Some mathematical problems in a neoclassical theory of electric charges

0.55 MB·English

by  Anatoli Babin

SOME MATHEMATICAL PROBLEMS IN A NEOCLASSICAL
THEORY OF ELECTRIC CHARGES
0
1 ANATOLIBABINANDALEXANDERFIGOTIN
0
2 Dedicated toRoger Temam on the occasion of his 70th birthday.

n
a Abstract. We study here a number of mathematical problems related to
J
our recently introduced neoclassical theory for electromagnetic phenomena in

1 which charges are represented by complex valued wave functions as in the
3 Schro¨dinger wave mechanics. In the non-relativistic case the dynamics of el-
ementary charges is governed by asystem of nonlinear Schro¨dinger equations

] coupledwiththeelectromagneticfields,andweprovethatifthewavefunctions
h
of charges are well separated and localized their centers converge to trajecto-
p
riesoftheclassicalpointchargesgovernedbytheNewton’sequationswiththe
-
h Lorentz forces. Wealsofoundexact solutionsintheformoflocalizedacceler-
t ating solitons. Our studies of a class of time multiharmonic solutions of the
a same fieldequations show that they satisfyPlanck-Einstein relation andthat
m
the energy levels of the nonlinear eigenvalue problem for the hydrogen atom
[ converge to the well-known energy levels of the linear Schr¨odinger operator
whenthefreechargesizeismuchlargerthantheBohrradius.

5
v
0
3
1. Introduction
6
3
It is well known that the concept of a point charge interacting with the electro-
.
0 magnetic (EM) field has fundamental problems. Indeed, in the classical electrody-
1 namicstheevolutionofapointchargeq ofamassminanexternalelectromagnetic
9
(EM) field is governedby Newton’s equation
0
: d 1
v [mv(t)]=q E(t,r(t))+ v(t) B(t,r(t)) (1)
i dt c ×
X (cid:20) (cid:21)
where r and v = r˙ =dr are respectively the charge’s position and velocity, E(t,r)
r dt
a and B(t,r) are the electric field and the magnetic induction, and the right-hand
side of the equation (1) is the Lorentz force. On other hand, if the charge’s time-
dependent position and velocity are respectively r and v then the corresponding
EM field is described by the Maxwell equations
1∂B
+ E=0, B=0, (2)
c ∂t ∇× ∇·
1∂E 4π
B= qδ(x r(t))v(t), E=4πqδ(x r(t)), (3)
c ∂t −∇× − c − ∇· −
where δ is the Dirac delta-function, v(t) = r˙(t), c is speed of light. If one would
like to consider equations (1)-(3) as a closed system ”charge-EM field” there is
a problem. Its origin is a singularity of the EM field exactly at the position of
1991 Mathematics Subject Classification. 35Q61;35Q55;35Q60;35Q70;35P30.
Key words and phrases. Maxwell equations, nonlinear Schro¨dinger equation, Newton’s law,
Lorentzforce,hydrogenatom,nonlineareigenvalueproblem.
SupportedbyAFOSRgrantnumberFA9550-04-1-0359.
1
2 ANATOLIBABINANDALEXANDERFIGOTIN
the point charge, as, for instance, for the electrostatic field E with the Coulomb’s
potential q with asingularityatx=r. Ifone wantsto stay within the classical
|x−r|
electromagentictheory,apossibleremedyistheintroductionofanextendedcharge
which, though very small, is not a point. There are two most well known models
for such an extended charge: the semi-relativistic Abraham rigid charge model (a
rigid sphere with spherically symmetric charge distribution), [35, Sections 2.4, 4.1,
10.2, 13], [32, Sections 2.2], and the Lorentz relativistically covariant model which
wasstudiedandadvancedin[1],[20,Sections16],[28],[30],[32,Sections2,6],[33],
[35,Sections2.5,4.2,10.1],[37]. Poincar´esuggestedin1905-1906,[31](seealso[20,
Sections16.4-16.6],[32,Sections2.3,6.1-6.3],[29,Section63],[33],[37,Section4.2]
andreferencestherein),toaddtotheLorentz-Abrahammodelnon-electromagnetic
cohesive forces which balance the charge internal repulsive electromagnetic forces
and remarkably restore also the covariance of the entire model.
Anotherwellknownproblemofaphysicalnaturewithpointchargesandtheclas-
sicalelectrodynamicsisrelatedtotheRutherfordplanetarymodelforthehydrogen
atom. The planetary model of the hydrogen atom is inconsistent with physically
observed phenomena such as the stability of atoms and the discreteness of the hy-
drogen energy levels. That inconsistency lead as we well know to introduction of
non-classicalmodels, such as the Bohr model and later on the Schr¨odinger’s model
of the hydrogen atom.
To address the above mentioned problems with the classicalelectrodynamics we
introduced recently in [5], [6] wave-corpuscle mechanics (WCM) for charges which
is conceived as one mechanics for macroscopic and atomic scales. This model is
neoclassical in the sense that it is based on the classicalconcept of the electromag-
netic field but an elementary charge is not a point but a wave-corpuscle described
byacomplex-valuedscalarwavefunctionψ withthedensity ψ(t,x)2 whichis not
| |
given a probabilistic interpretation. The dynamics and shape of a wave corpuscle
is governed by a nonlinear Klein-Gordon or a nonlinear Schr¨odinger equation in
relativisticandnonrelativisticcasesrespectively,andawave-corpuscleisdefinedas
a special type of solutions to these equations. Consequently an elementary charge
inthe WCMdoesnothaveaprescribedgeometrywhichdiffersitfromthe classical
AbrahamandLorentzmodelsandmorerecentdevelopments(see[7],[23],[25],[18],
[35] and references therein). Another important difference is that Newton’s equa-
tions are not postulated as in (1) but rather are derivedfrom the field equations in
a non-relativistic regime when charges are well separated and localized. Physical
aspects of the model are discussed in detail in [5], [6], where the Poincar´e type
cohesive forces are constructed somewhat differently then we propose here. The
primary focus of this paper is on studies of mathematical properties of the field
equations of the WCM.
The paper is organized as follows. In the first two subsections of this Intro-
duction we describe the system relativistic and non-relativistic Lagrangians and
the corresponding field equations, and in the third one we consider the relevant
nonlinearities and their examples. Importantly, the nonlinearity depends on a size
parametera>0associatedwithafreesolution(groundstate). Section2isdevoted
to remote interaction regimes for many charges. We provethere that in the case of
the non-relativistic field equations when a 0 (macroscopic limit) the centers of
→
the interacting charges defined by the formula
2
rℓ(t)= x ψℓ(t,x) dx,ℓ=1,...,N,
R3
Z (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
NEOCLASSICAL THEORY OF ELECTRIC CHARGES 3
convergetothesolutionsofNewton’sequationswiththeLorentzforcesifψℓ remain
localized. We also provide examples of exact solutions of the field equations in
the form of accelerating solitons for which the localization assumption holds. In
Section 3 we study systems of bound charges and time multiharmonic solutions to
the field equations. We consider there, in particular, a connection, discovered in
a different setting in [12], between the Planck-Einstein energy-frequency relation
and the logarithmic nonlinearity as well as some dynamical issues related to the
logarithmic nonlinearity closely related to results in [14], [16]. We continue then
withastudyofasystemoftwochargeswiththelogarithmicnonlinearityasamodel
forthehydrogenatom. Weprovethatifκ=a /a 0,wherea istheBohrradius,
1 1
→
then the lower energy levels associatedwith this model convergeto the well known
energy levels of the linear Schr¨odinger operator for the hydrogen atom. The proof
is based on an approachdeveloped in [9], [10] with certain modifications.
1.1. Relativistic Lagrangian and field equations. Let us consider a system
of N charges interacting directly only with the EM field described by its 4-vector
potential (ϕ,A). The charges are described by their wave functions ψℓ with the
superscript index ℓ = 1,...,N labeling them. The EM fields are related to the
potentials ϕ,A by the following standard relations
1
E= ϕ ∂ A, B= A. (4)
t
−∇ − c ∇×
We introduce now the system Lagrangian which is though similar to the one in-
L
troduced in [5], [6] but differs from it. A different Lagrangianis introduced here in
ordertogetthehydrogenatomfrequencyspectrumwithdesiredprecision. Namely,
to every ℓ-th charge is assigned an adjunct potential ϕℓ,Aℓ describing its addi-
tional degrees of freedom and our relativistic Lagrangianis defined by
(cid:0) (cid:1)
(ϕ,A), ψℓ N , ϕℓ,Aℓ N = 1 ϕ+ 1∂ A 2 ( A)2 (5)
L(cid:18) n oℓ=1 (cid:8)(cid:0) (cid:1)(cid:9)ℓ=1(cid:19) 8π "(cid:18)∇ c t (cid:19) − ∇× #
χ2 1 2 2 2
+ ∂˜ℓψℓ ˜ψℓ κ2 ψℓ Gℓ ψℓ∗ψℓ
2mℓ c2 t − ∇ − 0ℓ −
Xℓ (cid:20) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:16) (cid:17)(cid:21)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
1(cid:12) (cid:12)ϕℓ+(cid:12) 1∂(cid:12)Aℓ 2 (cid:12) (cid:12) Aℓ 2 ,
t
− ℓ 8π "(cid:18)∇ c (cid:19) − ∇× #
X (cid:0) (cid:1)
where (i) the covariant derivatives are defined by
∂˜ℓ =∂ + iqℓ ϕ−ϕℓ , ˜ℓ = iqℓ A−Aℓ , (6)
t t χ ∇ ∇− (cid:16) χc (cid:17)
(cid:0) (cid:1)
(ii)ψ∗ iscomplexconjugatetoψ;(iii)mℓ >0istheℓ-thchargemass,qℓ isthevalue
of the charge, κ = mℓc, and χ > 0 is a constant similar to the Planck constant
0ℓ χ
~= h ; (iv) Gℓ ψℓ∗ψℓ is a nonlinearity which will be described below.
2π
(cid:16) (cid:17)
4 ANATOLIBABINANDALEXANDERFIGOTIN
The Euler-Lagrange field equations for the Lagrangian defined in (5) include,
L
first of all, the Maxwell equations for the EM potentials
1
∂ A+ ϕ = 4π ρℓ, (7)
t
∇· c ∇ −
(cid:18) (cid:19) ℓ
X
1 1 4π
( A)+ ∂ ∂ A+ ϕ = Jℓ, (8)
t t
∇× ∇× c c ∇ c
(cid:18) (cid:19) ℓ
X
where the charge densities and currents are defined by
2
qℓ ψℓ ∂ ψℓ
ρℓ = χIm t +qℓ ϕ ϕℓ , (9)
− m(cid:12)(cid:12)ℓc2(cid:12)(cid:12) ψℓ − !
(cid:12) (cid:12) (cid:0) (cid:1)
2
qℓ ψℓ ψℓ qℓ A Aℓ
Jℓ = χIm∇ − . (10)
m(cid:12)(cid:12) ℓ(cid:12)(cid:12)  ψℓ − (cid:16) c (cid:17)
(cid:12) (cid:12)
 
The field equations include similar Maxwell equations for the adjunct potentials
1
∂ Aℓ+ ϕℓ = 4πρℓ, (11)
t
∇· c ∇ −
(cid:18) (cid:19)
1 1 4π
Aℓ + ∂ ∂ Aℓ+ ϕℓ = Jℓ, ℓ=1,...,N, (12)
t t
∇× ∇× c c ∇ c
(cid:18) (cid:19)
(cid:0) (cid:1)
and equations for the wave functions ψℓ in the form of nonlinear Klein-Gordon
equations
1 ∂˜ℓ∂˜ℓψℓ+ ˜ℓ2ψℓ Gℓ′ ψℓ∗ψℓ ψℓ κ2ψℓ =0, ℓ=1,...,N. (13)
− c2 t t ∇ − − 0
(cid:16) (cid:17)
Note that equations (13) for ψℓ are coupled to the equations for EM potentials via
the covariantderivatives.
The Lagrangian and equations are Lorentz and gauge invariant, and one can
verify that defined above ρℓ and Jℓ satisfy conservation/continuity equation
1
∂ ρℓ+ Jℓ =0.
t
c ∇·
If we choose the Lorentz gauge
1 1
∂ ϕ+ A=0, ∂ ϕℓ+ Aℓ =0 (14)
t t
c ∇· c ∇·
for all potentials, equations (7)-(12) take the form
N
1
ϕ ∂2ϕ= 4πρℓ, (15)
∇ − c2 t −
ℓ=1
X
N
1 4π
∂2A 2A= Jℓ, (16)
c2 t −∇ c
ℓ=1
X
1
2ϕℓ ∂2ϕℓ = 4πρℓ, (17)
∇ − c2 t −
1 4π
∂2Aℓ 2Aℓ = Jℓ,ℓ=1,...,N, (18)
c2 t −∇ c
NEOCLASSICAL THEORY OF ELECTRIC CHARGES 5
whereρℓ,Jℓ aredefinedby (9),(10). Basedonthe aboveequationsweassumethat
always
(ϕ,A)= ϕℓ,Aℓ . (19)
ℓ
X(cid:0) (cid:1)
Hence,fromnowonweassumethatintherelativisticcasethedynamicsofEMfields
and charges is determined by equations (17), (18), (9), (10), (13) andaccordingto
(19)and(6)thecovariantderivatives∂˜ℓand ˜ℓin(13)involveonlyfields ϕℓ′,Aℓ′
t ∇
with ℓ′ =ℓ, (cid:16) (cid:17)
6
∂˜ℓ =∂ + iqℓ ϕℓ′, ˜ℓ = iqℓ Aℓ′, (20)
t t χ ∇ ∇− χc
ℓ′6=ℓ ℓ′6=ℓ
X X
implying that the adjunct potentials completely compensate the EM self-actionfor
every charge and effectively there is no EM self-interaction (in [5], [6] this kind of
compensation was gained by using an additional nonlinear self-interaction).
Let us turn now to the nonlinearities G. We introduce the nonlinearities here
the same way as in [5], [6] with an intension to have a rest (ground) state for a
singlechargein the form ofa localizedwavefunction (form factor). Notice thatfor
a single charge evidently N =1, ϕ ϕℓ =0, A Aℓ = 0 and we look for the rest
− −
solution in the form
mc2
ψℓ(t,x)=e−iω0tψ(x), ω = =cκ (21)
0 0
χ
with Aℓ =0. Substituting (21) in (13) yields
2ϕ=4π ψ 2,
−∇ | |
2ψ+G′(ψ∗ψ)ψ =0.
−∇
Nowwechooseastrictlypositive,monotonicallydecreasingradialfunction˚ψ(ground
state)asaparameterforthemodelanddeterminethe nonlinearityG′ fromthe fol-
lowing charge equilibrium condition:
2
2˚ψ+G′ ˚ψ ˚ψ =0. (22)
−∇
(cid:18)(cid:12) (cid:12) (cid:19)
Theaboveequationallowstodeterminethe(cid:12)(cid:12)n(cid:12)(cid:12)onlinearityG′ aslongas˚ψ isastrictly
positive, smooth and monotonically decreasing function of r = x. In Section 1.3
| |
we consider the nonlinearity in more details providing also examples.
Note that using the Lorentz invariance of the system one can easily obtain a
solutionwhichrepresentsthe charge-fieldmovingwithaconstantvelocityv simply
by applyingto the restsolution ψℓ,ϕ,0 the Lorentztransformation(see [5], [6]).
(cid:16) (cid:17)
1.2. Non-relativistic Lagrangian and field equations. Our non-relativistic
model describes the case of charges moving with non-relativistic velocities and it
is set as follows. Using the frequency-shifting substitution (21) with more general
ψ =ψℓ (t,x) whichdepends on(t,x) we observethat the secondtime derivativein
ω
(13) can be written in the form
1 1 iqℓ 2 mℓ iqℓ
∂˜ℓ∂˜ℓψℓ = ∂ + ϕ ψℓ 2i ∂ + ϕ ψℓ +κ2ψℓ,
−c2 t t ω c2 t χ 6=ℓ ω− χ t χ 6=ℓ ω 0 ω
(cid:18) (cid:19) (cid:18) (cid:19)
6 ANATOLIBABINANDALEXANDERFIGOTIN
where
ϕ = ϕℓ′. (23)
6=ℓ
ℓ′6=ℓ
X
We neglect the term with the factor 1 and substitute 2imℓ ∂ + iqℓϕ ψℓ +
c2 − χ t χ 6=ℓ ω
κ2ψℓ fortheterm 1∂˜ℓ∂˜ℓψℓ in(13). Consequently,wereplace(cid:16)thenonlinear(cid:17)Klein-
0 ω −c2 t t ω
Gordon equation (13) by the following nonlinear Schr¨odinger equation (where we
denote the frequency shifted ψℓ by ψℓ)
ω
χi∂ ψℓ+ χ2 ˜ℓ 2ψℓ χ2 Gℓ′ ψℓ∗ψℓ ψℓ qℓϕ ψℓ =0, (24)
t 2mℓ ∇ − 2mℓ − 6=ℓ
(cid:16) (cid:17) (cid:16) (cid:17)
where ˜ℓ is given by (20). Since the magnetic fields generated by moving charges
∇
alsohavecoefficient 1 we neglectthem preservingonly the externalmagnetic fields
c
and replace ˜ℓ by the covariantgradient ˜ℓ defined by
∇ ∇ex
˜ℓ = iqℓAex. (25)
∇ex ∇− χc
So our non-relativistic Lagrangianis
ˆ ϕ, ψℓ N , ϕℓ N = |∇ϕ|2 + Lˆℓ ψℓ,ψℓ∗,ϕ , (26)
L0 ℓ=1 ℓ=1 8π
(cid:18) n o (cid:8) (cid:9) (cid:19) Xℓ (cid:16) (cid:17)
Lˆℓ = χi ψℓ∗∂ ψℓ ψℓ∂ ψℓ∗ χ2 ˜ℓ ψℓ 2+Gℓ ψℓ∗ψℓ
0 2 t − t − 2mℓ ∇ex −
h i (cid:26)(cid:12) (cid:12) (cid:16) (cid:17)(cid:27)
(cid:12) (cid:12)ϕℓ 2
qℓ ϕ+ϕ ϕℓ ψℓψℓ(cid:12)∗ ∇(cid:12) ,
− ex− − 8π
(cid:12) (cid:12)
(cid:0) (cid:1) (cid:12) (cid:12)
where A (t,x) and ϕ (t,x) are potentials of external EM fields, ψℓ∗ is complex
ex ex
conjugateofψℓ. TheEuler-Lagrangeequationsforthe electrostaticpotentialshave
the form
N
1
2ϕ= qℓψℓψℓ∗, (27)
− 4π∇
ℓ=1
X
1
2ϕℓ =qℓψℓψℓ∗,ℓ=1,...,N. (28)
− 4π∇
Assuming that that ϕ,ϕℓ vanish at infinity we obtain a reduced version of (19)
ϕ= ϕℓ. (29)
ℓ
X
Similarly to the relativistic case we assume now that nonrelativistic equations for
dynamics of charges intheexternalEMfieldwithpotentialsϕ ,A taketheform
ex ex
iχ∂ ψℓ = χ2 ˜ℓ 2ψℓ+qℓ ϕ +ϕ ψℓ+ χ2 Gℓ ′ ψℓ 2 ψℓ, (30)
t −2mℓ ∇ex 6=ℓ ex 2mℓ a
(cid:16) (cid:17) (cid:0) (cid:1) (cid:2) (cid:3) (cid:18)(cid:12)(cid:12) (cid:12)(cid:12) (cid:19)
where ϕ is given by (23) and ϕℓ is determined from the equatio(cid:12)ns(cid:12)
6=ℓ
2
2ϕℓ = 4πqℓ ψℓ ,ℓ=1,...,N. (31)
∇ −
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
NEOCLASSICAL THEORY OF ELECTRIC CHARGES 7
The solution of the above equation is given by the formula
2
ψℓ (t,y)
ϕℓ(t,x)=qℓ dy. (32)
(cid:12) y(cid:12) x
ZR3 (cid:12)(cid:12) | (cid:12)(cid:12)− |
The nonlinear self-interaction terms Gℓ in (30) are determined through the charge
a
equilibrium equation(22) and index a indicates the dependence on the size param-
eter a>0 which we introduce by the formula
G′ (s)=a−2G′ a3s . (33)
a 1
More detailed discussion of the nonlinearity is(cid:0)give(cid:1)n in the following section.
1.3. Nonlinearity, its basic properties and examples. As we have already
mentioned, the nonlinear self interaction function G is determined from the charge
equilibrium equation (22)basedontheformfactor(freegroundstate)˚ψ. Important
features ofour nonlinearity include: (i) the boundedness or slow subcriticalgrowth
of its derivative G′(s) for s with consequent boundedness from below of the
→ ∞
energy;(ii)slightlysingularbehaviorabouts=0,thatisforsmallwaveamplitudes.
InthissectionweconsidertheconstructionofthefunctionG,studyitsproperties
and provide examples for which the construction of G is carried out explicitly.
Throughout this section we have
ψ,˚ψ 0 and hence ψ =ψ.
≥ | |
We introduce explicitly the dependence of the free ground state ˚ψ on the size pa-
rameter a>0 through the following representation of the function˚ψ(r)
˚ψ(r)=˚ψ (r)=a−3/2˚ψ a−1r , (34)
a 1
where ˚ψ (r) is a function of the dimensionless var(cid:0)iable(cid:1)r 0. The dependence on
1 ≥
a is chosen so that L2-norm ˚ψ (r) does not depend on a, hence the function
a
˚ψ (r) satisfies the charge norm(cid:13)alizati(cid:13)on condition (53) for every a>0. Obviously,
a (cid:13) (cid:13)
definition (34) is consistentwit(cid:13)h (22)(cid:13)and (33). The size parameter a naturally has
the dimension of length. A properly defined spatial size of˚ψ , based, for instance,
a
onthevariance,isproportionaltoawithacoefficientdependingon˚ψ . Thecharge
1
equilibrium equation (22) can be written in the following form:
2˚ψ =G′ ˚ψ2 ˚ψ . (35)
∇ a a a a
(cid:16) (cid:17)
The function ˚ψ (r) is assumed to be a smooth (at least twice continuously dif-
a
ferentiable) positive monotonically decreasing function of r 0 which is square
integrablewith weight r2, we assume that its derivative˚ψ′ (r)≥is negativefor r>0
a
and we assume it to satisfy the charge normalization condition of the form (53);
such a function is usually called in literature a ground state.
Let us look first at the case a = 1, ˚ψ = ˚ψ , ˚ϕ = ˚ϕ , for which the equation
a 1 a 1
(35) yields the following representation for G′(˚ψ2) from (35)
1
G′ ˚ψ2(r) = (∇2˚ψ1)(r). (36)
1 1 ˚ψ (r)
(cid:16) (cid:17) 1
8 ANATOLIBABINANDALEXANDERFIGOTIN
Since ˚ψ2(r) is a monotonic function, we can find its inverse r =r ψ2 , yielding
1
G′ (s)= ∇2˚ψ1(r(s)), 0=˚ψ2( ) s ˚ψ2(0).(cid:0) (cid:1) (37)
1 ˚ψ (r(s)) 1 ∞ ≤ ≤ 1
1
Since ˚ψ (r) is smooth and ∂ ˚ψ < 0, G′(ψ 2) is smooth for 0 < ψ 2 < ˚ψ2(0). If
1 r 1 | | | | 1
we do not need G′(s) to be smooth, we extend G′(s) for s ˚ψ2(0) as a constant,
≥ 1
namely
G′ (s)=G′ ˚ψ2(0) if s ˚ψ2(0). (38)
1 1 1 ≥ 1
The first derivative of such an exten(cid:16)sion at(cid:17)s=˚ψ2(0) has a discontinuity point. If
1
˚ψ (r) is a smooth function of class Cn, n > 2, we always can define an extension
a
of G′(s) for s ˚ψ2(0) as a bounded function of class Cn−2 for all r >0 and
≥ 1
G′ (s)=G′ ˚ψ2(0) 1 if s ˚ψ2(0)+1. (39)
1 1 1 − ≥ 1
(cid:16) (cid:17)
Slowlygrowing(subcritical) functions G′(s)whichare notconstantfor larges also
can be used, see examples below.
In the case of arbitrary size parameter a > 0 we define G′ (s) by formula (33),
a
and this definition is consistent with (34) and (37).
Let us take a look at general properties of G′(s) as they follow from defining
them relations (37). In the examples below the function G′(s) is not differentiable
at s = 0, but if ˚ψ(r) decays exponentially or with a power law the nonlinearity
g(ψ) = G′(ψ 2)ψ as it enters the field equation (30) is differentiable for all ψ
| |
including zero, hence it satisfies the Lipschitz condition. For a Gaussian ˚ψ (r)
1
which decays superexponentially G′(ψ 2) is unbounded at zero and g(ψ) is not
| |
differentiable at zero. Since ˚ψ(x) > 0, the sign of G′ ψ 2 coincides with the
| | 1 | |
sign of 2˚ψ (x). At the originx=0 the function˚ψ ((cid:16)x) ha(cid:17)s its maximum and,
∇ 1 | | 1 | |
consequently, G′ (s) 0 for s close to s = ˚ψ2(0). The Laplacian applied to the
1 ≤ 1
radialfunction˚ψ takestheform 1 ∂2 r˚ψ x . Consequently,ifr˚ψ (r)isconvex
1 r∂r2 1| | 1
atr = x wehave 2˚ψ (x) 0. Sinc(cid:16)er2˚ψ (r(cid:17))isintegrable,wenaturallyassume
| | ∇ 1 | | ≥ 1
that x ˚ψ (x) 0 as x . Then if the second derivative of r˚ψ (r) has a
| | 1 | | → | | → ∞ 1
constant sign near infinity, it must be non-negative as well as G′ (s) for s 1. In
1 ≪
the examples we give below G′ (s) has exactly one zero on the half-axis.
1
Example 1. Consider a form factor˚ψ (r) decaying as a power law, namely
1
c
˚ψ (r)= pw , (40)
1 (1+r2)5/4
where c is the normalizationfactor, c =31/2/(4π)1/2. This function evidently
pw pw
is positive and monotonically decreasing. Let us find now G′(s) based on the
relations (37). An elementary computation of 2˚ψ shows that
∇ 1
15s2/5 45s4/5
G′(s)= , (41)
4c4/5 − 4c8/5
pw pw
75s7/5 25s9/5
G(s)= , for 0 s c2 .
28c4/5 − 4c8/5 ≤ ≤ pw
pw pw
NEOCLASSICAL THEORY OF ELECTRIC CHARGES 9
The extension for s c2 can be defined as a constant or the same formula (41)
≥ pw
can be used for all s 0 since corresponding nonlinearity has a subcritical growth.
≥
If we explicitly introduce size parameter a into the form factor using (34), we
define G′ (s) by (33). Notice that the variance of the form factor˚ψ2(x) decaying
a 1 | |
as a power law (40) is infinite.
Example 2. Now we consider an exponentially decaying form factor ˚ψ of the
1
form
∞ −1/2
˚ψ (r)=c e−(r2+1)1/2, c = 4π r2e−2(r2+1)1/2dr . (42)
1 e e
(cid:18) Z0 (cid:19)
Evidently ˚ψ (r) is positive and monotonically decreasing. The dependence r(s)
1
defined by the relation (42) is as follows:
r = ln2 c /√s 1 1/2, if √s ˚ψ (0)=c e−1. (43)
e − ≤ 1 e
An elementary comp(cid:2)utat(cid:0)ion sho(cid:1)ws th(cid:3)at
2˚ψ 2 1 1
∇ 1 = + + 1. (44)
− ˚ψ1 (r2+1)12 (r2+1) (r2+1)32 −
Combining (43) with (44) we readily obtain the following function for s c2e−2
≤ e
4 4 8
G′ (s)= 1 . (45)
1 − ln(c2/s) − ln2(c2/s) − ln3(c2/s)
(cid:20) e e e (cid:21)
We can extend it for larger s as follows:
G′ (s)=G′ c2e−2 = 3 if s c2e−2, (46)
1 1 e − ≥ e
orwecanuseasmoothextensiona(cid:0)sin(39(cid:1)). ThefunctionG′1(s)isnotdifferentiable
at s = 0. At the same time the function g(ψ) = G′ (ψ(r))ψ if we set g(0) = 0
1
is continuous and g(ψ) is continuously differentiable with respect to ψ at zero and
g(ψ) satisfies a Lipschitz condition. The variance of the exponential form factor
˚ψ (r) is obviously finite. To find G′ (s) for arbitrary a we use its representation
1 a
(33).
Example 3. Now we define a Gaussian form factor by the formula
˚ψ(r)=C e−r2/2, C = 1 . (47)
g g π3/4
Such a ground state is called gausson in [12]. Elementary computation shows that
∇2˚ψ(r) =r2 3= ln ˚ψ2(r)/C2 3.
˚ψ(r) − − g −
(cid:16) (cid:17)
Hence, we define the nonlinearity by the formula
G′ ψ 2 = ln ψ 2/C2 3, (48)
| | − | | g −
(cid:16) (cid:17) (cid:16) (cid:17)
andrefertoitaslogarithmic nonlinearity. Thenonlinearpotentialfunctionhasthe
form
s 1
G(s)= ln s′/C2 3 ds′ = slns+s ln 2 . (49)
− g − − π3/2 −
Z0 (cid:18) (cid:19)
Dependence on the si(cid:0)ze pa(cid:0)ramete(cid:1)r a>(cid:1)0 is given by the formula
G′ ψ 2 = a−2ln a3 ψ 2/C2 3a−2. (50)
a | | − | | g −
(cid:16) (cid:17) (cid:16) (cid:17)
10 ANATOLIBABINANDALEXANDERFIGOTIN
Obviouslyg(ψ)=G′ ψ 2 ψ is continuous for all ψ C if at zero we set g(0)=0
1 | | ∈
and is differentiable for every ψ =0 but is not differentiable at ψ =0 and does not
(cid:0) (cid:1) 6
satisfy the Lipschitz condition. Notice also that g(ψ) has a subcritical growth as
ψ .
| |→∞
2. Charges in remote interaction regimes
The primary focus of this section is to show that if the size parameter a 0
→
then the dynamics of centers of localized solutions is approximated by the Newton
equations with the Lorentz forces. This is done in the spirit of the well known in
the quantum mechanics Ehrenfest Theorem, [34, Sections 7, 23]. We also provide
as an example explicit wave-corpusclesolutions which have such a dynamics.
Asafirststepwedescribebasicpropertiesoftheclassicalsolutionsof(30),(31).
The Lagrangian ˆis gauge invariant and every ℓ-th charge has a 4-current ρℓ,Jℓ
L
defined by
(cid:0) (cid:1)
2 χqℓ ψℓ qℓ2A 2
ρℓ =q ψℓ , Jℓ = Im∇ ex ψℓ , (51)
mℓ ψℓ − mℓc !
(cid:12) (cid:12) (cid:12) (cid:12)
which satisfies the con(cid:12)(cid:12)tinu(cid:12)(cid:12)ity equations ∂tρℓ+ Jℓ =0 or (cid:12)(cid:12) (cid:12)(cid:12)
∇·
2 χ ψℓ 2 qℓ 2
∂ ψℓ + Im∇ ψℓ A ψℓ =0. (52)
t ∇· mℓ ψℓ − mℓc ex !
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
Note that Jℓ de(cid:12)fine(cid:12)d by (51) agrees with(cid:12)th(cid:12)e definition o(cid:12)f th(cid:12)e current (10) in the
Maxwell equations. Equations (52) can be obtained by multiplying (30) by ψℓ∗
and taking the imaginary part. Integrating the continuity equation we see that
2
ψℓ =const and we impose the following normalization condition:
(cid:13) (cid:13)
(cid:13)(cid:13) (cid:13)(cid:13) ψℓ 2 = ψℓ 2dx=1, t 0, ℓ=1,...,N. (53)
R3 ≥
(cid:13) (cid:13) Z (cid:12) (cid:12)
The motivation for(cid:13)thi(cid:13)s particu(cid:12)lar(cid:12)normalization is that this normalization allows
(cid:13) (cid:13) (cid:12) (cid:12)
2 2
ψℓ = ψℓ toconvergetoadelta-functionand,inaddition,forthisnormalization
a
(cid:12)the(cid:12)solut(cid:12)ion(cid:12)to (31)givenby the formula(32)convergesto the Coulomb’spotential
(cid:12) (cid:12) (cid:12) (cid:12)
w(cid:12) ith(cid:12) the(cid:12)val(cid:12)ueqℓofthechargeif ψℓ 2convergestoadeltafunction. Themomentum
a
density Pℓ for the Lagrangian ˆ(cid:12) in(cid:12) (26) is defined by the formula
L(cid:12)0 (cid:12)
(cid:12) (cid:12)
Pℓ = iχ ψℓ˜ℓ∗ψℓ∗ ψℓ∗˜ℓψℓ .
2 ∇ − ∇
(cid:16) (cid:17)
Note that so defined momentum density Pℓ is related with the current Jℓ in (51)
by the formula
mℓ
Pℓ(t,x)= Jℓ(t,x). (54)
qℓ
We introduce the total individual momenta Pℓ for ℓ-th charge by
Pℓ = Pℓdx, (55)
R3
Z
and obtain the following equations for the total individual momenta
dPℓ =qℓ Eℓ′ +E ψℓ 2+ 1vℓ B dx, (56)
dt ZR3(cid:20)(cid:16)Xℓ′6=ℓ ex(cid:17)(cid:12) (cid:12) c × ex(cid:21)
(cid:12) (cid:12)
(cid:12) (cid:12)