electromagnetism as a fundamental theory - nc … as a fundamental theory one theory for all spacial...

Electromagnetism as a Fundamental TheoryOne theory for all spacial scales

Alexander Figotin and Anatoli Babin

University of California at IrvineThe work was supported by AFOSR

March 11, 2011

A. Figotin and A. Babin (UCI) Electromagnetism as a Fundamental Theory March 11, 2011 1 / 46

Importance of Electromagnetic (EM) Interactions

It is the only interaction that is equally significant in all spacial scales:gigantic cosmic scales when electromagnetic radiation including lightpropagates through the Universe, macroscopic scales covering our lifeon Earth and microscopic atomic scales.

The EM interactions are fundamental to the chemistry andconsequently to the biology.

The EM interactions is a factual basis of the special relativityprinciple.


Einstein on the Electromagnetic Theory

"The most fascinating subject at the time that I was a student wasMaxwell’s theory. What made this theory appear revolutionary was thetransition from forces at a distance to fields as fundamental variables.

If one views this phase of the development of theory critically, one isstruck by the dualism which lies in the fact that the material point inNewton’s sense and the field as continuum are used as elementaryconcepts side by side.

The electron is a stranger in electrodynamics.

I feel that it is a delusion to think of the electrons and the fields astwo physically different, independent entities. Since neither can existwithout the other, there is only one reality to be described, whichhappens to have two different aspects; and the theory ought torecognize this from the start instead of doing things twice."


Einstein on the Electromagnetic Theory

"Before Maxwell people thought of physical reality-in so far as itrepresented events in nature-as material points, whose changesconsist only in motions which are subject to total differentialequations. After Maxwell they thought of physical reality asrepresented by continuous fields, not mechanically explicable, whichare subject to partial differential equations. This change in theconception of reality is the most profound and the most fruitful thatphysics has experienced since Newton; but it must also be grantedthat the complete realisation of the programme implied in this ideahas not by any means been carried out yet...."


Challenges in constructing a theory of EM Interactions

It has to be one thery for all spacial scales macroscopic andmicroscopic.

Clean separation of elementary and statistical aspects of theory.

An elementary charge can not be a mathematical point as it is todayin the classical and quantum mechanics, but it has to have astructure.


Maxwell’s (1831—1879) insights on constructing an EMtheory

"As the development of the ideas and methods of pure mathematicshas rendered it possible, by forming a mathematical theory ofdynamics, to bring to light many truths which could not have beendiscovered without mathematical training, so, if we are to formdynamical theories of other sciences, we must have our minds imbuedwith these dynamical truths as well as with mathematical methods.

We must therefore discover some method of investigation whichallows the mind at every step to lay hold of a clear physicalconception, without being committed to any theory founded on thephysical science from which that conception is borrowed, so that it isneither drawn aside from the subject in pursuit of analyticalsubtleties, nor carried beyond the truth by a favorite hypothesis."


Maxwell’s insights on constructing an EM theory

Maxwell compars his older and last versions of his theory in a letter tohis friend Peter Guthrie Tait:

“The former is built up to show that the phenomena [ofelectromagnetism] are such as can be explained by mechanism. Thenature of the mechanism is to the true mechanism what an orrery isto the Solar System. The latter is built on Lagrange’s DynamicalEquations and is not wise about vortices.”


Maxwell’s insights on constructing an EM theory

"The aim of Lagrange was to bring dynamics under the power of thecalculus.... Certain quantities (expressing the reactions between theparts of the system called into play by its physical connexions) appearin the equations of motion of the component parts of the system, andLagrange’s investigation, as seen from a mathematical point of view,is a method of eliminating these quantities from the final equations.

In following the steps of this elimination the mind is exercised incalculation, and should therefore be kept free from the intrusion ofdynamical ideas. Our aim, on the other hand, is to cultivate ourdynamical ideas. We therefore avail ourselves of the labours of themathematicians, and retranslate their results from the language of thecalculus into the language of dynamics, so that our words may call upthe mental image, not of some algebraical process, but of someproperty of moving bodies."


Some Mathematical Problems in a

Neoclassical Theory of Electric Charges

One theory for all spacial scales

"’Give me matter and motion,’he [Descartes] cried,’and I will construct the universe.’"

E. Whittaker.


Our papers on the subject

Babin A. and Figotin A., Some mathematical problems in aneoclassical theory of electric charges, Discrete and ContinuousDynamical Systems, 27, 4, 2010.Babin A. and Figotin A., Wave-Corpuscle Mechanics for ElectricCharges, Journal of Statistical Physics, November, 2009.

Babin A. and Figotin A., Electrodynamics of balanced charges,Foundations of Physics, 2010.


Some references

I. Bialynicki-Birula and J. Mycielski, Nonlinear Wave Mechanics,Annals of Physics, 100, 62-93, (1976); Gaussons: Solitons of theLogarithmic Schrödinger Equation, Physica Scripta, 20, 539-544(1979).

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations, Arch.Rational Mech. Anal. 82 (1983), 347—375.T. Cazenave, Stable solutions of the logarithmic Schrödingerequation, Nonlinear Anal. 7 (1983), 1127-1140. SemilinearSchrödinger equations, AMS, 2003.

T. Cazenave and P.-L. Lions, Orbital stability of standing waves forsome nonlinear Schrödinger equations, CMP, 85 (1982), no. 4,549—561.

A. Komech, Lectures on Quantum Mechanics (nonlinear PDE pointof view), Max Planck Institute, 2005.


Classical Electromagnetic Theory (CEM): Point Charge inEM Field

Point charge q of the mass m in an external EM field

ddt[mv (t)] = q

[E (r (t) , t) +

1cv (t)×B (r (t) , t)

]where r and v = dr

dt are respectively charge position and velocity,E (t, r) and B (t, r) are the electric field and the magnetic induction.EM field generated by a moving point charge

1c

∂B∂t+∇× E = 0, ∇ ·B = 0,

1c

∂E∂t−∇×B = −4π

cqδ (x− r (t)) v (t) ,

∇ · E = 4πqδ (x− r (t)) .


Closed Charge-"EM field" system

A closed system "charge-EM field" has a problem: divergence of theEM field exactly at the position of the point charge. Even for therelativistic motion equation

ddt[γmv (t)] = q

[E (r (t) , t) +

1cv (t)×B (r (t) , t)

],

where γ = 1/√1− v2 (t) /c2 is the Lorentz factor, and the system

has a Lagrangian the problem still persists.

This problem is well known and discussed in detail, for instance, in:I Durr D. and Teufel S., Bohmian Mechanics. The Physics andMathematics of Quantum Theory, Springer, 2009.

I Spohn H., Dynamics of Charged Particles and Their Radiation Field,Cambridge. 2004.


An "indulgence" from Hendric Lorentz (1853-1928)

An "indulgence" given by the great Hendric Lorentz (1853-1928) in1927 to future generations of physicist (from Section 42 "Structure ofthe Electron" of his monograph) reads:

"Therefore physicists are absolutely free to form any hypotheses onthe properties and size of electrons that may best suit them. You can,for instance, choose the old electron (a small sphere with chargeuniformly distributed over the surface) or Parson’s ring-shapedelectron, endowed with rotation and therefore with a magnetic field;you can also make different hypotheses about the size of the electron.In this connection I may mention that A. H. Compton’s experimentson the scattering of γ rays by electrons have led him to ascribe to theelectron a size considerably greater than it was formerly supposed tohave."


Electrodynamics of balanced charges (BEM)

The BEM theory is a relativistic Lagrangian theory. It is one theoryfor all spatial scales: macroscopic and atomic.

Balanced charge: a new concept for an elementary charge. It isdescribed by a complex-valued wave function as in the Schrodingerwave mechanics.

A b-charge does not interact with itself electromagnetically.

Every b-charge has its own elementary EM potential and thecorresponding EM field. It is naturally assigned a conservedelementary current via the Lagrangian.

B-charges interact with each other only through their elementary EMpotentials and fields.

The field equations for the elementary EM fields are exactly theMaxwell equations with the elementary conserved currents.

Force densities acting upon b-charges are described exactly by theLorentz formula.


BEM theory relation to the Classical EM and Quantumtheories?

When charges are well separated and move with nonrelativisticvelocities the BEM theory can be approximated by point chargesgoverned by the Newton equations with the Lorentz forces.

Radiative phenomena in the BEM theory are similar to those in theCEM theory in macroscopic scales.

The Hydrogen Atom spectrum and some other phenomena at atomicscales are described by the BEM theory similarly to the QuantumMechanics (QM).


However beautiful the strategy,

you should occasionally look at the results.

Sir Winston Churchill.


Electrodynamics of balanced charges (BEM)A system of N elementary b-charges

(ψ`,A`µ

), 1 ≤ ` ≤ N.

ψ` is the `-th b-charge wave function, A`µ and F `µν = ∂µA`ν − ∂νA`µ

are its elementary EM potential and field.The action upon the `-th charge by all other charges is described bythe `-th exterior EM potential and field:

A`µ6= = ∑`′ 6=`

A`′µ, A`µ6= =

(ϕ`6=,A`6=

), F `µν

6= = ∑`′ 6=`

F `′µν,

F `µν = ∂µA`ν − ∂νA`µ.

The total EM potential Aµ and field Fµν:

Aµ = ∑1≤`≤N

A`µ, Fµν = ∑1≤`≤N

F `µν.

The total EM field is just the sum of elementary ones, it has noindependent degrees of freedom which can carry EM energy.


Lagrangian for many interacting b-chargesLagrangian for the system of N b-charges:

L({

ψ`, ψ`;µ

},{

ψ`∗, ψ`∗;µ

},A`µ

)=

N

∑`=1

L`(

ψ`, ψ`;µ, ψ`∗, ψ`∗;µ

)+ LBEM,

LBEM = LCEM −Le, LCEM = −FµνFµν

16π, Le = − ∑

1≤`≤N

F `µνF `µν

16π,

where L` is the Lagrangian of the `-th bare charge, and the covariantderivatives are defined by the following formulas

ψ`;µ = ∂`µψ`, ψ`∗;µ = ∂`µ∗ψ`∗,

∂`µ = ∂µ +iq`A`µ6=

χc, ∂`µ∗ = ∂µ −

iq`A`µ6=χc

.

Covariant differentiation operators ∂µ and ∂∗µ provide for thecoupling between the charge and the EM field.


Lagrangian for many interacting b-chargesEM part LBEM can be obtained by the removal from the classical EMLagrangian LCEM all self-interaction contributions

LBEM = − ∑{`,`′}:`′ 6=`

F `µνF `′

µν

16π= − ∑

1≤`≤N

F `µνF `6=µν

16π.

The "bare" charge Lagrangians L` are

L`(

ψ`, ψ`;µ, ψ`∗, ψ`∗;µ

)=

χ2

2m`

{ψ`∗;µ ψ`;µ − κ`2ψ`∗ψ` − G `

(ψ`∗ψ`

)},

G ` is a nonlinear self-interaction function;m` > 0 is the charge mass; q` is the value of the charge;χ > 0 is a constant similar to the Planck constant h = h

2π and

κ` =ω`

c=m`c

χ, ω` =

m`c2

χ.


Euler-Lagrange field equations

From the gauge invariance via the Noether’s theorem we getelementary conserved currents,

J`ν = −iq`

χ

(∂L`

∂ψ`;νψ` − ∂L`

∂ψ∗`;νψ∗`)= −c

∂L`

∂A`6=ν

,

with the conservation law

∂νJ`ν = 0, ∂tρ` +∇ · J` = 0, J`ν =

(ρ`c, J`

).

Elementary wave equations[∂`µ∂`µ + κ`2 + G `′

(∣∣∣ψ`∣∣∣2)]ψ` = 0, ∂`µ = ∂µ +iq`A`µ6=

χc,

and similar equations for the conjugate ψ∗`.



Elementary Maxwell equations

∂µF `µν =4π

cJ`ν,

or in the familiar vector form

∇ · E` = 4π$`, ∇ ·B` = 0,

∇× E` + 1c

∂tB` = 0, ∇×B` − 1c

∂tE` =4π

cJ`.



Elementary currents:

J`ν = −q`χ

∣∣ψ`∣∣2m`

(Im

∂νψ`

ψ`+q`A`ν6=

χc

),

or in the vector form

ρ` = −q`∣∣ψ`∣∣2m`c2

(χ Im

∂tψ`

ψ`+ q`ϕ`6=

),

J` =q`∣∣ψ`∣∣2m`

(χ Im

∇ψ`

ψ`−q`A`6=

c

).


Single relativistic charge and the nonlinearityLagrangian

L0 =χ2

2m

{∣∣∂tψ∣∣2c2

−∣∣∇ψ

∣∣2 − κ20 |ψ|2 − G (ψ∗ψ)}

.

Without EM self-interaction L0 does not depend on the potentials ϕ,A! Though we can still find the potentials based on the elementaryMaxwell equations they have no role to play and carry no energy.Rest state of the b-charge

ψ (t, x) = e−iω0t ψ (x) , ω0 =mc2

χ= cκ0,

ϕ (t, x) = ϕ (x) , A (t, x) = 0,

where ψ (|x|) and ϕ = ϕ (|x|) are real-valued radial functions, and werefer to them, respectively, as form factor and form factor potential.Rest charge equations:

−∇2ψ+ G ′(∣∣ψ∣∣2) ψ = 0, −∇2 ϕ = 4π

∣∣ψ∣∣2 .


Charge equilibrium equation

Charge equilibrium equation for the resting charge:

−∇2ψ+ G ′(∣∣ψ∣∣2) ψ = 0.

It signifies a complete balance of the two forces: (i) internal elasticdeformation force −∆ψ; (ii) internal nonlinear self-interactionG ′(|ψ|2

)ψ.

We pick the form factor ψ considering it as the model parameter andthen the nonlinear self interaction function G is determined based onthe charge equilibrium equation .

We integrate the size of the b-charge into the model via sizeparameter a > 0:

G ′a (s) = a−2G ′1

(a3s)

, where G ′ (s) = ∂sG (s) .


Self-interaction nonlinearityThere is a certain degree of freedom in choosing the form factor andthe resulting nonlinearity.The proposed below choice is justified by its unique property: theenergies and the frequencies of the time-harmonic states of theHydrogen atom satisfy exactly the Einstein-Planck energy-frequencyrelation: E = hω (E = χω).The form factor is Gaussian and defined by

ψ (r) = Cg e−r2/2, Cg =

1π3/4 ,

implying

∇2ψ (r)ψ (r)

= r2 − 3 = − ln(

ψ2(r) /C 2g

)− 3.

ϕ (x) = q∫

R3

∣∣ψ (y)∣∣2|y− x| dy


Self-interaction nonlinearity: Gaussian form factor

Consequently, the nonlinearity reads

G ′ (s) = − ln(s/C 2g

)− 3,

implying

G (s) = −s ln s + s(ln

1π3/2 − 2

).

and we call it the logarithmic nonlinearity.

The nonlinearity explicit dependence on the size parameter a > 0 is

G ′a (s) = −a−2 ln(a3s/C 2g

)− 3.


Nonrelativistic BEM Lagrangian and its Field EquationsNon-relativistic Lagrangian

L0({

ψ`}N`=1

,{

ϕ`}N`=1

)=

∣∣∇∑` ϕ`∣∣2

8π+∑

`

L`(

ψ`, ψ`∗, ϕ)

,

L`0 =χi2

[ψ`∗∂tψ

` − ψ`∂tψ`∗]− χ2

2m`

{∣∣∣∇èxψ`∣∣∣2 + G ` (ψ`∗ψ`

)}−

−q`(

ϕ 6=` + ϕex

)ψ`ψ`∗ −

∣∣∇ϕ`∣∣2

8π,

where Aex (t, x) and ϕex (t, x) are potentials of external EM fields.The Euler-Lagrange field equations

iχ∂tψ` = − χ2

2m`

(∇èx

)2ψ`+q`

(ϕ 6=` + ϕex

)ψ`+

χ2

2m`

[G à]′ (∣∣∣ψ`∣∣∣2)ψ`,

∇2ϕ` = −4πq`∣∣∣ψ`∣∣∣2 , ` = 1, ...,N.

Consequently, potentials ϕ` can be represented as

ϕ` (t, x) = q`∫

R3

∣∣ψ`∣∣2 (t, y)|y− x| dy.


Exact wave-corpuscle solution for non-relativistic set upLet us assume a purely electric external EM field: Aex = 0,Eex (t, x) = −∇ϕex (t, x).We define the wave-corpuscle (soliton) ψ, ϕ by

ψ (t, x) = eiS/χψ, S = mv (t) · (x− r) + sp (t) ,

ψ = ψ (|x− r|) , ϕ = ϕ (|x− r|) , r = r (t) .

where ψ is the Gaussian form factor with the corresponding potentialϕ;r (t) is determined from the point charge equation

md2r (t)

dt2= qEex (t, r) ,

and v (t) , sp (t) are determined by formulas

v (t) =drdt

, sp (t) =∫ t

0

(mv2

2− qϕex (t, r (t))

)dt ′.


Exact wave-corpuscle (soliton) solutionTheoremSuppose that ϕex (t, x) is a continuous function which is linear withrespect to x. Then the wave-corpuscle wave function and its potential


ψ = ψ (|x− r|) , ϕ = ϕ (|x− r|) , r = r (t) .

is the exact solution to the Euler-Lagrange field equation provided r (t)satisfies the point charge equation

md2r (t)

dt2= qEex (t, r) ,

and v (t) , sp (t) are determined by formulas

v =drdt

, sp =∫ t

0

(mv2

2− qϕex (t, r (t))

)dt ′.


Multiharmonic solutions for a system of many charges

Field equations without external fields

iχ∂tψ`+

χ2

2m`∇2ψ`− q`ϕ 6=`ψ` =

χ2

2m`G ′`

(∣∣∣ψ`∣∣∣2)ψ`, ` = 1, ...,N,

ϕ 6=` = ∑`′ 6=`

ϕ`,14π∇2ϕ` = −q`

∣∣∣ψ`∣∣∣2 .

The total conserved energy

E = ∑`

E`,

E` =12

∫q`∣∣∣ψ`∣∣∣2 ϕ 6=` dx+

∫χ2

2m`

{∣∣∣∇ψ`∣∣∣2 + G ` (∣∣∣ψ`∣∣∣2)} dx.



`-th charge energy in the system’s field

E0` =∫q`∣∣∣ψ`∣∣∣2 ϕ 6=` dx+

∫χ2

2m`

{∣∣∣∇ψ`∣∣∣2 + G ` (∣∣∣ψ`∣∣∣2)} dx.

For a single charge E0` = E` = E . In the general case N ≥ 2 thetotal energy E does not equal the sum of E0`, and the differencebetween the sum of E0` and the total energy E coincides with thetotal energy of EM fields.

We assume

ψ` ∈ Ξ, where Ξ ={

ψ ∈ H1(R3) : ‖ψ‖2 =

∫|ψ|2 dx = 1

}.



Let us consider now multiharmonic solutions

ψ` (t, x) = e−iω`tψ` (x) , ϕ` (t, x) = ϕ` (x) .

14π∇2ϕ` = −q`

∣∣∣ψ`∣∣∣2 , or ϕ` (t, x) = q`∫

R3

∣∣ψ`∣∣2 (t, y)|y− x| dy.

Then ψ` (x) satisfy the following nonlinear eigenvalue problem,` = 1, ...,N

χω`ψ` +χ2

2m`∇2ψ` − q`ϕ 6=`ψ` −

χ2

2m`G ′`

(|ψ`|

2)

ψ` = 0.



Theorem

Let G`(s), ` = 1, ...,N be the logarithmic functions with a = a`. Suppose{ψσ`

}N`=1 ∈ ΞN , σ ∈ Σ is a set of solutions to the nonlinear eigenvalue

problem with the corresponding frequencies{

ωσ`

}N`=1 and finite energies{

Eσ0`

}N`=1. Then

E0` = χω` +χ2

2 (a`)2m`.

and any two solutions ψσ` = ψ` and ψσ1

` = ψ′` with σ, σ1 ∈ Σ satisfy thePlanck-Einstein relation

χ(ωσ` −ωσ1

`

)= Eσ

0` − Eσ10` , ` = 1, ...,N.


Newtonian Mechanics of point charges as an approximationWe introduce the `-th charge position r` (t) and velocity v` (t)

r` (t) = r`a (t) =∫

R3x∣∣∣ψ`a (t, x)∣∣∣2 dx, v` (t) =

1q`

∫R3J` (t, x) dx.

We find then that the positions and velocities are related exactly as inthe point charge mechanics:

dr` (t)dt

=∫

R3x∂t

∣∣∣ψ`∣∣∣2 dx = − 1q`

∫R3x∇·J` dx =

1q`

∫R3J`dx = v` (t) ,

We obtain the kinematic representation for the total momentumwhich is exactly the same as for the point charges mechanics

P` (t) =m`

q`

∫R3J` (t, x) dx = m`v` (t) ,


Newtonian Mechanics of point charges as an approximation

We obtain the following system of equations of motion for N charges:

m`d2r` (t)

d2t= q`

∫R3

[(∑`′ 6=` E

`′ + Eex

) ∣∣∣ψ`∣∣∣2 + 1cv` ×Bex

]dx, ` = 1, ...,N,

The derivation is similar to that of the Ehrenfest Theorem inquantum mechanics,

Suppose that for every `-th charge the densities∣∣ψ`∣∣2 and J` are

localized in a-vicinity of the position r` (t), and that∣∣∣r` (t)− r`′ (t)∣∣∣ ≥ γ > 0 with γ independent on a on time interval

[0,T ]. Then if a→ 0 we get∣∣∣ψ`∣∣∣2 (t, x)→ δ(x− r` (t)

), v` (t, x) = J`/q` → v` (t) δ

(x− r` (t)

),


Newtonian Mechanics of point charges as an approximation

We infer then

ϕ` (t, x)→ ϕ`0 (t, x) =q`

|x− r`| , ∇rϕ` (t, x)→

q`(x− r`

)|x− r`|3

as a→ 0.

Passing to the limit a→ 0 we get Newton’s equations of motion forpoint charges

m`d2r`

dt2= f`,

where f` are the Lorentz forces

f` = ∑`′ 6=`

q`E`′0 + q

`Eex

(r`)+1cv` ×Bex

(r`)

, ` = 1, ...,N,


Wave-corpuscle concept for an accelerating charge

We define the wave-corpuscle (soliton) ψ, ϕ by


ψ = ψ (|x− r|) , ϕ = ϕ (|x− r|) , r = r (t) .

where ψ is the Gaussian form factor with the corresponding potentialϕ;

If the potential ϕex (t, x) is linear in x then for any given trajectoryr (t)

ϕex (t, x) = ϕ0,ex (t) + ϕ′0,ex (t) · (x− r (t)) ,

where

ϕ′0,ex (t) = ∇xϕex (r (t) , t) , ϕ0,ex (t) = ϕex (t, r (t)) .


Wave-corpuscle concept for an accelerating charge

Observe then

∂tψ = exp(

iSχ

){[imχ(∂tv · (x− r)− v · ∂t r) +

ispχ

]ψ− ∂t r · ∇ψ

},

∇2ψ = exp(

iSχ

)[(imv

χ

)2ψ+ 2

imχv · ∇ψ+∇2ψ

].

Substituting above expressions into the field equations we obtainequations for functions v, r, sp:[

−m∂tv · (x− r)− v · ∂t r− sp]

ψ− iχ∂t r · ∇ψ

−m2v2ψ+ iχv · ∇ψ+

χ2

2m∇2ψ− qϕexψ− χ2

2mG ′ψ = 0.


Wave-corpuscle concept for an accelerating chargeUsing the charge equilibrium equation we eliminate G and ∇2:

−{m [∂tv · (x− r)− v · ∂t r] +

mv2

2+ sp + qϕex

}ψ

−iχ (∂t r− v)∇ψ = 0.

We equate then to zero the coeffi cients before ∇ψ and ψ above:

v = ∂t r, m [∂tv · (x− r)− v · ∂t r] +m2v2 + sp + qϕex = 0,

and recast the second equation as

m [∂tv · (x− r)− v · ∂t r]+ sp+mv2

2+q

[ϕ0,ex + ϕ′0,ex · (x− r)

]= 0.

We equate to zero the coeffi cient before (x− r) and the remainingcoeffi cient and obtain

m∂tv = −qϕ′0,ex (t) , sp −mv · ∂t r+mv2

2+ qϕ0,ex (t) = 0.


de Broglie factor for accelerating chargeFor purely electric external field we define

k (t) =∫

R3Im∇ψ (t, x)ψ (t, x)

∣∣ψ (t, x)∣∣2 dx.

The Fourier transform F of the wave-corpuscle ψ (t, x):

[Fψ] (t, k) = exp{

ir (t) k−isp (t)

χ

} (F[ψ]) (

k−mv (t)χ

),

implying k (t) =mv (t)

χ, v (t) =

drdt(t) .

The charge velocity v (t) equals the group velocity ∇kω (k (t)):

ω (k) =χk2

2m, ∇kω (k) =

χkm

, implying ∇kω (k (t)) = v (t) .


Planck-Einstein energy-frequency relationLet us consider now multiharmonic solutions

ψ` (t, x) = e−iω`tψ` (x) , ϕ` (t, x) = ϕ` (x) ,

14π∇2ϕ` = −q`

∣∣∣ψ`∣∣∣2 , or ϕ` (t, x) = q`∫

R3

∣∣ψ`∣∣2 (t, y)|y− x| dy.

Then {ψ` (x)}N`=1 satisfy the nonlinear eigenvalue problem

χω`ψ` +χ2

2m`∇2ψ` − q`ϕ 6=`ψ` −

χ2

2m`G ′`

(|ψ`|

2)

ψ` = 0.

This problem may have many solutions; every solution {ψ`}N`=1

determines a set of frequencies {ω`}N`=1 and energies {E0`}N`=1:

E0` =∫q`∣∣∣ψ`∣∣∣2 ϕ 6=` dx+

∫χ2

2m`

{∣∣∣∇ψ`∣∣∣2 + G ` (∣∣∣ψ`∣∣∣2)} dx.


Planck-Einstein energy-frequency relationWe seek nonlinearities G ` such that any two solutions {ψ`}

N`=1,{

ψ′`}N`=1 satisfy Planck-Einstein frequency-energy relation:

χ(ω` −ω′`

)= E0` − E′0`, ` = 1, ...,N.

Based on the nonlinear eigenvalue equations and the chargenormalization condition ‖ψ`‖ = 1 we obtain an integralrepresentation for the frequencies ω`:

χω` =∫ [

χ2

2m`|∇ψ`|

2 dy+ q`ϕ 6=` |ψ`|2 +

χ2

2m`G ′`(|ψ`|

2)|ψ`|

2]

dy.

Comparing the above with the integral for E0` we see that

χω` − E0` =χ2

2m`

∫ [G ′`(|ψ`|

2)|ψ`|

2 − G`(|ψ`|

2)]

dy.


Planck-Einstein energy-frequency relation

Consequently, we obtain for any two solutions{ψ`}N`=1 ,

{ψ′`}N`=1:

χ(ω` −ω′`

)−(E0` − E′0`

)=

=χ2

2m`

∫G`(∣∣ψ′`∣∣2)− G ′` (∣∣ψ′`∣∣2) ∣∣ψ′`∣∣2 dy−

− χ2

2m`

∫G`(|ψ`|

2)− G ′`

(|ψ`|

2)|ψ`|

2 dy.

Observe that for the above to hold it is suffi cient that for every |ψ`|2

with ‖ψ`‖ = 1, there exists constants C` such that∫ [G`(|ψ`|

2)− G ′`

(|ψ`|

2)|ψ`|

2]

dy = C`.


Planck-Einstein energy-frequency relationThe previous integral identities in turn will hold if the followingdifferential equations hold for some constants KG` :

sddsG` (s)− G` (s) = KG`s.

The above together with the normalization ‖ψ`‖ = 1 condition yield

χω`−E0` =χ2

2m`

∫ [G ′`(|ψ`|

2)|ψ`|

2 − G`(|ψ`|

2)]

dy = − χ2

2m`KG`

implying the Planck-Einstein energy-frequency relation.Solving the above differential equations we obtain the followingexplicit formula

G` (s) = KG`s ln s + C`s.

yielding KG` < 0 exactly the logarithmic nonlinearity whichcorresponds to the Gaussian factor.


Planck-Einstein energy-frequency relation

For the proper choice of constants KG` we obtain

G` (s) = G`,a (s) = −1

(a`)2s ln s +

1

(a`)2s(ln

1π3/2 − 2− 3 ln a

)where a` is the size parameter for `-th charge.

If KG = 0 then G`(|ψ`|

2)is quadratic and the eigenvalue equations

turn into the linear Schrödinger equations for which fulfillment of thePlanck-Einstein relation is a well-known fundamental property.

It is remarkable that the logarithmic nonlinearity which is singled outby the fulfillment of the Planck-Einstein relation has a second crucialproperty: it allows for a Gaussian localized soliton solution.

The above arguments continue to hold if there is an externaltime-independent electric field.


electromagnetism as a fundamental theory - nc … as a fundamental theory one theory for all spacial...

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