as quantum engineering. In the past two decades it has become increasingly clear that many (perhaps all) of the symptoms of classicality can be induced in quantum systems by their environments. Thus decoherence is caused by the interaction in which the environment in effect monitors certain observables of the system, destroying coherence between the pointer states corresponding to their eigenvalues. This leads to environment-induced superselection or einselection, a quantum process associated with selective loss of information. Einselected pointer states are stable. They can retain correlations with the rest of the universe in spite of the environment. Einselection enforces classicality by imposing an effective ban on the vast majority of the Hilbert space, eliminating especially the flagrantly nonlocal ''Schrodinger-cat states.'' The classical structure of phase space emerges from the quantum Hilbert space in the appropriate macroscopic limit. Combination of einselection with dynamics leads to the idealizations of a point and of a classical trajectory. In measurements, einselection replaces quantum entanglement between the apparatus and the measured system with the classical correlation. Only the preferred pointer observable of the apparatus can store information that has predictive power. When the measured quantum system is microscopic and isolated, this restriction on the predictive utility of its correlations with the macroscopic apparatus results in the effective ''collapse of the wave packet.'' The existential interpretation implied by einselection regards observers as open quantum systems, distinguished only by their ability to acquire, store, and process information. Spreading of the correlations with the effectively classical pointer states throughout the environment allows one to understand ''classical reality'' as a property based on the relatively objective existence of the einselected states. Effectively classical pointer states can be ''found out'' without being re-prepared, e.g, by intercepting the information already present in the environment. The redundancy of the records of pointer states in the environment (which can be thought of as their ''fitness'' in the Darwinian sense) is a measure of their classicality. A new symmetry appears in this setting. Environment-assisted invariance or envariance sheds new light on the nature of ignorance of the state of the system due to quantum correlations with the environment and leads to Born's rules and to reduced density matrices, ultimately justifying basic principles of the program of decoherence and einselection.

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Decoherence, einselection, and the quantum origins of the classical

Environment-induced decoherence and superselection have been a subject of intensive research over the past two decades, yet their implications for the foundational problems of quantum mechanics, most notably the quantum measurement problem, have remained a matter of great controversy. This paper is intended to clarify key features of the decoherence program, including its more recent results, and to investigate their application and consequences in the context of the main interpretive approaches of quantum mechanics.

http://faculty.up.edu/schlosshauer/publications/RevModPhys_76_001267.pdf

Decoherence, the measurement problem, and interpretations of quantum mechanics

Open quantum systems (OQSs) cannot always be described with the Markov approximation, which requires a large separation of system and environment time scales. An overview is given of some of the most important techniques available to tackle the dynamics of an OQS beyond the Markov approximation. Some of these techniques, such as master equations, Heisenberg equations, and stochastic methods, are based on solving the reduced OQS dynamics, while others, such as path integral Monte Carlo or chain mapping approaches, are based on solving the dynamics of the full system. The physical interpretation and derivation of the various approaches are emphasized, how they are connected is explored, and how different methods may be suitable for solving different problems is examined.

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Dynamics of non-Markovian open quantum systems

The origin of the phenomenological deterministic laws that approximately govern the quasiclassical domain of familiar experience is considered in the context of the quantum mechanics of closed systems such as the universe as a whole. A formulation of quantum mechanics is used that predicts probabilities for the individual members of a set of alternative coarse-grained histories that decohere, which means that there is negligible quantum interference between the individual histories in the set. We investigate the requirements for coarse grainings to yield decoherent sets of histories that are quasiclassical, i.e., such that the individual histories obey, with high probability, effective classical equations of motion interrupted continually by small fluctuations and occasionally by large ones. We discuss these requirements generally but study them specifically for coarse grainings of the type that follows a distinguished subset of a complete set of variables while ignoring the rest. More coarse graining is needed to achieve decoherence than would be suggested by naive arguments based on the uncertainty principle. Even coarser graining is required in the distinguished variables for them to have the necessary inertia to approach classical predictability in the presence of the noise consisting of the fluctuations that typical mechanisms of decoherence produce. We describe the derivation of phenomenological equations of motion explicitly for a particular class of models. Those models assume configuration space and a fundamental Lagrangian that is the difference between a kinetic energy quadratic in the velocities and a potential energy. The distinguished variables are taken to be a fixed subset of coordinates of configuration space. The initial density matrix of the closed system is assumed to factor into a product of a density matrix in the distinguished subset and another in the rest of the coordinates. With these restrictions, we improve the derivation from quantum mechanics of the phenomenological equations of motion governing a quasiclassical domain in the following respects: Probabilities of the correlations in time that define equations of motion are explicitly considered. Fully nonlinear cases are studied. Methods are exhibited for finding the form of the phenomenological equations of motion even when these are only distantly related to those of the fundamental action. The demonstration of the connection between quantum-mechanical causality and causality in classical phenomenological equations of motion is generalized. The connections among decoherence, noise, dissipation, and the amount of coarse graining necessary to achieve classical predictability are investigated quantitatively. Routes to removing the restrictions on the models in order to deal with more realistic coarse grainings are described.

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Classical equations for quantum systems

The environment surrounding a quantum system can, in effect, monitor some of the systems observables. As a result, the eigenstates of these observables continuously decohere and can behave like classical states.

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Decoherence and the Transition from Quantum to Classical—Revisited

We use the influence functional path-integral method to derive an exact master equation for the quantum Brownian motion of a particle linearly coupled to a general environment (ohmic, subohmic, or supraohmic) at arbitrary temperature and apply it to study certain aspects of the loss of quantum coherence.

Quantum Brownian motion in a general environment: Exact master equation with nonlocal dissipation and colored noise.

We address two basic issues in the theory of galaxy formation from fluctuations of quantum fields: 1) the nature and origin of noise and fluctuations and 2) the conditions for using a classical stochastic equation for their description. On the first issue, we derive the influence functional for a $\lambda \phi^4 $ field in a zero-temperature bath in de Sitter universe and obtain the correlator for the colored noises of vacuum fluctuations. This exemplifies a new mechanism we propose for colored noise generation which can act as seeds for galaxy formation with non-Gaussian distributions. For the second issue, we present a (functional) master equation for the inflaton field in de Sitter universe. By examining the form of the noise kernel we study the decoherence of the long-wavelength sector and the conditions for it to behave classically.

Quantum Origin of Noise and Fluctuations in Cosmology

We address two basic issues in the theory of galaxy formation from fluctuations of quantum fields: 1) the nature and origin of noise and fluctuations and 2) the conditions for using a classical stochastic equation for their description On the first issue, we derive the influence functional for a λo4 field in a zero-temperature bath in de Sitter universe and obtain the correlator for the colored noises of vacuum fluctuations This exemplifies a new mechanism we propose for colored noise generation which can act as seeds for galaxy formation with non-Gaussian distributions For the second issue, we present a (functional) master equation for the inflaton field in de Sitter universe By examining the form of the noise kernel we study the decoherence of the long-wavelength sector and the conditions for it to behave classically

The effective-action formalism used by Hartle, Hu, Fischetti, and Anderson for treating particle production and back reaction of free fields near the Planck time in the early Universe is extended here to interacting fields. We consider a massive (m) self-interacting \ensuremath{\lambda}${\ensuremath{\varphi}}^{4}$ scalar field coupled (\ensuremath{\xi}) to a Friedmann-Robertson-Walker spacetime and study the effect of interaction on particle production and on the dynamics of the background geometry. A background-field splitting is introduced in the one-loop effective action which is calculated by perturbative expansions up to the second order in \ensuremath{\lambda}, ${m}^{2}$, and \ensuremath{\xi}. Ultraviolet divergences from the fluctuation field are removed by introducing counterterms via dimensional regularization. From the regularized effective action equations governing the effective geometry and the background field are derived and their solutions sought. We consider separately the conformal versus the nonconformal (\ensuremath{\xi}), and the massless versus the massive (m) cases. We also contrast the cases with or without the background field (\ensuremath{\chi}) and those with or without interaction (\ensuremath{\lambda}). The dynamics of the scalar field and the scale factor and the probability of pair production in each case are calculated. In the discussion we present a qualitative explanation of these results, and draw the connection with related work. The classical limit of the present problem depicts the evolution of a Higgs field in curved space, its results being useful for the description of inflation and reheating processes in the grand-unified-theory epoch. The present problem is also the semiclassical limit of quantum cosmology. The results obtained here can be useful to tackling problems involving dynamical fields in curved space such as critical dynamics in the early Universe.

Yuhong Zhang

Papers

Quantum Brownian motion in a general environment: Exact master equation with nonlocal dissipation and colored noise.

Quantum Origin of Noise and Fluctuations in Cosmology

Quantum Origin of Noise and Fluctuations in Cosmology

Quantum effects of interacting fields in the early Universe