classical electrodynamics with point charges · the textbook story: lorentz electrodynamics ......

MotivationThe textbook story: Lorentz Electrodynamics

Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem

Summary and Outlook

To Detlef Dürr, for all he taught us!

Classical Electrodynamics with Point Charges

Michael K.-H. Kiessling

Department of MathematicsRutgers –New Brunswick

Nov. 1, 2014Ludwig Maximilian University

Michael K.-H. Kiessling Classical Electrodynamics with Point Charges

Summary and Outlook

Outline

1 Motivation

2 The textbook story: Lorentz Electrodynamics

3 Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman)E-Dynamics

4 The Electromagnetic Cauchy Problem

5 Summary and Outlook

Summary and Outlook

Electromagnetism and Atomism

A well-formulated Lorentz covariant electromagneticclassical theory of fully ionized plasma exists at thecontinuum level: “Magnetofluid-Mechanics”A well-formulated Lorentz covariant electromagneticclassical theory of fully ionized plasma exists at thekinetic level: “Vlasov theory”Goal: Well-formulated Lorentz covariant electromagneticclassical theory of fully ionized plasma at the atomisticlevel: “point charges accelerated by joint electromagneticfield which they generate”Why only fully ionized plasma? To get a fundamentalmodel of anything else requires Quantum Theory!

Summary and Outlook

The Maxwell-Lorentz field equations

Lorentz frame with space vector s ∈ R3 and time t ∈ RThe evolution equations

∂t B(t ,s) = −∇× E(t ,s)

∂t E(t ,s) = +∇× B(t ,s)− 4π∑

ek qk (t)δqk (t)(s)

The constraint equations

∇ · B(t ,s) = 0

∇ · E(t ,s) = 4π∑

ekδqk (t)(s)

NB: The velocities qk (t) are subluminal

Summary and Outlook

∂t B(t ,s) = −∇× E(t ,s)

∂t E(t ,s) = +∇× B(t ,s)− 4π∑

∇ · B(t ,s) = 0

∇ · E(t ,s) = 4π∑

ekδqk (t)(s)

Summary and Outlook

∂t B(t ,s) = −∇× E(t ,s)

∂t E(t ,s) = +∇× B(t ,s)− 4π∑

∇ · B(t ,s) = 0

∇ · E(t ,s) = 4π∑

ekδqk (t)(s)

Summary and Outlook

∂t B(t ,s) = −∇× E(t ,s)

∂t E(t ,s) = +∇× B(t ,s)− 4π∑

∇ · B(t ,s) = 0

∇ · E(t ,s) = 4π∑

ekδqk (t)(s)

Summary and Outlook

The Einstein-Newton equations of motion

Einstein’s velocity-momentum relation

qk (t) =pk (t)√

m2 + |pk (t)|2

Newton’s law for the rate of change of momentum

pk (t) = fk (t ,qk (t),pk (t))

Lorentz’ law for the electromagnetic force

fk (t ,qk ,pk ) = ek [E(t ,qk (t)) + qk (t)× B(t ,qk (t))]

Summary and Outlook

qk (t) =pk (t)√

m2 + |pk (t)|2

Summary and Outlook

qk (t) =pk (t)√

m2 + |pk (t)|2

Summary and Outlook

“Infinite in all directions”

Formally the equations of Lorentz Electrodynamics seemto pose a joint Cauchy problem for positions and momentaqk (t) and pk (t), and for the fields B(t ,s) and E(t ,s) withinitial data constrained by the divergence equations andthe subluminality of speeds.However, this Cauchy problem is rigorously ill defined!Reason: E(t ,qk (t)) and B(t ,qk (t)) “infinite in all directions”Can be “defined” through averaging, but result depends onhow the averaging is done.The upshot: The total Lorentz force is not well-definable!

Summary and Outlook

The Landau-Lifshitz equation

Nevertheless, though formally divergent, self-forces inrelativistic Lorentz Electrodynamics are traditionallyneglected “in leading order” and subsequently handledperturbatively (Landau-Lifshitz): 1 charge in external fields,

p(t) = e[Eext(t ,q(t)) + q(t)× Bext(t ,q(t))

3e2...q(t)

23e2...

q(t)−−replace −→ fLL(t ,q(t), q(t), q(t))

“Derivation” of...q uses infinite mass renormalization (Dirac)

fLL(t ,q(t), q(t), q(t)) through perturbative expansion (L-L).Works well for practical purposes.Rigorous derivation for Abraham model: Spohn et al.

Summary and Outlook

3e2...q(t)

23e2...

Summary and Outlook

3e2...q(t)

23e2...

Summary and Outlook

3e2...q(t)

23e2...

Summary and Outlook

3e2...q(t)

23e2...

Summary and Outlook

LorentzElectrodynamics of N Chargesw/oSelf-Forces

Q: Can we obtain a well-posed Cauchy problem also forN-body Lorentz electrodynamics without self-forces?A: Deckert and Hartenstein show: NOT globally!OK locally until the first particle worldline enters the forwardlightcone of another particle: VERY SHORT life span!Until then one has effectively N one-body problems, andone can even “radiation-reaction correct” them by addingLandau-Lifshitz terms.The upshot: Even when self-forces are purged, LorentzElectrodynamics does NOT supply an atomistic classicalmodel of a plasma for any relevant time scale.

Summary and Outlook

Fokker-Schwarzschild-Tetrode Electrodynamics

Newton-inspired Lorentz covariant N-body modelNo radiation field degrees of freedomNo self-interactionsCharges interact pairwise at a distance in an invariant way:

qk (t) =pk (t)√

m2 + |pk (t)|2

pk (t) = ek∑±

∑l 6=k

[ELW(t ,ql(t)) + ql(t)× BLW(t ,ql(t))

Advance-Delay-Differential Equations: no Cauchy problem.Rigorous works by Bauer, Dürr, Deckert, ....

Summary and Outlook

qk (t) =pk (t)√

m2 + |pk (t)|2

pk (t) = ek∑±

∑l 6=k

Summary and Outlook

qk (t) =pk (t)√

m2 + |pk (t)|2

pk (t) = ek∑±

∑l 6=k

Summary and Outlook

qk (t) =pk (t)√

m2 + |pk (t)|2

pk (t) = ek∑±

∑l 6=k

Summary and Outlook

qk (t) =pk (t)√

m2 + |pk (t)|2

pk (t) = ek∑±

∑l 6=k

Summary and Outlook

qk (t) =pk (t)√

m2 + |pk (t)|2

pk (t) = ek∑±

∑l 6=k

Summary and Outlook

Wheeler-Feynman Electrodynamics

FST electrodynamics PLUS absorber conditionExpresses advanced terms by retarded terms PLUS

...q k

Purging of...q k leads to Synge’s model

Recent work by Bauer, Dürr, DeckertNB: Fokker-Schwarzschild-Tetrode and Wheeler-Feynmanand Synge electrodynamics models all are very fascinatingin their own right, but also are very difficult to studyrigorously. Experts on differential-delay equations arestuck with much simpler models already.NB: I have difficulties imagining experimental tests.

Summary and Outlook

...q k

Summary and Outlook

...q k

Summary and Outlook

...q k

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...q k

Summary and Outlook

...q k

Summary and Outlook

Pre-metric Maxwell field equations

∂t B(t ,s) = −∇× E(t ,s)

∂t D(t ,s) = +∇× H(t ,s)− 4π∑

∇ · B(t ,s) = 0

∇ · D(t ,s) = 4π∑

ekδqk (t)(s)

NB: Need electromagnetic vacuum laws (B,D)↔ (H,E).

Summary and Outlook

∂t B(t ,s) = −∇× E(t ,s)

∂t D(t ,s) = +∇× H(t ,s)− 4π∑

∇ · B(t ,s) = 0

∇ · D(t ,s) = 4π∑

ekδqk (t)(s)

Summary and Outlook

∂t B(t ,s) = −∇× E(t ,s)

∂t D(t ,s) = +∇× H(t ,s)− 4π∑

∇ · B(t ,s) = 0

∇ · D(t ,s) = 4π∑

ekδqk (t)(s)

Summary and Outlook

∂t B(t ,s) = −∇× E(t ,s)

∂t D(t ,s) = +∇× H(t ,s)− 4π∑

∇ · B(t ,s) = 0

∇ · D(t ,s) = 4π∑

ekδqk (t)(s)

Summary and Outlook

Electromagnetic Vacuum Laws

Maxwell’s lawH = BE = D

Born-Infeld’s law

H =B− β3B× (B× D)√

1 + β2(|B|2 + |D|2) + β4|B× D|2

E =D− β3D× (D× B)√

1 + β2(|B|2 + |D|2) + β4|B× D|2

Bopp-Landé-Thomas(-Podolsky) law

H(t ,s) =(

1 + κ−2�)

B(t ,s)

D(t ,s) =(

1 + κ−2�)

E(t ,s) .Michael K.-H. Kiessling Classical Electrodynamics with Point Charges

Summary and Outlook

Born-Infeld’s law

H =B− β3B× (B× D)√

1 + β2(|B|2 + |D|2) + β4|B× D|2

E =D− β3D× (D× B)√

1 + β2(|B|2 + |D|2) + β4|B× D|2

H(t ,s) =(

1 + κ−2�)

B(t ,s)

D(t ,s) =(

1 + κ−2�)

Summary and Outlook

Born-Infeld’s law

H =B− β3B× (B× D)√

1 + β2(|B|2 + |D|2) + β4|B× D|2

E =D− β3D× (D× B)√

1 + β2(|B|2 + |D|2) + β4|B× D|2

H(t ,s) =(

1 + κ−2�)

B(t ,s)

D(t ,s) =(

1 + κ−2�)

Summary and Outlook

Action Principles

All three field theories derive from an action principle:

L(A,F,dF)d3sdt = 0

Maxwell-Lorentz field equations↔ Schwarzschild principleNB: Rigorously ill-defined with point charge sourcesbecause of infinite self-energiesMBI field equations↔ Born-Infeld principleNB: Expected to be well-definedMBLTP field equations↔ Bopp-Podolski principleNB: Expected to be well-defined

Summary and Outlook

Action Principles

L(A,F,dF)d3sdt = 0

Summary and Outlook

Action Principles

L(A,F,dF)d3sdt = 0

Summary and Outlook

Action Principles

L(A,F,dF)d3sdt = 0

Summary and Outlook

Conservation Laws −→ Equations of Motion

IF the field action principle is well-defined, adding

−∑

√1− |qk |2dt

to field action yields JOINT particle+field action principle.Now Noether’s theorem establishes energy-momentum(etc.) conservation for the stationary points.This leads to the fixed point equations: (For 1 pt charge)

q(t) = q(0) +∫ t

p√m2 + |p|2

(t)dt ,

p(t) = p(0)−∫R3

(Π(t ,s)−Π(0,s))d3s,

(NB: generalizes to N charges !)Michael K.-H. Kiessling Classical Electrodynamics with Point Charges

Summary and Outlook

−∑

√1− |qk |2dt

q(t) = q(0) +∫ t

p√m2 + |p|2

(t)dt ,

p(t) = p(0)−∫R3

(Π(t ,s)−Π(0,s))d3s,

Summary and Outlook

−∑

√1− |qk |2dt

q(t) = q(0) +∫ t

p√m2 + |p|2

(t)dt ,

p(t) = p(0)−∫R3

(Π(t ,s)−Π(0,s))d3s,

Summary and Outlook

The equations of motions

Π is field momentum density.For ML and for MBI field equations

4πΠ = D× B

For MBLTP field equations

4πΠMBLTP = D×B+E×H−E×B−κ−2(∇·E)(∇×B−κ E)

Key observation:

p(t) = p(0)−∫R3

(Π(t ,s)−Π(0,s))d3s = F (q(·),p(·))

Need to show that F (q(·),p(·)) is Lipschitz.

Summary and Outlook

4πΠ = D× B

Key observation:

p(t) = p(0)−∫R3

(Π(t ,s)−Π(0,s))d3s = F (q(·),p(·))

Summary and Outlook

4πΠ = D× B

Key observation:

p(t) = p(0)−∫R3

(Π(t ,s)−Π(0,s))d3s = F (q(·),p(·))

Summary and Outlook

4πΠ = D× B

Key observation:

p(t) = p(0)−∫R3

(Π(t ,s)−Π(0,s))d3s = F (q(·),p(·))

Summary and Outlook

4πΠ = D× B

Key observation:

p(t) = p(0)−∫R3

(Π(t ,s)−Π(0,s))d3s = F (q(·),p(·))

Summary and Outlook

Rigorous results

Very few rigorous results, namely:Global well-posedness of MBI field Cauchy problem withsmall data (Speck)Unique static finite-energy solutions of MBI field equationswith N point charges always exist and are regular awayfrom the pt charges (MK)Global well-posedness of MBLTP field Cauchy problemwith “arbitrary” data (standard)Partial results available for joint MBLTP - pt chargeavailable for static initial data.

Summary and Outlook

Rigorous results

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Rigorous results

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Rigorous results

Summary and Outlook

Rigorous results

Summary and Outlook

Summary

A well-formulated Lorentz covariant classicalelectrodynamics with point charges has yet to beestablished, but seems clearly feasible.Formulation as Cauchy problem requires either nonlinearor higher-order linear field equations to guarantee finiteself-energies and self-momenta.Equation of point charge motion is given by momentumconservation law (in concert with Einstein’svelocity-momentum relation) viewed as a fixed pointequation in the space of trajectories (given initial data).Progress is faster with MBLTP field equations thanks totheir linearity. Yet, the complexity is daunting.Formulation as action-at-a-distance problem availablesince early 20th century. Rigorous progress very slow.

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Outlook

Presumably in near future well-posedness of the static jointCauchy problem for MBLTP fields and a single point chargeHopefully in not too distance future the same (at leastlocally) dynamically for many point charges.Hopefully also eventually dynamical MBI field evolutionswith given point charge motions (technically very hard)Hopefully (at least locally) well-posedness of joint Cauchyproblem for MBI fields and their point charge sources (Ihope I live to see this!)Generalization of classical Cauchy problem to quantummotions via deformation to de-Broglie-Bohm type theory

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THANK ALL OF YOU FOR LISTENING!

classical electrodynamics with point charges · the textbook story: lorentz electrodynamics ......

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