56 found
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  1. On the Common Structure of Bohmian Mechanics and the Ghirardi-Rimini-Weber Theory: Dedicated to GianCarlo Ghirardi on the Occasion of His 70th Birthday.Valia Allori, Sheldon Goldstein, Roderich Tumulka & Nino Zanghi - 2008 - British Journal for the Philosophy of Science 59 (3):353 - 389.
    Bohmian mechanics and the Ghirardi-Rimini-Weber theory provide opposite resolutions of the quantum measurement problem: the former postulates additional variables (the particle positions) besides the wave function, whereas the latter implements spontaneous collapses of the wave function by a nonlinear and stochastic modification of Schrödinger's equation. Still, both theories, when understood appropriately, share the following structure: They are ultimately not about wave functions but about 'matter' moving in space, represented by either particle trajectories, fields on space-time, or a discrete set of (...)
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  2.  80
    On the Common Structure of Bohmian Mechanics and the Ghirardi–Rimini–Weber Theory Dedicated to GianCarlo Ghirardi on the Occasion of His 70th Birthday.Valia Allori, Sheldon Goldstein, Roderich Tumulka & Nino Zanghì - 2008 - British Journal for the Philosophy of Science 59 (3):353-389.
    Bohmian mechanics and the Ghirardi–Rimini–Weber theory provide opposite resolutions of the quantum measurement problem: the former postulates additional variables besides the wave function, whereas the latter implements spontaneous collapses of the wave function by a nonlinear and stochastic modification of Schrödinger's equation. Still, both theories, when understood appropriately, share the following structure: They are ultimately not about wave functions but about ‘matter’ moving in space, represented by either particle trajectories, fields on space-time, or a discrete set of space-time points. The (...)
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  3.  86
    On the Common Structure of Bohmian Mechanics and the Ghirardi–Rimini–Weber Theory: Dedicated to Giancarlo Ghirardi on the Occasion of His 70th Birthday.Valia Allori, Sheldon Goldstein, Roderich Tumulka & and Nino Zanghì - 2008 - British Journal for the Philosophy of Science 59 (3):353-389.
    Bohmian mechanics and the Ghirardi–Rimini–Weber theory provide opposite resolutions of the quantum measurement problem: the former postulates additional variables (the particle positions) besides the wave function, whereas the latter implements spontaneous collapses of the wave function by a nonlinear and stochastic modification of Schrödinger's equation. Still, both theories, when understood appropriately, share the following structure: They are ultimately not about wave functions but about ‘matter’ moving in space, represented by either particle trajectories, fields on space-time, or a discrete set of (...)
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  4.  94
    Boltzmann's Approach to Statistical Mechanics.Sheldon Goldstein - unknown
    In the last quarter of the nineteenth century, Ludwig Boltzmann explained how irreversible macroscopic laws, in particular the second law of thermodynamics, originate in the time-reversible laws of microscopic physics. Boltzmann’s analysis, the essence of which I shall review here, is basically correct. The most famous criticisms of Boltzmann’s later work on the subject have little merit. Most twentieth century innovations – such as the identification of the state of a physical system with a probability distribution on its phase space, (...)
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  5. Are All Particles Real?Sheldon Goldstein, James Taylor, Roderich Tumulka & Nino Zanghi - 2004 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 36 (1):103-112.
    In Bohmian mechanics elementary particles exist objectively, as point particles moving according to a law determined by a wavefunction. In this context, questions as to whether the particles of a certain species are real---questions such as, Do photons exist? Electrons? Or just the quarks?---have a clear meaning. We explain that, whatever the answer, there is a corresponding Bohm-type theory, and no experiment can ever decide between these theories. Another question that has a clear meaning is whether particles are intrinsically distinguishable, (...)
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  6. Bohmian Mechanics.Roderich Tumulka, Detlef Durr, Sheldon Goldstein & Nino Zanghi - 2009 - Compendium of Quantum Physics.
    Bohmian mechanics is a theory about point particles moving along trajectories. It has the property that in a world governed by Bohmian mechanics, observers see the same statistics for experimental results as predicted by quantum mechanics. Bohmian mechanics thus provides an explanation of quantum mechanics. Moreover, the Bohmian trajectories are defined in a non-conspiratorial way by a few simple laws.
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  7. Predictions and Primitive Ontology in Quantum Foundations: A Study of Examples.Valia Allori, Sheldon Goldstein, Roderich Tumulka & Nino Zanghi - 2012 - British Journal for the Philosophy of Science (2):axs048.
    A major disagreement between different views about the foundations of quantum mechanics concerns whether for a theory to be intelligible as a fundamental physical theory it must involve a ‘primitive ontology’ (PO), i.e. variables describing the distribution of matter in four-dimensional space–time. In this article, we illustrate the value of having a PO. We do so by focussing on the role that the PO plays for extracting predictions from a given theory and discuss valid and invalid derivations of predictions. To (...)
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  8. Bohmian Mechanics.Sheldon Goldstein - 2008 - Stanford Encyclopedia of Philosophy.
    Bohmian mechanics, which is also called the de Broglie-Bohm theory, the pilot-wave model, and the causal interpretation of quantum mechanics, is a version of quantum theory discovered by Louis de Broglie in 1927 and rediscovered by David Bohm in 1952. It is the simplest example of what is often called a hidden variables interpretation of quantum mechanics. In Bohmian mechanics a system of particles is described in part by its wave function, evolving, as usual, according to Schrödinger's equation. However, the (...)
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  9. Many Worlds and Schrodinger's First Quantum Theory.Valia Allori, Sheldon Goldstein, Roderich Tumulka & Nino Zanghi - 2011 - British Journal for the Philosophy of Science 62 (1):1-27.
    Schrödinger’s first proposal for the interpretation of quantum mechanics was based on a postulate relating the wave function on configuration space to charge density in physical space. Schrödinger apparently later thought that his proposal was empirically wrong. We argue here that this is not the case, at least for a very similar proposal with charge density replaced by mass density. We argue that when analyzed carefully, this theory is seen to be an empirically adequate many-worlds theory and not an empirically (...)
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  10. Quantum Equilibrium and the Origin of Absolute Uncertainty.Detlef Durr, Sheldon Goldstein & Nino Zanghi - 1992 - Journal of Statistical Physics 67:843-907.
     
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  11. Quantum Physics Without Quantum Philosophy.Detlef Dürr, Sheldon Goldstein & Nino Zanghì - 1995 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 26 (2):137-149.
    Quantum philosophy, a peculiar twentieth-century malady, is responsible for most of the conceptual muddle plaguing the foundations of quantum physics. When this philosophy is eschewed, one naturally arrives at Bohmian mechanics, which is what emerges from Schrodinger's equation for a nonrelativistic system of particles when we merely insist that 'particles' means particles. While distinctly non-Newtonian, Bohmian mechanics is a fully deterministic theory of particles in motion, a motion choreographed by the wave function. The quantum formalism emerges when measurement situations are (...)
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  12.  45
    Quantum Theory Without Observers.Sheldon Goldstein - unknown
    Despite its extraordinary predictive successes, quantum mechanics has, since its inception some seventy years ago, been plagued by conceptual di culties. The basic problem, plainly put, is this: It is not at all clear what quantum mechanics is about. What, in fact, does quantum mechanics describe?
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  13. Typicality and Notions of Probability in Physics.Sheldon Goldstein - 2012 - In Yemima Ben-Menahem & Meir Hemmo (eds.), Probability in Physics. Springer. pp. 59--71.
  14. Bohmian Mechanics and Quantum Theory: An Appraisal.James T. Cushing, Arthur Fine & Sheldon Goldstein - 1996 - Springer.
     
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  15.  6
    Predictions and Primitive Ontology in Quantum Foundations: A Study of Examples.Valia Allori, Sheldon Goldstein, Roderich Tumulka & Nino Zanghi - 2014 - British Journal for the Philosophy of Science 65 (2):323-352.
    A major disagreement between different views about the foundations of quantum mechanics concerns whether for a theory to be intelligible as a fundamental physical theory it must involve a ‘primitive ontology’ (PO), i.e. variables describing the distribution of matter in four-dimensional space–time. In this article, we illustrate the value of having a PO. We do so by focusing on the role that the PO plays for extracting predictions from a given theory and discuss valid and invalid derivations of predictions. To (...)
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  16.  67
    Naive Realism About Operators.Martin Daumer, Detlef Dürr, Sheldon Goldstein & Nino Zanghì - 1996 - Erkenntnis 45 (2-3):379 - 397.
    A source of much difficulty and confusion in the interpretation of quantum mechanics is a naive realism about operators. By this we refer to various ways of taking too seriously the notion of operator-as-observable, and in particular to the all too casual talk about measuring operators that occurs when the subject is quantum mechanics. Without a specification of what should be meant by measuring a quantum observable, such an expression can have no clear meaning. A definite specification is provided by (...)
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  17.  24
    Reality and the Role of the Wavefunction in Quantum Theory.Sheldon Goldstein & Nino Zanghi - unknown
    The most puzzling issue in the foundations of quantum mechanics is perhaps that of the status of the wave function of a system in a quantum universe. Is the wave function objective or subjective? Does it represent the physical state of the system or merely our information about the system? And if the former, does it provide a complete description of the system or only a partial description? We shall address these questions here mainly from a Bohmian perspective, and shall (...)
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  18.  87
    Opposite Arrows of Time Can Reconcile Relativity and Nonlocality.Sheldon Goldstein - manuscript
    We present a quantum model for the motion of N point particles, implying nonlocal (i.e., superluminal) influences of external fields on the trajectories, that is nonetheless fully relativistic. In contrast to other models that have been proposed, this one involves no additional space-time structure as would be provided by a (possibly dynamical) foliation of space-time. This is achieved through the interplay of opposite microcausal and macrocausal (i.e., thermodynamic) arrows of time. PACS numbers 03.65.Ud; 03.65.Ta; 03.30.+p..
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  19. Bohmian Trajectories as the Foundation of Quantum Mechanics.Sheldon Goldstein - unknown
    Bohmian trajectories have been used for various purposes, including the numerical simulation of the time-dependent Schr¨ odinger equation and the visualization of time-dependent wave functions. We review the purpose they were invented for: to serve as the foundation of quantum mechanics, i.e., to explain quantum mechanics in terms of a theory that is free of paradoxes and allows an understanding that is as clear as that of classical mechanics. Indeed, they succeed in serving that purpose in the context of a (...)
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  20.  70
    Revised Proof of the Uniqueness Theorem for ‘No Collapse’ Interpretations of Quantum Mechanics.Jeffrey Bub, Rob Clifton & Sheldon Goldstein - 2000 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 31 (1):95-98.
    We show that the Bub-Clifton uniqueness theorem (1996) for 'no collapse' interpretations of quantum mechanics can be proved without the 'weak separability' assumption.
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  21.  84
    What Does the Free Will Theorem Actually Prove?Sheldon Goldstein - unknown
    Conway and Kochen have presented a “free will theorem” [4, 6] which they claim shows that “if indeed we humans have free will, then [so do] elementary particles.” In a more precise fashion, they claim it shows that for certain quantum experiments in which the experimenters can choose between several options, no deterministic or stochastic model can account for the observed outcomes without violating a condition “MIN” motivated by relativistic symmetry. We point out that for stochastic models this conclusion is (...)
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  22. The Message of the Quantum?Martin Daumer, Detlef Duerr, Sheldon Goldstein, Tim Maudlin, Roderich Tumulka & Nino Zanghi - unknown
    We criticize speculations to the effect that quantum mechanics is fundamentally about information. We do this by pointing out how unfounded such speculations in fact are. Our analysis focuses on the dubious claims of this kind recently made by Anton Zeilinger.
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  23.  47
    Review Essay: Bohmian Mechanics and the Quantum Revolution. [REVIEW]Sheldon Goldstein - 1996 - Synthese 107 (1):145 - 165.
  24.  30
    On the Role of Density Matrices in Bohmian Mechanics.Detlef Dürr, Sheldon Goldstein, Roderich Tumulka & Nino Zanghí - 2005 - Foundations of Physics 35 (3):449-467.
  25.  74
    The Quantum Formalism and the Grw Formalism.Sheldon Goldstein - unknown
    The Ghirardi–Rimini–Weber (GRW) theory of spontaneous wave function collapse is known to provide a quantum theory without observers, in fact two different ones by using either the matter density ontology (GRWm) or the flash ontology (GRWf). Both theories are known to make predictions different from those of quantum mechanics, but the difference is so small that no decisive experiment can as yet be performed. While some testable deviations from quantum mechanics have long been known, we provide here something that has (...)
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  26. Bohmian Mechanics and Quantum Theory: An Appraisal.James T. Cushing, Arthur Fine & Sheldon Goldstein - 1998 - British Journal for the Philosophy of Science 49 (2):332-337.
     
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  27.  51
    Long-Time Behavior of Macroscopic Quantum Systems: Commentary Accompanying the English Translation of John Von Neumann's 1929 Article on the Quantum Ergodic Theorem.Sheldon Goldstein & Roderich Tumulka - unknown
    The renewed interest in the foundations of quantum statistical mechanics in recent years has led us to study John von Neumann’s 1929 article on the quantum ergodic theorem. We have found this almost forgotten article, which until now has been available only in German, to be a treasure chest, and to be much misunderstood. In it, von Neumann studied the long-time behavior of macroscopic quantum systems. While one of the two theorems announced in his title, the one he calls the (...)
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  28.  84
    On the Time Scales in the Approach to Equilibrium of Macroscopic Quantum Systems.Hal Tasaki, Sheldon Goldstein & Takashi Hara - unknown
    The recent renewed interest in the foundation of quantum statistical mechanics and in the dynamics of isolated quantum systems has led to a revival of the old approach by von Neumann to investigate the problem of thermalization only in terms of quantum dynamics in an isolated system [1, 2]. It has been demonstrated in some general or concrete settings that a pure initial state evolving under quantum dynamics indeed approaches an equilibrium state [3–9]. The underlying idea that a single pure (...)
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  29. Bohmian Mechanics and Quantum Information.Sheldon Goldstein - 2010 - Foundations of Physics 40 (4):335-355.
    Many recent results suggest that quantum theory is about information, and that quantum theory is best understood as arising from principles concerning information and information processing. At the same time, by far the simplest version of quantum mechanics, Bohmian mechanics, is concerned, not with information but with the behavior of an objective microscopic reality given by particles and their positions. What I would like to do here is to examine whether, and to what extent, the importance of information, observation, and (...)
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  30. Boltzmann Entropy for Dense Fluids Not in Local Equilibrium.Sheldon Goldstein - manuscript
    Using computer simulations, we investigate the time evolution of the (Boltzmann) entropy of a dense fluid not in local equilibrium. The macrovariables M describing the system are the (empirical) particle density f = {f(x,v)} and the total energy E. We find that S(ft,E) is a monotone increasing in time even when its kinetic part is decreasing. We argue that for isolated Hamiltonian systems monotonicity of S(Mt) = S(MXt) should hold generally for ‘‘typical’’ (the overwhelming majority of) initial microstates (phase points) (...)
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  31.  31
    On the Uniqueness of Quantum Equilibrium in Bohmian Mechanics.Sheldon Goldstein & W. Struyve - manuscript
    In Bohmian mechanics the distribution |ψ|2 is regarded as the equilibrium distribution. We consider its uniqueness, finding that it is the unique equivariant distribution that is also a local functional of the wave function ψ.
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  32.  80
    Seven Steps Toward the Classical World.Valia Allori, Detlef Duerr, Nino Zanghi & Sheldon Goldstein - 2002 - Journal of Optics B 4:482–488.
    Classical physics is about real objects, like apples falling from trees, whose motion is governed by Newtonian laws. In standard quantum mechanics only the wave function or the results of measurements exist, and to answer the question of how the classical world can be part of the quantum world is a rather formidable task. However, this is not the case for Bohmian mechanics, which, like classical mechanics, is a theory about real objects. In Bohmian terms, the problem of the classical (...)
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  33. Quantum Physics Without Quantum Philosophy.Detlef Dürr, Sheldon Goldstein & Nino Zanghì - 1995 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 26 (2):137-149.
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  34.  75
    On a Realistic Theory for Quantum Physics.Sheldon Goldstein - unknown
    future evolution of the field. These ideas thou h old 'th k oug o, are ei er un nown oz misunderstood, Our point here is that a stron realistic os". g ' ' posi'.ion has consequences: it offers a completely natural..
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  35.  3
    On the Common Structure of Bohmian Mechanics and the Ghirardi-Rimini-Weber Theory.Valia Allori, Sheldon Goldstein, Roderich Tumulka & Nino Zanghi - unknown
    Bohmian mechanics and the Ghirardi-Rimini-Weber theory provide opposite resolutions of the quantum measurement problem: the former postulates additional variables besides the wave function, whereas the latter implements spontaneous collapses of the wave function by a nonlinear and stochastic modification of Schrodinger's equation. Still, both theories, when understood appropriately, share the following structure: They are ultimately not about wave functions but about ``matter'' moving in space, represented by either particle trajectories, fields on space-time, or a discrete set of space-time points. The (...)
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  36.  40
    Contents.Hal Tasaki, Sheldon Goldstein & Takashi Hara - unknown
    We study the problem of the approach to equilibrium in a macroscopic quantum system in an abstract setting. We prove that, for a typical choice of “nonequilibrium subspace”, any initial state (from the energy shell) thermalizes, and in fact does so very quickly, on the order of the Boltzmann time τ B := h/(k B T ). This apparently unrealistic, but mathematically rigorous, conclusion has the important physical implication that the moderately slow decay observed in reality is not typical in (...)
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  37.  74
    On the Weak Measurement of Velocity in Bohmian Mechanics.Sheldon Goldstein - unknown
    In a recent article [1], Wiseman has proposed the use of so-called weak measurements for the determination of the velocity of a quantum particle at a given position, and has shown that according to quantum mechanics the result of such a procedure is the Bohmian velocity of the particle. Although Bohmian mechanics is empirically equivalent to variants based on velocity formulas different from the Bohmian one, and although it has been proven that the velocity in Bohmian mechanics is not measurable, (...)
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  38.  69
    Bell-Type Quantum Field Theories.Sheldon Goldstein - manuscript
    In [3] John S. Bell proposed how to associate particle trajectories with a lattice quantum field theory, yielding what can be regarded as a |Ψ|2-distributed Markov process on the appropriate configuration space. A similar process can be defined in the continuum, for more or less any regularized quantum field theory; such processes we call Bell-type quantum field theories. We describe methods for explicitly constructing these processes. These concern, in addition to the definition of the Markov processes, the efficient calculation of (...)
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  39.  54
    Bohmian Mechanics and the Quantum Revolution. [REVIEW]Sheldon Goldstein - 1996 - Synthese 107 (1):145 - 165.
    When I was young I was fascinated by the quantum revolution: the transition from classical definiteness and determinism to quantum indeterminacy and uncertainty, from classical laws that are indifferent, if not hostile, to the human presence, to quantum laws that fundamentally depend upon an observer for their very meaning. I was intrigued by the radical subjectivity, as expressed by Heisenberg’s assertion [3] that “The idea of an objective real world whose smallest parts exist objectively in the same sense as stones (...)
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  40.  55
    Normal Typicality and Von Neumann's Quantum Ergodic Theorem.Sheldon Goldstein & Roderich Tumulka - unknown
    We discuss the content and significance of John von Neumann’s quantum ergodic theorem (QET) of 1929, a strong result arising from the mere mathematical structure of quantum mechanics. The QET is a precise formulation of what we call normal typicality, i.e., the statement that, for typical large systems, every initial wave function ψ0 from an energy shell is “normal”: it evolves in such a way that |ψt ψt| is, for most t, macroscopically equivalent to the micro-canonical density matrix. The QET (...)
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  41. Bohmian Mechanics and Quantum Equilibrium.Sheldon Goldstein, D. Dürr & N. Zanghì - manuscript
    in Stochastic Processes, Physics and Geometry II, edited by S. Albeverio, U. Cattaneo, D. Merlini (World Scientific, Singapore, 1995) pp. 221-232.
     
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  42.  49
    Long-Time Behavior of Macroscopic Quantum Systems: Commentary Accompanying the English Translation of John von Neumann’s 1929 Article on the Quantum Ergodic Theorem.Sheldon Goldstein, Roderich Tumulka, Joel L. Lebowitz & Nino Zangh`ı - unknown
    The renewed interest in the foundations of quantum statistical mechanics in recent years has led us to study John von Neumann’s 1929 article on the quantum ergodic theorem. We have found this almost forgotten article, which until now has been available only in German, to be a treasure chest, and to be much misunderstood. In it, von Neumann studied the long-time behavior of macroscopic quantum systems. While one of the two theorems announced in his title, the one he calls the (...)
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  43.  50
    Paradoxical Reflection in Quantum Mechanics.Pedro L. Garrido, Sheldon Goldstein, Jani Lukkarinen & Roderich Tumulka - unknown
    This article concerns a phenomenon of elementary quantum mechanics that is quite counter-intuitive, very non-classical, and apparently not widely known: a quantum particle can get reflected at a potential step downwards. In contrast, classical particles get reflected only at upward steps. As a consequence, a quantum particle can be trapped for a long time (though not forever) in a region surrounded by downward potential steps, that is, on a plateau. Said succinctly, a quantum particle tends not to fall off a (...)
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  44.  51
    Are All Particles Identical?Sheldon Goldstein - manuscript
    We consider the possibility that all particles in the world are fundamentally identical, i.e., belong to the same species. Different masses, charges, spins, flavors, or colors then merely correspond to different quantum states of the same particle, just as spin-up and spin-down do. The implications of this viewpoint can be best appreciated within Bohmian mechanics, a precise formulation of quantum mechanics with particle trajectories. The implementation of this viewpoint in such a theory leads to trajectories different from those of the (...)
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  45.  49
    Quantum Equilibrium and the Role of Operators as Observables in Quantum Theory.Sheldon Goldstein - manuscript
    Bohmian mechanics is arguably the most naively obvious embedding imaginable of Schr¨ odinger’s equation into a completely coherent physical theory. It describes a world in which particles move in a highly non-Newtonian sort of way, one which may at first appear to have little to do with the spectrum of predictions of quantum mechanics. It turns out, however, that as a consequence of the defining dynamical equations of Bohmian mechanics, when a system has wave function ψ its configuration is typically (...)
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  46.  44
    On the Approach to Thermal Equilibrium of Macroscopic Quantum Systems.Sheldon Goldstein & Roderich Tumulka - unknown
    We consider an isolated, macroscopic quantum system. Let H be a microcanonical “energy shell,” i.e., a subspace of the system’s Hilbert space spanned by the (finitely) many energy eigenstates with energies between E and E + δE. The thermal equilibrium macro-state at energy E corresponds to a subspace Heq of H such that dim Heq/ dim H is close to 1. We say that a system with state vector ψ H is in thermal equilibrium if ψ is “close” to Heq. (...)
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  47.  36
    Quantum Mechanics in Multiply-Connected Spaces.Sheldon Goldstein, D. Dürr, J. Taylor, R. Tumulka & and N. Zanghì - manuscript
    J. Phys. A, to appear, quant-ph/0506173.
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  48.  37
    Topological Factors Derived From Bohmian Mechanics.Sheldon Goldstein - manuscript
    We derive for Bohmian mechanics topological factors for quantum systems with a multiply-connected configuration space Q. These include nonabelian factors corresponding to what we call holonomy-twisted representations of the fundamental group of Q. We employ wave functions on the universal covering space of Q. As a byproduct of our analysis, we obtain an explanation, within the framework of Bohmian mechanics, of the fact that the wave function of a system of identical particles is either symmetric or anti-symmetric.
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  49.  19
    A Global Equilibrium as the Foundation of Quantum Randomness.Detlef Dürr, Sheldon Goldstein & Nino Zanghí - 1993 - Foundations of Physics 23 (5):721-738.
    We analyze the origin of quantum randomness within the framework of a completely deterministic theory of particle motion—Bohmian mechanics. We show that a universe governed by this mechanics evolves in such a way as to give rise to the appearance of randomness, with empirical distributions in agreement with the predictions of the quantum formalism. Crucial ingredients in our analysis are the concept of the effective wave function of a subsystem and that of a random system. The latter is a notion (...)
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  50.  35
    Quantum Hamiltonians and Stochastic Jumps.Sheldon Goldstein - manuscript
    With many Hamiltonians one can naturally associate a |Ψ|2-distributed Markov process. For nonrelativistic quantum mechanics, this process is in fact deterministic, and is known as Bohmian mechanics. For the Hamiltonian of a quantum field theory, it is typically a jump process on the configuration space of a variable number of particles. We define these processes for regularized quantum field theories, thereby generalizing previous work of John S. Bell [3] and of ourselves [11]. We introduce a formula expressing the jump rates (...)
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