Relativistic Hamilton-Jacobi Equation

5 Reaction-Transport Fronts Propagating into Unstable States 

The analogy with the relativistic mechanics allows us to derive the explicit expression for the function G(x,t) and thereby to find the reaction Ifont position and its velocity. 

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In this section we consider the problem of front propagation for the three-dimensional reaction-telegraph equation involving KPP kinetics. The goal is to derive the equation governing the evolution of the front in the long-time large-scale limit as in the previous sections and to show that this equation is identical in form to the relativistic Hamilton-Jacobi equation. 

Section VI shows the power of the modulus-phase formalism and is included in this chapter partly for methodological purposes. In this formalism, the equations of continuity and the Hamilton-Jacobi equations can be naturally derived in both the nonrelativistic and the relativistic (Dirac) theories of the electron. It is shown that in the four-component (spinor) theory of electrons, the two exha components in the spinor wave function will have only a minor effect on the topological phase, provided certain conditions are met (nearly nonrelativistic velocities and external fields that are not excessively large). 

The aim of this section is to show how the modulus-phase formulation, which is the keytone of our chapter, leads very directly to the equation of continuity and to the Hamilton-Jacobi equation. These equations have formed the basic building blocks in Bohm s formulation of non-relativistic quantum mechanics [318]. We begin with the nonrelativistic case, for which the simplicity of the derivation has... 

The terms before the square brackets give the nonrelativistic part of the Hamilton-Jacobi equation and the continuity equation shown in Eqs. (142) and (141), while the term with the squaie brackets contribute relativistic corrections. All terms from are of the nonmixing type between components. There are further relativistic terms, to which we now turn. 

This part of our chapter has shown that the use of the two variables, moduli and phases, leads in a direct way to the derivation of the continuity and Hamilton-Jacobi equations for both scalar and spinor wave functions. For the latter case, we show that the differential equations for each spinor component are (in the nearly nomelativistic limit) approximately decoupled. Because of this decoupling (mutual independence) it appears that the reciprocal relations between phases and moduli derived in Section III hold to a good approximation for each spinor component separately, too. For velocities and electromagnetic field strengths that ate nomrally below the relativistic scale, the Berry phase obtained from the Schrddinger equation (for scalar fields) will not be altered by consideration of the Dirac equation. 

The first is a relativistic Hamilton-Jacobi (HJ) equation for a particle with variable rest mass [101]... 

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