Phase Space

This book covers the theory and applications of the Wigner phase space distribution function and its symmetry properties. The book explains why the phase space picture of quantum mechanics is needed, in addition to the conventional Schrödinger or Heisenberg picture. It is shown that the uncertainty relation can be represented more accurately in this picture. In addition, the phase space picture is shown to be the natural representation of quantum mechanics for modern optics and relativistic quantum mechanics of extended objects. Contents:Phase Space in Classical MechanicsForms of Quantum MechanicsWigner Phase- Space Distribution FunctionsLinear Canonical Transformations in Quantum MechanicsCoherent and Squeezed StatesPhase-Space Picture of Coherent and Squeezed StatesLorentz TransformationsCovariant Harmonic OscillatorsLorentz-Squeezed HadronsSpace-Time Geometry of Extended Particles Readership: Physicists, applied physicists and mathematical physicists. keywords:Lorentz Transformations;Wigner's Little Groups;Quantum Optics;Relativistic Quantum Mechanics;Phase Space;Wigner Function;Squeezed States;Feynman's Parton Picture;Covariant Harmonic Oscillators;Space-Time Geometry;Hadrons;Group Theory “… if Casimir invariants and Lorentz groups excite you, you'll be at home in Kim and Noz's lecture notes…” Contemporary Physics

Tied in to Baxter's masterful Manifold trilogy, these thematically linked stories are drawn from the vast graph of possibilities across which the lives of hero Reid Malenfant have been scattered. It is the year 2025. Reid Malenfant is the commander of a NASA earth-orbiting science platform. The platform is intended to probe the planets of the nearest star system by bouncing laser pulses off them. But no echoes are returned... and Malenfant's reality begins to crumble around him. Huddling with his family, awaiting the end -- or an unknown new beginning -- Malenfant tells stories of other possibilities, other realities. The linked stories encompass the myriad possibilities that might govern our relationship with the universe: are we truly alone, or will we eventually meet other lifeforms? Perhaps intelligent species decide to turn their back on the stars, or maybe expansionist species are destined to fail. The final possibility -- that the Universe as we know it is in fact an elaborate illusion designed to protect us from the fearful reality -- is brilliantly explored in the tour de force novella that ends the volume.

Quantum Optics in Phase Space provides a concise introduction to the rapidly moving field of quantum optics from the point of view of phase space. Modern in style and didactically skillful, Quantum Optics in Phase Space prepares students for their own research by presenting detailed derivations, many illustrations and a large set of workable problems at the end of each chapter. Often, the theoretical treatments are accompanied by the corresponding experiments. An exhaustive list of references provides a guide to the literature. Quantum Optics in Phase Space also serves advanced researchers as a comprehensive reference book. Starting with an extensive review of the experiments that define quantum optics and a brief summary of the foundations of quantum mechanics the author Wolfgang P. Schleich illustrates the properties of quantum states with the help of the Wigner phase space distribution function. His description of waves ala WKB connects semi-classical phase space with the Berry phase. These semi-classical techniques provide deeper insight into the timely topics of wave packet dynamics, fractional revivals and the Talbot effect. Whereas the first half of the book deals with mechanical oscillators such as ions in a trap or atoms in a standing wave the second half addresses problems where the quantization of the radiation field is of importance. Such topics extensively discussed include optical interferometry, the atom-field interaction, quantum state preparation and measurement, entanglement, decoherence, the one-atom maser and atom optics in quantized light fields. Quantum Optics in Phase Space presents the subject of quantum optics as transparently as possible. Giving wide-ranging references, it enables students to study and solve problems with modern scientific literature. The result is a remarkably concise yet comprehensive and accessible text- and reference book - an inspiring source of information and insight for students, teachers and researchers alike.

Wigner's quasi-probability distribution function in phase space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum transport in quantum optics; nuclear physics; decoherence, quantum computing, and quantum chaos. It is also important in signal processing and the mathematics of algebraic deformation. A remarkable aspect of its internal logic, pioneered by Groenewold and Moyal, has only emerged in the last quarter-century: it furnishes a third, alternative, formulation of quantum mechanics, independent of the conventional Hilbert space, or path integral formulations.In this logically complete and self-standing formulation, one need not choose sides ? coordinate or momentum space. It works in full phase space, accommodating the uncertainty principle, and it offers unique insights into the classical limit of quantum theory. This invaluable book is a collection of the seminal papers on the formulation, with an introductory overview which provides a trail map for those papers; an extensive bibliography; and simple illustrations, suitable for applications to a broad range of physics problems. It can provide supplementary material for a beginning graduate course in quantum mechanics.

This book covers the theory and applications of the Wigner phase space distribution function and its symmetry properties. The book explains why the phase space picture of quantum mechanics is needed, in addition to the conventional Schr”dinger or Heisenberg picture. It is shown that the uncertainty relation can be represented more accurately in this picture. In addition, the phase space picture is shown to be the natural representation of quantum mechanics for modern optics and relativistic quantum mechanics of extended objects.

In this monograph, we shall present a new mathematical formulation of quantum theory, clarify a number of discrepancies within the prior formulation of quantum theory, give new applications to experiments in physics, and extend the realm of application of quantum theory well beyond physics. Here, we motivate this new formulation and sketch how it developed. Since the publication of Dirac's famous book on quantum mechanics [Dirac, 1930] and von Neumann's classic text on the mathematical foundations of quantum mechanics two years later [von Neumann, 1932], there have appeared a number of lines of development, the intent of each being to enrich quantum theory by extra polating or even modifying the original basic structure. These lines of development have seemed to go in different directions, the major directions of which are identified here: First is the introduction of group theoretical methods [Weyl, 1928; Wigner, 1931] with the natural extension to coherent state theory [Klauder and Sudarshan, 1968; Peremolov, 1971]. The call for an axiomatic approach to physics [Hilbert, 1900; Sixth Problem] led to the development of quantum logic [Mackey, 1963; Jauch, 1968; Varadarajan, 1968, 1970; Piron, 1976; Beltrametti & Cassinelli, 1981], to the creation of the operational approach [Ludwig, 1983-85, 1985; Davies, 1976] with its application to quantum communication theory [Helstrom, 1976; Holevo, 1982), and to the development of the C* approach [Emch, 1972]. An approach through stochastic differential equations ("stochastic mechanics") was developed [Nelson, 1964, 1966, 1967].

Wigner's quasi-probability distribution function in phase space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum transport in quantum optics; nuclear physics; decoherence, quantum computing, and quantum chaos. It is also important in signal processing and the mathematics of algebraic deformation. A remarkable aspect of its internal logic, pioneered by Groenewold and Moyal, has only emerged in the last quarter-century: it furnishes a third, alternative, formulation of quantum mechanics, independent of the conventional Hilbert space, or path integral formulations. In this logically complete and self-standing formulation, one need not choose sides — coordinate or momentum space. It works in full phase space, accommodating the uncertainty principle, and it offers unique insights into the classical limit of quantum theory. This invaluable book is a collection of the seminal papers on the formulation, with an introductory overview which provides a trail map for those papers; an extensive bibliography; and simple illustrations, suitable for applications to a broad range of physics problems. It can provide supplementary material for a beginning graduate course in quantum mechanics. Contents:The Wigner FunctionSolving for the Wigner FunctionThe Uncertainty PrincipleEhrenfest's TheoremIllustration: The Harmonic OscillatorTime EvolutionNondiagonal Wigner FunctionsStationary Perturbation TheoryPropagatorsCanonical TransformationsThe Weyl CorrespondenceAlternate Rules of AssociationThe Groenwold–van Hove Theorem and the Uniqueness of MBs and ∗-ProductsOmitted MiscellanySelected Papers: Brief Historical Outline Readership: Advanced undergraduates, beginning graduate students and researchers in physics, quantum computing, chemistry and information processing. Keywords:Phase Space Quantization;Wigner Functions;Star Products;DeformationsReviews:“… the authors have struck the right note in their choice of presentation and also their decision as to what to omit, since the subject matter covers a very broad range … the authors have performed an excellent job in presenting a timely and very useful resource for investigators, in potentially many areas requiring quantum physics, who wish to use quasi-probability functions, particularly the Wigner function. I highly recommend it.”International Journal of Quantum Information

The Standard Model of elementary particles, although very successful, contains various elements that are put in by hand. Understanding their origin requires going beyond the model and searching for “new physics”. The present book elaborates on one particular proposal concerning such physics. While the original conception is 50 years old, it has not lost its appeal over time. Its basic idea is that space — an arena of events treated in the Standard Model as a classical background — is a concept which emerges from a strictly discrete quantum layer in the limit of large quantum numbers. This book discusses an extension of this view by replacing space with phase space. It combines the results of the author's research papers and places them in much broader philosophical and phenomenological contexts, thus providing further arguments in favor of the proposed alternative. The book should be of interest to the philosophically-minded readers who are willing to contemplate unorthodox ideas on the very nature of the world. Contents:IntroductionPhilosophy and Physics:Reality and Its DescriptionClassical and Quantum Aspects of RealityTime for a ChangeElementary Particles:The Standard Model and the Subparticle ParadigmThe Problem of MassConstituent Quarks and Spacetime PointsElementary Particles and Macroscopic SpacePhase Space and Quantum:Phase Space and Its SymmetriesQuantizing Phase SpaceElementary Particles from a Phase-Space PerspectiveGeneralizing the Concept of MassOverview Readership: Professionals and researchers in theoretical physics, philosophy of science, particle physics and quantum physics. Keywords:Elementary Particles;Classical-Quantum Tension;Emergent Spacetime;Process Philosophy;Being and Becoming;Time and Change;Standard Model;Problem of Mass;Current and Constituent Quarks;Low-Energy Hadron Phenomenology;Space Quantization;Born's Reciprocity;Phase-Space Symmetries;Dirac Linearization;Clifford Algebra;Internal Quantum Numbers;Hariri-Shupe Preon Model;Hadron and Quark CompositenessKey Features:The main idea is supported by a rare, coherent combination of philosophical, theoretical and phenomenological argumentsThe book uniquely links the structure of a single Standard Model generation of elementary particles to the symmetries of nonrelativistic phase spaceThe book suggests a novel way of looking at the concept of quark mass, as well as that of mass in general