Projective Invariance and Einstein's Equations

It is shown that a projectively invariant Lagrangian field theory based on linear non-symmetric connections in space-time and arbitrary source fields is equivalent to Einstein's standard theory of gravitation coupled to a source Lagrangian depending solely on the original source fields. A key point is that, as in the case of Lagrangian field theories based on symmetric connections in space-time, the Euler-Lagrange field equations uniquely determine the projective invariant part of the linear connection in terms of the metric, their first-order derivatives, the source fields, and their conjugate momenta.     

Covariant Hamiltonian Field Theory: Path Integral Quantization

The Hamiltonian counterpart of classical Lagrangian field theory is covariant Hamiltonian field theory where momenta correspond to derivatives of fields with respect to all world coordinates. In particular, classical Lagrangian and covariant Hamiltonian field theories are equivalent in the case of a hyperregular Lagrangian, and they are quasi-equivalent if a Lagrangian is almost-regular. In order to quantize covariant Hamiltonian field theory, one usually attempts to construct and quantize a multisymplectic generalization of the Poisson bracket. In the present work, the path integral quantization of covariant Hamiltonian field theory is suggested. We use the fact that a covariant Hamiltonian field system is equivalent to a certain Lagrangian system on a phase space which is quantized in the framework of perturbative quantum field theory. We show that, in the case of almost-regular quadratic Lagrangians, path integral quantizations of associated Lagrangian and Hamiltonian field systems are equivalent.     

A new integrable system on the sphere and conformally equivariant quantization

Taking full advantage of two independent projectively equivalent metrics on the ellipsoid leading to Liouville integrability of the geodesic flow via the well-known Jacobi–Moser system, we disclose a novel integrable system on the sphere Sⁿ$S^n$, namely the dual Moser system. The latter falls, along with the Jacobi–Moser and Neumann–Uhlenbeck systems, into the category of (locally) Stäckel systems. Moreover, it is proved that quantum integrability of both Neumann–Uhlenbeck and dual Moser systems is ensured by means of the conformally equivariant quantization procedure.     

The Post-NEWTONian Effect of Gravitation and the Retardation of the Gravitational Potential in Classical and Relativistic Theories of Gravitation

For planetary motions the post-NEWTON ian approximations of classical, special-relativistic, and general-covariant theories are compared. It is shown that, in this approximation, the anisotropy terms, which occur in the effective interaction potential in classical and special-relativistic theories, suggest a retardation of gravitation. In the post-NEWTON ian approximation of general-covariant theories the fixation of a retardation velocity is equivalent to coordinate conditions. – All post-NEWTON ian corrections are dipole-like ones, while, according to GAUSS , the classical perturbation theory generally leads to quadrupole-like corrections of the perturbation potential.     

Slightly Bimetric Gravitation

The inclusion of a flat metric tensor in gravitation permits the formulation of a gravitational stress-energy tensor and the formal derivation of general relativity from a linear theory in flat spacetime. Building on the works of Kraichnan and Deser, we present such a derivation using universal coupling and gauge invariance.Next we slightly weaken the assumptions of universal coupling and gauge invariance, obtaining a larger "slightly bimetric" class of theories, in which the Euler-Lagrange equations depend only on a curved metric, matter fields, and the determinant of the flat metric. The theories are equivalent to generally covariant theories with an arbitrary cosmological constant and an arbitrarily coupled scalar field, which can serve as an inflaton or dark matter.The question of the consistency of the null cone structures of the two metrics is addressed.     

Mining metrics for buried treasure

The same but different: That might describe two metrics. On the surface CLASSI may show two metrics are locally equivalent, but buried beneath may be a wealth of further structure. This was beautifully described in a paper by Malcolm MacCallum in 1998. Here I will illustrate the effect with two flat metrics — one describing ordinary Minkowski spacetime and the other describing a threeparameter family of Gal'tsov-Letelier-Tod spacetimes. I will dig out the beautiful hidden classical singularity structure of the latter (a structure first noticed by Tod in 1994) and then show how quantum considerations can illuminate the riches. I will then discuss how quantum structure can help us understand classical singularities and metric parameters in a variety of exact solutions mined from the Exact Solutions book.     

The Role of Time in Relational Quantum Theories

We propose a solution to the problem of time for systems with a single global Hamiltonian constraint. Our solution stems from the observation that, for these theories, conventional gauge theory methods fail to capture the full classical dynamics of the system and must therefore be deemed inappropriate. We propose a new strategy for consistently quantizing systems with a relational notion of time that does capture the full classical dynamics of the system and allows for evolution parametrized by an equitable internal clock. This proposal contains the minimal temporal structure necessary to retain the ordering of events required to describe classical evolution. In the context of shape dynamics (an equivalent formulation of general relativity that is locally scale invariant and free of the local problem of time) our proposal can be shown to constitute a natural methodology for describing dynamical evolution in quantum gravity and to lead to a quantum theory analogous to the Dirac quantization of unimodular gravity.     

Projective synchronization of chaotic systems with bidirectional nonlinear coupling

This paper presents a new scheme for constructing bidirectional nonlinear coupled chaotic systems which synchronize projectively. Conditions necessary for projective synchronization (PS) of two bidirectionally coupled chaotic systems are derived using Lyapunov stability theory. The proposed PS scheme is discussed by taking as examples the so-called unified chaotic model, the Lorenz–Stenflo system and the nonautonomous chaotic Van der Pol oscillator. Numerical simulation results are presented to show the efficiency of the proposed synchronization scheme.     

Contrasting Classical and Quantum Vacuum States in Non-inertial Frames

Classical electron theory with classical electromagnetic zero-point radiation (stochastic electrodynamics) is the classical theory which most closely approximates quantum electrodynamics. Indeed, in inertial frames, there is a general connection between classical field theories with classical zero-point radiation and quantum field theories. However, this connection does not extend to noninertial frames where the time parameter is not a geodesic coordinate. Quantum field theory applies the canonical quantization procedure (depending on the local time coordinate) to a mirror-walled box, and, in general, each non-inertial coordinate frame has its own vacuum state. In particular, there is a distinction between the “Minkowski vacuum” for a box at rest in an inertial frame and a “Rindler vacuum” for an accelerating box which has fixed spatial coordinates in an (accelerating) Rindler frame. In complete contrast, the spectrum of random classical zero-point radiation is based upon symmetry principles of relativistic spacetime; in empty space, the correlation functions depend upon only the geodesic separations (and their coordinate derivatives) between the spacetime points. The behavior of classical zero-point radiation in a noninertial frame is found by tensor transformations and still depends only upon the geodesic separations, now expressed in the non-inertial coordinates. It makes no difference whether a box of classical zero-point radiation is gradually or suddenly set into uniform acceleration; the radiation in the interior retains the same correlation function except for small end-point (Casimir) corrections. Thus in classical theory where zero-point radiation is defined in terms of geodesic separations, there is nothing physically comparable to the quantum distinction between the Minkowski and Rindler vacuum states. It is also noted that relativistic classical systems with internal potential energy must be spatially extended and can not be point systems. The classical analysis gives no grounds for the “heating effects of acceleration through the vacuum” which appear in the literature of quantum field theory. Thus this distinction provides (in principle) an experimental test to distinguish the two theories.     

Generalized Projective Synchronization Between Rössler System and New Unified Chaotic System

Based on symbolic computation system Maple and Lyapunov stability theory, an active control method is used to projectively synchronize two different chaotic systems — Lorenz-Chen-Lü system (LCL) and Rössler system, which belong to different dynamic systems. In this paper, we achieve generalized projective synchronization between the two different chaotic systems by directing the scaling factor onto the desired value arbitrarily. To illustrate our result, numerical simulations are used to perform the process of the synchronization and successfully put the orbits of drive system (LCL) and orbits of the response system (Rössler system) in the same plot for understanding intuitively.     

Non-perturbative renormalization group for simple fluids

We present a new non-perturbative renormalization group for classical simple fluids. The theory is built in the Grand Canonical ensemble and also in the framework of two equivalent scalar field theories. The exact mapping between the three renormalization flows is established rigorously. In the Grand Canonical ensemble the theory may be seen as an extension of the Hierarchical Reference Theory [Adv. Phys. 44, 211 (1995)] but, however, does not suffer from its shortcomings at subcritical temperatures. In the framework of a new canonical field theory for the liquid state developed with that aim, our construction identifies with the effective average action approach developed recently [Phys. Rep. 363 (2002)].     

Evolution of Classical and Quantum States in the Groupoid Picture of Quantum Mechanics

The evolution of states of the composition of classical and quantum systems in the groupoid formalism for physical theories introduced recently is discussed. It is shown that the notion of a classical system, in the sense of Birkhoff and von Neumann, is equivalent, in the case of systems with a countable number of outputs, to a totally disconnected groupoid with Abelian von Neumann algebra. The impossibility of evolving a separable state of a composite system made up of a classical and a quantum one into an entangled state by means of a unitary evolution is proven in accordance with Raggio’s theorem, which is extended to include a new family of separable states corresponding to the composition of a system with a totally disconnected space of outcomes and a quantum one.     

The Mathematical Role of Time and Space-Time in Classical Physics

We use Padoa's principle of independence of primitive symbols in axiomatic systems in order to discuss the mathematical role of time and spacetime in some classical physical theories. We show that time is eliminable in Newtonian mechanics and that spacetime is also dispensable in Hamiltonian mechanics, Maxwell's electromagnetic theory, the Dirac electron, classical gauge fields, and general relativity.     

Fundamental theories of waves and particles formulated without classical mass

Quantum and classical mechanics are two conceptually and mathematically different theories of physics, and yet they do use the same concept of classical mass that was originally introduced by Newton in his formulation of the laws of dynamics. In this paper, physical consequences of using the classical mass by both theories are explored, and a novel approach that allows formulating fundamental (Galilean invariant) theories of waves and particles without formally introducing the classical mass is presented. In this new formulation, the theories depend only on one common parameter called ‘wave mass’, which is deduced from experiments for selected elementary particles and for the classical mass of one kilogram. It is shown that quantum theory with the wave mass is independent of the Planck constant and that higher accuracy of performing calculations can be attained by such theory. Natural units in connection with the presented approach are also discussed and justification beyond dimensional analysis is given for the particular choice of such units.     

Applications of periodic orbit theory toN-particle systems

In recent years a number of new techniques have become available in nonequilibrium statistical mechanics, all derived from dynamical system theory, especially from the thermodynamic formalism of Ruelle. We focus here on periodic orbit theory, and we compare it with a novel approach proposed by Evans, Cohen, and Morriss, and developed further by Gallavotti and Cohen. We argue that the two approaches based on such theories are equivalent for systems of many particles if the underlying dynamics is similar to that of Anosov systems, and that such equivalence should remain in more general situations. We extend our previous explanation of irreversibility in the thermostatted Lorentz gas toN-particle diffusion and shearing systems.     

Miller型超高速摄影系统经典设计理论的研究

对Miller型超高速摄影系统的经典设计理论进行了系统研究.这些设计理论,即离焦设计理论、共轴设计理论和等速设计理论,都有原理误差,只能做到误差的最小化,而且任何两种设计理论都不能同时在同一个系统中实现,但是可以根据系统的具体要求给出最佳设计.     

Combinatorial Space from Loop Quantum Gravity

The canonical quantization of diffeomorphism invariant theories of connections in terms of loop variables is revisited. Such theories include general relativity described in terms of Ashtekar-Barbero variables and extension to Yang-Mills fields (with or without fermions) coupled to gravity. It is argued that the operators induced by classical diffeomorphism invariant or covariant functions are respectively invariant or covariant under a suitable completion of the diffeomorphism group. The canonical quantization in terms of loop variables described here, yields a representation of the algebra of observables in a separable Hilbert space. Furthermore, the resulting quantum theory is equivalent to a model for diffeomorphism invariant gauge theories which replaces space with a manifestly combinatorial object.