V. E. Shapiro

so the terms

and I of (3) are. But the evolution of

by Equation (3) and these terms are in effect

invariant only in the case of the ideal where I comes down to a Poisson bracket

with

a func tion of

z, t, thus making the sum

a generalized potential (a dressed Hamiltonian) governing the system.

Obviously it is not the patches beyond this ideal that are rare in reality, it is the patches of ideal. Beyond this

ideal, the terms

, I and

are hardly invariant and cause persistent flows already in the very

stati ona r y c o ndi tions of energ y co nse r va ti o n, wher e Equa tion (2) is a utonomo us ( see [2]). Such au t onomy offer s

a clearer view of the vortex energy realm versus the quantum postulates by (1), (2). Below we shall concentrate

on this case.

The realm of the classical ideal in point is then the domain of stationary vortex-free phase fluid flows of

density

with

and H bound from below for stability. This necessitates rigid

constraints on I: the irreversible drift and diffusion must then be in detailed balance-tend to zero in self-

compensation for each of all n degrees of system freedom. As diffusion means sources, the irreversible drift is to

be sinks, dissipatio n, cause relaxation to the state o f ideal. In conditions of e nergy conservation be yond detailed

balance, the irreversible drift and diffusion inevitably ca use persistent (not entrai ned in stationary chaos on the

average) phase fluid flows.

An e xotic exa mple of s uch i mb alanc e is t he s yst ems with no n-holonomic constraints that do no work on each

of n degrees of systems freedom; it can be viewed as a limit of zeroth diffusio n and utmost stron g frictio n, i.e.,

zero-time r ela xa t io n ca u si ng t he non -holonomy. Both this limit and a general imbalance between the irreversible

dri ft and d iffu sio n of fi nite rat es, whic h do work o n the syst e m, but in a wa y of ze ro th tot al work o n the syste m,

are within the realm of vortex energy. This encompasses all conditions of conserved vortex energy realm with

its persistent flows of phase f luid within autonomic Equation (3).

Let us now juxtapose the outlined realm of vortex energy with that of density distribution

for the

quantum-ideal proble m statem e nt.

4. Fundamental Contradictions

The quantum theory is indissolubly related to the uncertainty principle, hence, its domain is to be beyond the

purely dynamical non-holonomic systems of zeroth diffusion. The quantum phenomena to be observed as

recurrent against ambient chaos suppose conditions of relaxation, stability and ergodicity, and all that, if this

were the case, would be only in the vortex energy realm beyond the ideal nonholonomy. But there is no niche

for the systems having discrete energy spectrum of states in the vortex energy realm of diffusion and relaxation

of finite rates: finite rates inevitably cause spectra of finite line-widths.

So, the inherent hallmarks that together constitute the notion of quantum ideal have no niche in whatever

autonomic conditions in question. It means that the quantum ideal construction fails as a rigorously defined

bridge to energy conservation observations, appears without footing beyond the classic ideal. These things are

verities when treating the states of quantum systems as a mixture of pure states each given a statistical weight in

the sense used in classics, i.e., reso rting to a semi-cla ssical appr oach to the irreversible kinetics and the concept

of entropy.

Along with the incompa tibility set-forth abo ve, the followi ng basic inconsiste ncy takes place. Its crucial point

is the density distr ibu tion

of the quantum ideal, its pure states, is independent of t in autono mic co nditions.

The solutions to autonomic (3) in detailed balance are also independent of t as then

, and it is

assumed so in the semi-classics modeling in question. However, the conditions of energy conservation of

autonomic Equation (3) beyond detailed balance is another story and makes a difference.

Thereat, it is not just that

in (3) links to imbalance, it is the variations (correlations) between

densities of phase fl uid at different t’s is a matter of princip le, signifie s persistent flo ws of fluid as observable in

the

space of systems in the stationary realm of vortex energy. So, while t-dependent phenomena in

stationary outer conditions are usually referred to non-equilibria, this is not the case of vortex energy realm,

where it is just the equilibriu m states of motio n with conse rved energy [2]. The persistent flows in point can be

fast and stable depending on conditions of self-sustained imbalance. It is largely admitted of a test.

The same should be expected from the quantum phenomena, including its ideal by (1) and (2) for

independent of t. However, since

reduces then to a sum

(5)