Relational Evolution of the Degrees of Freedom of Generally Covariant Quantum Theories

We study the classical and quantum dynamics of generally covariant theories with vanishing Hamiltonian and with a finite number of degrees of freedom. In particular, the geometric meaning of the full solution of the relational evolution of the degrees of freedom is displayed, which means the determination of the total number of evolving constants of motion required. Also a method to find evolving constants is proposed. The generalized Heisenberg picture needs M time variables, as opposed to the Heisenberg picture of standard quantum mechanics where one time variable t is enough. As an application, we study the parametrized harmonic oscillator and the SL(2, R) model with one physical degree of freedom that mimics the constraint structure of general relativity where a Schrödinger equation emerges in its quantum dynamics.     

Information in asset pricing: a wave function approach

This paper introduces a quantum‐like wave function as an information wave function. We show how the option pricing partial differential equation can be re‐written when we account for such information wave function. We use two stochastic differential equations, one of which relates to Nelson's hypothesis of Universal Brownian motion. We also provide for two examples which further highlight the proposed theory.     

Constants of Motion for Several One-Dimensional Systems and Problems Associated with Getting Their Hamiltonians

The constants of motion of the following systems are deduced: a relativistic particle with linear dissipation; a no-relativistic particle with a time explicitly depending force; a no-relativistic particle with a constant force and time depending mass; and a relativistic particle under a conservative force with position depending mass. The Hamiltonian for these systems, which is determined by getting the velocity as a function of position and generalized linear momentum, can be found explicitly at first approximation for the first system. The Hamiltonians for the other systems are kept implicitly in their expressions for their constants of motion.     

Noether's Theorem and the Conservation Laws of the KORTEWEG-DE VRIES Equation

A method developed recently by the author to derive a continuum of conservation laws by Noether's theorem from the so-called extended Bäcklund transformations is applied to the KORTEWEG -DE VRIES equation that describes various nonlinear dispersive wave phenomena in hydrodynamics, plasma physics and solid state physics. Further applications of Noether's theorem concerning this equation are given. It is shown that the Galilean transformation in the present case has an analogous function as Lie's transformation has with respect to the sine-Gordon equation.     

The gauge theory of dislocations: conservation and balance laws

We derive conservation and balance laws for the translational gauge theory of dislocations by applying Noether's theorem. We present an improved translational gauge theory of dislocations including the dislocation density tensor and the dislocation current tensor. The invariance of the variational principle under the continuous group of transformations is studied. Through Lie's infinitesimal invariance criterion we obtain conserved translational and rotational currents for the total Lagrangian made up of an elastic and dislocation part. We calculate the broken scaling current. Looking only on one part of the whole system, the conservation laws are changed into balance laws. Because of the lack of translational, rotational and dilatation invariance for each part, a configurational force, moment and power appears. The corresponding J , L and M integrals are obtained. Only isotropic and homogeneous materials are considered and we restrict ourselves to a linear theory. We choose constitutive laws for the most general linear form of material isotropy. Also we give the conservation and balance laws corresponding to the gauge symmetry and the addition of solutions. From the addition of solutions we derive a reciprocity theorem for the gauge theory of dislocations. Also, we derive the conservation laws for stress-free states of dislocations.     

相对运动变质量完整系统的共形不变性与守恒量

研究了相对运动变质量完整系统的共形不变性与守恒量,提出了该系统共形不变性的概念,推导出了相对运动变质量完整系统的运动微分方程具有共形不变性并且是 Lie 对称性的充要条件,借助规范函数满足的结构方程导出系统相应的守恒量,并给出应用算例. 关键词： 变质量 相对运动 共形不变性 守恒量     

相空间中相对运动完整力学系统的共形不变性与守恒量

研究了相空间中相对运动完整力学系统的共形不变性与守恒量.给出了该系统共形不变性的定义,并推导出相空间中相对运动完整力学系统的运动微分方程具有共形不变性并且是Lie对称性的充分必要条件.利用规范函数满足的结构方程导出该系统相应的守恒量,并给出应用算例.     

Construction of exact dynamical invariants of two-dimensional classical system

A general method is used for the construction of second constant of motion of fourth order in momenta using the complex coordinates (z, _z ^- ). A fourth-order potential equation is obtained whose solutions directly provide a large class of integrable systems. The potential equation is tested with an interesting example which admits second constants of motion.     

Gauge invariance based on the extrinsic geometry in the rigid string

The string model with the extrinsic curvature is studied which is a gauge invariant field theory with higher order derivatives. We present an equivalent action without any higher order derivatives which keeps the gauge invariance. We point out the difficulty caused by the second class constraints in Dirac's canonical method. Following a new method for dynamical systems with second class constraints, we construct an equivalent model which has no second class constrants but as a new gauge invariance. This gauge invariance guarantees the equivalence between the original model and the new one. We show that the model can be quantized in this formalism. We find the unitarity violation of the model.     

Emmy Noether and Her Theorems

Today Noether's principal theorem occupies a prominent place in theoretical physics, though for a long time its significance was largely overlooked. Even now, relatively few physicists realize that Emmy Noether's original paper from 1918 contains two fundamental theorems. Moreover, both theorems are essential for understanding her original motivation, namely to distinguish between proper and improper conservation laws in physics.     

The Fluctuation Theorem

The question of how reversible microscopic equations of motion can lead to irreversible macroscopic behaviour has been one of the central issues in statistical mechanics for more than a century. The basic issues were known to Gibbs. Boltzmann conducted a very public debate with Loschmidt and others without a satisfactory resolution. In recent decades there has been no real change in the situation. In 1993 we discovered a relation, subsequently known as the Fluctuation Theorem (FT), which gives an analytical expression for the probability of observing Second Law violating dynamical fluctuations in thermostatted dissipative non-equilibrium systems. The relation was derived heuristically and applied to the special case of dissipative non-equilibrium systems subject to constant energy 'thermostatting'. These restrictions meant that the full importance of the Theorem was not immediately apparent. Within a few years, derivations of the Theorem were improved but it has only been in the last few of years that the generality of the Theorem has been appreciated. We now know that the Second Law of Thermodynamics can be derived assuming ergodicity at equilibrium, and causality. We take the assumption of causality to be axiomatic. It is causality which ultimately is responsible for breaking time reversal symmetry and which leads to the possibility of irreversible macroscopic behaviour. The Fluctuation Theorem does much more than merely prove that in large systems observed for long periods of time, the Second Law is overwhelmingly likely to be valid. The Fluctuation Theorem quantifies the probability of observing Second Law violations in small systems observed for a short time. Unlike the Boltzmann equation, the FT is completely consistent with Loschmidt's observation that for time reversible dynamics, every dynamical phase space trajectory and its conjugate time reversed 'anti-trajectory', are both solutions of the underlying equations of motion. Indeed the standard proofs of the FT explicitly consider conjugate pairs of phase space trajectories. Quantitative predictions made by the Fluctuation Theorem regarding the probability of Second Law violations have been confirmed experimentally, both using molecular dynamics computer simulation and very recently in laboratory experiments.     

Correlations in classical and quantum systems far from thermal equilibrium

We consider classical systems described by a Fokker-Planck equation or a generalized Fokker-Planck equation and quantum systems described by a density matrix equation or by a generalized Fokker-Planck equation using the principle of quantum classical correspondence. We split the corresponding operators of the equation of motion into a part which refers to the proper system and another one which describes the coupling of the proper system to the external world (reservoirs). We demonstrate that by use of conservation laws, referring to the proper systems, exact relations hold for certain moments, valid for all temperatures and coupling constants of the reservoirs. Using the concepts of a previous paper we describe then a perturbation theoretical approach which allows in a simple manner to determine a number of important correlation functions (moments of the total system). The time dependent case is briefly discussed. The applicability and usefulness of the present procedure is demonstrated by the example of the single-mode laser yielding e.g. expressions for the atom-field correlation.     

变质量Chetaev型非完整系统Appell方程Mei对称性的共形不变性与守恒量

研究了变质量Chetaev型非完整系统Appell方程Mei对称性的共形不变性和守恒量.在群的无限小变换下,定义了变质量Chetaev型非完整系统Appell方程Mei对称性和共形不变性,给出了该系统Mei对称性的共形不变性确定方程,并推导出系统相应的守恒量表达式.最后,给出了应用算例.     

Ableitung einer kontinuierlichen Mannigfaltigkeit von Erhaltungssätzen der modifizierten Korteweg-de-Vries-Gleichung nach dem Noetherschen Satz

Derivation of a Continuous Set of Conservation Laws for the Modified Korteweg-de Vries Equation by Noether's Theorem A method developed recently to derive a continuous set of conservation laws from extended Bäcklund transformations by means of Noether's theorem is applied to the modified Korteweg-de Vries (m. KdV) equation that describes Alfvén waves in a plasma. The corresponding conserved currents are equivalent to those found by WADATI , SANUKI and KONNO . It is shown that the extended Bäcklund transformation B?_α for the m. KdV equation, which coincides with that for the sine-Gordon equation, by MIURA'S transformation becomes the extended Bäcklund transformation β_x for the Korteweg-de Vries equation where x = 1/2α.     

A unified approach to quantum and classical TTW systems based on factorizations

A unifying method based on factorization properties is introduced for finding symmetries of quantum and classical superintegrable systems using the example of the Tremblay–Turbiner–Winternitz (TTW) model. It is shown that the symmetries of the quantum system can be implemented in a natural way to its classical version. Besides, by this procedure we get also other type of constants of motion depending explicitly on time that allow to find directly the motion of the system whose corresponding trajectories coincide with those obtained previously by using its symmetries.     

原子-辐射场相互作用系统 Heisenberg运动方程的相干态表述

用Heisenberg-Weyle(简称H-W)群直积SU(2)群上的相干态表述了原 子-辐射场相互作用系统的Heisenberg运动方程。引入一个算符振幅函数,将Heisenberg绘 景中算符运动方程转化成可分离变量的偏微分方程,给出了方程的形式解以及时间演化算符 在该表述下的表示。     

On symmetries,reductions, conservation laws and conserved quantities of optical solitons with inter-modal dispersion

This paper studies the compressional dispersive Alfvén (CDA) waves where Noether symmetries will be calculated from which the corresponding conservation laws will be obtained via Noether's theorem. Furthermore, one case of double reduction is performed via the association of a conserved vector with a Noether symmetry (with zero gauge). The conserved quantities of optical solitons in the presence of intermodal dispersion that is governed by the perturbed nonlinear Schrödinger's equation with Kerr law nonlinearity. The invariance-multiplier method is adopted to carry out the analysis, from which the conserved densities are then retrieved. Finally, the conserved quantities are obtained using the 1-soliton solution of the governing equation.     

Mach Like Principle from Conserved Charges

We study models where the gauge coupling constants, masses and the gravitational constant are functions of some conserved charge in the universe, and furthermore a cosmological constant that depends on the total charge of the universe. We first consider the standard Dirac action, but where the mass and the electromagnetic coupling constant are a function of the charge in the universe and afterwards extend this to curved spacetime and consider gauge coupling constants, the gravitational constant and the mass as a function of the charge of the universe, which represent a sort of Mach principle for all the constants of nature. In the flat space formulation, the formalism is not manifestly Lorentz invariant, however Lorentz invariance can be restored by performing a phase transformation of the Dirac field. One interesting model of this type is one where the action is invariant under rescalings of the Dirac wave function. In the curved space time formulation, there is the additional feature that some of the equations of motion break the general coordinate invariance also, but in a way that can be understood as a coordinate choice only, so the equations are still of the General Relativity type, but with a certain natural coordinate choice, where there is no current of the charge. We have generalized what we have done and also constructed a cosmological constant which depends on the total charge of the universe. We discuss how these ideas work when the space where the charges live is finite. If we were to use some only approximately conserved charge for these constructions, like say baryon number (in the context of the standard model), this will lead to corresponding violations of Lorentz symmetry in the early universe for example. We also briefly discuss another non-local formulations where the coupling constants are functions of the Pontryagin index of some non-abelian gauge field configurations. The construction of charge dependent contributions can also be motivated from the structure of the “infra-red counter terms” needed to cancel infra red divergences for example in three dimensions.     

Quantization of Brownian Motion

Following our work on the quantization of nonconservative systems using fractional calculus, the canonical quantization of a system with Brownian motion is carried out according to the Dirac method. A suitable Lagrangian corresponding to the Langevin equation is set up. Further, a Hamiltonian is constructed and is transformed to Schrödinger's equation which is solved.