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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
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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!
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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!
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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!
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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!
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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∑
k
ek qk (t)δqk (t)(s)
The constraint equations
∇ · B(t ,s) = 0
∇ · E(t ,s) = 4π∑
k
ekδqk (t)(s)
NB: The velocities qk (t) are subluminal
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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∑
k
ek qk (t)δqk (t)(s)
The constraint equations
∇ · B(t ,s) = 0
∇ · E(t ,s) = 4π∑
k
ekδqk (t)(s)
NB: The velocities qk (t) are subluminal
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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∑
k
ek qk (t)δqk (t)(s)
The constraint equations
∇ · B(t ,s) = 0
∇ · E(t ,s) = 4π∑
k
ekδqk (t)(s)
NB: The velocities qk (t) are subluminal
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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∑
k
ek qk (t)δqk (t)(s)
The constraint equations
∇ · B(t ,s) = 0
∇ · E(t ,s) = 4π∑
k
ekδqk (t)(s)
NB: The velocities qk (t) are subluminal
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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))]
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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))]
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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))]
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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!
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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!
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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!
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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!
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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))
]+ 2
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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))
]+ 2
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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))
]+ 2
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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))
]+ 2
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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))
]+ 2
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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∑±
12
∑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, ....
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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∑±
12
∑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, ....
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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∑±
12
∑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, ....
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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∑±
12
∑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, ....
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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∑±
12
∑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, ....
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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∑±
12
∑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, ....
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
Summary and Outlook
Pre-metric Maxwell field equations
Lorentz frame with space vector s ∈ R3 and time t ∈ RThe evolution equations
∂t B(t ,s) = −∇× E(t ,s)
∂t D(t ,s) = +∇× H(t ,s)− 4π∑
k
ek qk (t)δqk (t)(s)
The constraint equations
∇ · B(t ,s) = 0
∇ · D(t ,s) = 4π∑
k
ekδqk (t)(s)
NB: Need electromagnetic vacuum laws (B,D)↔ (H,E).
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
Summary and Outlook
Pre-metric Maxwell field equations
Lorentz frame with space vector s ∈ R3 and time t ∈ RThe evolution equations
∂t B(t ,s) = −∇× E(t ,s)
∂t D(t ,s) = +∇× H(t ,s)− 4π∑
k
ek qk (t)δqk (t)(s)
The constraint equations
∇ · B(t ,s) = 0
∇ · D(t ,s) = 4π∑
k
ekδqk (t)(s)
NB: Need electromagnetic vacuum laws (B,D)↔ (H,E).
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
Summary and Outlook
Pre-metric Maxwell field equations
Lorentz frame with space vector s ∈ R3 and time t ∈ RThe evolution equations
∂t B(t ,s) = −∇× E(t ,s)
∂t D(t ,s) = +∇× H(t ,s)− 4π∑
k
ek qk (t)δqk (t)(s)
The constraint equations
∇ · B(t ,s) = 0
∇ · D(t ,s) = 4π∑
k
ekδqk (t)(s)
NB: Need electromagnetic vacuum laws (B,D)↔ (H,E).
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
Summary and Outlook
Pre-metric Maxwell field equations
Lorentz frame with space vector s ∈ R3 and time t ∈ RThe evolution equations
∂t B(t ,s) = −∇× E(t ,s)
∂t D(t ,s) = +∇× H(t ,s)− 4π∑
k
ek qk (t)δqk (t)(s)
The constraint equations
∇ · B(t ,s) = 0
∇ · D(t ,s) = 4π∑
k
ekδqk (t)(s)
NB: Need electromagnetic vacuum laws (B,D)↔ (H,E).
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
Summary and Outlook
Action Principles
All three field theories derive from an action principle:
δ
∫ T
0
∫R3
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
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
Summary and Outlook
Action Principles
All three field theories derive from an action principle:
δ
∫ T
0
∫R3
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
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
Summary and Outlook
Action Principles
All three field theories derive from an action principle:
δ
∫ T
0
∫R3
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
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
Summary and Outlook
Action Principles
All three field theories derive from an action principle:
δ
∫ T
0
∫R3
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
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
Summary and Outlook
Conservation Laws −→ Equations of Motion
IF the field action principle is well-defined, adding
−∑
k
∫ T
0
√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
0
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
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
Summary and Outlook
Conservation Laws −→ Equations of Motion
IF the field action principle is well-defined, adding
−∑
k
∫ T
0
√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
0
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
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
Summary and Outlook
Conservation Laws −→ Equations of Motion
IF the field action principle is well-defined, adding
−∑
k
∫ T
0
√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
0
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
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
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.
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
Summary and Outlook
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
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
Summary and Outlook
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
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
Summary and Outlook
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
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
Summary and Outlook
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
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
MotivationThe textbook story: Lorentz Electrodynamics
Fokker-Schwarzschild-Tetrode-(Wheeler-Feynman) E-DynamicsThe Electromagnetic Cauchy Problem
Summary and Outlook
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
Michael K.-H. Kiessling Classical Electrodynamics with Point Charges
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