Within the density matrix formalism, it is shown that a simple way to get decoherence is through the introduction of a "quantum" of time (chronon): which implies replacing the differential Liouville--von Neumann equation with a finite-difference version of it. In this way, one is given the possibility of using a rather simple quantum equation to describe the decoherence effects due to dissipation. Namely, the mere introduction (not of a "time-lattice", but simply) of a "chronon" allows us to go on from differential to finite-difference equations; and in particular to write down the quantum-theoretical equations (Schroedinger equation, Liouville--von Neumann equation,...) in three different ways: "retarded", "symmetrical", and "advanced". One of such three formulations --the retarded one-- describes in an elementary way a system which is exchanging (and losing) energy with the environment; and in its density-matrix version, indeed, it can be easily shown that all non-diagonal terms go to zero very rapidly. [A much larger presentation of the theoretical ground on which this paper is based appeared in the e-print quant-ph/9706059, and in the preprint IC/98/74, ICTP; Trieste, 1998].
Within the density matrix formalism, it is shown that a simple way to get decoherence is through the introduction of a "quantum" of time (chronon): which implies replacing the differential Liouville--von Neumann equation with a finite-difference version of it. In this way, one is given the possibility of using a rather simple quantum equation to describe the decoherence effects due to dissipation. Namely, the mere introduction (not of a "time-lattice", but simply) of a "chronon" allows us to go on from differential to finite-difference equations; and in particular to write down the quantum-theoretical equations (Schroedinger equation, Liouville--von Neumann equation,...) in three different ways: "retarded", "symmetrical", and "advanced". One of such three formulations --the retarded one-- describes in an elementary way a system which is exchanging (and losing) energy with the environment; and in its density-matrix version, indeed, it can be easily shown that all non-diagonal terms go to zero very rapidly. [A much larger presentation of the theoretical ground on which this paper is based appeared in the e-print quant-ph/9706059, and in the preprint IC/98/74, ICTP; Trieste, 1998].
In this paper, within the density matrix formalism, it is shown that a simple way to get decoherence is through the introduction of a “quantum” of time(or rather of a {em chronon/}): thus replacing the differential Liouville-von~Neumann equation with a finite-difference version of it. In this way, one is given the possibility of using a very simple quantum equation to describe the decoherence effects due to dissipation, and of partially solving the measurement-problem in quantum mechanics (avoiding any recourse to the wave-function collapse, without having recourse to any statistical approach). Namely, the mere introduction (not of a time-lattice, but simply) of the ``chronon allows us to go on from differential to finite-difference equations; and in particular to write down the Schroedinger equation (as well as the Liouville-von~Neumann equation) in three different ways: “retarded”, “symmetrical”, and “advanced”. One of such three formulations --the em retarded one-- describes in an elementary way a system which is exchanging (and losing) energy with the environment. In its density-matrix version, indeed, it can be easily shown that all non-diagonal terms go to zero very rapidly.
In this review-article, we discuss the consequences of the introduction of a quantum of time tau_0 in the formalism of non-relativistic quantum mechanics (QM) by referring ourselves in particular to the theory of the chronon as proposed by P.Caldirola. Such an interesting finite difference theory, forwards --at the classical level-- a solution for the motion of a particle endowed with a non-negligible charge in an external electromagnetic field, overcoming all the known difficulties met by Abraham-Lorentz's and Dirac's approaches (and even allowing a clear answer to the question whether a free falling charged particle does or does not emit radiation), and --at the quantum level-- yields a remarkable mass spectrum for leptons. After having briefly reviewed Caldirola's approach, we compare one another the new Schroedinger, Heisenberg and density-operator (Liouville-von Neumann) pictures resulting from it. Moreover, for each representation, three (retarded, symmetric and advanced) formulations are possible, which refer either to times t and t-tau_0, or to times t-tau_0/2 and t+tau_0/2, or to times t and t+tau_0, respectively. It is interesting to notice that, e.g., the retarded QM does naturally appear to describe QM with friction, i.e., to describe dissipative quantum systems (like a particle moving in an absorbing medium). In this sense, discretized QM is much richer than the ordinary one. When the density matrix formalism is applied to the solution of the measurement problem in QM, very interesting results are met, so as a natural explication of decoherence.
In this review-article, we discuss the consequences of the introduction of a quantum of time tau_0 in the formalism of non-relativistic quantum mechanics (QM) by referring ourselves in particular to the theory of the "chronon" as proposed by P.Caldirola. Such an interesting "finite difference" theory, forwards --at the classical level-- a solution for the motion of a particle endowed with a non-negligible charge in an external electromagnetic field, overcoming all the known difficulties met by Abraham-Lorentz's and Dirac's approaches (and even allowing a clear answer to the question whether a free falling charged particle does or does not emit radiation), and --at the quantum level-- yields a remarkable mass spectrum for leptons. After having briefly reviewed Caldirola's approach, we compare one another the new Schroedinger, Heisenberg and density-operator (Liouville-von Neumann) pictures resulting from it. Moreover, for each representation, three (retarded, symmetric and advanced) formulations are possible, which refer either to times t and t-tau_0, or to times t-tau_0/2 and t+tau_0/2, or to times t and t+tau_0, respectively. It is interesting to notice that, e.g., the "retarded" QM does naturally appear to describe QM with friction, i.e., to describe dissipative quantum systems (like a particle moving in an absorbing medium). In this sense, discretized QM is much richer than the ordinary one. When the density matrix formalism is applied to the solution of the measurement problem in QM, very interesting results are met, so as a natural explication of "decoherence".
This thesis examines dissipative forces in physical systems in both microscopic and quantum level. The system-environment interaction is modeled by assuming that the environment is composed of simple harmonic oscillators and that there is a linear interaction between these oscillators and the system. The classical case is solved by determining the retarded Green’s function, which yields the equation of motion of the system. For quantum dissipation, the Lindblad master equation is used to describe the time evolution of the reduced density matrix of the system, and the propagator is investigated to identify important parameters, including the term of Newton's friction force.
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 Brownian dynamics of the density operator for a quantum system interacting with a classical heat bath is described using a stochastic, non-linear Liouville equation obtained from a variational principle. The rate of the noise-induced transitions is expressed as a function of the environmental spectral density, and is discussed for the case of white noise and blackbody radiation. The time-scale of decoherence and dissipation is obtained numerically for a system of two quantum states. These are the ground and first excited state of the center of mass vibrations for a single ion confined in a harmonic trap. PACS numbers: 42.50.Lc,05.40.+j,32.80.Pj 1 Introduction The evolution of a quantum system surrounded by a thermal environment represents a fundamental problem of the quantum theory, increasingly important for applications. In the classical mechanics, a particle interacting with a heat bath has a Brownian evolution towards thermalization, which reduces to a purely dissipative moti...