Wigner's function and other distribution functions in mock phase spaces

Balázs, N. L.; Jennings, B.K.

doi:10.1016/0370-1573(84)90151-0

Cited by 408 publications

(216 citation statements)

References 29 publications

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“…The present more abstract approach is also more general, as it encompasses other representations, termed as Liouville representations [75,301,302] in which the product is implemented differently; an example of these, the Wigner representation, is useful in the semi-classical limit. The structure of the set of observables, a C * -algebra [156], involves addition, multiplication by complex c-numbers, hermitean conjugation, and non-commutative product 92 .…”

Section: Physical Quantities: Observablesmentioning

confidence: 99%

“…The formalism of density operators is more flexible than that of pure states: It affords the possibility not only of changing the basis in the Hilbert space, but also of performing linear transformations in the vector space of observables, which mix the left and right indices of observables i|Ô| j and of density matrices i|D| j . The resulting Liouville representations of quantum mechanics [75,301,302] are useful in many circumstances. They include for instance the Wigner representation, suited to study the semi-classical limit, and the polarization representation for a spin, currently used by experimentalists, in which any operator is represented by its coordinates on the basis (3.1) of the space of operators; in the present work, the parametrization of the stateD of S + M by P dis M (m) and C dis a (m) enters this framework (Eqs.…”

Section: G Density Operators For Beginnersmentioning

confidence: 99%

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Understanding quantum measurement from the solution of dynamical models

Allahverdyan¹,

Balian²,

2013

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The quantum measurement problem, to wit, understanding why a unique outcome is obtained in each individual experiment, is currently tackled by solving models. After an introduction we review the many dynamical models proposed over the years for elucidating quantum measurements. The approaches range from standard quantum theory, relying for instance on quantum statistical mechanics or on decoherence, to quantum-classical methods, to consistent histories and to modifications of the theory. Next, a flexible and rather realistic quantum model is introduced, describing the measurement of the z-component of a spin through interaction with a magnetic memory simulated by a Curie-Weiss magnet, including N 1 spins weakly coupled to a phonon bath. Initially prepared in a metastable paramagnetic state, it may transit to its up or down ferromagnetic state, triggered by its coupling with the tested spin, so that its magnetization acts as a pointer. A detailed solution of the dynamical equations is worked out, exhibiting several time scales. Conditions on the parameters of the model are found, which ensure that the process satisfies all the features of ideal measurements. Various imperfections of the measurement are discussed, as well as attempts of incompatible measurements. The first steps consist in the solution of the Hamiltonian dynamics for the spin-apparatus density matrixD(t). Its off-diagonal blocks in a basis selected by the spin-pointer coupling, rapidly decay owing to the many degrees of freedom of the pointer. Recurrences are ruled out either by some randomness of that coupling, or by the interaction with the bath. On a longer time scale, the trend towards equilibrium of the magnet produces a final stateD(t f ) that involves correlations between the system and the indications of the pointer, thus ensuring registration. AlthoughD(t f ) has the form expected for ideal measurements, it only describes a large set of runs. Individual runs are approached by analyzing the final states associated with all possible subensembles of runs, within a specified version of the statistical interpretation. There the difficulty lies in a quantum ambiguity: There exist many incompatible decompositions of the density matrixD(t f ) into a sum of sub-matrices, so that one cannot infer from its sole determination the states that would describe small subsets of runs. This difficulty is overcome by dynamics due to suitable interactions within the apparatus, which produce a special combination of relaxation and decoherence associated with the broken invariance of the pointer. Any subset of runs thus reaches over a brief delay a stable state which satisfies the same hierarchic property as in classical probability theory; the reduction of the state for each individual run follows. Standard quantum statistical mechanics alone appears sufficient to explain the occurrence of a unique answer in each run and the emergence of classicality in a measurement process. Finally, pedagogical exercises are proposed and lessons for future works on m...

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Section: Physical Quantities: Observablesmentioning

confidence: 99%

Section: G Density Operators For Beginnersmentioning

confidence: 99%

Understanding quantum measurement from the solution of dynamical models

Allahverdyan¹,

Balian²,

2013

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“…It is on this object that we shall later implement the relevant kinematical approximations. To this aim, we introduce the Wigner transform of ρ(r, r ′ , t) [48,49] :…”

Section: Kinetic Equations For Quantum Particlesmentioning

confidence: 99%

“…To give a crude estimate of these collisional effects, one may use the relaxation time approximation, and write the kinetic equation as 48) where τ col is a typical relaxation time. It is important here to distinguish between colour and colourless excitations.…”

Section: Effect Of Collisionsmentioning

confidence: 99%

“…In the limit p ≫ ω 0 , the branches ±ω + (p) describe the "normal" (anti)fermion, with a dispersion relation 48) describing the propagation of a massive particle with the "asymptotic" mass M ∞ = √ 2ω 0 . In the same limit, the "abnormal" branches ±ω − (p) disappear from the spectrum since their residues are exponentially suppressed [150].…”

Section: B) For Large Momentamentioning

confidence: 99%

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The quark–gluon plasma: collective dynamics and hard thermal loops

Blaizot¹,

Iancu²

2002

Physics Reports

511

714

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We present a unified description of the high temperature phase of QCD, the so-called quark-gluon plasma, in a regime where the effective gauge coupling g is sufficiently small to allow for weak coupling calculations. The main focuss is the construction of the effective theory for the collective excitations which develop at a typical scale gT , which is well separated from the typical energy of single particle excitations which is the temperature T . We show that the short wavelength thermal fluctuations, i.e., the plasma particles, provide a source for long wavelength oscillations of average fields which carry the quantum numbers of the plasma constituents, the quarks and the gluons. To leading order in g, the plasma particles obey simple gauge-covariant kinetic equations, whose derivation from the general Dyson-Schwinger equations is outlined. By solving these equations, we effectively integrate out the hard degrees of freedom, and are left with an effective theory for the soft collective excitations. As a by-product, the "hard thermal loops" emerge naturally in a physically transparent framework. We show that the collective excitations can be described in terms of classical fields, and develop for these a Hamiltonian formalism. This can be used to estimate the effect of the soft thermal fluctuations on the correlation functions. The effect of collisions among the hard particles is also studied. In particular we discuss how the collisions affect the lifetimes of quasiparticle excitations in a regime where the mean free path is comparable with the range of the relevant interactions. Collisions play also a decisive role in the construction a Member of CNRS. E-mail: blaizot@spht.saclay.cea.fr b Member of CNRS. E-mail: iancu@spht.saclay.cea.fr c Laboratoire de la Direction des Sciences de la Matière du Commissariatà l'Energie Atomique of the effective theory for ultrasoft excitations, with momenta ∼ g 2 T , a topic which is briefly addressed at the end of this paper.

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Classical Chaos Versus Quantum Dynamics: Kam Tori and Cantori as Dynamical Barriers

Radons

Geisel

1989

Advances in Chemical Physics

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Wigner's function and other distribution functions in mock phase spaces

Cited by 408 publications

References 29 publications

Understanding quantum measurement from the solution of dynamical models

Understanding quantum measurement from the solution of dynamical models

The quark–gluon plasma: collective dynamics and hard thermal loops

Classical Chaos Versus Quantum Dynamics: Kam Tori and Cantori as Dynamical Barriers

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