Transition of the Heisenberg equation for h?o to the dynamic equation of a monoatomic ideal gas and quantization of relativistic hydrodynamics

Abstract

When expressions of the type ~leiS/h and q~2 e-iS/h are inserted (such expressions are used in the Wentzel-Kramers -Brillouin (WKB) technique for solving the linear Klein-Gordon- Foek equation), only two particular waves are obtained. Ill the case of linear superpositions of these waves, a solution can be constructed which satisfies any initial condition of the Cauchy type. In the nonlinear case, the superposition principle is not applicable, and the substitution of qheiS/h can result in particular solutions only. However, we note that both waves qheiS/h and c22eiS/h are "accommodated" by a single action function S.~ i.e., the general solution depends upon a single phase. As we will see, this remark is also applicable to the case of the nonlinear Heisenberg equation. Therefore, we will use the Bogolyubov-Krylov averaging technique [11. In the following, we will consider the general equation l~ 2 [] ~ + k~t'(,~) = o. (a) We obtain the Heisenberg equation when F(u)=u 3. In principle, we will follow the scheme developed by S~ Ao Lomov [2] (though this scheme was developed for completely different problems), but use the modification which is close to the WKB technique and the methods outlined in [3]. We introduce an auxiliary parameter T, i.e., we consider a solution to Eq. (i) which depends upon an additional parameter 7-. In other words, we will consider a solution u(x, t, ~-) as a vector u( t) (with x=x I, x 2, x3) of the space ~ ix] of whole bound functions of T: u(x, t) ~ ,(~, t, "0. We consider the operator~['c] in the space ~=iO/O'r. In analogy to the usual WKB technique, we will look for a solution to Eq. (1) in the form J~ S(x, t) u (x, t) = e ,~ (x, t), (2) where ~o(x, t) denotes a function with an infinite number of derivatives with respect to x and t in ~q[,]. We note that