Papers by Sheldon Goldstein
Springer eBooks, 2009
ABSTRACT
Jerusalem studies in philosophy and history of science, 2022
The Great Divide in metaphysical debates about laws of nature is between Humeans, who think that ... more The Great Divide in metaphysical debates about laws of nature is between Humeans, who think that laws merely describe the distribution of matter, and non-Humeans, who think that laws govern it. The metaphysics can place demands on the proper formulations of physical theories. It is sometimes assumed that the governing view requires a fundamental / intrinsic direction of time: to govern, laws must be dynamical, producing later states of the world from earlier ones, in accord with the fundamental direction of time in the universe. In this paper, we propose a minimal primitivism about laws of nature (MinP) according to which there is no such requirement. On our view, laws govern by constraining the physical possibilities. Our view captures the essence of the governing view without taking on extraneous commitments about the direction of time or dynamic production. Moreover, as a version of primitivism, our view requires no reduction / analysis of laws in terms of universals, powers, or dispositions. Our view accommodates several potential candidates for fundamental laws, including the principle of least action, the Past Hypothesis, the Einstein equation of general relativity, and even controversial examples found in the Wheeler-Feynman theory of electrodynamics and retrocausal theories of quantum mechanics. By understanding governing as constraining, non-Humeans who accept MinP have the same freedom to contemplate a wide variety of candidate fundamental laws as Humeans do.
Cambridge University Press eBooks, Sep 5, 2016
The Monist, Sep 10, 2019
I will contrast the two main approaches to the foundations of statistical mechanics: the individu... more I will contrast the two main approaches to the foundations of statistical mechanics: the individualist (Boltzmannian) approach and the ensemblist approach (associated with Gibbs). I will indicate the virtues of each, and argue that the conflict between them is perhaps not as great as often imagined.
Foundations of Physics, May 7, 2022
Fundamental theories of physics, Oct 3, 2020
We dedicate this paper to the memory of Giancarlo Ghirardi, who devoted his life to understanding... more We dedicate this paper to the memory of Giancarlo Ghirardi, who devoted his life to understanding quantum mechanics. He was a friend of John Bell, who was inspired by Giancarlo's work. He was also a friend of two of us (J.B. and S.G.).
Contemporary mathematics, 1985
We present an invariance principle for antisymmetric functions of a reversible Markov process whi... more We present an invariance principle for antisymmetric functions of a reversible Markov process which immediately implies convergence to Brownian motion for a wide class of random motions in random environments. We apply it to establish convergence to Brownian motion (i) for a walker moving in the infinite cluster of the two-dimensional bond percolation model, (ii) for a d-dimensional walker moving in a symmetric random environment under very mild assumptions on the distribution of the environment, (iii) for a tagged particle in a d-dimensional symmetric lattice gas which allows interchanges, (iv) for a tagged particle in a d-dimensional system of interacting Brownian particles. Our formulation also leads naturally to bounds on the diffusion constant.
arXiv (Cornell University), Nov 18, 2021
Relational mechanics is a reformulation of mechanics (classical or quantum) for which space is re... more Relational mechanics is a reformulation of mechanics (classical or quantum) for which space is relational. This means that the configuration of an Nparticle system is a shape, which is what remains when the effects of rotations, translations and dilations are quotiented out. This reformulation of mechanics naturally leads to a relational notion of time as well, in which a history of the universe is just a curve in shape space without any reference to a special parametrization of the curve given by an absolute Newtonian time. When relational mechanics (classical or quantum) is regarded as fundamental, the usual descriptions in terms of absolute space and absolute time emerge merely as corresponding to the choice of a gauge. This gauge freedom forces us to recognize that what we have traditionally regarded as fundamental in physics might in fact be imposed by us through our choice of gauge. It thus imparts a somewhat Kantian aspect to physical theory.
arXiv (Cornell University), May 2, 2020
We give a conceptually simple proof of nonlocality (based on the previous work of [32, 33, 17, 18... more We give a conceptually simple proof of nonlocality (based on the previous work of [32, 33, 17, 18]) using only the perfect correlations between results of measurements on distant systems discussed by Einstein, Podolsky and Rosen-correlations that EPR thought proved the incompleteness of quantum mechanics. Our argument relies on an extension of EPR by Schrödinger.
arXiv (Cornell University), May 2, 2020
We give a conceptually simple proof of nonlocality (based on the previous work of [32, 33, 17, 18... more We give a conceptually simple proof of nonlocality (based on the previous work of [32, 33, 17, 18]) using only the perfect correlations between results of measurements on distant systems discussed by Einstein, Podolsky and Rosen-correlations that EPR thought proved the incompleteness of quantum mechanics. Our argument relies on an extension of EPR by Schrödinger.
Journal of Statistical Physics, Aug 20, 2019
Dedicated to our mentor, Joel L. Lebowitz, a master of statistical physics who, when it came to f... more Dedicated to our mentor, Joel L. Lebowitz, a master of statistical physics who, when it came to foundational issues of quantum mechanics, was always willing to listen.
Publications mathématiques et informatique de Rennes, 1975
Journal of Statistical Physics, Aug 15, 2023
Springer eBooks, Nov 7, 2012
Bohmian mechanics is the most naively obvious embedding imaginable of Schrödinger's equation into... more Bohmian mechanics is the most naively obvious embedding imaginable of Schrödinger'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 random, with probability density ρ given by |ψ| 2 , the quantum equilibrium distribution. It also turns out that the entire quantum formalism, operators as observables and all the rest, naturally emerges in Bohmian mechanics from the analysis of "measurements." This analysis reveals the status of operators as observables in the description of quantum phenomena, and facilitates a clear view of the range of applicability of the usual quantum mechanical formulas. * Dedicated to Elliott Lieb on the occasion of his 70th birthday. Elliott will be (we fear unpleasantly) surprised to learn that he bears a greater responsibility for this paper than he could possibly imagine. We would of course like to think that our work addresses in some way the concern suggested by the title of his recent talks, The Quantum-Mechanical World View: A Remarkably Successful but Still Incomplete Theory, but we recognize that our understanding of incompleteness is much more naive than Elliott's. He did, however, encourage us in his capacity as an editor of the Reviews of Modern Physics to submit a paper on the role of operators in quantum theory. That was 12 year ago. Elliott is no longer an editor there and the paper that developed is not quite a review.
Journal of Statistical Physics, Jun 13, 2019
Recently, there has been progress in developing interior-boundary conditions (IBCs) as a techniqu... more Recently, there has been progress in developing interior-boundary conditions (IBCs) as a technique of avoiding the problem of ultraviolet divergence in nonrelativistic quantum field theories while treating space as a continuum and electrons as point particles. An IBC can be expressed in the particle-position representation of a Fock vector ψ as a condition on the values of ψ on the set of collision configurations, and the corresponding Hamiltonian is defined on a domain of vectors satisfying this condition. We describe here how Bohmian mechanics can be extended to this type of Hamiltonian. In fact, part of the development of IBCs was inspired by the Bohmian picture. Particle creation and annihilation correspond to jumps in configuration space; the annihilation is deterministic and occurs when two particles (of the appropriate species) meet, whereas the creation is stochastic and occurs at a rate dictated by the demand for the equivariance of the |ψ| 2 distribution, time reversal symmetry, and the Markov property. The process is closely related to processes known as Bell-type quantum field theories.
arXiv (Cornell University), Sep 16, 2021
To illustrate Boltzmann's construction of an entropy function that is defined for a microstate of... more To illustrate Boltzmann's construction of an entropy function that is defined for a microstate of a macroscopic system, we present here the simple example of the free expansion of a one dimensional gas of non-interacting point particles. The construction requires one to define macrostates, corresponding to macroscopic variables. We define a macrostate M by specifying the fraction of particles in rectangular boxes ∆x∆v of the single particle position-velocity space {x, v}. We verify that when the number of particles is large the Boltzmann entropy, SB(t), of a typical microstate of a nonequilibrium ensemble coincides with the Gibbs entropy of the coarse-grained time-evolved oneparticle distribution associated with this ensemble. SB(t) approaches its maximum possible value for the dynamical evolution of the given initial state. The rate of approach depends on the size of ∆v in the definition of the macrostate, going to zero at any fixed time t when ∆v → 0. Surprisingly the different curves SB(t) collapse when time is scaled with ∆v as: t ∼ τ /∆v. We find an explicit expression for SB(τ) in the limit ∆v → 0. We also consider a different, more hydrodynamical, definition of macrostates for which SB(t) is monotone increasing, unlike the previous one which has small decaying oscillations near its maximum value. Our system is non-ergodic, non-chaotic and non-interacting; our results thus illustrate that these concepts are not as relevant as sometimes claimed, for observing macroscopic irreversibility and entropy increase. Rather, the notions of initial conditions, typicality, large numbers and coarse-graining are the important factors. We demonstrate these ideas through extensive simulations as well as analytic results.
arXiv (Cornell University), Dec 24, 2017
Thermodynamics makes definite predictions about the thermal behavior of macroscopic systems in an... more Thermodynamics makes definite predictions about the thermal behavior of macroscopic systems in and out of equilibrium. Statistical mechanics aims to derive this behavior from the dynamics and statistics of the atoms and molecules making up these systems. A key element in this derivation is the large number of microscopic degrees of freedom of macroscopic systems. Therefore, the extension of thermodynamic concepts, such as entropy, to small (nano) systems raises many questions. Here we shall reexamine various definitions of entropy for nonequilibrium systems, large and small. These include thermodynamic (hydrodynamic), Boltzmann, and Gibbs-Shannon entropies. We shall argue that, despite its common use, the last is not an appropriate physical entropy for such systems, either isolated or in contact with thermal reservoirs: physical entropies should depend on the microstate of the system, not on a subjective probability distribution. To square this point of view with experimental results of Bechhoefer we shall argue that the Gibbs-Shannon entropy of a nano particle in a thermal fluid should be interpreted as the Boltzmann entropy of a dilute gas of Brownian particles in the fluid.
Journal of Mathematical Physics, Aug 1, 2022
We describe the extremal translation invariant stationary (ETIS) states of the facilitated exclus... more We describe the extremal translation invariant stationary (ETIS) states of the facilitated exclusion process on Z. In this model all particles on sites with one occupied and one empty neighbor jump at each integer time to the empty neighbor site, and if two particles attempt to jump into the same empty site we choose one randomly to succeed. The ETIS states are qualitatively different for densities ρ < 1/2, ρ = 1/2, and 1/2 < ρ < 1, but in each density region we find states which may be grouped into families, each of which is in natural correspondence with the set of all ergodic measures on {0, 1} Z. For ρ < 1/2 there is one such family, containing all the ergodic states in which the probability of two adjacent occupied sites is zero. For ρ = 1/2 there are two families, in which configurations translate to the left and right, respectively, with constant speed 2. For the high density case there is a continuum of families. We show that all ETIS * Dedicated to the memory of Freeman Dyson, friend and teacher.
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Papers by Sheldon Goldstein