LOCALIZATION AND PATTERN FORMATION

IN QUANTUM PHYSICS.

I. PHENOMENA OF LOCALIZATION

Antonina N. Fedorova, Michael G. Zeitlin

IPME RAS, St. Petersburg, V.O. Bolshoj pr., 61, 199178,

Russia

e-mail: zeitlin@math.ipme.ru

e-mail: anton@math.ipme.ru

http://www.ipme.ru/zeitlin.html

http://www.ipme.nw.ru/zeitlin.html

Submitted to Proc. of SPIE Meeting, The Nature of Light:

What is a Photon?

Optics & Photonics, SP200, San Diego, CA, July-August, 2005

Abstract

In these two related parts we present a set of methods, an-alytical and numerical, which can illuminate the behaviour ofquantum system, especially in the complex systems, e.g., where

the standard ”coherent-states” approach cannot be applied. Thekey points demonstrating advantages of this approach are: (i)

effects of localization of possible quantum states, more properthan ”gaussian-like states”; (ii) effects of non-perturbative mul-

tiscales which cannot be calculated by means of perturbation ap-proaches; (iii) effects of formation of complex quantum patterns

from localized modes or classification and possible control of thefull zoo of quantum states, including (meta) stable localized pat-terns (waveletons). In this first part we consider the applica-

tions of numerical-analytical technique based on local nonlinearharmonic analysis to quantum/quasiclassical description of non-

linear (polynomial/rational) dynamical problems which appearin many areas of physics. We’ll consider calculations of Wigner

functions as the solution of Wigner-Moyal-von Neumann equa-tion(s) corresponding to polynomial Hamiltonians. Modeling

demonstrates the appearance of (meta) stable patterns gener-ated by high-localized (coherent) structures or entangled/chaoticbehaviour. We can control the type of behaviour on the level of

reduced algebraical variational system. At the end we presentedthe qualitative definition of the Quantum Objects in compari-

son with their Classical Counterparts, which natural domain ofdefinition is the category of multiscale/multiresolution decompo-

sitions according to the action of internal/hidden symmetry ofthe proper realization of scales of functional spaces (the multi-scale decompositions of the scales of Hilbert spaces of states). It

gives rational natural explanation of such pure quantum effectsas “self-interaction” (self-interference) and instantaneous quan-

tum interaction (transmission of information).

keywords: localization, pattern formation, multiscales, mul-

tiresolution, waveletons, generic symmetry, Wigner-Moyal

dynamics

2

1 INTRODUCTION

It is obviously that the current (and the nearest future) level

of possible experiments in the area of quantum physics as a

whole and quantum computers as some quintessence and im-

portant region of applications, as well as the level of the phe-

nomenological modeling overtake the present level of math-

ematical/theoretical description [1]. Considering, for exam-

ple, the problem of description of possible/realizable states,

one hardly believes that plane waves or/and (squeezed) gaus-

sians/coherent states exhausted all needful things, demanded,

e.g., for full description/construction of the possible quantum

CPU-like devices. Complexity of the proper zoo of states, in-

cluding entangled/chaotic states, is still far from clear under-

standing which can provide practical controllable realization.

It seems that the first thing, one needs to do in this direction,

is related to a possible reasonable extension of understanding

of the quantum dynamics in a whole volume. One needs to

sketch up the underlying ingredients of the theory (spaces

of states, observables, measures, classes of smothness, quan-

tization set-up etc) in an attempt to provide the maximally

extendable but at the same time really calculable and realiz-

able description of the dynamics of quantum world. Generic

idea is very trivial: it is well known that the idea of “sym-

metry” is the key ingredient of any reasonable physical the-

ory from classical (in)finite dimensional (integrable) Hamil-

tonian dynamics to different sub-planckian models based on

strings/branes/orbifolds etc. During century kinematical,

dynamical and hidden symmetries played the key role in

our understanding of physical process. Roughly speaking,

the representation theory of underlying symmetry (classical

or quantum, groups or (bi)algebras, finite or infinite dimen-

3

sional, continuous or discrete) is a proper instrument for de-

scription of (orbital) dynamics. Starting point for us is a

possible model for (continuous) “qubit” with subsequent de-

scription of the whole zoo of possible realizable/controllable

states/patterns which may be useful from the point of view of

quantum experimentalists and/or engineers. The proper rep-

resentation theory is well known as “local nonlinear harmonic

analysis”, in particular case of simple underlying symmetry–

affine group–aka wavelet analysis. From our point of view

the advantages of such approach are as follows:

i) natural realization of localized states in any proper func-

tional relalization of (Hilbert) space of states.

ii) hidden symmetry of choosen realization of proper func-

tional model provides the (whole) spectrum of possible states

via the so-called multiresolution decomposition.

So, indeed, the hidden symmetry (non-abelian affine group

in the simplest case) of the space of states via proper rep-

resentation theory generates the physical spectrum and this

procedure depends on the choice of the functional realization

of the space of states. It explicitly demonstrates that the

structure and properties of the functional realization of the

space of states are the natural properties of physical world at

the same level of importance as a particular choice of Hamil-

tonian, or equation of motion, or action principle (variational

method).

At the next step we need to consider the consequences of

our choice i)-ii) for the algebra of observables. In this di-

rection one needs to mention the class of operators we are

interested in to present proper description for a class of max-

imally generalized but reasonable class of problems. It seems

that it must be the pseudodifferential operators, especially if

4

we underline that in the spirit of points i)-ii) above we need

to take Wigner-Weyl framework as generic for constructing

basic quantum equations of motions. It is obviously, that con-

sideration of symbols of operators insted of operators them-

selves is the starting point as for the mathematical theory of

pseudodifferential operators as for quantum dynamics formu-

lated in the language of Wigner-like equations. It should be

noted that in such picture we can naturally include the effects

of self-interaction on the way of construction and subsequent

analysis of nonlinear quantum models.

So, our consideration will be in the framework of (Nonlin-

ear) Pseudodifferential Dynamics (ΨDOD).

As a result of i)–ii), we’ll have:

iii) most sparse, almost diagonal, representation for wide

class of operators included in the set-up of the whole prob-

lems.

It’s possible by using the so-called Fast Wavelet Transform

representation for algebra of observables..

Then points i)–iii) provide us by

iv) natural (non-perturbative) multiscale decomposition for

all dynamical quantities, as states as observables.

The simplest and proper case we’ll have in Wigner-Weyl

representation.

Existence of such internal multiscales with different dy-

namics at each scale and transitions, interactions, and in-

termittency between scales demonstrates that quantum me-

chanics, nevertheless its linear structure, is really a serious

part of physics. It seems, that this underlying quantum com-

plexity is a result of transition by means of (still unclear)

procedure of quantization from complexity related to nonlin-

5

earity of classical counterpart to the rich pseudodifferential

(more exactly, microlocal) structure on the quantum side.

For the reasons, explained later, we prefer to divide all pos-

sible configurations related to possible solutions of our quan-

tum equation of motion (Wigner-like equations, mostly) into

two classes:

(a) standard solutions (b) controllable solutions.

The meaning of the latter will be explained in part II,

but, anyway, the whole zoo of solutions consists of possible

patterns, including very important ones from the point of

view of underlying physics:

v) localized modes (basis modes, eigenmodes) and con-

structed from them chaotic or/and entangled, decoherent (if

we change Wigner equation for (master) Lindblad one) pat-

terns.

It should be noted that these bases modes are (non) linear

in contrast with usual ones because they come from (non)

abelian basis group while the usual Fourier (commutative)

analysis starts from U(1) abelian modes (plane waves). They

are really “eigen” but in sense of decomposition of represen-

tation of the underlying hidden symmetry group which gen-

erates the multiresolution decomposition. Zoo of patterns is

built from these modes by means of variational procedures

(there is a lot of) more or less standard in mathematical

physics. It allows to control the convergence from one side

but, what is more important,

vi) to consider the problem of the control of patterns (types

of behaviour) on the level of reduced (variational) algebraical

equations.

We need to mention that it’s possible to change the sim-

6

plest generic group of hidden internal symmetry from the

affine (translation and dilations) to much more general, but,

in any case, this generic symmetry will produce the proper

natural high localized eigenmodes, as well as the decompo-

sition of the functional realization of space of states into the

proper orbits; and all that allows to compute dynamical con-

sequence of this procedure, i.e. pattern formation, and, as

a result, to classify the whole spectrum of proper states.

For practical reasons controllable patterns (with prescribed

behaviour) are the most useful. We mention the so-called

waveleton-like pattern, as the most important one. We use

the following allusion in the space of words:

waveleton:=soliton⊔

wavelet

It means:

vii) waveleton ≈ (meta)stable localized (controllable) pat-

tern

To summarize, we may say that approach described below

allows us

viii) to solve wide classes of general ΨDOD problems, in-

cluding generic for quantum physics Wigner-like equations,

and

ix) to present the analytical/numerical realization for phys-

ically interesting patterns.

It should be noted the effectiveness of numerical realization

of this program (minimal complexity of calculations) as addi-

tional advantage. So, items i)-ix) point out all main features

of our approach [2]-[8].

7

2 MOTIVES

2.1 CLASS OF MODELS

Here we describe a class of problems which can be analysed

by methods described in Introduction. We start from indi-

vidual dynamics and finish by (non)-equilibrium ensembles.

All models are belonged to ΨDOD class and described by

finite or infinite (named hierarchies in such cases) system of

ΨDOD equations.

a). Individual classical/quantum mechanics (cM/qM) (lin-

ear/nonlinear; cM ⊂ qM),

(∗ - Quantization of) Polynomial Hamiltonians:

H(p, q, t) =∑

i.j

aij(t)piqj (1)

b). QFT-like models in framework of the second quantization

(dynamics in Fock spaces).

c). Classical (non) equilibrium ensembles via BBGKY Hi-

erarchy. (with reductions to different forms of Vlasov-

Maxwell/Poisson equations).

d). Wignerization of a): Wigner-Moyal-Weyl-von Neumann-

Lindblad.

e). Wignerization of c): Quantum (Non) Equilibrium Ensem-

bles.

Important remark, points a)-e), are considered in ΨDO

picture of (Non)Linear ΨDO Dynamics (surely, all qM ⊂

8

ΨDOD). Dynamical variables/observables are the symbols

of operators or functions; in case of ensembles, the main set

of dynamical variables consists of partitions (n-particle par-

tition functions).

2.2 EFEECTS (what we are interested in)

i). Hierarchy of internal/hidden scales (time, space, phase

space).

ii). Non-perturbative multiscales: from slow to fast contribu-

tions, from the coarser to the finer level of resolution/de-

composition.

iii). Coexistence of hierarchy of multiscale dynamics with tran-

sitions between scales.

iv). Realization of the key features of the complex quantum

world such as the existence of chaotic and/or entangled

states with possible destruction in “open/dissipative” regimes

due to interactions with quantum/classical environment

and transition to decoherent states.

At this level we may interpret the effect of misterious en-

tanglement or “quantum interaction” as a result of simple

interscale interaction or intermittency (with allusion to hy-

drodynamics) as the mixing of orbits generated by multireso-

lution representation of hidden underlying symmetry. Surely,

the concrete realization of such a symmetry is a natural phys-

ical property of the physical model as well as the space of

representation and its proper functional realization. So, in-

stantaneous interactions (or transmission of “quantum bits”

or “teleportation”) materialize not in the physical space-time

9

variety but in the space of representation of hidden symme-

try along the orbits/scales constructed by proper representa-

tions. Dynamical/kinematical principles of usual space-time

varieties, definitely, don’t cover kinematics of internal quan-

tum space of state or, in more weak formulation, we still

havn’t such explicit relations.

One additional important comment: as usually in mod-

ern physics, we have the hierarchy of underlying symmetries;

so our internal symmetry on functional realization of space

of states is really not more than kinematical, because much

more rich algebraic structure, related to operator Cuntz al-

gebra and quantum groups, is hidden inside. The proper

representations can generate much more interesting effects

than ones described above. We’ll consider it elsewhere but

mention here only how it can be realized by the existing

functorial maps between proper categories:

QMF −→ Loop groups −→ Cuntz operator algebra −→

Quantum Group structure,

where QMF are the so-called quadratic mirror filters

generating the realization of multiresolution decomposition/

representation in any functional space; loop group is well

known in many areas of physics, e.g. soliton theory, strings

etc, roughly speaking, its algebra coincides with Virasoro al-

gebra; Cuntz operator algebra is universal C∗ algebra gener-

ated by N elements with two relations between them; Quan-

tum group structure (bialgebra, Hopf algebra, etc) is well

known in many areas because of its universality. It should

be noted the appearance of natural Fock structure inside this

functorial sequence above with the creation operator realized

as some generalization of Cuntz-Toeplitz isometries. Surely,

all that can open a new vision of old problems and bring new

10

possibilities.

We finish this part by the following qualitative definitions

of objects, which description and understanding in different

physical models is our main goal in these two papers.

• By a localized state (localized mode) we mean the corre-

sponding (particular) solution (or generating mode) which

is localized in maximally small region of the phase (as in

c- as in q-case) space.

• By an entangled/chaotic pattern we mean some solution

(or asymptotics of solution) which has random-like dis-

tributed energy (or information) spectrum in a full do-

main of definition. In quantum case we need to con-

sider additional entangled-like patterns, roughly speak-

ing, which cannot be separated into pieces of sub-systems.

• By a localized pattern (waveleton) we mean (asymptoti-

cally) (meta) stable solution localized in relatively small

region of the whole phase space (or a domain of defi-

nition). In this case the energy is distributed during

some time (sufficiently large) between only a few local-

ized modes (from point 1). We believe, it is a good im-

age for plasma in a fusion state (energy confinement) or

model for quantum continuous “qubit” or a result of the

process of decoherence in open quantum system when the

full entangled state degenerates into localized (quasiclas-

sical) pattern.

11

2.3 Methods

i). Representation theory of internal/hidden/underlying sym-

metry, Kinematical, Dynamical, Hidden.

ii). Arena (space of representation): proper functional real-

ization of (Hilbert) space of states.

iii). Harmonic analysis on (non)abelian group of internal sym-

metry. Local/Nonlinear (non-abelian) Harmonic Analysis

(e.g, wavelet/gabor etc. analysis) instead of linear non-

localized U(1) Fourier analysis. Multiresolution (multi-

scale) representation. Dynamics on proper orbit/scale

(inside the whole hierarchy of multiscales) in functional

space. The key ingredients are the appearance of multi-

scales (orbits) and the existence of high-localized natural

(eigen)modes [10].

iv). Variational formulation (control of convergence, reduc-

tions to algebraic systems, control of type of behaviour).

3 SET-UP/FORMULATION

Let us consider the following ΨDOD dynamical problem

LjOpiΨ = 0, (2)

described by a finite or infinite number of equations which

include general classes of operators Opi such as differential,

integral, pseudodifferential etc

Surely, all Wigner-like equations/hierarchies are inside.

The main objects are:

12

i). (Hilbert) space of states, H = Ψ, with a proper func-

tional realization, e.g.,: L2, Sobolev, Schwartz, C0, Ck, ...

C∞, ...;

Definitely, L2(R2), L2(S2), L2(S1× S1), L2(S1× S1 nZn)

are different objects.

ii). Class of smoothness. The proper choice determines nat-

ural consideration of dynamics

with/without Chaos/Fractality property.

iii). Decompositions

Ψ ≈∑

i

aiei (3)

via high-localized bases (wavelet families, generic wavelet

packets etc), frames, atomic decomposition (Fig. 1) with

the following main properties: (exp) control of conver-

gence, maximal rate of convergence for any Ψ in any H.

iv). Observables/Operators (ODO, PDO, ΨDO, SIO,..., Mi-

crolocal analysis of Kashiwara-Shapira (with change from

functions to sheafs)) satisfy the main property – the ma-

trix representation in localized bases

< Ψ|Opi|Ψ > (4)

is maximum sparse:

D11 0 0 . . .

0 D22 0 . . .

0 0 D33 . . .... ... ... . . .

This almost diagonal structure is provided by the so-

called Fast Wavelet Transform.

13

Figure 1: Localized modes.

v). Measures: multifractal wavelet measures µi together

with the class of smothness are very important for anal-

ysis of complicated analytical behaviour.

vi). Variational/Pojection methods, from Galerkin to Rabi-

nowitz minimax, Floer (in symplectic case of Arnold-

Weinstein curves with preservation of Poisson/symplectic

structures). Main advantages are the reduction to alge-

braic systems, which provides a tool for the smart subse-

quent control of behaviour and control of convergence.

vii). Multiresolution or multiscale decomposition, MRA (or

wavelet microscope) consists of the understanding and

choosing of

1). (internal) symmetry structure, e.g.,

affine group = translations, dilations or many oth-

ers... and construction of

14

2). representation/action of this symmetry on H = Ψ.

As a result of such hidden coherence together with

using point v) we’ll have:

a). LOCALIZED BASES b). EXACT MULTI-

SCALE DECOMPOSITION with the best conver-

gence properties and real evaluation of the rate of

convergence via proper “multi-norms”

Figures 2, 3, 5, 6 demonstrate MRA decompositions for

one- and multi-kicks while Figures 4 and 7 present the

same for the case of the generic simple fractal model,

Riemann–Weierstrass function.

viii). Effectiveness of proper numerics: CPU-time, HDD-space,

minimal complexity of algorithms, and (Shannon) entropy

of calculations, are provided by points i)-vii) above.

ix). Quantization via ∗ star product or Deformation Quanti-

zation. We’ll use it on Wigner-Weyl-Moyal naive level.

Finally, such Variational-Multiscale approach based on points

i)-viii) provides us by the full ZOO of PATTERNS:

LOCALIZED, CHAOTIC/ENTANGLED, etc. In next Sec-

tion and in the Part II we’ll consider details for important

cases of Wigner-like equations.

4 PATTERN FORMATION IN WIGNER-MOYAL DYNAM-

ICS

So, in this Section shortly and in the Part II [9] in details,

we’ll consider the applications of numerical-analytical tech-

nique based on local nonlinear harmonic analysis (wavelet

analysis [10]) to (quasiclassical) quantization of nonlinear (poly-

nomial/rational) dynamical problems which appear in many

15

areas of physics [1]. Our starting point is the general point of

view of deformation quantization approach at least on naive

Moyal level [1]. So, for quantum calculations we need to find

an associative (but non-commutative) star product ∗ on the

space of formal power series in ~ with coefficients in the space

of smooth functions such that

f ∗ g = fg + ~f, g+∑

n≥2

~nBn(f, g). (5)

In this paper we consider calculations of Wigner functions

W (p, q, t) (WF) as the solution of some sort of Wigner equa-

tions [1]:

i~∂

∂tW = H ∗W −W ∗H (6)

corresponding to polynomial HamiltoniansH(p, q, t). In gen-

eral, equation (6) is nonlocal (pseudodifferential) for arbi-

trary Hamiltonians but in case of polynomial Hamiltonians

we have only a finite number of terms in the corresponding

series. For example, in stationary case, after Weyl-Wigner

mapping we have the following equation on WF in c-numbers

[1]:

( p2

2m+

~

2i

p

m

∂

∂q−

~2

8m

∂2

∂q2

)

W (p, q) +

U(

q −~

2i

∂

∂p

)

W (p, q) = εW (p, q) (7)

and after expanding the potential U into the Taylor series

we have two real finite partial differential equations. Our

approach is based on extension of our variational-wavelet ap-

proach [2]-[8]. Wavelet analysis is some set of mathemat-

ical methods, which gives us the possibility to work with

well-localized bases in functional spaces and gives maximum

16

sparse forms for the general type of operators (differential, in-

tegral, pseudodifferential) in such bases. We decompose the

solutions of (6), (7) according to underlying hidden scales as

W (t, q, p) =

∞⊕

i=ic

δiW (t, q, p) (8)

where value ic corresponds to the coarsest level of resolution

c in the full multiresolution decomposition

Vc ⊂ Vc+1 ⊂ Vc+2 ⊂ . . . . (9)

As a result, the solution of any Wigner-like equations has the

following multiscale/multiresolution decomposition via non-

linear high-localized eigenmodes, which corresponds to the

full multiresolution expansion in all underlying scales start-

ing from coarsest one (polynomial tensor bases, variational

machinery etc will be considered in details in Part II [9];

x = (x, y, px, py, ...)):

W (t, x) =∑

(i,j)∈Z2

aijUi ⊗ V j(t, x), (10)

V j(t) = V j,slowN (t) +

∑

l≥N

V jl (ωlt), ωl ∼ 2l

Ui(x) = Ui,slowM (x) +

∑

m≥M

Uim(kmx), km ∼ 2m

Representation (10) gives the expansion into the slow part

and fast oscillating parts for arbitrary N, M. So, we may

move from coarse scales of resolution to the finest one, along

the orbits generated by actions of hidden internal symmetry

group, to obtain more detailed information about our dynam-

ical process. In contrast with different approaches represen-

tation (10) doesn’t use perturbation technique or lineariza-

tion procedures. So, by using wavelet bases with their best

17

(phase) space/time localization properties we can describe

complex structures in quantum systems with complicated be-

haviour. Modeling demonstrates the formation of different

patterns, including chaotic/entangled from high-localized co-

herent structures. Our (nonlinear) eigenmodes are more re-

alistic for the modeling of nonlinear classical/quantum dy-

namical process than the corresponding linear gaussian-like

coherent states. Here we mention only the best convergence

properties of expansions based on wavelet packets, which re-

alize the minimal Shannon entropy property and exponential

control of convergence based on the norm introduced in [7],

[8]. Fig. 9 and Fig. 10, 11 show the contributions to WF

from localized eigenmodes in dimensions one and two, re-

spectively, while Fig. 11–14 gives the representations for full

solutions, constructed from the first 6 scales via MRA and

demonstrate stable localized pattern formation (waveleton)

and chaotic/entangled behaviour. For comparison, Fig. 8,

demonstrates direct modeling [10] of WF for the case of

three localized modes which are belonged to generic family

of wavelet packets.

Let us finish with some phenomenological description which

can be considered as an introduction to Part 2 [9] and at the

same time as an attempt of qualitative description of the

quantum dynamics as a whole and in comparison with its

classical counterpart. It’s possible to take for reminiscence

the famous Dirac’s phrase that “electron can interact only

itself via the process of quantum interference”.

LetG be the hidden/internal symmetry group on the spaces

of quantum states which generates via MRA (9) the multi-

scale/multiresolution representation for all dynamical quan-

tities, unified in object O(t), such as states, observables, par-

titions: Oi(t) = ψi(t), Opi(t),W in(t), where i is the proper

18

scale index. Then, the following (commutative) diagram rep-

resents the details of quantum life from the point of view of

representations of G on the choosen functional realization

which leads to decomposition of the whole quantum evolu-

tion into the proper orbits or scales corresponding to the

proper level of resolution. MorphismsW (t) describe Wigner-

Weyl evolution in the algebra of symbols, while the processes

of interactions with open World, such as the measurement

or decoherence, correspond to morphisms (or even functors)

which transform the infinite set of scales characterizing the

quantum object into finite ones, sometimes consisting of one

element (demolition/destructive measurement).

W (t)

Oi(t1) −→ Oj(t2)

↓ m(t1) ↓ m(t2)

W (t)

Oic(t1) −→ Ojc(t2)

where reduced morphisms W (t) correspond to (semi)classical

or quasiclassical evolution.

So, qualitatively,

Quantum Objects can be represented by an infinite or

enough large set of coexisting and interacting subsets like

(8), (9) (Figures 1-7 for some allusion).

while

Classical Objects can be described by one or few only

levels of resolution with (almost) supressed interscale self-

interaction.

It’s possible to consider Wigner functions as some measure

of the quantum character of the system: as soon as it will be

19

positive, we arrive to classical regime and no more reasons

to have the full hierarchy decomposition in the representa-

tions like (8). So, Dirac’s self-interference is nothing but

the multiscale mixture/intermittency. Definitely, the degree

of this self-interaction leads to different qualitative types of

behaviour, such as localized quasiclassical states, separable,

entangled, chaotic etc. At the same time the instantaneous

quantum interaction or transmission of information from Al-

ice to Bob takes place not in the physical kinematical space-

time but in Hilbert spaces of states in their proper functional

realization where there is a different kinematic life. To de-

scribe a set of Quantum Objects we need to realize our Space

of States (Hilbert space) not as one functional space but as

the so-called and well known in mathematics scale of spaces,

e.g. Bsp,q, F

sp,q [11]. The proper multiscale decomposition for

the scale of space provides us by the method of description

of the set of quantum objects in case if the “size” of one

Hilbert space is not enough to describe the complicated in-

ternal World. We’ll consider it elsewhere, while in the second

part [9] we consider the one-scale case (to avoid possible mis-

understanding we need take into account that one-scale case

is also described by an infinite scale of spaces (9), but it’s in-

ternal decomposition of the unique, attached to the problem,

Hilbert space).

20

Acknowledgments

We are very grateful to Prof. Chandrasekhar Roychoudhuri,

University of Connecticut, Ms. Jenny Woods, Mrs. Kristi

Kelso, SPIE (International Society for Optical Engineering)

for kind attention and help, and United States Air Force Of-

fice for Scientific Research, and Nippon Sheet Glasses Co.

Ltd., and SPIE for financial support provided our participa-

tion in SPIE Meeting, San Diego, July-August, 2005.

21

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[7] A.N. Fedorova and M.G. Zeitlin, Progress in Nonequilib-

rium Green’s Functions II, Ed. M. Bonitz, 481, World Sci-

entific, Singapore, 2003; arXiv: physics/0212066; quant-

phys/0306197.

[8] A.N. Fedorova and M.G. Zeitlin, Quantum Aspects of

Beam Physics, 22, Eds. Pisin Chen, K. Reil, World Scien-

tific, 2004, SLAC-R-630, Nuclear Instruments and Meth-

22

ods in Physics Research Section A, Volume 534, Issues 1-

2, 309, 314, 2004; quant-ph/0406009, quant-ph/0406010,

quant-ph/0306197.

[9] A.N. Fedorova and M.G. Zeitlin, Localization and Pat-

tern Formation in Quantum Physics. II. Waveletons in

Quantum Ensembles, this Volume.

[10] Y. Meyer, Wavelets and Operators, Cambridge Univ. Press,

1990; D. Donoho, WaveLab, Stanford, 2000; F. Auger

e.a., Time-Frequency Toolbox, CNRS, 1996;

[11] H. Triebel, Theory of Functional Spaces, Birkhauser, 1983.

23

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

500

1000

1500

2000

2500

Figure 2: Kick.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

Figure 3: Multi-Kicks.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4x 10

−3

Figure 4: RW-fractal.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

Figure 5: MRA for Kick.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

Figure 6: MRA for Multi-Kicks.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

Figure 7: MRA for RW-fractal.

24

−0.1

0

0.1

Rea

l par

t

Signal in time

024

Linear scaleE

nerg

y sp

ectr

al d

ensi

ty

50 100 150 200 2500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

WV, lin. scale, Threshold=5%

Time [s]

Fre

quen

cy [H

z]

50100

150200

250

0

0.1

0.2

0.3

0.4

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Am

plitu

de

WV, lin. scale, Threshold=5%

Time [s]Frequency [Hz]

Figure 8: Wigner function for 3 wavelet packets.

25

Figure 9: 1-dimensional MRA modes.

0

10

20

30

0

10

20

300

0.2

0.4

0.6

0.8

momentum (p)coordinate (q)

dist

ribut

ion

func

tion

F(q

,p)

Figure 10: 2-dimensional MRA mode.

010

2030

4050

60

0

20

40

60−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Figure 11: Contribution to WF at scale 1.

0

20

40

60

0

20

40

600

0.2

0.4

0.6

0.8

1

momentum (p)coordinate (q)

dist

ribut

ion

func

tion

F(q

,p)

Figure 12: Chaotic-like pattern.

020

4060

80100

120

0

50

100

150−0.2

−0.1

0

0.1

0.2

0.3

Figure 13: Entangled state/pattern.

0

20

40

60

0

20

40

600

0.2

0.4

0.6

0.8

1

momentum (p)coordinate (q)

dist

ribut

ion

func

tion

F(q

,p)

Figure 14: Localized pattern: waveleton.

26

LOCALIZATION AND PATTERN FORMATION

IN QUANTUM PHYSICS.

II. WAVELETONS IN QUANTUM ENSEMBLES

Antonina N. Fedorova, Michael G. Zeitlin

IPME RAS, St. Petersburg, V.O. Bolshoj pr., 61, 199178,

Russia

e-mail: zeitlin@math.ipme.ru

e-mail: anton@math.ipme.ru

http://www.ipme.ru/zeitlin.html

http://www.ipme.nw.ru/zeitlin.html

Submitted to Proc. of SPIE Meeting, The Nature of Light:

What is a Photon?

Optics & Photonics, SP200, San Diego, CA, July-August, 2005

Abstract

In this second part we present a set of methods, analyticaland numerical, which can describe behaviour in (non) equilib-rium ensembles, both classical and quantum, especially in the

complex systems, where the standard approaches cannot be ap-plied. The key points demonstrating advantages of this approach

are: (i) effects of localization of possible quantum states; (ii) ef-fects of non-perturbative multiscales which cannot be calculated

by means of perturbation approaches; (iii) effects of formationof complex/collective quantum patterns from localized modes

and classification and possible control of the full zoo of quantumstates, including (meta) stable localized patterns (waveletons).We demonstrate the appearance of nontrivial localized (meta)

stable states/patterns in a number of collective models coveredby the (quantum)/(master) hierarchy of Wigner-von Neumann-

Moyal-Lindblad equations, which are the result of “wigneriza-tion” procedure (Weyl-Wigner-Moyal quantization) of classical

BBGKY kinetic hierarchy, and present the explicit construc-tions for exact analytical/numerical computations. Our fast and

efficient approach is based on variational and multiresolutionrepresentations in the bases of polynomial tensor algebras ofgeneralized localized states (fast convergent variational-wavelet

representation). We construct the representations for hierar-chy/algebra of observables(symbols)/distribution functions via

the complete multiscale decompositions, which allow to considerthe polynomial and rational type of nonlinearities. The solu-

tions are represented via the exact decomposition in nonlinearhigh-localized eigenmodes, which correspond to the full multires-olution expansion in all underlying hidden time/space or phase

space scales. In contrast with different approaches we do not useperturbation technique or linearization procedures. Numerical

modeling shows the creation of different internal structures fromlocalized modes, which are related to the localized (meta) sta-

ble patterns (waveletons), entangled ensembles (with subsequentdecoherence) and/or chaotic-like type of behaviour

keywords: localization, pattern formation, multiscales, mul-

tiresolution, waveletons, (non) equilibrium ensembles

28

5 INTRODUCTION: CLASSICAL AND QUANTUM EN-

SEMBLES

In this paper we consider the applications of a numerical-ana-

lytical technique based on local nonlinear harmonic analysis

to the description of quantum ensembles. The corresponding

class of individual Hamiltonians has the form

H(p, q) =p2

2m+ U(p, q), (11)

where U(p, q) is an arbitrary polynomial function on p, q, and

plays the key role in many areas of physics [1]. Many cases,

related to some physics models, are considered in [2]-[8]. It is

a continuation of our more qualitative approach considered

in part I [9]. In this part our goals are some attempt of classi-

fication and the constructions of explicit numerical-analytical

representations for the existing quantum states in the class

of models described (non) equilibrium ensembles and related

collective models. There is a hope on the understanding of

relation between the structure of initial Hamiltonians and the

possible types of quantum states and the qualitative type of

their behaviour. Inside the full spectrum there are at least

three possibilities which are the most important from our

point of view: localized states, chaotic-like or/and entan-

gled patterns, localized (stable) patterns (qualitative defini-

tions/descriptions can be found in part I [9]). All such states

are interesting in the different areas of physics [1]. Our start-

ing point is the general point of view of a deformation quan-

tization approach at least on the naive Moyal/Weyl/Wigner

level. The main point of such approach is based on ideas

from [1], which allow to consider the algebras of quantum

observables as the deformations of commutative algebras of

classical observables (functions). So, if we have the classical

29

counterpart of Hamiltonian (1) as a model for classical dy-

namics and the Poisson manifold M (or symplectic manifold

or Lie coalgebra, etc) as the corresponding phase space, then

for quantum calculations we need first of all to find an asso-

ciative (but non-commutative) star product ∗ on the space

of formal power series in ~ with coefficients in the space of

smooth functions on M such that

f ∗ g = fg + ~f, g+∑

n≥2

~nBn(f, g), (12)

where f, g is the Poisson brackets, Bn are bidifferential

operators. In the naive calculations we may use the simple

formal rule:

∗ ≡ exp (i~

2(←−∂ q−→∂ p −

←−∂ p−→∂ q)) (13)

In this paper we consider the calculations of the Wigner

functionsW (p, q, t) (WF) corresponding to the classical poly-

nomial Hamiltonian H(p, q, t) as the solution of the Wigner-

von Neumann equation [1]:

i~∂

∂tW = H ∗W −W ∗H (14)

and related Wigner-like equations for different ensembles.

According to the Weyl transform, a quantum state (wave

function or density operator ρ) corresponds to the Wigner

function, which is the analogue in some sense of classical

phase-space distribution [1]. We consider the following op-

erator form of differential equations for time-dependent WF,

W = W (p, q, t):

Wt =2

~sin [

~

2(∂Hq ∂

Wp − ∂

Hp ∂

Wq )] ·HW (15)

which is a result of the Weyl transform or “wignerization” of

30

von Neumann equation for density matrix:

i~∂ρ

∂t= [H,ρ] (16)

5.1 BBGKY Ensembles

We start from set-up for kinetic BBGKY hierarchy (as c-

counterpart of proper q-hierarchy). We present the explicit

analytical construction for solutions of both hierarchies of

equations, which is based on tensor algebra extensions of

multiresolution representation and variational formulation.

We give explicit representation for hierarchy of n-particle re-

duced distribution functions in the base of high-localized gen-

eralized coherent (regarding underlying generic symmetry

(affine group in the simplest case)) states given by polynomial

tensor algebra of our basis functions (wavelet families), which

takes into account contributions from all underlying hidden

multiscales from the coarsest scale of resolution to the finest

one to provide full information about stochastic dynamical

process. The difference between classical and quantum case

is concetrated in the structure of the set of operators in-

cluded in the set-up and, surely, depends on the method of

quantization. But, in the naive Wigner-Weyl approach for

quantum case the symbols of operators play the same role as

usual functions in classical case. In some sense, our approach

resembles Bogolyubov’s one and related approaches but we

don’t use any perturbation technique (like virial expansion)

or linearization procedures. Most important, that numerical

modeling in both cases shows the creation of different in-

ternal (coherent) structures from localized modes, which are

related to stable (equilibrium) or unstable type of behaviour

and corresponding pattern (waveletons) formation.

Let M be the phase space of ensemble of N particles (dimM =

31

6N) with coordinates

xi = (qi, pi), i = 1, ...,N, qi = (q1i , q

2i , q

3i ) ∈ R

3,

pi = (p1i , p

2i , p

3i ) ∈ R

3, q = (q1, . . . , qN) ∈ R3N . (17)

Individual and collective measures are:

µi = dxi = dqidpi, µ =

N∏

i=1

µi (18)

Distribution functionDN(x1, . . . , xN ; t) satisfies Liouville equa-

tion of motion for ensemble with Hamiltonian HN and nor-

malization constraint:

∂DN

∂t= HN ,DN,

∫

DN(x1, . . . , xN ; t)dµ = 1 (19)

where Poisson brackets are:

HN , DN =N∑

i=1

(∂HN

∂qi

∂DN

∂pi−∂HN

∂pi

∂DN

∂qi) (20)

Our constructions can be applied to the following general

Hamiltonians:

HN =

N∑

i=1

(p2i

2m+ Ui(q)) +

∑

1≤i≤j≤N

Uij(qi, qj) (21)

where potentials Ui(q) = Ui(q1, . . . , qN) and Uij(qi, qj) are

not more than rational functions on coordinates. Let Ls and

Lij be the Liouvillean operators (vector fields)

Ls =s∑

j=1

(pj

m

∂

∂qj−∂uj

∂q

∂

∂pj)−

∑

1≤i≤j≤s

Lij, (22)

Lij =∂Uij

∂qi

∂

∂pi+∂Uij

∂qj

∂

∂pj

32

For s=N we have the following representation for Liouvillean

vector field LN = HN , · and the corresponding ensemble

equation of motion:

∂DN

∂t+ LNDN = 0 (23)

LN is self-adjoint operator regarding standard pairing on the

set of phase space functions. Let

FN(x1, . . . , xN ; t) =∑

SN

DN(x1, . . . , xN ; t) (24)

be the N-particle distribution function (SN is permutation

group of N element s). Then we have the hierarchy of reduced

distribution functions (V s is the corresponding normalized

volume factor)

Fs(x1, . . . , xs; t) = V s

∫

DN(x1, . . . , xN ; t)∏

s+1≤i≤N

µi (25)

After standard manipulations we arrived to c-BBGKY hier-

archy:

∂Fs

∂t+ LsFs =

1

υ

∫

dµs+1

s∑

i=1

Li,s+1Fs+1 (26)

It should be noted that we may apply our approach even to

more general formulation. As in the general as in particular

situations (cut-off, e.g.) we are interested in the cases when

Fk(x1, . . . , xk; t) =

k∏

i=1

F1(xi; t) +Gk(x1, . . . , xk; t), (27)

where Gk are correlators, really have additional reductions as

in the simplest case of one-particle truncation (Vlasov/Boltzmann-

like systems). So, the proper dynamical formulation is re-

duced to the (infinite) set of equations for correlators/partition

33

functions. Then by using physical motivated reductions or/and

during the corresponding cut-off procedure we obtain, in-

stead of linear and pseudodifferential (in general case) equa-

tions, their finite-dimensional but nonlinear approximations

with the polynomial type of nonlinearities (more exactly,

multilinearities). Our key point in the following consider-

ation is the proper generalization of naive perturbative mul-

tiscale Bogolyubov’s structure restricted by the set of addi-

tional physical hypotheses.

5.2 Quantum ensembles

Let us start from the second quantized representation for an

algebra of observables

A = (A0, A1, . . . , As, ...) (28)

in the standard form

A = A0 +

∫

dx1Ψ+(x1)A1Ψ(x1) +

. . .+ (s!)−1

∫

dx1 . . . dxsΨ+(x1)

. . .Ψ+(xs)AsΨ(xs) . . .Ψ(x1) + . . . (29)

N-particle Wigner functions

Ws(x1, . . . , xs) = (30)∫

dk1 . . . dksexp(− is∑

i=1

kipi)TrρΨ+(q1 −1

2~k1) . . .

Ψ+(qs −1

2~ks)Ψ(qs +

1

2~ks) . . .Ψ(q1 +

1

2~ks)

allow us to consider them as some quasiprobabilities and pro-

vide useful bridge between c- and q-cases:

34

< A >= TrρA = (31)∞∑

s=0

(s!)−1

∫ s∏

i=1

dµiAs(x1, . . . , xs)Ws(x1, . . . , xs)

The full description for quantum ensemble can be done by

the whole hierarchy of functions (symbols):

W = Ws(x1, . . . , xs), s = 0, 1, 2 . . . (32)

So, we may consider the following q-hierarchy as the result

of “wignerization” procedure for c-BBGKY one:

∂tWs(t, x1, . . . , xs) = (33)s∑

j=1

L0jWs(x1, . . . , xs) +

∑

j<n

s∑

n=1

Lj,nWs(x1, . . . , xs)

+

s∑

j=1

∫

dxs+1δ(ks+1)Lj,s+1Ws+1(x1, . . . , xs+1)

where

L0j = −(

i

m)kjpj (34)

Lj,n = (i~)−1

∫

d`Vl

[

exp(

−1

2~`(

∂

∂pj−

∂

∂pn))

− (35)

exp(1

2~`(

∂

∂pj−

∂

∂pn))]

exp(

− `(∂

∂kj−

∂

∂kn))

In quantum statistics the ensemble properties are described

by the density operator

ρ(t) =∑

i

wi|Ψi(t) >< Ψi(t)|,∑

i

wi = 1 (36)

35

After Weyl transform we have the following decomposition

via partial Wigner functions Wi(p, q, t) for the whole ensem-

ble Wigner function:

W (p, q, t) =∑

i

wiWi(p, q, t) (37)

where the partial Wigner functions

Wn(q, p, t) ≡ (38)1

2π~

∫

dξexp(−i

~pξ)Ψ∗n(q −

1

2ξ, t)Ψn(q +

1

2ξ, t)

are solutions of proper Wigner equations:

∂Wn

∂t= −

p

m

∂Wn

∂q+∞∑

`=0

(−1)`(~/2)2`

(2`+ 1)!

∂2`+1Un(q)

∂q2`+1

∂2`+1Wn

∂p2`+1(39)

Our approach, presented below, in some sense has allu-

sion on the analysis of the following standard simple model

considered in [1]. Let us consider model of interaction of

nonresonant atom with quantized electromagnetic field:

H =p2x

2m+ U(x), U(x) = U0(z, t)g(x)a+a (40)

where potential U depends on creation/annihilation opera-

tors and some polynomial on x operator function (or approx-

imation) g(x). It is possible to solve Schroedinger equation

i~d|Ψ >

dt= H|Ψ > (41)

by the simple ansatz

|Ψ(t) >=

∞∑

−∞

wn

∫

dxΨn(x, t)|x > ⊗|n > (42)

36

which leads to the hierarchy of analogous equations with po-

tentials created by n-particle Fock subspaces

i~∂Ψn(x, t)

∂t=

p2x

2m+ U0(t)g(x)nΨn(x, t) (43)

where Ψn(x, t) is the probability amplitude of finding the

atom at the time t at the position x and the field in the

n Fock state. Instead of this, we may apply the Wigner

approach starting with proper full density matrix

ρ = |Ψ(t) >< Ψ(t)| = (44)∑

n′,n′′

wn′w∗n′′

∫

dx′∫

dx′′Ψn′(x′, t)Ψ∗n′′(x

′′, t)

|x′ >< x′′| ⊗ |n′ >< n′′|

Standard reduction gives pure atomic density matrix

ρa ≡

∫ ∞

n=0

< n|ρ|n >= (45)

∑

|wn|2

∫

dx′∫

dx′′Ψn(x′, t)Ψ∗n(x

′′, t)|x′ >< x′′|

Then we have incoherent superposition

W (x, p, t) =∞∑

n=0

|wn|2Wn(x, p, t) (46)

of the atomic Wigner functions (28) corresponding to the

atom motion in the potential Un(x) (which is not more than

polynomial in x) generated by n-level Fock state. They are

solutions of proper Wigner equations (29).

The next case describes the important decoherence pro-

cess. Let us have collective and environment subsystems with

their own Hilbert spaces

H = Hc ⊗He (47)

37

Relevant dynamics is described by three parts including in-

teraction

H = Hc ⊗ Ie + Ic ⊗He +Hint (48)

For analysis, we can choose Lindblad master equation [1]

ρ =1

i~[H,ρ]−

∑

n

γn(L+nLnρ+ ρL+

nLn − 2LnρL+n ) (49)

which preserves the positivity of density matrix and it is

Markovian but it is not general form of exact master equa-

tion. Other choice is Wigner transform of master equation

and it is more preferable for us

W = H,WPB + (50)∑

n≥1

~2n(−1)n

22n(2n+ 1)!∂2n+1q U(q)∂2n+1

p W (q, p) +

2γ∂ppW +D∂2pW

In the next section we consider the variational-wavelet ap-

proach for the solution of all these Wigner-like equations (4),

(5), (6), (23), (29), (40) for the case of an arbitrary polyno-

mial U(q, p), which corresponds to a finite number of terms in

the series expansion in (5), (29), (40) or to proper finite order

of ~. Analogous approach can be applied to classical counter-

part (16) also. Our approach is based on the extension of our

variational-wavelet approach [2]-[8]. Wavelet analysis is some

set of mathematical methods, which gives the possibility to

take into account high-localized states, control convergence of

any type of expansions and gives maximum sparse forms for

the general type of operators in such localized bases. These

bases are the natural generalization of standard coherent,

squeezed, thermal squeezed states [1], which correspond to

38

quadratical systems (pure linear dynamics) with Gaussian

Wigner functions. The representations of underlying sym-

metry group (affine group in the simplest case) on the proper

functional space of states generate the exact multiscale ex-

pansions which allow to control contributions to the final re-

sult from each scale of resolution from the whole underlying

infinite scale of spaces. Numerical calculations according to

methods of part I [9] explicitly demonstrate the quantum in-

terference of generalized localized states, pattern formation

from localized eigenmodes and the appearance of (stable) lo-

calized patterns (waveletons).

6 VARIATIONAL MULTIRESOLUTION REPRESENTA-

TION

6.1 Multiscale Decomposition for Space of States: Functional Realiza-tion and Metric Structure

We obtain our multiscale/multiresolution representations for

solutions of Wigner-like equations via a variational-wavelet

approach. We represent the solutions as decomposition into

localized eigenmodes (regarding action of affine group, i.e.

hidden symmetry of the underlying functional space of states)

related to the hidden underlying set of scales [10]:

Wn(t, q, p) =∞⊕

i=ic

W in(t, q, p), (51)

where value ic corresponds to the coarsest level of resolution

c or to the internal scale with the number c in the full mul-

tiresolution decomposition of the underlying functional space

(L2, e.g.) corresponding to the problem under consideration:

Vc ⊂ Vc+1 ⊂ Vc+2 ⊂ . . . (52)

39

and p = (p1, p2, . . .), q = (q1, q2, . . .), xi = (p1, q1, . . . , pi, qi)

are coordinates in phase space. In the following we may con-

sider as fixed as variable numbers of particles.

We introduce the Fock-like space structure (in addition to

the standard one, if we consider second-quantized case) on

the whole space of internal hidden scales.

H =⊕

i

⊗

n

Hni (53)

for the set of n-partial Wigner functions (states):

W i = W i0,W

i1(x1; t), . . . ,W

iN(x1, . . . , xN ; t), . . ., (54)

where Wp(x1, . . . , xp; t) ∈ Hp, H0 = C, Hp = L2(R6p) (or

any different proper functional space), with the natural Fock

space like norm:

(W,W ) = W 20 +

∑

i

∫

W 2i (x1, . . . , xi; t)

i∏

`=1

µ`. (55)

First of all, we consider W = W (t) as a function of time

only, W ∈ L2(R), via multiresolution decomposition which

naturally and efficiently introduces the infinite sequence of

the underlying hidden scales [10]. We have the contribution

to the final result from each scale of resolution from the whole

infinite scale of spaces (16). The closed subspace Vj(j ∈ Z)

corresponds to the level j of resolution, or to the scale j and

satisfies the following properties: let Dj be the orthonormal

complement of Vj with respect to Vj+1: Vj+1 = Vj⊕

Dj. Then

we have the following decomposition:

W (t) =⊕

−∞<j<∞

Dj = Vc

∞⊕

j=0

Dj, (56)

in case when Vc is the coarsest scale of resolution. The sub-

group of translations generates a basis for the fixed scale

40

number: spank∈Z2j/2Ψ(2jt − k) = Dj. The whole basis is

generated by action of the full affine group:

spank∈Z,j∈Z2j/2Ψ(2jt− k) = spank,j∈ZΨj,k = W (t)(57)

6.2 Tensor Product Structure

Let sequence

V tj , V t

j ⊂ L2(R) (58)

correspond to multiresolution analysis on time axis and

Vxij , V

xij ⊂ L

2(R) (59)

correspond to multiresolution analysis for coordinate xi, then

V n+1j = V x1

j ⊗ . . .⊗ Vxnj ⊗ V

tj (60)

corresponds to multiresolution analysis for n-particle distri-

bution fuction Wn(x1, . . . , xn; t). E.g., for n = 2:

V 20 = f : f(x1, x2) =

∑

k1,k2

ak1,k2φ2(x1 − k1, x2 − k2),

ak1,k2∈ `2(Z2), (61)

where

φ2(x1, x2) = φ1(x1)φ2(x2) = φ1 ⊗ φ2(x1, x2), (62)

and φi(xi) ≡ φ(xi) form a multiresolution basis correspond-

ing to Vxij . If

φ1(x1 − `), ` ∈ Z (63)

form an orthonormal set, then

φ2(x1 − k1, x2 − k2) (64)

form an orthonormal basis for V 20 . Action of affine group pro-

vides us by multiresolution representation of L2(R2). After

introducing detail spaces D2j , we have, e.g.

V 21 = V 2

0 ⊕D20. (65)

41

Then 3-component basis for D20 is generated by translations

of three functions

Ψ21 = φ1(x1)⊗Ψ2(x2),

Ψ22 = Ψ1(x1)⊗ φ

2(x2), (66)

Ψ23 = Ψ1(x1)⊗Ψ2(x2)

Also, we may use the rectangle lattice of scales and one-

dimentional wavelet decomposition :

f(x1, x2) =∑

i,`;j,k

< f,Ψi,` ⊗Ψj,k > Ψj,` ⊗Ψj,k(x1, x2), (67)

where bases functions

Ψi,` ⊗Ψj,k (68)

depend on two scales 2−i and 2−j.

After construction the multidimensional bases we obtain

our multiscale/multiresolution representations for observables

(operators, symbols), states, partitions via the variational

approaches [2]-[8] as for c-BBGKY as for its quantum coun-

terpart and related reductions but before we need to con-

struct reasonable multiscale decomposition for all operators

included in the set-up.

6.3 —Large FWT Decomposition for Observables

One of the key point of wavelet analysis approach, the so

called Fast Wavelet Transform (FWT), demonstrates that for

a large class of operators the wavelet functions are good ap-

proximation for true eigenvectors; and the corresponding ma-

trices are almost diagonal. FWT gives the maximum sparse

form for wide classes of operators [10]. So, let us denote our

(integral/differential) operator from equations under consid-

eration as T (L2(Rn) → L2(Rn)) and its kernel as K. We

42

have the following representation:

< Tf, g >=

∫ ∫

K(x, y)f(y)g(x)dxdy. (69)

In case when f and g are wavelets ϕj,k = 2j/2ϕ(2jx−k), (21)

provides the standard representation for operator T . Let

us consider multiresolution representation . . . ⊂ V2 ⊂ V1 ⊂

V0 ⊂ V−1 ⊂ V−2 . . .. The basis in each Vj is ϕj,k(x), where in-

dices k, j represent translations and scaling respectively. Let

Pj : L2(Rn)→ Vj (j ∈ Z) be projection operators on the sub-

space Vj corresponding to level j of resolution: (Pjf)(x) =∑

k < f,ϕj,k > ϕj,k(x). Let Qj = Pj−1 − Pj be the projec-

tion operator on the subspace Dj (Vj−1 = Vj ⊕Dj), then we

have the following representation of operator T which takes

into account contributions from each level of resolution from

different scales starting with the coarsest and ending to the

finest scales [10]:

T =∑

j∈Z

(QjTQj +QjTPj + PjTQj). (70)

We need to remember that this is a result of presence of

affine group inside this construction. The non-standard form

of operator representation is a representation of operator T as

a chain of triples T = Aj, Bj,Γjj∈Z, acting on the subspaces

Vj and Dj: Aj : Dj → Dj, Bj : Vj → Dj,Γj : Dj → Vj, where

operators Aj, Bj,Γjj∈Z are defined as Aj = QjTQj, Bj =

QjTPj, Γj = PjTQj. The operator T admits a recursive

definition via

Tj =

(

Aj+1 Bj+1

Γj+1 Tj+1

)

, (71)

where Tj = PjTPj and Tj acts on Vj : Vj → Vj. So, it is

possible to provide the following “sparse” action of operator

43

Tj on elements f of functional realization of our space of

states H:

(Tjf)(x) =∑

k∈Z

(

2−j∑

`

r`fj,k−`

)

ϕj,k(x), (72)

in the wavelet basis ϕj,k(x) = 2−j/2ϕ(2−jx− k), where

fj,k−1 = 2−j/2∫

f(x)ϕ(2−jx− k + `)dx (73)

are wavelet coefficients and r` are the roots of some additional

linear system of equations related to the “type of localiza-

tion” [10]. So, we have the simple linear parametrization of

matrix representation of our operators in localized wavelet

bases and of the action of this operator on arbitrary vec-

tor/state in proper functional space.

6.4 Variational Approach

Now, after preliminary work with (functional) spaces, states

and operators, we may apply our variational approach from

[2]-[8].

Let L be an arbitrary (non)linear differential/integral op-

erator with matrix dimension d (finite or infinite), which acts

on some set of functions from L2(Ω⊗n): Ψ ≡ Ψ(t, x1, x2, . . .) =

(Ψ1(t, x1, x2, . . .), . . ., Ψd(t, x1, x2, . . .)), xi ∈ Ω ⊂ R6, n is

the number of particles:

LΨ ≡ L(Q, t, xi)Ψ(t, xi) = 0, (74)

Q ≡ Qd0,d1,d2,...(t, x1, x2, . . . , ∂/∂t, ∂/∂x1, ∂/∂x2,

. . . ,

∫

µk)

=

d0,d1,d2,...∑

i0,i1,i2,...=1

qi0i1i2...(t, x1, x2, . . .)

44

( ∂

∂t

)i0( ∂

∂x1

)i1( ∂

∂x2

)i2. . .

∫

µk

Let us consider now the N mode approximation for the so-

lution as the following ansatz:

ΨN(t, x1, x2, . . .) = (75)N∑

i0,i1,i2,...=1

ai0i1i2...Ai0 ⊗Bi1 ⊗ Ci2 . . . (t, x1, x2, . . .)

We shall determine the expansion coefficients from the fol-

lowing conditions (related to proper choosing of variational

approach):

`Nk0,k1,k2,...≡ (76)

∫

(LΨN)Ak0(t)Bk1

(x1)Ck2(x2)dtdx1dx2 . . . = 0

Thus, we have exactly dNn algebraical equations for dNn un-

knowns ai0,i1,.... This variational approach reduces the initial

problem to the problem of solution of functional equations at

the first stage and some algebraical problems at the second

one. It allows to unify the multiresolution expansion with

variational construction[2]-[8].

As a result, the solution is parametrized by the solutions of

two sets of reduced algebraical problems, one is linear or non-

linear (depending on the structure of the generic operator L)

and the rest are linear problems related to the computation

of the coefficients of reduced algebraic equations. It is also re-

lated to the choice of exact measure of localization (including

class of smothness) which are proper for our set-up. These

coefficients can be found by some functional/algebraic meth-

ods by using the compactly supported wavelet basis functions

or any other wavelet families [10].

As a result the solution of the equations/hierarchies from

Section 1, as in c- as in q-region, has the following multiscale

45

or multiresolution decomposition via nonlinear high-localized

eigenmodes

W (t, x1, x2, . . .) =∑

(i,j)∈Z2

aijUi ⊗ V j(t, x1, . . .),

V j(t) = V j,slowN (t) +

∑

l≥N

V jl (ωlt), ωl ∼ 2l, (77)

U i(xs) = U i,slowM (xs) +

∑

m≥M

U im(ksmxs), k

sm ∼ 2m,

which corresponds to the full multiresolution expansion in

all underlying time/space scales. The formulae (67) give the

expansion into a slow part and fast oscillating parts for ar-

bitrary N,M . So, we may move from the coarse scales of

resolution to the finest ones for obtaining more detailed in-

formation about the dynamical process. In this way one ob-

tains contributions to the full solution from each scale of

resolution or each time/space scale or from each nonlinear

eigenmode. It should be noted that such representations give

the best possible localization properties in the correspond-

ing (phase)space/time coordinates. Formulae (67) do not use

perturbation techniques or linearization procedures. Numer-

ical calculations are based on compactly supported wavelets

and wavelet packets and on evaluation of the accuracy on

the level N of the corresponding cut-off of the full system

regarding norm (45):

‖WN+1 −WN‖ ≤ ε. (78)

7 MODELING OF PATTERNS

To summarize, the key points are:

1. The ansatz-oriented choice of the (multidimensional)

bases related to some polynomial tensor algebra.

46

2. The choice of proper variational principle. A few proje-

ction/ Galerkin-like principles for constructing (weak) solu-

tions are considered. The advantages of formulations related

to biorthogonal (wavelet) decomposition should be noted.

3. The choice of bases functions in the scale spaces Dj

from wavelet zoo. They correspond to high-localized (non-

linear) oscillations/excitations, nontrivial local (stable) dis-

tributions/fluctuations, etc. Besides fast convergence prop-

erties it should be noted minimal complexity of all underlying

calculations, especially in case of choice of wavelet packets

which minimize Shannon entropy.

4. Operator representations providing maximum sparse

representations for arbitrary (pseudo) differential/ integral

operators df/dx, dnf/dxn,∫

T (x, y)f(y)dy), etc.

5. (Multi)linearization. Besides the variation approach we

can consider also a different method to deal with (polyno-

mial) nonlinearities: para-products-like decompositions.

To classify the qualitative behaviour we apply standard

methods from general control theory or really use the con-

trol. We will start from a priori unknown coefficients, the ex-

act values of which will subsequently be recovered. Roughly

speaking, we will fix only class of nonlinearity (polynomial in

our case) which covers a broad variety of examples of possi-

ble truncation of the systems. As a simple model we choose

band-triangular non-sparse matrices (aij). These matrices

provide tensor structure of bases in (extended) phase space

and are generated by the roots of the reduced variational

(Galerkin-like) systems. As a second step we need to re-

store the coefficients from these matrices by which we may

classify the types of behaviour. We start with the localized

mode, which is a base mode/eigenfunction, (Fig. 1, 9, 10

from Part I), corresponding to definitions from Section 2.2,

47

Part I, which was constructed as a tensor product of the

two Daubechies functions. Fig. 1, 4 below demonstrate the

result of summation of series (67) up to value of the dila-

tion/scale parameter equal to four and six, respectively. It’s

done in the bases of symmlets [10] with the corresponding

matrix elements equal to one. The size of matrix of “Fourier-

wavelet coefficients” is 512x512. So, different possible distri-

butions of the root values of the generical algebraical sys-

tems (66) provide qualitatively different types of behaviour.

Generic algebraic system (66), Generalized Dispersion Rela-

tion (GDR), provide the possibility for algebraic control. The

above choice provides us by a distribution with chaotic-like

equidistribution. But, if we consider a band-like structure

of matrix (aij) with the band along the main diagonal with

finite size ( 512) and values, e.g. five, while the other val-

ues are equal to one, we obtain localization in a fixed finite

area of the full phase space, i.e. almost all energy of the sys-

tem is concentrated in this small volume. This corresponds

to waveleton states and is shown in Fig. 2, constructed by

means of Daubechies-based wavelet packets. Depending on

the type of solution, such localization may be conserved dur-

ing the whole time evolution (asymptotically-stable) or up to

the needed value from the whole time scale (e.g. enough for

plasma fusion/confinement in the case of fusion modeling by

means of c-BBGKY hierarchy for dynamics of partitions).

8 CONCLUSION

So, by using wavelet bases with their best (phase) space/time

localization properties we can describe the localized (coher-

ent) structures in quantum systems with complicated be-

haviour (Fig. 1, 4). The modeling demonstrates the forma-

48

tion of different (stable) pattern or orbits generated by in-

ternal hidden symmetry from high-localized structures. Our

(nonlinear) eigenmodes are more realistic for the modelling

of nonlinear classical/quantum dynamical process than the

corresponding linear gaussian-like coherent states. Here we

mention only the best convergence properties of the expan-

sions based on wavelet packets, which realize the minimal

Shannon entropy property and the exponential control of

convergence of expansions like (67) based on the norm (45).

Fig. 2 corresponds to (possible) result of superselection (eins-

election) [1] after decoherence process started from entangled

state (Fig. 5); Fig. 3 and Fig. 6 demonstrate the steps of mul-

tiscale resolution during modeling of entangled states leading

to the growth of degree of entanglement. It should be noted

that we can control the type of behaviour on the level of the

reduced algebraical variational system, GDR (66).

References

[1] D. Sternheimer, arXiv: math. QA/9809056; W. P. Schle-

ich, Quantum Optics in Phase Space, Wiley, 2000; S.

de Groot, L Suttorp, Foundations of Electrodynam-

ics, North-Holland, 1972; W. Zurek, arXiv: quant-

ph/0105127.

[2] A.N. Fedorova and M.G. Zeitlin, Math. and Comp. in Sim-

ulation, 46, 527, 1998; New Applications of Nonlinear and

Chaotic Dynamics in Mechanics, Ed. F. Moon, 31, 101 Klu-

wer, Boston, 1998.

[3] A.N. Fedorova and M.G. Zeitlin, American Institute of

Physics, Conf. Proc. v. 405, 87, 1997; arXiv: physics/-

9710035.

49

[4] A.N. Fedorova, M.G. Zeitlin, and Z. Parsa, AIP Conf.

Proc. v. 468, 48, 69, 1999; arXiv: physics/990262,

990263.

[5] A.N. Fedorova and M.G. Zeitlin, The Physics of High Bright-

ness Beams, Ed. J. Rosenzweig, 235, World Scientific, Sin-

gapore, 2001; arXiv: physics/0003095.

[6] A.N. Fedorova and M.G. Zeitlin, Quantum Aspects of Beam

Physics, Ed. P. Chen, 527, 539, World Scientific, Singa-

pore, 2002; arXiv: physics/0101006, 0101007.

[7] A.N. Fedorova and M.G. Zeitlin, Progress in Nonequilib-

rium Green’s Functions II, Ed. M. Bonitz, 481, World Sci-

entific, Singapore, 2003; arXiv: physics/0212066; quant-

phys/0306197.

[8] A.N. Fedorova and M.G. Zeitlin, Quantum Aspects of

Beam Physics, Eds. Pisin Chen, K. Reil, 22, World Scien-

tific, 2004, SLAC-R-630, Nuclear Instruments and Meth-

ods in Physics Research Section A, Volume 534, Issues 1-

2, 309, 314, 2004; quant-ph/0406009, quant-ph/0406010,

quant-ph/0306197.

[9] A.N. Fedorova and M.G. Zeitlin, Localization and Pat-

tern Formation in Quantum Physics. I. Phenomena of

Localization, this Volume.

[10] Y. Meyer, Wavelets and Operators, Cambridge Univ. Press,

1990.

50

05

1015

2025

30

0

10

20

30−0.1

−0.05

0

0.05

0.1

Figure 15: Level 4 MRA.

05

1015

2025

30

0

10

20

30−2

0

2

4

6

8

momentum (p)coordinate (q)

W(q

,p)

Figure 16: Localized pattern, (waveleton) Wignerfunction.

100 200 300 400 500 600 700 800 900 1000

100

200

300

400

500

600

700

800

900

1000

momentum (p)

coor

dina

te (

q)

Figure 17: Interference picture on the level 4 ap-proximation for Wigner function.

05

1015

2025

30

0

10

20

30−0.4

−0.2

0

0.2

0.4

0.6

Figure 18: Level 6 MRA.

0

50

100

150

0

50

100

150−2

−1

0

1

2

3

momentum (p)coordinate (q)

W(q

,p)

Figure 19: Entangled-like Wigner function.

100 200 300 400 500 600 700 800 900 1000

100

200

300

400

500

600

700

800

900

1000

momentum (p)

coor

dina

te (

q)

Figure 20: Interference picture on the level 6 ap-proximation for Wigner function.

51