Non-zero rest mass

In 1923 de Broglie made the bold suggestion that matter, like light, has a dual nature in that it sometimes behaves like particles and sometimes like waves. He suggested that material (i.e., non-zero-rest mass) particles with a momentum p = mv should have wave properties and a corresponding wavelength given by... 

Finally, we note that the following "classical consistency relations hold. For particles with a non-zero rest mass, the energy law gives, using Eqs. (32) and (33)... 

Neutrinos no longer count as "dark matter candidates" since they are known experimentally to oscillate among flavors and so to have non-zero rest masses. The implied masses are such that the ones we know and love contribute less than Qv = 0.01 (ApOO, Sect. 12.4.2). Observed large scale structure and models for its formation are happiest with Qv = 0.006 or thereabouts. (57)... 

Assuming that the gravitational source is a spherical black hole-like object at rest, i.e. that the angular momentum of the non-zero rest-mass particle is a constant of motion one obtains... 

It can be seen from eq. (1.6.7) that it is impossible to accelerate a body with a non-zero resting mass by a finite force F to a speed equal to the fight speed in vacuum. In order to explain this circumstance Einstein had to introduce the speed dependence of the particle mass (1.62). 

As a result, quasi-particles in graphene exhibit the linear dispersion relation E = hvp, where vp is the Fermi velocity ( 10 m/s), as if they were massless relativistic particles. Thus, graphene s quasiparticles behave differently from those in conventional metals and semiconductors, where the energy spectrum can be approximated by a parabolic dispersion relation. Electron transport in all known condensed-matter systems is described by the (non-relativistic) Schrodinger equation and relativistic effects are usually negligible. In contrast, the electrons of graphene are described by the (relativistic) Dirac equation, i.e. they mimic relativistic charged particles with zero rest mass and constant velocity [10]. 

The feed vector F0 is similar to F, and contains mass fractions, which are added to the circuit after eveiy transition. In the case shown in Fig. 2a, it contains only the first non-zero vector of the three, which is the fraction size distribution /o in the feed to the circuit. The rest two vectors are the zero vectors of the same size. 

From the point of view of the parhcle-wave dualism, the electron that has the rest mass and charge can move in a certain direction with a certain velocity. A plane wave xp [x, t) = sin(kx — art) extends to infinity in both spatial direchons, so it cannot represent a parhcle whose wave function is non-zero in a limited region of space. 

One note that Eq. 1.85 is decoherence protected for the case mo = 0 since dJiConj is non-diagonal with the determinant equal to zero for all values of r. For mo 0 the only singular point is at r = Rls. We also stress that the operator Es replaces the classical notion of rest-mass. Thus we obtain an invariant rf(H/onj = —c d, which is zero (null-vector) for photons, and the line element expression (in the spherical case)... 

To align the zero of the relativistic energy scale with that of the non-relativistic one, the rest mass of the electron is subtracted through the substitution... 

The Zimm model rests upon the Langevin equation for over-damped motion of the monomers, i.e., it applies for times larger than the Brownian time scale Tb 2> OTm/where is Stokes friction coefficient [12]. On such time scales, velocity correlation functions have decayed to zero and the monomer momenta are in equilibrium with the solvent Moreover, hydrodynamic interactions between the various parts of the polymer are assumed to propagate instantaneously. This is not the case in our simulations. First of all, the monomer inertia term is taken into account, which implies non-zero velocity autocorrelation functions. Secondly, the hydrodynamic interactions build up gradually. The center-of-mass velocity autocorrelation function displayed in Fig. 9 reflects these aspects. The correlation function exhibits a long-time tail, which decays as (vcm(t)vcm(O)) on larger time scales. The... 

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