Quantum mechanics of space and time

A formulation of relativistic quantum mechanics is presented independent of the theory of Hilbert space and also independent of the hypothesis of spacetime manifold. A hierarchy is established in the nondistributive lattice of physical ensembles, and it is shown that the projections relating different members of the hierarchy form a semigroup. It is shown how to develop a statistical theory based on the definition of a statistical operator. Involutions defined on the matrix representations of the semigroup are interpreted in terms ofCPT conjugations. The theory of particles of spin one-half and systems with higher spin is developed from first principles. Methods are also developed for defining energy, momentum, orbital angular momentum, and weighted spacetime coordinates without reference to a manifold.     

Quantum mechanics and relativity: attempt at a new start

The approach which led Louis de Broglie to the assertion, for particles with nonzero rest mass, of the two correlated relationsp =h/ andW =mc ² =hv, is reexamined. A modified approach is then developed. This leads to a set of mutually coherent new relations with respect to which de Broglie's relationsp =h/ andW =mc ² =hv appear as certain approximations. The mentioned set of new relations entails the prediction of specific effects which can be verified experimentally. If it is confirmed, this set of new relations might constitute the germ of a theory able to accomplish a veritable unification of relativity and microphysics.     

Quantum mechanics and the direction of time

In recent papers the authors have discussed the dynamical properties of large Poincaré systems (LPS), that is, nonintegrable systems with a continuous spectrum (both classical and quantum). An interesting example of LPS is given by the Friedrichs model of field theory. As is well known, perturbation methods analytic in the coupling constant diverge because of resonant denominators. We show that this Poincaré catastrophe can be eliminated by a natural time ordering of the dynamical states. We obtain then a dynamical theory which incorporates a privileged direction of time (and therefore the second law of thermodynamics). However, it is only in very simple situations that this time ordering can be performed in an extended Hilbert space. In general, we need to go to the Liouville space (superspace) and introduce a time ordering of dynamical states according to the number of particles involved in correlations. This leads then to a generalization of quantum mechanics in which the usual Heisenberg's eigenvalue problem is replaced by a complex eigenvalue problem in the Liouville space.     

Brownian local time and quantum mechanics

LetV be any (sufficiently regular) attractive potential in one and two dimensions. We make rigorous an argument of M. Kac [1], relating the recurrence of the Brownian motion to the existence of at least one bound state for the quantum HamiltonianH = –(/2)+ V.     

Violation of signal locality and unitarity in a merger of quantum mechanics and general relativity

It is shown that a violation of signal locality and unitarity occur in a particular merger of quantum mechanics and general relativity.     

On Classical and Quantum Objectivity

We propose a conceptual framework for understanding the relationship between observables and operators in mechanics. To do so, we introduce a postulate that establishes a correspondence between the objective properties permitting to identify physical states and the symmetry transformations that modify their gauge dependant properties. We show that the uncertainty principle results from a faithful—or equivariant—realization of this correspondence. It is a consequence of the proposed postulate that the quantum notion of objective physical states is not incomplete, but rather that the classical notion is overdetermined.     

Causality, symmetries and quantum mechanics

It is argued that there is no evidence for causality as a metaphysical relation in quantum phenomena. The assumptions that there are no causal laws, but only probabilities for physical processes constrained by symmetries, leads naturally to quantum mechanics. In particular, an argument is made for why there are probability amplitudes that are complex numbers. This argument generalizes the Feynman path integral formulation of quantum mechanics to include all possible terms in the action that are allowed by the symmetries, but only the lowest order terms are observable at the presently accessible energy scales, which is consistent with observation. The notion of relational reality is introduced in order to give physical meaning to probabilities. This appears to give rise to a new interpretation of quantum mechanics.     

Invariance Analysis and Variational Conservation Laws for the Wave Equation on Some Manifolds

In this paper we discuss symmetries of a nonlinear wave equation that arises as a consequence of some Riemannian metrics of signature −2. The objective of this study is to show how geometry can be responsible in giving rise to a nonlinear inhomogeneous wave equation rather than assuming nonlinearities in the wave equation from physical considerations. We find Lie point symmetries of the corresponding wave equations and give their solutions in two cases. Some interesting physical conclusions relating to conservation laws such as energy, linear and angular momenta are also determined.     

Quantum mechanics,knot theory,and quantum doubles

In previous papers we have described quantum mechanics as a matrix symplectic geometry and showed the existence of a braiding and Hopf algebra structure behind our lattice quantum phase space. The first aim of this work is to give the defining commutation relations of the quantum Weyl-Schwinger-Heisenberg group associated with our ℜ-matrix solution. The second aim is to describe the knot formalism at work behind the matrix quantum mechanics. In this context, the quantum mechanics of a particle-antiparticle system (pˉp) moving in the quantum phase space is viewed as a quantum double.     

Classical and quantum general relativity: A new paradigm

We argue that recent developments in discretizations of classical and quantum gravity imply a new paradigm for doing research in these areas. The paradigm consists in discretizing the theory in such a way that the resulting discrete theory has no constraints. This solves many of the hard conceptual problems of quantum gravity. It also appears as a useful tool in some numerical simulations of interest in classical relativity. We outline some of the salient aspects and results of this new framework. Fifth Award in the 2005 Essay Competition of the Gravity Research Foundation. - Ed.     

General relativity and quantum mechanics in five dimensions

In 5D, I take the metric in canonical form and define causality by null-paths. Then spacetime is modulated by a factor equivalent to the wave function, and the 5D geodesic equation gives the 4D Klein-Gordon equation. These results effectively show how general relativity and quantum mechanics may be unified in 5D.     

The local Hurst exponent of the financial time series in the vicinity of crashes on the Polish stock exchange market

We investigate the local fractal properties of the financial time series based on the whole history evolution (1991–2007) of the Warsaw Stock Exchange Index (WIG), connected with the largest developing financial market in Europe. Calculating the so-called local time-dependent Hurst exponent for the WIG time series we find the dependence between the behavior of the local fractal properties of the WIG time series and the crashes’ appearance on the financial market. We formulate the necessary conditions based on the behavior which have to be satisfied if the rupture or crash point is expected soon. As a result we show that the signal to sell or the signal to buy on the stock exchange market can be translated into evolution pattern. We also find a relation between the rate of the drop and the total correction the WIG index gains after the crash. The current situation on the market, particularly related to the recent Fed intervention in September ’07, is also discussed.     

String theory, scale relativity and the generalized uncertainty principle

Extensions (modifications) of the Heisenberg uncertainty principle are derived within the framework of the theory of special scale-relativity proposed by Nottale. In particular, generalizations of the stringy uncertainty principle are obtained where the size of the strings is bounded by the Planck scale and the size of the universe. Based on the fractal structures inherent with two dimensional quantum gravity, which has attracted considerable interest recently, we conjecture that the underlying fundamental principle behind string theory should be based on an extension of the scale relativity principle whereboth dynamics as well as scales are incorporated in the same footing.     

牛顿力学中的惯性质量与狭义相对论中的惯性质量

分别详细说明了在牛顿力学中和在狭义相对论中,惯性和惯性质量的概念是如何引入的.明确地阐述了狭义相对论同牛顿力学相类似,物体(可视为质点或粒子)的固有质量(或静止质量)就是其惯性质量.通过分析,指出并强调了运动质量只是个规定,并非物体惯性的大小真的随运动发生了改变.最后还对静止质量为零、速度为光速的粒子只遵从狭义相对论而...     

Some speculations on a causal unification of relativity,gravitation, and quantum mechanics

Some speculations on a causal model that could provide a common conceptual foundation for relativity, gravitation, and quantum mechanics are presented. The present approach is a unification of three theories, the first being the repulsive theory of gravitational forces first proposed by Lesage in the eighteenth century. Lesage attempted to explain gravitational forces from the principle of conservation of momentum of some hypothetical particles, which we shall call gravitons. These gravitons, whose density is assumed homogenous, are constantly colliding with objects. The gravitational force is caused by a shielding effect of bodies when they are near each other. One also can make a clear physical distinction between an accelerating and a nonaccelerating object from this viewpoint. The second of these theories is the Brownian motion theory of quantum mechanics or stochastic mechanics, which treats the nondeterministic nature of quantum mechanics as being due to a Brownian motion of all objects. This Brownian motion being caused by the statistical variation in the graviton flux. The above two theories are unified in this article with the causal theory of special relativity. Within the present context, the time dilations (and other effects) of relativity are explained by assuming that the rate of a clock is a function of the total number or intensity of gravitons and the average frequency or energy of the gravitons that the clock receives. Two clocks having some relative velocity in the same intensity gravitational field would then have a different rate because the average frequency of the gravitons would be different for each clock owing to the Doppler effect. That is, they would essentially be in different fields considering both the frequency and intensity. The special theory would then be the special case of the general theory where the intensity is constant but the average frequency varies. In all the previous it is necessary to assume a particular model of the creation of the universe, namely the Big Bang theory. This assumption gives us the existence of a preferred reference frame, the frame in which the Big Bang explosion was at rest. The above concepts of graviton distribution and real time dilations become meaningful by assuming the Big Bang theory along with this preferred frame. An experimental test is proposed.     

Inconsistency of Quantum—Classical Dynamics, and What it Implies

A new proof of the impossibility of a universal quantum-classical dynamics is given. It has at least two consequences. The standard paradigm “quantum system is measured by a classical apparatus” is untenable, while a quantum matter can be consistently coupled only with a quantum gravity.     

Time,Clocks and Parametric Invariance

In the context of a parametric theory (with the time being a dynamical variable) we consider here the coupling between the quantum vacuum and the background gravitation that pervades the universe (unavoidable because of the universality and long range of gravity). We show that this coupling, combined with the fourth Heisenberg relation, would break the parametric invariance of the gravitational equations, introducing thus a difference between the marches of the atomic and the astronomical clocks. More precisely, they would be progressively and adiabatically desynchronized with respect to one another in such a way that the latter would lag behind the former. This would produce a discrepancy between gravitational theory and observations, which use astronomical and atomic time respectively. It turns out that this result, surprising at it might be, is fully compatible with current physics, since it does not conflict with any known physical law or principle. We argue that this phenomenon must be studied, since it could have cosmological consequences.     

Dark optical solitons and conservation laws for parabolic and dual-power law nonlinearities in (2 + 1)-dimensions

This paper studies the dynamics of optical solitons with parabolic and dual-power law nonlinearities. The dark 1-soliton solution is first obtained by the ansatz method along with the necessary constraint conditions, for both of these nonlinearities. Subsequently, the invariance, conservation laws and double reductions of the governing nonlinear Schrödinger's equation are studied and the conserved densities are thus revealed.