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Register a new userThomas-Wigner rotation and Thomas precession: actualized approach
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Added by
Alexander Kholmetskii
Publication date
2014
fields
Physics
and research's language is
English
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We show that the explanation of Thomas-Wigner rotation (TWR) and Thomas precession (TP) in the framework of special theory of relativity (STR) contains a number of points of inconsistency, in particular, with respect to physical interpretation of the Einstein velocity composition law in successive space-time transformations. In addition, we show that the common interpretation of TP falls into conflict with the causality principle. In order to eliminate such a conflict, we suggest considering the velocity parameter, entering into expression for the frequency of TP, as being always related to a rotation-free Lorentz transformation. Such an assumption (which actually resolves any causal paradoxes with respect to TP), comes however to be in contradiction with the spirit of STR. The results obtained are discussed.
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We review why the Thomas rotation is a crucial facet of special relativity, that is just as fundamental, and just as unintuitive and paradoxical, as such traditional effects as length contraction, time dilation, and the ambiguity of simultaneity. We show how this phenomenon can be quite naturally introduced and investigated in the context of a typical introductory course on special relativity, in a way that is appropriate for, and completely accessible to, undergraduate students. We also demonstrate, in a more advanced section aimed at the graduate student studying the Dirac equation and relativistic quantum field theory, that careful consideration of the Thomas rotation will become vital as modern experiments in particle physics continue to move from unpolarized to polarized cross-sections.
Kinetic theory of Dirac fermions is studied within the matrix valued differential forms method. It is based on the symplectic form derived by employing the semiclassical wave packet build of the positive energy solutions of the Dirac equation. A satisfactory definition of the distribution matrix elements imposes to work in the basis where the helicity is diagonal which is also needed to attain the massless limit. We show that the kinematic Thomas precession correction can be studied straightforwardly within this approach. It contributes on an equal footing with the Berry gauge fields. In fact in equations of motion it eliminates the terms arising from the Berry gauge fields.
We provide a transformation formula of non-commutative Donaldson-Thomas invariants under a composition of mutations. Consequently, we get a description of a composition of cluster transformations in terms of quiver Grassmannians. As an application, we give an alternative proof of Fomin-Zelevinskys conjectures on $F$-polynomials and $g$-vectors.
The Thomas-Fermi approach to galaxy structure determines selfconsistently the fermionic warm dark matter (WDM) gravitational potential given the distribution function f(E). This framework is appropriate for macroscopic quantum systems: neutron stars, white dwarfs and WDM galaxies. Compact dwarf galaxies follow from the quantum degenerate regime, while dilute and large galaxies from the classical Boltzmann regime. We find analytic scaling relations for the main galaxy magnitudes as halo radius r_h, mass M_h and phase space density. The observational data for a large variety of galaxies are all well reproduced by these theoretical scaling relations. For the compact dwarfs, our results show small deviations from the scaling due to quantum macroscopic effects. We contrast the theoretical curves for the circular velocities and density profiles with the observational ones. All these results are independent of any WDM particle physics model, they only follow from the gravity interaction of the WDM particles and their fermionic nature. The theory rotation and density curves reproduce very well for r < r_h the observations of 10 different and independent sets of data for galaxy masses from 5x10^9 Msun till 5x10^{11} Msun. Our normalized circular velocity curves turn to be universal functions of r/r_h for all galaxies and reproduce very well the observational curves for r < r_h. Conclusion: the Thomas-Fermi approach correctly describes the galaxy structures (Abridged).
Given the Thomas-Fermi equation sqrt(x)phi=phi*(3/2), this paper changes first the dependent variable by defining y(x)=sqrt(x phi(x)). The boundary conditions require that y(x) must vanish at the origin as sqrt(x), whereas it has a fall-off behaviour at infinity proportional to the power (1/2)(1-chi) of the independent variable x, chi being a positive number. Such boundary conditions lead to a 1-parameter family of approximate solutions in the form sqrt(x) times a ratio of finite linear combinations of integer and half-odd powers of x. If chi is set equal to 3, in order to agree exactly with the asymptotic solution of Sommerfeld, explicit forms of the approximate solution are obtained for all values of x. They agree exactly with the Majorana solution at small x, and remain very close to the numerical solution for all values of x. Remarkably, without making any use of series, our approximate solutions achieve a smooth transition from small-x to large-x behaviour. Eventually, the generalized Thomas-Fermi equation that includes relativistic, non-extensive and thermal effects is studied, finding approximate solutions at small and large x for small or finite values of the physical parameters in this equation.
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