noises. Such systems are parametrized by certain Lie algebras of observables of the system (Sec. 10). With appropriate choices of these Lie algebras one can deduce generalizations and corrections of several phenomenological equations previously introduced at different times to explain different phenomena. Two examples are considered: the Lie algebra [q, p]=i (Sec. 12), which is shown to lead to the equations of the damped harmonic oscillator; and the Lie algebra of SO(3) (Sec. 13) which is shown to lead to the Bloch equations. In both cases the equations obtained are independent of the model of noise. Moreover, in the former case, it is proved that the only possible noises which preserve the commutation relations of p, q are the quantum Brownian motions, commonly used in laser theory and solid state physics.
It is shown that a stochastic process can be viewed as a set of states (normalized positive linear functionals) over an Abelian C*-algebra. Alternatively, the stochastic process can be associated with a set of representations of the algebra as a subalgebra of the (noncommutative) C*-algebra of bounded operators in a Hilbert space. Then, an operator equation can be associated with every stochastic equation in some general conditions. The formalism is applied to Brownian motion. Then, we study the nonrelativistic motion of a single particle in stochastic electrodynamics, a theory which has been proposed as a possible alternative to quantum mechanics. The equations of motion, which are derived, coincide with the basic ones of quantum mechanics. The differences between this theory and quantum mechanics are summarized.
In Paper I of this investigation a new calculus for functions of noncommuting operators was developed, based on the notion of mapping of operators onto c-number functions. With the help of this calculus, a general theory is formulated, in the present paper, of phase-space representation of quantum-mechanical systems. It is shown that there is a whole class of such representations, one associated with each type of mapping, the simplest one being the well-known representation due to Weyl. For each representation, the quantum-mechanical expectation value of an operator is found to be expressible in the form of a phase-space average of classical statistical mechanics. The phase-space distribution functions are, however, not true probabilities, in general. The phase-space forms of the main quantum-mechanical equations of motion are obtained and are found to have the form of a generalized Liouville equation. The phase-space form of the Bloch equation for the density operator of a quantum system in thermal equilibrium is also derived. The generalized characteristic functions of boson systems are defined and their main properties are studied. The equations of motion for the characteristic functions are also derived. As an illustration of the theory, a generalized stochastic description of a quantized electromagnetic field is obtained.