A.P. Kirilyuk
where coefficients ci should be found by state-function matching at the boundary where
effective interaction vanishes. The observed (generalised) density, ρ(ξ,Q) , is obtained
as state-function squared modulus, ρ(ξ,Q)=| Ψ(ξ,Q) ∣2 (for “wave-like” complexity
levels), or as state-function itself, ρ(ξ,Q) = Ψ (ξ,Q) (for “particle-like” levels) [9].
Although the EP expression of a problem, eqs. (8)-(12), is formally equivalent to
its initial version, eqs. (1), (2), (5), only the former reveals, due to its “dynamically rich”
structure, the essential features designated as dynamic multivaluedness (or redundance)
and entanglement and remaining hidden in the conventional formalism and especially
its perturbative form of “exact” (or closed) solutions. Dynamic multivaluedness appears
as redundant number of locally complete, and therefore incompatible, but equally real
problem solutions, called realisations, while dynamic entanglement describes the
related “cohesion” between interacting components within each realisation, expressing
system “nonseparability”. Because of equal reality and incompatibility of realisations,
the system is forced, by the driving interaction itself, to permanently change them in a
causally random order, forming each time a new version of component entanglement.
The total number of eigen-solutions can be estimated by the maximum power of the
characteristic equation for eq. (8). IfNξandNQ are the numbers of terms in the sums
over i and n in eq. (9b), equal to the numbers of system components (N) and their
internal states, then the eigenvalue number is Nmax = Nξ(NξNQ + 1)=(Nξ)2NQ +Nξ,
which gives the Nξ -fold redundance of usual “complete” set of NξNQ eigen-solutions
of eqs. (5) plus an additional, “incomplete” set of Nξ eigen-solutions. The number of
“regular” realisations is Nи = Nξ = N, whereas the truncated set of solutions forms a
specific, “intermediate” realisation that plays the role of transitional state during chaotic
system jumps between “regular” realisations and provides thus the universal, causally
complete extension of the quantum wavefunction and classical distribution function [9-
13]. Note that dynamic multivaluedness is obtained only in the unreduced EP version
(starting from the genuine quantum chaos description [14,15]), whereas practically all
scholar applications of this well-known approach (see e.g. [16]) resort to its perturbative
reduction that kills inevitably all manifestations of complex (chaotic) dynamics and is
equivalent to the dynamically single-valued, effectively zero-dimensional (point-like)
model of reality, containing only one, “averaged” system realisation (or projection).
The discovered multivaluedness of the unreduced solution and the ensuing
chaoticity of unceasing realisation change are expressed by the truly complete general
solution of a problem presenting the observed density ρ(ξ,Q) (or a similar quantity) as
the causally probabilistic sum of individual realisation densities, {ρr(ξ,Q)}:
Nи^ +
ρ(ξ,Q )=∑+ Pr (ξ, Q ), (13)
r=1
where summation over r includes all observed realisations, while the sign + designates
the causally probabilistic sum. The dynamically probabilistic general solution of eq.
(13) is accompanied by the dynamically derived values of realisation probabilities αr :
αr=
Nr
nи
(
Nr = 1,..., Nи;
ї
∑ Nr = Nи
r J
, ∑αr=1
r
(14)
where Nr is the number of elementary realisations grouped in the r-th “compound” re-
alisation, but remaining unresolved in a general case. It is important that eqs. (13), (14)