相对论时空观--"低速到高速"之二

1奇特的光速 牛顿力学的失效首先来源于光速的奇特性,即光速与光源的运动无关.例如,在匀速前进的车厢内,在中央吊一光源(图1),它向四外发光.在车厢内测量,光源是静止的,向各方向传播的光的速率是相等的.如果在地面上测量,光源是运动的.实测结果,不但各个方向光的速率相等,而且这一速率和在车厢内测得的光的速率也相等,都是3×108m/s.光速的这一特性也可以说成是光速与参考系无关.这一特性完全背离了伽利略速度变换,从而根本上动摇了牛顿力学的基础而使之失效.     

From time operator to chronons

A time operator, which incorporates the idea of time as a dynamical variable, was first introduced in the context of a theory of irreversible evolution. The existence of a time operator has interesting implications in several areas of physics. Here we demonstrate a close link between the existence of the time operator for relativistic particles and the existence of an indivisible time interval or chronons for dynamical evolution. More explicitly, we consider a Klein-Gordon particle and require the existence of a time operator for its evolution. We also make a natural choice of the form of the time operator which expresses it in terms of the generators of the Poincaré group. These then imply that the physical time evolution group must be the discrete subgroup U_n (n integers) of the originally given evolution group U_t of the Klein-Gordon particle and the constant is given by =h/2mc². This means that the requirement of the existence of a time operator implies that the time evolution cannot be followed to time intervals smaller than and, as such, emerges as a chronon for the dynamical evolution. Expecting that the same results hold for a Dirac particle also, we conclude that the so-called Zitterbewegungdoes not occur in reality. Thus, possible confirmation of the existence of chronons would result if no observableconsequence of Zitterbewegungis actually realized in nature. This calls for a search of observable consequences of the Zitterbewegungand a re-examination of their agreement (if any) with experiments. A possible consequence of Zitterbewegung,the so-called Darwin term present in the Dirac Hamiltonian in an electric field, is briefly considered.     

How to introduce time operator

Time operator can be introduced by three different approaches: by pertaining it to dynamical variables; by quantizing the classical expression of time; and taken as the restriction of energy shift generator to the Hilbert space of a physical system.     

A time operator in quantum mechanics

The lower bound on a continuous energy spectrum suffices t⊙ mathematically preclude the construction of a hermitian time operator canonically conjugate to the Hamiltonian. This problem is overcome by enlarging the Hilbert space in such a way as to have either an unbound spectrum or a doubly degenerate positive spectrum. In the enlarged space, the eigenvalue spectrum θ of such an operator ranges from minus to plus infinity and constitutes the conjugate variable of the energy E. On the other hand, the evolution parameter t of the dynamical equations is related to the expectation value of the time operator. Both extensions yield the usual dynamics for state vectors restricted to the physical subspace. Vectors in the complement subspaces describe either negative energy wave packets with forward evolution or positive energy wave packets with backwards evolution. The time energy uncertainty relation is discussed.     

Spatio-temporal invariants of the time reversal operator

The decomposition of the time reversal operator, known by the French acronym DORT, is a technique to extract point scatterers' monochromatic Green's functions from a medium. It is used to detect, locate, and focus on scatterers in various domains such as underwater acoustics, medical ultrasound, and nondestructive evaluation. A limitation of the method arises from its single-frequency nature, when the signals used in acoustics are often broadband. Reconstruction of the broadband Green's functions from the single-frequency Green's functions can be very difficult when numerous scatterers are present in the medium. Moreover, the method does not take advantage of the axial resolution associated with broadband signals. Time domain methods are investigated here as an answer to these problems. It is shown that the time reversal operator in the time domain takes the form of a tensor. The properties of the invariants are discussed. It is shown they do not have all the expected properties. Another method is proposed that requires a priori information on the medium.     

The time reversal operator for semigroup evolutions

A quantum theory combining an irreversible time evolution semigroup with a time reversal operator is presented.     

The molecular vibration-rotation kinetic-energy operator for general internal coordinates

The molecular kinetic-energy operator for general internal coordinates is formulated in terms of simple generalisations of the matrices A, B, α, and β of Crawford. A new matrix γ of the gradients of the translational and rotational constraints facilitates the calculation of the β matrix. It is shown that the kinetic pseudo-potential U is most conveniently calculated as an atomic sum, and results are given for valence coordinates in various types of molecules.     

Relativistic time delays in the Dirac approach to nucleon-nucleus scattering

In connection with a characteristic feature of the effective optical potential in the Dirac approach two types of time delays are considered in the relativistic eikonal approximation. One is obtained from the scattering amplitude and the other given by the wave packet motion in the interaction region. These time delays turn out to differ in sign at intermediate energies, in contrast to the agreement between corresponding nonrelativistic time delays.     

Relativistic corrections to quantum mechanical concept ofJ/ψ formation time

The effect ofJ/ψ dissolution in quark-gluon plasma is calculated within quantum mechanical approach taking into account also relativistic corrections. It is found that relativistic effects do not influence significantly the time dependence of theJ/ψ dissolution. Consequently the rate ofJ/ψ suppression is not dramatically changed.     

Analytic treatment of time fractional nonlinear operator equation with applications

In this paper, analytic treatment of time fractional nonlinear operator equation has been presented by using Adomian’s decomposition method (ADM). To illustrate the procedure, the nonlinear Schrodinger (NLS) equation of fractional order has been solved. The solution is expressed in terms of Mittag-Leffler function. The present method performs extremely well in terms of accuracy, efficiency and simplicity.     

Shift operator method and Runge-Kutta exponential time differencing method for plasma

In this paper, the shift operator finite difference time domain (SO-FDTD) method and Runge-Kutta exponential time differencing (RKETD) method are introduced. The high accuracy and efficiency of the two methods are verified by calculating the reflection and transmission coefficients of electromagnetic waves through a collisional plasma slab. A comparison of computational efficiency of the two methods is presented by simulating the electromagnetic wave propagation in homogeneous non-magnetized plasma. The numerical results indicate that the calculation time using SO-FDTD method is less than that using RKETD method with almost the same accuracy.     

An inversion formula for a Prandtl-Ishlinskii operator with time dependent thresholds

We extend the well-known inversion result for Prandtl-Ishlinskii operators to the case of time dependent thresholds. The exact inversion formula is shown to hold under the condition that the distances between the thresholds do not decrease in time. Examples show that this threshold dilation condition cannot be completely omitted.     

Relativistic equipartition

We develop the analogue of the law of equipartition of energy for relativistic systems and examine some of its consequences. For mixtures of special relativistic ideal gases of different rest masses we find that the energy gets partitioned in an unexpected way, the ultra-relativistic component taking essentially twice as much energy per degree of freedom as the more massive non-relativistic component. This conclusion may have an application when considering thermalized photons in the interior of the sun. With an eye toward the possible extension of our considerations to general relativity, we show that, in the presence of a distinguished time-like Killing field, the global parameter which characterized the equilibrium state of an ideal gas can be determined by local measurements employing a suitably calibrated thermometer.