Chaos is often explained in terms of random behaviour; and having positive Kolmogorov–Sinai entropy (KSE) is taken to be indicative of randomness. Although seemly plausible, the association of positive KSE with random behaviour needs justification since the definition of the KSE does not make reference to any notion that is connected to randomness. A common way of justifying this use of the KSE is to draw parallels between the KSE and ShannonÕs information theoretic entropy. However, as it stands this no (...) more than a heuristic point, because no rigorous connection between the KSE and ShannonÕs entropy has been established yet. This paper fills this gap by proving that the KSE of a Hamiltonian dynamical system is equivalent to a generalized version of ShannonÕs information theoretic entropy under certain plausible assumptions. Ó 2005 Elsevier Ltd. All rights reserved. (shrink)

MaxEnt inference algorithm and information theory are relevant for the time evolution of macroscopic systems considered as problem of incomplete information. Two different MaxEnt approaches are introduced in this work, both applied to prediction of time evolution for closed Hamiltonian systems. The first one is based on Liouville equation for the conditional probability distribution, introduced as a strict microscopic constraint on time evolution in phase space. The conditional probability distribution is defined for the set of microstates associated with the (...) set of phase space paths determined by solutions of Hamilton’s equations. The MaxEnt inference algorithm with Shannon’s concept of the conditional information entropy is then applied to prediction, consistently with this strict microscopic constraint on time evolution in phase space. The second approach is based on the same concepts, with a difference that Liouville equation for the conditional probability distribution is introduced as a macroscopic constraint given by a phase space average. We consider the incomplete nature of our information about microscopic dynamics in a rational way that is consistent with Jaynes’ formulation of predictive statistical mechanics, and the concept of macroscopic reproducibility for time dependent processes. Maximization of the conditional information entropy subject to this macroscopic constraint leads to a loss of correlation between the initial phase space paths and final microstates. Information entropy is the theoretic upper bound on the conditional information entropy, with the upper bound attained only in case of the complete loss of correlation. In this alternative approach to prediction of macroscopic time evolution, maximization of the conditional information entropy is equivalent to the loss of statistical correlation, and leads to corresponding loss of information. In accordance with the original idea of Jaynes, irreversibility appears as a consequence of gradual loss of information about possible microstates of the system. (shrink)

This is a substantially expanded version of a chapter-contribution to "The Springer Handbook of Spacetime", edited by Abhay Ashtekar and Vesselin Petkov, published by Springer Verlag in 2014. This contribution introduces the reader to the reformulation of Einstein's field equations of General Relativity as a constrained evolutionary system of Hamiltonian type and discusses some of its uses,together with some technical and conceptual aspects. Attempts were made to keep the presentation self contained and accessible to first-year graduate students. This implies (...) a certain degree of explicitness and occasional reviews of background material. (shrink)

The connection between quantum mechanics and classical statistical mechanics has motivated in the past the representation of the Schrödinger quantum-wave equation in terms of “projections” onto the quantum configuration space of suitable phase-space asymptotic kinetic models. This feature has suggested the search of a possible exact super-dimensional classical dynamical system, denoted as Schrödinger CDS, which uniquely determines the time-evolution of the underlying quantum state describing a set of N like and mutually interacting quantum particles. In this paper the realization of (...) the same CDS in terms of a coupled set of Hamiltonian systems is established. These are respectively associated with a quantum-hydrodynamic CDS advancing in time the quantum fluid velocity and a further one the RD-CDS, describing the relative dynamics with respect to the quantum fluid. (shrink)

Hamiltonian light-front dynamics of quantum fields may provide a useful approach to systematic nonperturbative approximations to quantum field theories. We investigate inequivalent Hilbert-space representations of the light-front field algebra in which the stability group of the light front is implemented by unitary transformations. The Hilbert space representation of states is generated by the operator algebra from the vacuum state. There is a large class of vacuum states besides the Fock vacuum which meet all the invariance requirements. The light-front (...) Hamiltonian must annihilate the vacuum and have a positive spectrum. We exhibit relations of the Hamiltonian to the nontrivial vacuum structure. (shrink)

A theorem is derived that enables a systematic enumeration of all the linear superoperators ℒ (associated with a two-level quantum system) that generate, via the law of motion ℒρ= $\dot \rho$ , mappings ρ(0) → ρ(t) restricted to the domain of statistical operators. Such dynamical evolutions include the usual Hamiltonian motion as a special case, but they also encompass more general motions, which are noncyclic and feature a destination state ρ(t → ∞) that is in some cases independent of (...) ρ(0). (shrink)

I show how Sir William Rowan Hamilton’s philosophical commitments led him to a causal interpretation of classical mechanics. I argue that Hamilton’s metaphysics of causation was injected into his dynamics by way of a causal interpretation of force. I then detail how forces are indispensable to both Hamilton’s formulation of classical mechanics and what we now call Hamiltonian mechanics (i.e., the modern formulation). On this point, my efforts primarily consist of showing that the contemporary orthodox interpretation of potential (...) energy is the interpretation found in Hamilton’s work. Hamilton called the potential energy function the “force-function” because he believed that it represents forces at work in the world. Various non-historical arguments for this orthodox interpretation of potential energy are provided, and matters are concluded by showing that in classical Hamiltonian mechanics, facts about the potential energies of systems are grounded in facts about forces. Thus, if one can tolerate the view that forces are causes of motion, then Hamilton provides one with a road map for transporting causation into one of the most mathematically sophisticated formulations of classical mechanics, viz., Hamiltonian mechanics. (shrink)

The analysis of the Helmholtz equation is shown to lead to an exact Hamiltonian system describing in terms of ray trajectories, for a stationary refractive medium, a very wide family of wave-like phenomena (including diffraction and interference) going much beyond the limits of the geometrical optics (“eikonal”) approximation, which is contained as a simple limiting case. Due to the fact, moreover, that the time independent Schrödinger equation is itself a Helmholtz-like equation, the same mathematics holding for a classical optical (...) beam turns out to apply to a quantum particle beam moving in a stationary force field, and leads to a system of Hamiltonian equations providing exact and deterministic particle trajectories and dynamical laws, and containing the laws of Classical Mechanics in the eikonal limit. (shrink)

Dissipative systems are described by a Hamiltonian, combined with a “dynamical matrix” which generalizes the simplectic form of the equations of motion. Criteria for dissipation are given and the examples of a particle with friction and of the Lotka-Volterra model are presented. Quantization is first introduced by translating generalized Poisson brackets into commutators and anticommutators. Then a generalized Schrödinger equation expressed by a dynamical matrix is constructed and discussed.

Representation of quantum states by statistical ensembles on the quantum phase space in the Hamiltonian form of quantum mechanics is analyzed. Various mathematical properties and some physical interpretations of the equivalence classes of ensembles representing a mixed quantum state in the Hamiltonian formulation are examined. In particular, non-uniqueness of the quantum phase space probability density associated with the quantum mixed state, Liouville dynamics of the probability densities and the possibility to represent the reduced states of bipartite systems (...) by marginal distributions are discussed in detail. These considerations are used to study ensembles of hybrid quantum-classical systems. In particular, nonlinear evolution of a single hybrid system in a pure state and unequal evolutions of initially equivalent ensembles are discussed in the context of coupled hybrid systems. (shrink)

Various processes are often classified as both deterministic and random or chaotic. The main difficulty in analysing the randomness of such processes is the apparent tension between the notions of randomness and determinism: what type of randomness could exist in a deterministic process? Ergodic theory seems to offer a particularly promising theoretical tool for tackling this problem by positing a hierarchy, the so-called ‘ergodic hierarchy’, which is commonly assumed to provide a hierarchy of increasing degrees of randomness. However, that notion (...) of randomness requires clarification. The mathematical definition of EH does not make explicit appeal to randomness; nor does the usual way of presenting EH involve a specification of the notion of randomness that is supposed to underlie the hierarchy. In this paper we argue that EH is best understood as a hierarchy of random behaviour if randomness is explicated in terms of unpredictability. We then show that, contrary to common wisdom, EH is useful in characterising the behaviour of Hamiltonian dynamical systems. (shrink)

The Hamiltonian structure of General Relativity (GR), for both metric and tetrad gravity in a definite continuous family of space-times, is fully exploited in order to show that: i) the "Hole Argument" can be bypassed by means of a specific "physical individuation" of point-events of the space-time manifold M^4 in terms of the "autonomous degrees of freedom" of the vacuum gravitational field (Dirac observables), while the "Leibniz equivalence" is reduced to differences in the "non-inertial appearances" (connected to gauge variables) (...) of the same phenomena. ii) the chrono-geometric structure of a solution of Einstein equations for given, gauge-fixed, initial data (a "3-geometry" satisfying the relevant constraints on the Cauchy surface), can be interpreted as an "unfolding" in mathematical global time of a sequence of "achronal 3-spaces" characterized by "dynamically determined conventions" about distant simultaneity. This result stands out as an important conceptual difference with respect to the standard chrono-geometrical view of Special Relativity (SR) and allows, in a specific sense, for an "endurantist" interpretations of ordinary physical objects in GR. (shrink)

L. Hardy has formulated an axiomatization program of quantum mechanics and generalized probability theories that has been quite influential. In this paper, properties of typical Hamiltonian dynamical systems are used to argue that there are applications of probability in physical theories of systems with dynamical complexity that require continuous spaces of pure states. Hardy’s axiomatization program does not deal with such theories. In particular Hardy’s fifth axiom does not differentiate between such applications of classical probability and quantum probability.

In General Relativity in Hamiltonian form, change has seemed to be missing, defined only asymptotically, or otherwise obscured at best, because the Hamiltonian is a sum of first-class constraints and a boundary term and thus supposedly generates gauge transformations. By construing change as essential time dependence, one can find change locally in vacuum GR in the Hamiltonian formulation just where it should be. But what if spinors are present? This paper is motivated by the tendency in space-time (...) philosophy tends to slight fermionic/spinorial matter, the tendency in Hamiltonian GR to misplace changes of time coordinate, and the tendency in treatments of the Einstein-Dirac equation to include a gratuitous local Lorentz gauge symmetry along with the physically significant coordinate freedom. Spatial dependence is dropped in most of the paper, both restricting the physical situation and largely fixing the spatial coordinates. In the interest of including all and only the coordinate freedom, the Einstein-Dirac equation is investigated using the Schwinger time gauge and Kibble-Deser symmetric triad condition are employed as a \ version of the DeWitt-Ogievetsky-Polubarinov nonlinear group realization formalism that dispenses with a tetrad and local Lorentz gauge freedom. Change is the lack of a time-like stronger-than-Killing field for which the Lie derivative of the metric-spinor complex vanishes. An appropriate \-friendly form of the Rosenfeld-Anderson-Bergmann-Castellani gauge generator G, a tuned sum of first class-constraints, is shown to change the canonical Lagrangian by a total derivative, implying the preservation of Hamilton’s equations. Given the essential presence of second-class constraints with spinors and their lack of resemblance to a gauge theory, it is useful to have an explicit physically interesting example. This gauge generator implements changes of time coordinate for solutions of the equations of motion, showing that the gauge generator makes sense even with spinors. (shrink)

Energy is a crucial concept within classical and quantum physics. An essential tool to quantify energy is the Hamiltonian. Here, we consider how to define a Hamiltonian in general probabilistic theories—a framework in which quantum theory is a special case. We list desiderata which the definition should meet. For 3-dimensional systems, we provide a fully-defined recipe which satisfies these desiderata. We discuss the higher dimensional case where some freedom of choice is left remaining. We apply the definition to (...) example toy theories, and discuss how the quantum notion of time evolution as a phase between energy eigenstates generalises to other theories. (shrink)

One can (for the most part) formulate a model of a classical system in either the Lagrangian or the Hamiltonian framework. Though it is often thought that those two formulations are equivalent in all important ways, this is not true: the underlying geometrical structures one uses to formulate each theory are not isomorphic. This raises the question of whether one of the two is a more natural framework for the representation of classical systems. In the event, the answer is (...) yes: I state and sketch proofs of two technical results—inspired by simple physical arguments about the generic properties of classical systems—to the effect that, in a precise sense, classical systems evince exactly the geometric structure Lagrangian mechanics provides for the representation of systems, and none provided by Hamiltonian. The argument not only clarifies the conceptual structure of the two systems of mechanics, but also their relations to each other and their respective mechanisms for representing physical systems. It also shows why naïvely structural approaches to the representational content of physical theories cannot work. [Lagrange] grasped that he had gained a method of stating dynamical truths in a way, which is perfectly indifferent to the particular methods of measurement employed in fixing the positions of the various parts of the system. Accordingly, he went on to deduce equations of motion, which are equally applicable whatever quantitative measurements have been made, provided that they are adequate to fix positions. The beauty and almost divine simplicity of these equations is such that these formulae are worthy to rank with those mysterious symbols which in ancient times were held directly to indicate the Supreme Reason at the base of all things. (Whitehead [1948], p. 63)1. Introduction2.Classical Systems3. The Possible Interactions of a Classical System and the Structure of Its Space of States4. Classical Systems Are Lagrangian5. Classical Systems Are Not Hamiltonian6. How Lagrangian and Hamiltonian Mechanics Represent Classical Systems7. The Conceptual Structure of Classical Mechanics. (shrink)

The Dirac generator formalism for relativistic Hamiltonian dynamics is reviewed along with its extension to constraint formalism. In these theories evolution is with respect to a dynamically defined parameter, and thus time evolution involves an eleventh generator. These formulations evade the No-Interaction Theorem. But the incorporation of separability reopens the question, and together with the World Line Condition leads to a second no-interaction theorem for systems of three or more particles. Proofs are omitted, but the results of recent (...) research in this area is highlighted. (shrink)

The Lagrangian and Hamiltonian properties of classical electrodynamics models and their associated Dirac quantizations are studied. Using the vacuum field theory approach developed in (Prykarpatsky et al. Theor. Math. Phys. 160(2): 1079–1095, 2009 and The field structure of a vacuum, Maxwell equations and relativity theory aspects. Preprint ICTP) consistent canonical Hamiltonian reformulations of some alternative classical electrodynamics models are devised, and these formulations include the Lorentz condition in a natural way. The Dirac quantization procedure corresponding to the (...) class='Hi'>Hamiltonian formulations is developed. The crucial importance of the rest reference systems, with respect to which the dynamics of charged point particles is framed, is explained and emphasized. A concise expression for the Lorentz force is derived by suitably taking into account the duality of electromagnetic field and charged particle interactions. Finally, a physical explanation of the vacuum field medium and its relativistic properties fitting the mathematical framework developed is formulated and discussed. (shrink)

We have studied the dynamics and symmetries of a particle constrained to move in a torus knot. The Hamiltonian system turns out to be Second Class in Dirac’s formulation and the Dirac brackets yield novel noncommutative structures. The equations of motion are obtained for a path in general where the knot is present in the particle orbit but it is not restricted to a particular torus. We also study the motion when it is restricted to a specific torus. (...) The rotational symmetries are studied as well. We have also considered the behavior of small fluctuations of the particle motion about a fixed torus knot. (shrink)

Quantum frameworks for modeling competitive ecological systems and self-organizing structures have been investigated under multiple perspectives yielded by quantum mechanics. These comprise the description of the phase-space prey–predator competition dynamics in the framework of the Weyl–Wigner quantum mechanics. In this case, from the classical dynamics described by the Lotka–Volterra (LV) Hamiltonian, quantum states convoluted by statistical gaussian ensembles can be analytically evaluated. Quantum modifications on the patterns of equilibrium and stability of the prey–predator dynamics can then (...) be identified. These include quantum distortions over the equilibrium point drivers of the LV dynamics which are quantified through the Wigner current fluxes obtained from an onset Hamiltonian background. In addition, for gaussian ensembles highly localized around the equilibrium point, stability properties are shown to be affected by emergent topological quantum domains which, in some cases, could lead either to extinction and revival scenarios or to the perpetual coexistence of both prey and predator agents identified as quantum observables in microscopic systems. Conclusively, quantum and gaussian statistical driving parameters are shown to affect the stability criteria and the time evolution pattern for such microbiological-like communities. (shrink)

Emergence has traditionally been described as satisfying specific properties, notably nonreducibility of the emergent object or properties to their substrate, novelty, and unpredictability from the properties of the substrate. Sometimes more mysterious properties such as independence from the substrate, separate substances and teleological properties are invoked. I will argue that the latter are both unnecessary and unwarranted. The descriptive properties can be analyzed in more detail in logical terms, but the logical conditions alone do not tell us how to identify (...) the conditions through interactions with the world. In order to do that we need dynamical properties – properties that do something. This paper, then, will be directed at identifying the dynamical conditions necessary and sufficient for emergence. Emergent properties and objects all result or are maintained by dissipative and radically nonholonomic processes. Emergent properties are relatively common in physics, but have been ignored because of the predominant use of Hamiltonian methods assuming energy conservation. Emergent objects are all dissipative systems, which have been recognized as special only in the past fifty years or so. Of interest are autonomous systems, including living and thinking systems. They show functionality and are self governed. (shrink)

Concentrating upon applications that are most relevant to modern physics, this valuable book surveys variational principles and examines their relationship to dynamics and quantum theory. Stressing the history and theory of these mathematical concepts rather than the mechanics, the authors provide many insights into the development of quantum mechanics and present much hard-to-find material in a remarkably lucid, compact form. After summarizing the historical background from Pythagoras to Francis Bacon, Professors Yourgrau and Mandelstram cover Fermat's principle of least time, (...) the principle of least action of Maupertuis, development of this principle by Euler and Lagrange, and the equations of Lagrange and Hamilton. Equipped by this thorough preparation to treat variational principles in general, they proceed to derive Hamilton's principle, the Hamilton-Jacobi equation, and Hamilton's canonical equations. An investigation of electrodynamics in Hamiltonian form covers next, followed by a resume of variational principles in classical dynamics. The authors then launch into an analysis of their most significant topics: the relation between variational principles and wave mechanics, and the principles of Feynman and Schwinger in quantum mechanics. Two concluding chapters extend the discussion to hydrodynamics and natural philosophy. Professional physicists, mathematicians, and advanced students with a strong mathematical background will find this stimulating volume invaluable reading. Extremely popular in its hardcover edition, this volume will find even wider appreciation in its first fine inexpensive paperbound edition. (shrink)

In the literature on the interpretation of quantum mechanics, not many works attempt to adopt a proactive perspective aimed at seeing how different interpretations can enrich each other through a productive dialogue. In particular, few proposals have been devised to show that different approaches can be clarified by comparing them, and can even complement each other, improving or leading to a more fertile overall approach. The purpose of this paper is framed within this perspective of complementation and mutual enrichment. In (...) particular, the Relational Quantum Mechanics and the Modal-Hamiltonian Interpretation are compared, highlighting their differences and points of contact. The final purpose is to show that, in spite of their disagreements, they are not contradictory but, on the contrary, they can be made compatible from an overarching perspective and can even complement each other in a fruitful way. (shrink)

The symplectic quantization scheme proposed for matter scalar fields in the companion paper (Gradenigo and Livi, arXiv:2101.02125, 2021) is generalized here to the case of space–time quantum fluctuations. That is, we present a new formalism to frame the quantum gravity problem. Inspired by the stochastic quantization approach to gravity, symplectic quantization considers an explicit dependence of the metric tensor gμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{\mu \nu }$$\end{document} on an additional time variable, named intrinsic time at variance (...) with the coordinate time of relativity, from which it is different. The physical meaning of intrinsic time, which is truly a parameter and not a coordinate, is to label the sequence of gμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{\mu \nu }$$\end{document} quantum fluctuations at a given point of the four-dimensional space–time continuum. For this reason symplectic quantization necessarily incorporates a new degree of freedom, the derivative g˙μν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{g}}_{\mu \nu }$$\end{document} of the metric field with respect to intrinsic time, corresponding to the conjugated momentum πμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\mu \nu }$$\end{document}. Our proposal is to describe the quantum fluctuations of gravity by means of a symplectic dynamics generated by a generalized action functional A[gμν,πμν]=K[gμν,πμν]-S[gμν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}[g_{\mu \nu },\pi _{\mu \nu }] = {\mathcal {K}}[g_{\mu \nu },\pi _{\mu \nu }] - S[g_{\mu \nu }]$$\end{document}, playing formally the role of a Hamilton function, where S[gμν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S[g_{\mu \nu }]$$\end{document} is the standard Einstein–Hilbert action while K[gμν,πμν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {K}}[g_{\mu \nu },\pi _{\mu \nu }]$$\end{document} is a new term including the kinetic degrees of freedom of the field. Such an action allows us to define an ensemble for the quantum fluctuations of gμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{\mu \nu }$$\end{document} analogous to the microcanonical one in statistical mechanics, with the only difference that in the present case one has conservation of the generalized action A[gμν,πμν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}[g_{\mu \nu },\pi _{\mu \nu }]$$\end{document} and not of energy. Since the Einstein–Hilbert action S[gμν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S[g_{\mu \nu }]$$\end{document} plays the role of a potential term in the new pseudo-Hamiltonian formalism, it can fluctuate along the symplectic action-preserving dynamics. These fluctuations are the quantum fluctuations of gμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{\mu \nu }$$\end{document}. Finally, we show how the standard path-integral approach to gravity can be obtained as an approximation of the symplectic quantization approach. By doing so we explain how the integration over the conjugated momentum field πμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\mu \nu }$$\end{document} gives rise to a cosmological constant term in the path-integral approach. (shrink)

If the ordinary quantal Liouville equation ℒρ= $\dot \rho $ is generalized by discarding the customary stricture that ℒ be of the standard Hamiltonian commutator form, the new quantum dynamics that emerges has sufficient theoretical fertility to permit description even of a thermodynamically irreversible process in an isolated system, i.e., a motion ρ(t) in which entropy increases but energy is conserved. For a two-level quantum system, the complete family of time-independent linear superoperators ℒ that generate such motions is (...) derived; and a physically interesting example is presented in detail. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy, Psychiatry, & Psychology 12.3 (2005) 229-234 [Access article in PDF] Nonlinear Dynamics at the Cutting Edge of Modernity: A Postmodern View Gordon Globus Keywords nonlinear dynamics, modernity, postmodernity, quantum brain theory, free will, self-organization, autopoiesis, autorhoesis Although nonlinear dynamical conceptu-alizations have been applied to psychia-try for over 20 years,1 they have not had significant impact on the field. Unfortunately Heinrichs' very thoughtful contribution to the discussion (...) is unlikely to move such disinterest. Pragmatically successful paradigms in the phase of Kuhnian "normal science" are highly resistant to change. There is no crisis that besets triumphalist biological psychiatry that might lead to clinical or scientific revolution. At the same time, as I suggest at the end of this commentary, the ideas of nonlinear dynamics may not be revolutionary enough, remaining staunchly within the framework of modernity.In what follows I discuss four issues related to Heinrichs' article. First, I try to enrich his presentation of nonlinear dynamics by bringing in the role of self-organization in the evolution of nonlinear dynamical brain systems and by making the nonlinear dynamics "tunable." Second, I discuss whether Heinrichs succeeds in avoiding the dualism of the biopsychosocial model that he criticizes. Third, I look at the issue of "free will," which Heinrichs would understandably like to avoid. Finally, I mention a new form of dynamics, namely thermofield dynamics, which I think is more promising than nonlinear dynamics. Self-Organizing Self-Tuning Nonlinear Dynamical Systems In the case of psychiatry, we are not considering nonlinear dynamical systems like the weather or an ecosystem, but the brain's neural networks in terms of nonlinear dynamics with all its chaos. There is a principle of such systems of neural networks that Heinrichs does not bring out, although it is extremely important for them. These systems are self-organizing under a Hamiltonian principle of "least energy." To use Hopfield and Tank's (1986) terms, nonlinear dynamical neural networks self-organize toward attractors that minimize the "computational energy" of the system, or for Smolensky (1988), self-organize toward "harmony," or for Globus (1995, 2003), in a Heideggerian vein, self-organize to maximize "belonging-together." In this elaborated framework, rational deliberation is one constraint among many on a self-organizing nonlinear dynamical process. [End Page 229]Heinrichs brings out beautifully the nonlinear dynamical way of thinking clinically. This is not the dynamics of mechanistic psychodynamics, which features psychological forces in vectorial conflict—id versus superego and their compromises—but a flowing evolution of states in which small changes in initial state may have large effects (nonlinearity) on the state trajectory. To live a life is often to feel surprise, which nonlinear dynamics with its chaotic regimes well portrays.But there is a way that classical Freud and nonlinear dynamical neural networks do come together—in the "economics," discharges and conservations of libidinal energy, expenditures and credits in Derrida's (1978) accounting. Heinrichs does not mention this similarity. The economic principles differ between Freud's late nineteenth-century thermodynamics and Heinrichs' late twentieth-century nonlinear dynamics, it is true, but it is all economics.Freud called this economic functioning the "pleasure principle" in which "libido" seeks to discharge its instinctual energy store, reducing the level of instinctual tension. This "primary process" proceeds to an attractor by the quickest, most direct way possible, whereas the "secondary process" defers reduction in instinctual tension, subjecting the primary process to thought. Self-organizing directly to the attractor is the primary process with which the secondary process interferes, introducing deferral.The economy of nonlinear dynamics pictures a "landscape," a variegated topography with peaks and valleys. Each point on the topography represents a state. Peak points are repellors—for example, states in which the phobic object is approached—and the lowest points of the valleys are attractors—for example, obsessions and compulsions. The path of occupied states self-organizes away from repellors and toward attractors. This topography is an energy landscape. Its dynamics spontaneously tends—"self-organizes"—toward least energy/high-constraint satisfaction. Freud's pleasure principle is just the Hamiltonian least energy... (shrink)

In the covariant Hamiltonian mechanics with action-at-a-distance, we compare the proper time and dynamical time representations of the coordinate space world line using the differential geometry of nongeodesic curves in 3+1 Minkowski spacetime. The covariant generalization of the Serret-Frenet equations for the point particle with interaction are derived using the arc length representation. A set of invariant point particle kinematical properties are derived which are equivalent to the solutions of the equations of motion in coordinate space and which are (...) functions of either the proper time or the dynamical time. Expressions for the quantities are given for the example of the covariant harmonic oscillator and comments are offered regarding the measurability of the dynamical time. (shrink)

This communication is part I of a series of papers which explore the theoretical possibility of generalizing quantum dynamics in such a way that the predicted motions of an isolated system would include the irreversible (entropy-increasing) state evolutions that seem essential if the second law of thermodynamics is ever to become a theorem of mechanics. In this first paper, the general mathematical framework for describing linear but not necessarily Hamiltonian mappings of the statistical operator is reviewed, with particular (...) attention to detailed representations of the Kossakowski conditions for the case of a two-level system. (shrink)

The supposition of the manifest covariance of average trajectory world lines is violated in Hamiltonian formulations of relativistic quantum mechanics. This is due to the nonlinear appearance of particle dynamical variable operators in the Heisenberg picture boosted position, velocity, and momentum operators. The magnitude of this deviation from world line manifest covariance is found to be exceedingly small for a number of common time of flight experiments.

On an influential account, chaos is explained in terms of random behaviour; and random behaviour in turn is explained in terms of having positive Kolmogorov-Sinai entropy (KSE). Though intuitively plausible, the association of the KSE with random behaviour needs justification since the definition of the KSE does not make reference to any notion that is connected to randomness. I provide this justification for the case of Hamiltonian systems by proving that the KSE is equivalent to a generalized version of (...) Shannon's communication-theoretic entropy under certain plausible assumptions. I then discuss consequences of this equivalence for randomness in chaotic dynamical systems. Introduction Elements of dynamical systems theory Entropy in communication theory Entropy in dynamical systems theory Comparison with other accounts Product versus process randomness. (shrink)

We treat the classical dynamics of the hydrogen atom in perpendicular electric and magnetic fields as a celestial mechanics problem. By expressing the Hamiltonian in appropriate action–angle variables, we separate the different time scales of the motion. The method of averaging then allows us to reduce the system to two degrees of freedom, and to classify the most important periodic orbits.

In the paper we construct a new kind of fractional dynamical model, i.e. the fractional relativistic Yamaleev oscillator model, and explore its dynamical behaviors. We will find that the fractional relativistic Yamaleev oscillator model possesses Lie algebraic structure and satisfies generalized Poisson conservation law. We will also give the Poisson conserved quantities of the model. Further, the relation between conserved quantities and integral invariants of the model is studied and it is proved that, by using the Poisson conserved quantities, we (...) can construct integral invariants of the model. Finally, the stability of the manifold of equilibrium states of the fractional relativistic Yamaleev oscillator model is studied. The paper provides a general method, i.e. fractional generalized Hamiltonian method, for constructing a family of fractional dynamical models of an actual dynamical system. (shrink)

If one demystifies entropy the second law of thermodynamics comes out as an emergent property entirely based on the simple dynamic mechanical laws that govern the motion and energies of system parts on a micro-scale. The emergence of the second law is illustrated in this paper through the development of a new, very simple and highly efficient technique to compare time-averaged energies in isolated conservative linear large scale dynamical systems. Entropy is replaced by a notion that is much more transparent (...) and more or less dual called ectropy. Ectropy has been introduced before but we further modify the notion of ectropy such that the unit in which it is expressed becomes the unit of energy. The second law of thermodynamics in terms of ectropy states that ectropy decreases with time on a large enough time-scale and has an absolute minimum equal to zero. Zero ectropy corresponds to energy equipartition. Basically we show that by enlarging the dimension of an isolated conservative linear dynamical system and the dimension of the system parts over which we consider time-averaged energy partition, the tendency towards equipartition increases while equipartition is achieved in the limit. This illustrates that the second law is an emergent property of these systems. Finally from our large scale linear dynamic model we clarify Loschmidt’s paradox concerning the irreversible behavior of ectropy obtained from the reversible dynamic laws that govern motion and energy at the micro-scale. (shrink)

Here is an accurate and readable translation of a seminal article by Henri Poincaré that is a classic in the study of dynamical systems popularly called chaos theory. In an effort to understand the stability of orbits in the solar system, Poincaré applied a Hamiltonian formulation to the equations of planetary motion and studied these differential equations in the limited case of three bodies to arrive at properties of the equations' solutions, such as orbital resonances and horseshoe orbits. Poincaré (...) wrote for professional mathematicians and astronomers interested in celestial mechanics and differential equations. Contemporary historians of math or science and researchers in dynamical systems and planetary motion with an interest in the origin or history of their field will find his work fascinating. (shrink)

In the classical Stueckelberg-Horwitz-Piron relativistic Hamiltonian mechanics, a significant aspect of evolution of the classical n-body particle system with mutual interaction is the method by which events along distinct particle world lines are put into correspondence as a dynamical state. Approaches to this procedure are discussed in connection with active and passive symmetry principles.

Contrary to the predominant way of doing physics, we claim that the geometrical structure of a general differentiable space-time manifold can be determined from purely dynamical considerations. Anyn-dimensional manifoldV a has associated with it a symplectic structure given by the2n numbersp andx of the2n-dimensional cotangent fiber bundle TVn. Hence, one is led, in a natural way, to the Hamiltonian description of dynamics, constructed in terms of the covariant momentump (a dynamical quantity) and of the contravariant position vectorx (a (...) geometrical quantity). That is, the Hamiltonian description furnishes a natural way of relating dynamics and geometry. Thus, starting from the Hamiltonian state function (for a particle)—taken as the fundamental dynamical entity—we show that general relativistic physics implies a general pseudo-Riemannian geometry, whereas the physics of the special theory of relativity is tied up with Minkowski space-time, and nonrelativistic dynamics is bound up to Newton-Cartan space-time. (shrink)

The extension of quantum mechanics to a general functional space (“rigged Hilbert space”), which incorporates time-symmetry breaking, is applied to construct extract dynamical models of entropy production and entropy flow. They are illustrated by using a simple conservative Hamiltonian system for multilevel atoms coupled to a time-dependent external force. The external force destroys the monotonicity of the ℋ-function evolution. This leads to a model of the entropy flow that allows a steady nonequilibrium structure of the emitted field around the (...) unstable particles. (shrink)

We study the question of how to decompose Hilbert space into a preferred tensor-product factorization without any pre-existing structure other than a Hamiltonian operator, in particular the case of a bipartite decomposition into "system" and "environment." Such a decomposition can be defined by looking for subsystems that exhibit quasi-classical behavior. The correct decomposition is one in which pointer states of the system are relatively robust against environmental monitoring (their entanglement with the environment does not continually and dramatically increase) and (...) remain localized around approximately-classical trajectories. We present an in-principle algorithm for finding such a decomposition by minimizing a combination of entanglement growth and internal spreading of the system. Both of these properties are related to locality in different ways. This formalism could be relevant to the emergence of spacetime from quantum entanglement. (shrink)

In quantum relativistic Hamiltonian dynamics, the time evolution of interacting particles is described by the Hamiltonian with an interaction-dependent term (potential energy). Boost operators are responsible for (Lorentz) transformations of observables between different moving inertial frames of reference. Relativistic invariance requires that interaction-dependent terms (potential boosts) are present also in the boost operators and therefore Lorentz transformations depend on the interaction acting in the system. This fact is ignored in special relativity, which postulates the universality of Lorentz (...) transformations and their independence of interactions. Taking into account potential boosts in Lorentz transformations allows us to resolve the “no-interaction” paradox formulated by Currie, Jordan, and Sudarshan [Rev. Mod. Phys. 35, 350 (1963)] and to predict a number of potentially observable effects contradicting special relativity. In particular, we demonstrate that the longitudinal electric field (Coulomb potential) of a moving charge propagates instantaneously. We show that this effect as well as superluminal spreading of localized particle states is in full agreement with causality in all inertial frames of reference. Formulas relating time and position of events in interacting systems reduce to the usual Lorentz transformations only in the classical limit (ħ→0) and for weak interactions. Therefore, the concept of Minkowski space-time is just an approximation which should be avoided in rigorous theoretical constructions. (shrink)

In the previous papers, it was demonstrated that applying the principle of maximum information entropy by maximizing the conditional information entropy, subject to the constraint given by the Liouville equation averaged over the phase space, leads to a definition of the rate of entropy change for closed Hamiltonian systems without any additional assumptions. Here, we generalize this basic model and, with the introduction of the additional constraints which are equivalent to the hydrodynamic continuity equations, show that the results obtained (...) are consistent with the known results from the nonequilibrium statistical mechanics and thermodynamics of irreversible processes. In this way, as a part of the approach developed in this paper, the rate of entropy change and entropy production density for the classical Hamiltonian fluid are obtained. The results obtained suggest the general applicability of the foundational principles of predictive statistical mechanics and their importance for the theory of irreversibility. (shrink)

Neither question (1) nor question (2) posed on page 446 have been adequately answered in this paper. Regarding (1) we have merely given functor maps onto the object languages of physical theories and regarding (2) we have merely described the algebraic structure of observables. A more satisfactory treatment will most likely involve (1) a generalization to algebraic categories, universal algebra and model theory in such a way as to capture the full inference structure of (perhaps van Fraassen's modal) quantum logic, (...) and (2) the introduction of the category of differentiable manifolds in order to capture the full symplectic structure of (perhaps Chernoff-Marsden's) Hamiltonian dynamics. (See van Fraassen [4] and Chernoff-Marsden [5] in Other References.)Finally, the intriguing suggestion of Finkelstein that “category theory is to metaphysics what group theory is to physics” deserves serious consideration. Indeed, although the theme has not been touched on explicitly in this paper it was central in motivating the project initially. However, it is more properly to be developed in a context external to a symposium on quantum logic. (shrink)

Based on Newton’s laws reformulated in the Hamiltonian dynamics combined with statistical mechanics, we formulate a statistical mechanical theory supporting the hypothesis of a closed universe oscillating in phase-space. We find that the behavior of this universe as a whole can be represented by a free entropic oscillator whose lifespan is nonhomogeneous, thus implying that time is shorter or longer according to the state of this universe given through its entropy. We conclude that time reduces to the entropy (...) production of this universe and that a nonzero entropy production means that local fluctuations could exist giving rise to the appearance of masses and to the curvature of the space. (shrink)

In the usual procedure of deriving equilibrium thermodynamics from classical statistical mechanics, Gibbsian fine-grained entropy is taken as the analogue of thermodynamical entropy. However, it is well known that the fine-grained entropy remains constant under the Hamiltonian flow. In this paper it is argued that we need not search for alternatives for fine-grained entropy, nor do we have to reject Hamiltonian dynamics, in order to solve the problem of the constancy of fine-grained entropy and, more generally, to (...) account for the non-equilibrium part of the laws of thermodynamics. Rather, we have to weaken the requirement that equilibrium be identified with a stationary probability distribution. (shrink)

We show that the strong form of Heisenberg’s inequalities due to Robertson and Schrödinger can be formally derived using only classical considerations. This is achieved using a statistical tool known as the “minimum volume ellipsoid” together with the notion of symplectic capacity, which we view as a topological measure of uncertainty invariant under Hamiltonian dynamics. This invariant provides a right measurement tool to define what “quantum scale” is. We take the opportunity to discuss the principle of the symplectic (...) camel, which is at the origin of the definition of symplectic capacities, and which provides an interesting link between classical and quantum physics. (shrink)

The paper aims to spell out the relevance of the Berry phase in view of the question what the minimal mathematical structure is that accounts for all observable quantum phenomena. The question is both of conceptual and of ontological interest. While common wisdom tells us that the quantum structure is represented by the structure of the projective Hilbert space, the appropriate structure rich enough to account for the Berry phase is the U(1) bundle over that projective space. The Berry phase (...) is ultimately rooted in the curvature of this quantum bundle, it cannot be traced back to the Hamiltonian dynamics alone. This motivates the ontological claim in the final part of the paper that, if one strives for a realistic understanding of quantum theory including the Berry phase, one should adopt a form of ontic structural realism. (shrink)

There are at least three different notions of degrees of freedom that are important in comparison of quantum and classical dynamical systems. One is related to the type of dynamical equations and inequivalent initial conditions, the other to the structure of the system and the third to the properties of dynamical orbits. In this paper, definitions and comparison in classical and quantum systems of the tree types of DF are formulated and discussed. In particular, we concentrate on comparison of the (...) number of the so called dynamical DF in a quantum system and its classical model. The comparison involves analyzes of relations between integrability of the classical model, dynamical symmetry and separability of the quantum and the corresponding classical systems and dynamical generation of appropriately defined quantumness. The analyzes is conducted using illustrative typical systems. A conjecture summarizing the observed relation between generation of quantumness by the quantum dynamics and dynamical properties of the classical model is formulated. (shrink)

The purpose of this paper is to review relativistic quantum theories with an invariant evolution parameter. Parametrized relativistic quantum theories (PRQT) have appeared under such names as constraint Hamiltonian dynamics, four-space formalism, indefinite mass, micrononcausal quantum theory, parametrized path integral formalism, relativistic dynamics, Schwinger proper time method, stochastic interpretation of quantum mechanics and stochastic quantization. The review focuses on the fundamental concepts underlying the theories. Similarities as well as differences are highlighted, and an extensive bibliography is provided.

Newtonian gravity is modified minimally to obtain a Lorentz covariant theory of gravity in a background flat space. Gravity is assumed to appear as a potential. Constraint Hamiltonian dynamics is used to determine particle trajectories in a manifestly covariant fashion. The resulting theory is significantly different from the general theory of relativity. However, all known experimental results (precession of planetary orbits, bending of the path of light near the sun, and gravitational spectral shift) are still explained by this (...) theory. (shrink)

The dynamics-from-permutations of classical Ising spins is generalized here for an arbitrarily long chain. This serves as an ontological model with discrete dynamics generated by pairwise exchange interactions defining the unitary update operator. The model incorporates a finite signal velocity and resembles in many aspects a discrete free field theory. We deduce the corresponding Hamiltonian operator and show that it generates an exact terminating Baker–Campbell–Hausdorff formula. Motivation for this study is provided by the Cellular Automaton Interpretation of (...) Quantum Mechanics. We find that our ontological model, which is classical and deterministic, appears as if of quantum mechanical kind in an appropriate formal description. However, it is striking that small deformations of the model turn it into a genuine quantum theory. This supports the view that quantum mechanics stems from an epistemic approach handling physical phenomena. (shrink)

Here, in our approximation of polaron theory, we examine the importance of introducing theT product, which turn out to be a very convenient theoretical approach for the calculation of thermodynamical averages.We focus attention on the investigation of the so-called linear polaron Hamiltonian and present in detail the calculation of the correlation function, spectral function, and Green function for such a linear system.It is shown that the linear polaron Hamiltonian provides an exactly solvable model of our system, and the (...) result obtained with this approach holds true for an arbitrary coupling constant which describes the strength of interaction between the electron and the lattice vibrations. Then, with the help of a variational technique, we show the possibility of reducing the real polaron Hamiltonian to a socalled trial or approximate linear model Hamiltonian.We also consider the exact calculation of free energy with a special technique that reduces calculations with the help of the T product, which, in our opinion, works much better and is easier than other analogous considerations, for example, the path-integral or Feynman-integral method.(1,2)Here we furthermore recall our own work,(4) where it was shown that the results of Refs. 7 and 8 concerning the impedance calculation in the polaron model may be obtained directly without the use of the path-integral method.The study of the polaron system's thermodynamics is carried out by us in the framework of the functional method. A calculation of the free energy and the momentum distribution function is proposed.Note also that the polaron systems with strong coupling(9) proved to be useful in different quantum field models in connection with the construction of dynamical models of composite particles. A rigorous solution of the special strong-coupling polaron problem, describing the interaction of a nonrelativistic particle with a quantum field, was given by Bogolubov.(3) The works of Tavkhelidze, Fedyanin, Khrustalev, and others(10–13) are dedicated to the further development and generalization of the Bogolubov method.Notice, too, that the electron-photon interaction effects play an important part in many problems of modern solid state theory (see, e.g., Refs. 7 and 14–19).The present paper summarizes a set of lectures delivered as a special course in the physics department of Moscow State University. (shrink)