Classical relativistic

The functions h (x, p) are the classical relativistic Hamiltonians of spinless particles with positive and negative energies, respectively. 

As is well known (Chirikov, 1979 Izrailev, 1990), the phase-space evolution of the norelativistic classical kicked rotor is described by nonrelativistic standard map. The analysis of this map shows that the motion of the nonrelativistic kicked rotor is accompanied by unlimited diffusion in the energy and momentum. However, this diffusion is suppressed in the quantum case (Casati et.al., 1979 Izrailev, 1990). Such a suppression of diffusive growth of the energy can be observed when one considers the (classical) relativistic extention of the classical standard map (Nomura et.al., 1992) which was obtained recently by considering the motion of the relativistic electron in the field of an electrostatic wave packet. The relativistic generalization of the standard map is obtained recently (Nomura et.al., 1992)... 

It should be noted that there is a limited number of works on classical relativistic dynamical chaos (Chernikov et.al., 1989 Drake and et.al., 1996 Matrasulov, 2001). However, the study of the relativistic systems is important both from fundamental as well as from practical viewpoints. Such systems as electrons accelerating in laser-plasma accelerators (Mora, 1993), heavy and superheavy atoms (Matrasulov, 2001) and many other systems in nuclear and particle physics are essentially relativistic systems which can exhibit chaotic dynamics and need to be treated by taking into account relativistic dynamics. Besides that interaction with magnetic field can also strengthen the role of the relativistic effects since the electron gains additional velocity in a magnetic field. 

On the classical relativistic level, the starting point is the Einstein equation... 

This is the classical relativistic expression for the interaction of an electron or proton with the classical electromagnetic held. The quantized version of Eq. (275) is the van der Waerden equation [1] as described by Sakurai [68] in his Eq. (3.24). The RFR term in relativistic classical physics is contained within the term e2fit1 1 fit2 1, a result that can be demonstrated by expanding this term as follows... 

Noether s theorem will be proved here for a classical relativistic theory defined by a generic field , which may have spinor or tensor indices. The Lagrangian density (, 9/x) is assumed to be Lorentz invariant and to depend only on scalar forms defined by spinor or tensor fields. It is assumed that coordinate displacements are described by Jacobi s theorem S(d4x) = d4x 9/xgeneral variation of the action integral, evaluated over a closed space-time region 2, is... 

Salamin and Faisal (1996,1997) analysed harmonic generation and the ponderomotive scattering of electrons in intense laser fields based on a classical relativistic... 

The results of this classical relativistic calculations (Salamin and Faisal 1997) are presented in Figure 1.5 and compared with experimentally deduced ponderomotive scattering angles 0 (Meyerhofer et al. 19% Moore et al. 1995) as a function of the escape kinetic energy AT of the electron. 

An equation of state (EOS) describes relations between these properties, e.g. pressure, density and temperature. The form of the equation of state depends on whether the fluid particles can be treated as classical or non-classical, relativistic or non-relativistic. 

Fig. 16. A map of density-temperature space illustrating the different regimes for the equation of state, including the classical, relativistic, and degenerate limits...

The result about the velocity obtained above is not what we expect from classical relativistic kinematics. In classical mechanics, the connection between the kinietic energy E, the momentum p, and the velocity v can be described by the formula... 

The various terms (99) can be interpreted physically. Expanding the classical relativistic kinetic energy into powers of 1 /c gives the power series... 

If Eq. (93a) could be solved with Eq. (93b), the solution to the Dirac equation can be obtained exactly. However, Eq. (93a) has the total and potential energies in the denominator, and an appropriate approximation is needed. In our strategy, E — V in the denominator is replaced by the classical relativistic kinetic energy (relativistic substitutive correction)... 

The classical relativistic Lagrange function for one particle (electron) due to the kinetic energy, the vector and scalar potentials is... 

The second volume of the Landau-Lifshitz series on theoretical physics continues directly after volume I and covers the classical theory of fields. It starts with an introduction of Einstein s principle of relativity and a discussion of the special theory of relativity for mechanics. It follows a rather complete presentation of classical electrodynamics including radiation phenomena and scattering of waves of different energy. It concludes with an introduction of gravitational fields, the theory of general relativity and classical relativistic cosmology. 

In the last chapter the basic framework of classical nonrelativistic mechanics has been developed. This theory crucially relies on the Galilean principle of relativity (cf. section 2.1.2), which does not match experimental results for high velocities and therefore has to be replaced by the more general relativity principle of Einstein. It will directly lead to classical relativistic mechanics and electrodynamics, where again the term classical is used to distinguish this theory from the corresponding relativistic quantum theory to be presented in later chapters. 

The classical relativistic energy of a free particle moving in space is given as... 

The first term, Hj, is the spin-orbit (one electron term) and spin-other-orbit (two electron term) couplings, which are the topic of the following subsection. The second term Hf contains the spin-spin coupling term and Fermi contact interaction. Both the Hj and f/ can lift degeneracy in multiplets. The parameter Hf is the Dirac correction term for electron spin and Ff is the classical relativistic correction to the interaction between electrons due to retardation of the electromagnetic field produced by an electron. The parameter H is the so-called mass-velocity effect, due to the variation of electron mass with velocity. Finally, H is the effect of external electric and magnetic fields. 

The purpose of this chapter is to introduce the Dirac equation, which will provide us with a basis for developing the relativistic quantum mechanics of electronic systems. Thus far we have reviewed some basic features of the classical relativistic theory, which is the foundation of relativistic quantum theory. As in the nonrelativistic case, quantum mechanical equations may be obtained from the classical relativistic particle equations by use of the correspondence principle, where we replace classical variables by operators. Of particular interest are the substitutions... 

In going from a classical relativistic description to relativistic quantum mechanics, we require that the equations obtained are invariant under Lorentz transformations. Other basic requirements, such as gauge invariance, must also apply to the equations of relativistic quantum mechanics. 

We now turn to the quantization of the classical relativistic Hamiltonian, and in particular to its representation in the Dirac equation. Quantization of the classical relativistic Hamiltonian has been treated in detail in many texts, such as Rose (1961),... 

While this provides us with an equation for the relativistic electron, the u and matrices arose from the mathematical treatment, and only indirectly from the physics. It would be nice if we could also give these quantities a physical interpretation. In order to find some classical operator or quantity corresponding to a and p, we compare the Dirac Hamiltonian with the classical relativistic Hamiltonian. For this purpose we use the classical relativistic expression for the energy from (2.60) for a field-free system. 

From the discussion of classical relativistic electromagnetic interactions in chapter 3, we know that the Coulomb interaction between charged particles is not Lorentz invariant, and that we need to take into account the finite transmission speed of the signal between particles (the retardation). If we make the replacement u = ca in (3.84), we get an interaction for low velocities that might well be suitable for relativistic quantum chemistry ... 

The first of the relativistic correction terms is called the mass-velocity operator. If we expand the square root operator in the classical relativistic Hamiltonian for a free particle, we find... 

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