The Pauli objection

Schroedinger's equation says that the Hamiltonian is the generator of time translations. This seems to imply that any reasonable definition of time operator must be conjugate to the Hamiltonian. Then both time and energy must have the same spectrum since conjugate operators are unitarily equivalent. Clearly this is not always true: normal Hamiltonians have lower bounded spectrum and often only have discrete eigenvalues, whereas we typically desire that time can take any real value. Pauli concluded that constructing a general a time operator is impossible (although clearly it can be done in specific cases). Here we show how the Pauli argument fails when one uses an external system (a "clock") to track time, so that time arises as correlations between the system and the clock (conditional probability amplitudes framework). In this case, the time operator is not conjugate to the system Hamiltonian, but its eigenvalues still satisfy the Schroedinger equation for arbitrary Hamiltonians.Comment: 6 page

Time-Evolution of the Power Spectrum of the Black Hole X-ray Nova XTE J1550-564

We have studied the time evolution of the power spectrum of XTE J1550-564, using X-ray luminosity time series data obtained by the Rossi X-Ray Timing Explorer satellite. A number of important practical fundamental issues arise in the analysis of these data, including dealing with time-tagged event data, removal of noise from a highly non-stationary signal, and comparison of different time-frequency distributions. We present two new methods to understand the time frequency variations, and compare them to the dynamic power spectrum of Homan et al. All of the approaches provide evidence that the QPO frequency varies in a systematic way during the time evolution of the signal.Comment: 4 pages, 3 figures; 2001 IEEE - EURASIP Workshop on Nonlinear Signal and Image Processing (June 3-6, 2001), and to appear in the proceeding

Nonlinear Transformation of Differential Equations into Phase Space

Time-frequency representations transform a one-dimensional function into a two-dimensional function in the phase-space of time and frequency. The transformation to accomplish is a nonlinear transformation and there are an infinite number of such transformations. We obtain the governing differential equation for any two-dimensional bilinear phase-space function for the case when the governing equation for the time function is an ordinary differential equation with constant coefficients. This connects the dynamical features of the problem directly to the phase-space function and it has a number of advantages

Approximation of the Wigner Distribution for Dynamical Systems Governed by Differential Equations

A conceptually new approximation method to study the time-frequency properties of dynamical systems characterized by linear ordinary differential equations is presented. We bypass solving the differential equation governing the motion by writing the exact Wigner distribution corresponding to the solution of the differential equation. The resulting equation is a partial differential equation in time and frequency. We then show how it lends itself to effective approximation methods because in the time frequency plane there is a high degree of localization of the signal. Numerical examples are given and compared to exact solutions

Instantaneous spectrum estimation of event-based densities

We present a method for obtaining a time-varying spectrum that is particularly suited when the data are in event-based form. This form arises in many areas of science and engineering, and especially in astronomy, where one has photon counting detectors. The method presented consists of three procedures. First, estimating the density using the kernel method; second, highpass filtering the manifestly positive density; finally, obtaining the time-frequency distribution with a modified Welch′s method. For the sake of validation event-based data are generated from a given distribution and the proposed method is used to construct the time-frequency spectrum and is compared to the original density. The results demonstrate the effectiveness of the method

Abundance and guild structure of grasshoppers (Orthoptera: Acridoidea) in communally grazed and protected savanna

This study was conducted to determine how savanna grass sward modifications caused by heavy grazing pressure influenced the abundance and guild structure of grasshoppers. Heavily grazed communal land was compared with a lightly grazed area and a mowed airstrip, in adjacent protected land, in the Mpumalanga lowveld, South Africa. Plant species composition, height, aerial cover and greenness of grass in the herbaceous stratum were measured in representative sites. Total grasshopper abundance and relative abundance of grasshopper species were also assessed in each site. Grasshoppers were assigned to feeding and habitat functional groups for comparison among the three areas. The heavily grazed area, characterised by short vegetation and low aerial cover, high greenness of grass, and high frequency of forbs, was inhabited by grasshopper species associated with bare ground or short and/or sparse grass, that were non-graminivorous or soft grass feeders. The lightly grazed area, characterised by tall vegetation and high aerial cover, low greenness of grass, and low frequency of forbs, was inhabited by grasshopper species associated with long and/or thick grass, that were mixed feeders or tough grass feeders. The mowed area, characterised by short vegetation and low aerial cover, low greenness of grass, and low frequency of forbs, exhibited lower grasshopper abundance, species richness, and diversity than either of the grazed areas