The decoherence effect, which may be held responsible for the vanishing of macroscopic quantum interferences, has only been exhibited in special models though it is presumed to be universal (at least when there is a possibility of dissipation). A more general model encompassing the already known ones is proposed in the present work. A master equation for the reduced density operator is established by means of two different methods. A direct method, which is given first, is simple but too specific. A second method relies on the projection method in the theory of irreversible processes. It is in principle very general and not confined to a special model. It shows that decoherence is a typical irreversible process from a theoretical standpoint. The master equation is worked out in detail when there exist convenient, so-called microstable collective observables (for which a criterion is given), selecting a preferred basis in which diagonalization occurs. For macroscopic mechanics, this basis is given by the position coordinates in three-dimensional space of relatively small parts in an object. Previously proposed models, including the quantum diffusion model, are found to be special cases of the present theory.
The decoherence effect, which may be held responsible for the vanishing of macroscopic quantum interferences, has only been exhibited in special models though it is presumed to be universal (at least when there is a possibility of dissipation). A more general model encompassing the already known ones is proposed in the present work. A master equation for the reduced density operator is established by means of two different methods. A direct method, which is given first, is simple but too specific. A second method relies on the projection method in the theory of irreversible processes. It is in principle very general and not confined to a special model. It shows that decoherence is a typical irreversible process from a theoretical standpoint. The master equation is worked out in detail when there exist convenient, so-called microstable collective observables (for which a criterion is given), selecting a preferred basis in which diagonalization occurs. For macroscopic mechanics, this basis is given by the position coordinates in three-dimensional space of relatively small parts in an object. Previously proposed models, including the quantum diffusion model, are found to be special cases of the present theory.
A theory of the decoherence effect can be obtained by applying the projection method in the theory of irreversible processes. The procedure is shown specifically on a model, already general enough for covering all the previously known ones. Decoherence is found to be governed by a master equation, which is particulary simple when the existence of convenient collective observables yields a preferred basis for diagonalization. A criterion is given for the existence of such “microstable” observables. In the case of mechanical macroscopic bodies, this basis exits and is given by the position coordinates in ordinary three-dimensional space for the various parts of the bodies.
The problem investigated in this paper is einselection, i. e. the selection of mutually exclusive quantum states with definite probabilities through decoherence. Its study is based on a theory of decoherence resulting from the projection method in the quantum theory of irreversible processes, which is general enough for giving reliable predictions. This approach leads to a definition (or redefinition) of the coupling with the environment involving only fluctuations. The range of application of perturbation calculus is then wide, resulting in a rather general master equation. Two distinct cases of decoherence are then found: (i) A ``degenerate'' case (already encountered with solvable models) where decoherence amounts essentially to approximate diagonalization; (ii) A general case where the einselected states are essentially classical. They are mixed states. Their density operators are proportional to microlocal projection operators (or ``quasi projectors'') which were previously introduced in the quantum expression of classical properties. It is found at various places that the main limitation in our understanding of decoherence is the lack of a systematic method for constructing collective observables.
The problem investigated in this paper is einselection, i.e., the selection of mutually exclusive quantum states with definite probabilities through decoherence. Its study is based on a theory of decoherence resulting from the projection method in the quantum theory of irreversible processes, which is general enough for giving reliable predictions. This approach leads to a definition (or redefinition) of the coupling with the environment involving only fluctuations. The range of application of perturbation calculus is then wide, resulting in a rather general master equation. Two distinct cases of decoherence are then found: (i) a ``degenerate'' case (already encountered with solvable models) where decoherence amounts essentially to approximate diagonalization; (ii) a general case where the einselected states are essentially classical. They are mixed states. Their density operators are proportional to microlocal projection operators (or ``quasiprojectors'') that were previously introduced in the quantum expression of classical properties. It is found at various places that the main limitation in our understanding of decoherence is the lack of a systematic method for constructing collective observables.
The act of measurement on a quantum state is supposed to "dephase" (dephasing refers to the phenomenon that the states lose phase coherence; then the phases get randomized in interaction with a bath of other oscillators, which is referred to as "decoherence"), then "decohere" and "collapse" (or more precisely "register" and "reduce") the state into one of several eigenstates of the operator corresponding to the observable being measured. This measurement process is sometimes described as outside standard quantum-mechanical evolution and not calculable from Schrödinger's equation. Progress has, however, been made in studying this problem with two main calculation tools-one uses a time-independent Hamiltonian, while a rather more general approach proving that decoherence occurs under some generic conditions. The two general approaches to the study of wave-function collapse are as follows. The first approach, called the "consistent" or "decoherent"' histories approach, studies microscopic histories that diverge probabilistically and explains collapse as an event in our particular history. The other, referred to as the "environmental decoherence" approach studies the effect of the environment upon the quantum system, to explain wave-function decoherence which is produced by irreversible effects of various sorts. However, as we know, wave-function collapse is not related to thermal connection with the environment, rather, it is inherent to how measurements are performed by macroscopic apparata. In the "environmental decoherence" approach, one studies decoherence using a Markovian-approximated Master equation to study the time-evolution of the reduced density matrix (post dephasing) and obtains the long-time dependence of the off-diagonal elements of this matrix. The calculation in this paper studies the evolution of a quantum system starting with "dephasing" followed by the effects of the environment with some differences from prior analyses. We start from the Schrödinger equation for the state of the system, with a time-dependent Hamiltonian that reflects the actual microscopic interactions that are occurring. Then we systematically solve (exactly) for the time-evolved state, without invoking a Markovian approximation when writing out the effective time-evolution equation, i.e., keeping the evolution unitary until the end. This approach is useful, and it shows that the system wave-function will explicitly "un-collapse" if the measurement apparatus is sufficiently small. However, in the limit of a macroscopic system, this "dephasing" quickly leads to "decoherence"-collapse is a temporary state that will simply take extremely long (of the order of multiple universe lifetimes) to reverse. This has been attempted previously and our calculation is particularly simple and calculable. We make some connections to the work by Linden et al. while doing so. The calculation in this paper has interesting implications for the interpretation of the Wigner's friend experiment, as well as the Mott experiment, which is explored in "Connection to some general theoretical results" and "Recurrence times" (especially the enumerated points in "Recurrence times"). The upshot is that as long as Wigner's friend is macroscopically large (or uses a macroscopically large measuring instrument), no one needs to worry that Wigner would see something different from his friend. Indeed, Wigner's friend does not even need to be conscious during the measurement that she conducts. It also allows one to reasonably interpret some of the more recent thought experiments proposed. In particular, as a result of the mathematical analysis, the short-time behavior of a collapsing system, at least the one considered in this paper, is not exponential. Instead, it is the usual Fermi-golden rule result. The long-term behavior is, of course, still exponential. This is a second novel feature of the paper-we connect the short-term Fermi-golden rule (quadratic-in-time behavior) transition probability to the exponential long-time behavior of a collapsing wave-function in one continuous mathematical formulation.
The conventional interpretation of quantum mechanics, though it permits a correspondence to classical physics, leaves the exact mechanism of transition unclear. Though this was only of philosophical importance throughout the twentieth century, over the past decade new technological developments, such as quantum computing, require a more thorough understanding of not just the result of quantum emergence, but also its mechanism. Quantum decoherence theory is the model that developed out of necessity to deal with the quantum-classical transition explicitly, and without external observers. In this thesis, we present a self-contained and rigorously argued full derivation of the master equation for quantum Brownian motion, one of the key results in quantum decoherence theory. We accomplish this from a foundational perspective, only assuming a few basic axioms of quantum mechanics and deriving their consequences. We then consider a physical example of the master equation and show that quantum decoherence successfully represents the transition from a quantum to classical system.
Quantum decoherence represents a fundamental challenge in the development and application of quantum technologies. This paper investigates the theoretical foundations of quantum decoherence, moving beyond the traditional view of it as a simple, isolated process. We argue that decoherence is a complex phenomenon arising from the interplay of multiple physical mechanisms at various scales. The core of this work is the development of a multi-scale model that integrates microscopic quantum effects – such as displacement, impurities, and time evolution – with macroscopic decoherence effects related to circuit design and material selection. We propose a framework for analyzing and simulating decoherence, recognizing the need for a more comprehensive mathematical description. This paper lays the groundwork for future research aimed at refining our understanding of decoherence and ultimately improving the robustness of quantum systems. ---