Projective invariance and the fifth coordinate

The paper considers the consequences of the principle of parametric (projective) invariance in the context of the general variational problem for stationary curves in a four-dimensional space-time. The necessity is shown of introducing a parameter along the trajectory as a fifth coordinate. The condition of cylindricality along it then acquires an obvious significance. The projective invariants are calculated for the trajectories of charged test particles in a gravielectromagnetic field. It is shown that one of these coincides with the density of the electromagnetic-field Lagrangian.     

Scale invariance and inflation

A scale invariant model for early universe inflationary cosmology is developed. In order to realize dilatation invariance and spontaneous symmetry breaking we introduce two scalar fields, a dilaton and an inflaton. The scale invariant theory encompasses the Brans-Dicke and induced-gravity models as limiting cases. The model is solved numerically for a wide class of initial conditions. We find that the inflationary epoch is generically characterized by a two phase evolution of the universe: A single or double exponential era and a power-law expansion. Onset of gravity triggers double exponential evolution of the scale factor. We further examine inflation in the Brans-Dicke theory and find that scale invariance is restored in the course of spontaneous symmetry breaking.     

Extended relativity: Mass and the fifth dimension

A self-consistent relativistic formalism is presented which postulates that mass is the eigenvalue of a fifth momentum operator component. Lorentz covariance is generalized so that a systematic program for covariant wave equations can be formed. The fifth dimension is identified with cosmic time, resulting in a bias toward matter over antimatter for the universe. A distinction between μ ande also seems possible through the space-time extension.     

Scale invariance,unimodular gravity and dark energy

We demonstrate that the combination of the ideas of unimodular gravity, scale invariance, and the existence of an exactly massless dilaton leads to the evolution of the universe supported by present observations: inflation in the past, followed by the radiation and matter dominated stages and accelerated expansion at present. All mass scales in this type of theories come from one and the same source.     

Scale invariance and superfluid turbulence

We construct a Schroedinger field theory invariant under local spatial scaling. It is shown to provide an effective theory of superfluid turbulence by deriving, analytically, the observed Kolmogorov 5/3 law and to lead to a Biot–Savart interaction between the observed filament excitations of the system as well.     

Quasi invariants and generalized killing vectors

The connection between quasi-invariants (invariants of a Hamiltonian system defined only on a single constant energy hypersurface) and generalized Killing vector fields associated with the corresponding Jacobi metric is investigated. The results are used to deduce a generalised form of the classical Whittaker problem in two degrees of freedom.     

Scale invariance and spontaneous symmetry breaking

Spontaneous breaking of gauge symmetries is studied in theories with nonlinearly realized scale invariance. The classically sliding vacuum expectation values are fixed through quantum corrections. The anomaly of the dilatation current determines the vacuum energy density as well as the dilaton mass. The coupling of gravity to matter is modified in such a way that the cosmological constant vanishes.     

Gradient conformal killing vectors and exact solutions

We establish a connection between conformally related Einstein spaces and conformai killing vectors (CKV). We begin with the conformal map and prove that (a) under the conformal mapping¯g _ik=^–2g_ik, the necessary and sufficient condition for the tracefree part of the Ricci tensor (S _ik=R_ik–(R/4)g _ik) to remain invariant is that _i is a CKV ofg _ik, and (b) the most general form for for conformally flat Einstein space, which is the de Sitter space, is composed of three terms each of which alone represents a flat space. The existence of gradient CKV (GCKV) is examined in relation to vacuum and perfect fluid spacetimes.     

Scale invariance analysis of the premature ECG signals

The multifractal detrended fluctuation analysis and detrending moving average algorithm were introduced in detail and applied to the study of the multifractal characteristics of the normal signals, the atrial premature beat (APB) signals and the premature ventricular contraction (PVC) signals. By analyzing the generalized Hurst exponents, Renyi exponents and multifractal spectrum and comparing the relation of h∼h(q)$h\sim h\left(q\right)$ for original signals and their shuffled time series, the result indicated that the three signals have multifractality and present long-range correlation in a certain range. According to the mean value of Δα$\Delta \alpha$, we found that the strength of the multifractality is varying. The PVC signals is the strongest, and the Normal signals is the weakest. It is useful for clinical practice of medicine to distinguish APB signals with PVC signals.     

Scale invariance in percolation theory and fractals

The properties of the similarity transformation in percolation theory in the complex plane of the percolation probability are studied. It is shown that the percolation problem on a two-dimensional square lattice reduces to the Mandelbrot transformation, leading to a fractal behavior of the percolation probability in the complex plane. The hierarchical chains of impedances, reducing to a nonlinear mapping of the impedance space onto itself, are studied. An infinite continuation of the procedure leads to a fixed point. It is shown that the number of steps required to reach a neighborhood of this point has a fractal distribution. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 6, 427–432 (25 September 1996)     

Scale invariance and lack of self-averaging in fragmentation

We derive exact statistical properties of a recursive fragmentation process. We show that introducing a fragmentation probability 0相似文献   

Scale invariance in dynamic fragmentation of quartz

In the work, we studied statistical regularities in fragmentation of cylindrical quartz specimens under dynamic loading. The original equipment used in the study ensured integrity of the specimens after impact for the determination of fragment size distribution (spatial scaling) and allowed recording fractoluminescence pulses at newly formed fracture surfaces for the determination of pulse spacing distribution (temporal scaling). The results of experimental data processing show that the size distributions for both the spatial parameter (fragment size) and the temporal parameter (fractoluminescence pulse spacing) are described by a power function. This enables us to refer dynamic fragmentation of quartz to phenomena exhibiting self-organized criticality.     

Scale and conformal invariance in quantum field theory

We study the relation between invariance under rigid and local changes of length scale. In two dimensions, we complete an argument of Zamolodchikov showing that the rigid invariance implies the local under broad conditions. In three or more dimensions we are unable to find either a general proof or a counterexample, but we find some new conformally invariant systems.     

Scale invariance and viscosity of a two-dimensional Fermi gas

We investigate collective excitations of a harmonically trapped two-dimensional Fermi gas from the collisionless (zero sound) to the hydrodynamic (first sound) regime. The breathing mode, which is sensitive to the equation of state, is observed with an undamped amplitude at a frequency 2 times the dipole mode frequency for a large range of interaction strengths and different temperatures. This provides evidence for a dynamical SO(2,1) scaling symmetry of the two-dimensional Fermi gas. Moreover, we investigate the quadrupole mode to measure the shear viscosity of the two-dimensional gas and study its temperature dependence.     

Scale invariance in plastic flow of crystalline solids

From the traditional viewpoint of continuum plasticity, plastic deformation of crystalline solids is, at least in the absence of so-called plastic instabilities, envisaged as a smooth and quasi-laminar flow process. Recent theoretical and experimental investigations, however, demonstrate that crystal plasticity is characterized by large intrinsic spatio-temporal fluctuations with scale-invariant characteristics: In time, deformation proceeds through intermittent bursts with power-law size distributions; in space, deformation patterns and deformation-induced surface morphology are characterized by long-range correlations, self-similarity and/or self-affine roughness. We discuss this scale-invariant behaviour in terms of robust scaling associated with a non-equilibrium critical point (‘yielding transition’).

Scale invariance in plastic flow of crystalline solids

All authors

Michael Zaiser

https://doi.org/10.1080/00018730600583514

Published online:

01 February 2007

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