Premetric Classical Electrodynamics:
Achievements and Prospects1
Friedrich W. Hehl† ∗ and Jonathan Lux†
† Universität
zu Köln and ∗ University of Missouri, Columbia
Bad Honnef, Monday, 21 March 2011, 10:00h
Problems and Developments of Classical Electrodynamics
475th Wilhelm & Else Heraeus Seminar
We thank Nico Giulini and Volker Perlick for the invit. to this talk.
Supported by GIF, Jerusalem-München
1
file Sun/EldynHeraeus08.tex
Partially based on joint work with Yuri Obukhov and Yakov Itin.
Contents
Part A. Fundamental facts
1. Definition of premetric electrodynamics (pmED)
2. Experimental basis
3. The three fundamental axioms
4. Dimensional analysis
5. Fouth axiom, electric/magnetic reciprocity
6. Maxwell-Lenz and Maxwell-anti-Lenz systems
[7. Magnetic monopoles?]
8. “Maxwell in chains”
9. Post’s topological quantum superstructure for the Maxwell equations
[10. (1+2) dimensional electrodynamics, quantum Hall effect (QHE)]
Part B. 5th axiom: Constitutive laws H = H[F ], selected applics.
11. Some plausibility remarks
12. Nonlocal linear (linear connection of spacetime is needed a priori)
13. Local nonlinear (metric of spacetime is needed a priori)
14. Local linear (no a priori metric given, light cone is derived)
15. Metamaterials and the four families of (1+3)d Maxwell equations
Was it a god who wrote these signs?
(“War es ein Gott, der diese Zeichen schrieb?”)
Boltzmann on Maxwell’s equations
quoting Goethe’s Faust
Part A. 1. Definition of pmED
Premetric classical electrodynamics2 (pmED) is a classical theory of
electromagnetism that is based on electric charge and magnetic flux
conservation and on an unspecified constitutive law for vacuum and for
matter.
A consequence are the premetric Maxwell equations dH = J, dF = 0,
with the twisted excitation 2-form H = D − H ∧ dt, the twisted current
3-form J = ρ − j ∧ dt, and the unwisted field strength 2-form
F = B + E ∧ dt. All these notions depend neither on the metric nor on
the connection of spacetime, that is, the gravitational field is not involved
in the foundations of classical electrodynamics.
2
H. & Obukhov, Foundations of Class. Electrodynamics. Boston 2003
2. Experimental basis
• Electric charge conservation: verified in millions of elementary particle
reactions. No deviation ever found, specific experiments were run on a
search for possible violations, negative outcome. A test theory for
violations of this law was developed by Claus Lämmerzahl, see his lecture.
Today single electrons can be counted experimentally, no need for metric
length scale, decide what is inside and outside a volume, exterior calculus
and integration, topological meaning of Stokes’ law. Charge as new
physical dimension, use quantity equations that are valid in all system of
units. Theory of dimensions of Wallot, Dorgelo & Schouten, Toupin,
Post, etc. Don’t start electromagnetics with Coulomb law, only sensible
if charge is conserved, often tacitly assumed and not spelled out.
• Magnetic flux conservation: verified in millions of of working electric
motors (induction law), also Abrikosov lines in sperconductors of type II
don’t emerge spoantaneously, they move in or out via the boundary of
the domain, they can be counted.
dB = 0 and Ḃ + d E = 0 are 2 projections of dF = 0; comp. ρ̇ + d j = 0.
{z
}
|
contin. eq. for B
B is 2-form
(area) and
R ρ is 3-form (volume). The integrals over both,
R
Q = ω3 ρ and Φ = ω2 B, are conserved! Magnetic [U(1)] monopoles
6 0 but dB = 0 ).
never found. Those in solids are a misnomer (dH =
Direct observation of individual Abrikosov flux lines in type II
superconductors acc. to Essmann & Träuble (1967). Niobium disc
(diameter 4 mm, thickness 1 mm) was exposed at 1.2 K to
H = 78 kA/m. Flux lines decorated by ferromagnetic particles, fixed by
replica technique. Replica observed by electron microscope. The
parameter of the flux-line lattice was 170 nm
3. The three fundamental axioms
Minimalistic axiomatic approach, n dims. (1st, 2nd, and 3rd axioms):
I
I
J = 0,
fα = (eα ⌋F ) ∧ J ,
F = 0,
|
{z
}
Ωn−1
Ω2
Lorentz force, eα = frame
with ∂Ωn−1 = 0 and ∂Ω2 = 0. The differential versions read:
dJ
J
= 0,
= dH,
fα = (eα ⌋F ) ∧ J ,
dF = 0 ,
F = dA .
Certain asymmetry built in: H is measurable, A is not measurable.
Diffeomorphism inv. eqs., do not depend on metric nor on connection.
(
dD =ρ
(∂a Da = ρ) ,
dH =J
a
abc
− Ḋ + d H = j
(−Ḋ + ǫ ∂b Hc = j a ) .
dF =0
(
dB =0
Ḃ + d E = 0
(∂a B a = 0) ,
(Ḃ + ǫ ∂b Ec = 0) .
a
abc
The exterior form version is valid in all dimensions n > 1. The vector
calculus version is only valid for n = 1 + 3.
H
D
E
B
Faraday-Schouten pictograms of the electromagnetic field.
D
ic
ctr
ele
d st
gn
eti
c
H
ma
fiel
on
2−forms
E
ati
cit
ex
1−forms
ren
gth
B
Different aspects of the elmg. field. The four quantities H, D, E , B
constitute the elmg. field. They are all of a microscopic nature.
4. Dimensional analysis
Abolute (relative) dimensions of a physical quantity (of its components).
Physical laws as quantity (not numerical) eqs. From mechanics:
ℓ = length, t = time, h = action ;
for pmED add. q = electric charge;
see new SI concepts 2011. No indep. magn. dim.; derived are φ = magn.
flux = h/q and r = resistance = q 2 /h (c doesn’t play a role here!)
[J] = [H] = q,
[F ] = [A] = φ;
[H] × [F ] = h,
[H]
[J]
1
=
=
[F ]
[A]
r
We learn from pmED that (q, φ; h, r) are 4d scalars and fundamentally
important quantities (in SI: C, Wb; Js, Ω). Microscopically, these
quantities may be quantized (quantization does not require a metric):
◮
elementary charge e,
◮
flux quantum Φ0 = h/(2e) ≈ 2 × 10−15 Wb (“Abrikosov
charge”=1/KJ (Josephson constant ≈ 0.5 × 1015 Hz/V),
◮
spin ~/2,
◮
vacuum resistance 1/λ0 ; von-Klitzing constant RK = h/e 2
≈ 25 813 Ω (QHE), in metrology KJ , RK of fundamental importance
I invite you to gain this insight from the ‘zoological’ units, emus etc.
5. Fouth axiom, electric/magnetic reciprocity
The first three axioms more or less uniquely yield the expression for the
energy-momentum densities and their fluxes. Take the 2nd axiom,
substitute the 1st one and integrate partially:
tensor
z}|{
1
fα = (eα ⌋F ) ∧ J = d k Σα − (F ∧ Leα H − H ∧ Leα F ) .
| {z } 2
no tensor
(Kinematik) energy-momentum 3-form density (Minkowski):
k
Σα :=
1
[F ∧ (eα ⌋H) − H ∧ (eα ⌋F )]
2
(4th axiom) .
The theory of k Σα is further developed in the lecture of Yuri Obukhov.
Σα is traceless: ϑα ∧ k Σα = 0. It is electric/magnetic reciprocal, that is,
k
1
H → ζF , F → − H =⇒ k Σα → k Σα ,
ζ
(ζ twisted scalar function) .
Electric/magnetic reciprocity does not require magn. charge. Decomp.:
(
(
E → ζ1 H ,
1
H → −ζE ,
F →− H
H → ζF
ζ
D → ζB ,
B → − ζ1 D .
6. Maxwell-Lenz and Maxwell-anti-Lenz
Lenz rule determines sign in dE ⊕ Ḃ = 0, in contrast to dH ⊖ Ḋ = j.
Introduce open factors (iT , iS ; hT , hS ; fT , fS ) ∈ {+1, −1} and fΛ := fS /fT :
J = iT j ∧ dσ + iS ρ ,
H = hT H ∧ dσ + hS D ,
F = fT E ∧ dσ + fS B .
Insert the usual conventions, like the sign of the electric charge, counting
out-flowing current as positive etc.. Find that only one sign is left open:
dE + fΛ Ḃ = 0 ,
fΛ ∈ {+1, −1}.
Consequently, in pmED the 4d Maxwell eqs. bifurcate in Maxwell-Lenz
and Maxwell-anti-Lenz. Which should we take as the physical one?
The energy-momentum 3-form yields
mg. energy d.
el. energy d.
Poynting 2-form
z }| {
z }| {
z }| {
1
1
k
Σ0̂ = − E ∧ H ∧dt + D ∧ E +fΛ B ∧ H .
2
2
Apply el./mg. reciprocity:
-Poynting 2-form
k
Σ0̂ = +
z }| {
H∧E
mg. energy d.
el. energy d.
z }| {
z }| {
1
1
∧dt + B ∧ H +fΛ D ∧ E .
2
2
Electric/magnetic reciprocity rules out the anti-Lenz option!
7. Magnetic monopoles?
Make the Maxwell eqs. ‘symmetric’. Besides d H = J, have
d F = Jmg
(rev. 3rd axiom) .
Contradicts experiments, most recent experiments last year. Still, could
our formalism survive? Adapt Lorentz force density:
fα = (eα ⌋F ) ∧ J − (eα ⌋H) ∧ Jmg
(rev. 2nd axiom) .
R.h.s. is of the type ∼ ρE + j ∧ B − ρmg H − jmg ∧ D. Cf. original
definition of H. Magnetic charge would destroy the straightforward
operational meaning of the 2nd axiom.
Edelen, Kaiser, Obukhov & H. have shown that, amazingly, k Σα survives
untouched! Consequently, magnetic charge doesn’t contribute explicitly
to the energy-momentum of the field.
From now on, in accordance with experiment, we put Jmg = 0!
8. “Maxwell in chains”
◮
Discrete formulation discover topological structure. Found around
1930 by Lefshetz, de Rham, Veblen, Weyl, Whitehead... Reviewed
by Griffith & Harris 1978, e.g.
◮
Appl. to ED (motivation for de Rham) since ∼1950, in particular
...Bossavit and Tonti have established the subject, Zirnbauer 2003
(see subtitle), still an object of mathematical research, see Harrison
2011 and papers by S. Kurz and B. Auchmann, see lecture today by
Alain Bossavit who extends to electro-elasto-dynamic phenomena
◮
Here pmED with time left continuous, space n = 3:
k-cells: points (k = 0), line segments (1), surf. seg. (2), vols. (3)
k-chains as lin. comb. of k-cells with real coeff. ⇒ vector space Ck
k-cochains in dual vector space Ck∗
P
Example: Charge density 0-form ρ0 = Pj qj Pj , points Pj , 3-cochain
total charge in a volume U is Q[U] = Pj ∈U qj .
◮
◮
◮
Define inner (interior) and outer (exterior) orientation, see figures
Boundary operator ∂ as a linear map from k-chains to
(k − 1)-chains:
inn/out
∂ : Ckinn/out → Ck−1
.
2
3
1
Figure: Oriented chains in three dimensions
It preserves the type of orientation consistently and specifies the inuitive
picture of assigning a boundary to its interior. An example:
Figure: Boundary of a 3-chain
A specific and also very intuitive feature of a boundary can be proven
rigorously: the boundary of a boundary is zero/empty, which means
∂ ◦ ∂ = 0.
Intersection pairing I that assigns a real number to a pair of a k-chain c
with inner orientation and a (3 − k)-chain ω with outer orientation:
out
I : Ckinn × C3−k
→ R.
Figure: Intersection pairing of two chains
R
out
) . A fundamental
Use the suggestive notation c ω := I(ckinn , ω3−k
relation between boundary operator and intersection pairing is a discrete
version of Stokes’ theorem
out
out
I(∂ckinn , ω3−k
) = (−1)k+1 I(ckinn , ∂ω3−k
).
Define dω := (−1)Rdeg(ω) ∂ω.
R Then, find a look-alike of the continuous
ω = d ω.
Stokes’ theorem:
c
∂c
• Maxwell equations
Charge density ρinn
is a 0-chain. Charge carries no handedness, it is not
0
endowed with an orientation, the inn is just for bookkeeping. Combine
ρinn
with a 3-chain U3out via the intersection
pairing. It defines the
0
out
3-cochain of electric charge: Q U3out = I(ρinn
0 , U3 ).
inn
out
Current
density j1 evaluated on S2 yields the electric current
I S2out = I(j1inn , S2out ). Charge conservation in global form:
∂ Q U + I ∂U = 0
∀ U ∈ C3out ;
∂t
after inserting the definitions, it reads
Z
Z
Z
Z
Z
ρ̇ +
j=
ρ̇ +
dj = (ρ̇ + dj) = 0
∀ U ∈ C3out .
U
∂U
U
U
U
Thus we get the spatially discrete version of the continuity equation:
inn
ρ̇inn
= 0.
0 + d j1
Chains of electric and magnetic excitation, should carry an inner
orientation. The global version of the electric excitation is the electric
flux Ψ; thus it should be a 1-chain with inner orientation D1inn and,
accordingly, the electric flux is defined via Ψ[S2out ] = I(D1inn , S2out ).
Analogously, the magn. excitation is a 2-chain with inner orientation
H2inn , its global version, called magn. voltage, is F [γ1out ] = I(H2inn , γ1out ).
Inhomogeneous Maxwell equations
In global form:
Ψ ∂U = Q U
∂ F ∂S
= I S + Ψ S
∂t
∀ U ∈ C3out ,
∀ S ∈ C2out .
Locally, in chains:
d D1inn
=
ρinn
0 ,
d H2inn
=
j1inn + Ḋ1inn .
According to their definitions, the electric and magnetic field strengths
are endowed with an outer orientation. The global version of the
magnetic field strength is the magnetic flux; thus it sould be a 1-chain
outer orientation B1out and, accordingly, the magnetic flux is defined via
Φ[S2inn ] = I(B1out , S2inn ). Analogously, the electric field strength is a
2-chain with outer orientation E2out , its global version, called electric
voltage is E[γ1inn ] = I(E2out , γ1inn ).
Homogeneous Maxwell equations
In global form:
Φ ∂U = 0
∂ = 0
E ∂S + Φ S
∂t
∀ U ∈ C3inn ,
∀ S ∈ C2inn ,
Locally, in chains:
d
E2out
d B1out
=
0,
Ḃ1out
=
0.
+
This is a purely topological and spatially discrete formulation of the
Maxwell eqs., spatial derivatives are replaced by boundary operation.
These eqs. can be taken as a starting point for numerical evaluations.
9. Post’s topological quantum superstructure
for Maxwell eqs. (Post 1995 and priv. comm. Feb. 2010)
From 3rd and 1st axiom and Künneth’s formula he arrives at (Z1 , Z2 , Z3
are 1d, 2d, 3d integration cycles, n, s are integers)
XI
XI
XI
h
h
A = n
H = |{z}
A ∧ H = ns
,
,
se
.
e
2
|{z}
|{z}
Z
Z2 Z
Z1 Z
3
encl. el. charge
Z3
2
1
encl. mg. flux
| {z }
| {z }
|
{z
} encl. action
Aharonov-Bohm
Gauss-Ampère
Kiehn-Post
Macrosc., phenomenol., and topological information stored in these
R
’s.
Quantum theory: ‘action’ is quantized with h/2, charge with e. Thus GA
and KP should be valid universally. In supercond. mg. flux can be quant.
So far, no information from the constitutive law was fed into AB, GA, KP.
Apply AB, GA, KP to the Quantum Hall effect (QHE). Since the charges
are 2-dimensionally spread, we have to go to (1+2)-dimensional pmED:
XI
XI
XI
h
h
H = se ,
A ∧ H = ns .
A=n ,
e
2
Z1 Z
1
Z1 Z
1
Z2 Z
2
2
We can read off H = −σH A, with σH = rat. number × eh . Quantized!
Kiehn (n − 1)-form A ∧ H is fund. diff. from Chern-Simons 3-form A ∧ F .
10. 1+2 dim. electrodyn., QHE
(J. Fröhlich et al.)
The charge density is now a 2-form. Thus,
(2)
(2)
d J = 0,
(1)
J = dH ,
(2)
(2)
(1)
F =dA.
dF = 0,
We expect, see above, the constitutive law
J = −σH F .
1+2 decomp.: j = σH E and ρ = −σH B. Metric-free! Toplogical.
[σH ] = q 2 /h = 1/resistance, dJ = 0 and dF = 0 ⇒ σH = const..
Integrate Maxwell:
dH = −σH F
or
H = −σH A .
(1+2 D Kiehn or Chern-Simons electrodynamics with Lagrangian
L3D =
σH
1
A ∧ F − A ∧ J)
H ∧dH −A∧dH ∼
==−
2σH
2
I
I
gate
source
SiO2
drain
Al
SiO2
Al
SiO2
Al
n+
n+
p
Si
z
UH
y
2DEG
x
Mosfet with semidconductor Si and insulator SiO2 , adapted form
E. Braun (1992)
jx
jy
y
UH
Uy
Ux
Q
Bx y
jx
P
x
I
Schematic view of the quantum Hall experiment with a 2D electron gas
(2DEG)
Part B: 11. Some plausibility remarks
◮
We have then dH = J , dF = 0, formulated on a (1 + 3)d ‘naked’
manifold, and nothing more. We need a constitutive law. Starting
point a functional H = H[F ]. Such a law embodies (one version of)
the 5th axiom.
◮
Standard Maxwell-Lorentz vacuum electrodynamics (5th axiom at
its simplest): Introduce the conv. comps. of the excitation tensor
Hij := 12 ǫijkl Hkl (recall H = 21 Hij dx i ∧ dx j , F = 21 Fij dx i ∧ dx j )
r
ε0 ⋆
1 ⋆
1 √
H=
F =
F or, in components, Hij =
−g g ik g jl Fkl .
µ0
Ω0
Ω0
Vacuum impedance Ω0 ≈ 377 ohm, g ij metric, ⋆ Hodge star. This
is conv. wisdom. But here vacuum electrodynamics is immediately
implemented on an arbitrary Riemannian (‘Lorentzian’) manifold.
No covariant derivatives needed, no comma goes to semicolon rule...
◮
Macrophysics is treated with the 6th axiom:
J = J mat + J free , dJ mat = 0 , see our book, we skip it here.
◮
Constitutive law can be applied to the vacuum (see MaxwellLorentz ED, axion ED, classical string vacuum in the case of
Born-Infeld ED), to matter, or to metamaterials.
Provided we have no metric, the simplest ansatz is Hij ∼ Fij ; it doesn’t
work. The next option is a general linear homogeneous relationship:
1
Hij = κij kl Fkl , with κij kl = − κji kl = − κij lk .
2
Equivalently,
1
1
Hij = χijkl Fkl with χijkl := ǫijmn κmn kl = −χjikl = −χijlk .
2
2
Thus a linear spacetime relation postulates the existence of 6 × 6
functions κij kl (x) or χijkl (x), respectively, depending on time and space.
Explicitly, in terms of a (1+3)-decomposition, we find
Da = εab − ǫabc nc Eb
+ ( β a b + sb a − δba sc c ) B b + α B a ,
b
c
B
Ha = µ−1
ab − ǫ̂abc m
+ −β b a + sa b − δab sc c Eb − α Ea .
εab permittivity and µ−1
ab impermeability tensors; magnetoelectric
cross-term β a b (Fresnel-Fizeau effect). nc , mc el. & mg. Faraday effects,
and sa b optical activity; α is 4D pseudoscalar with dimension of an
admittance. Comparison with experiment, for Cr2 O3 see PRA’08.
12. Nonlocal linear constitutive law
Because the locality pronciple should break down at high accelerations,
Mashhoon has proposed a nonlocal electrodynamics in which nonlocality
is induced by the noninertial dynamics of observers; i.e., it results from
the observers’ accelerations. Then, instead of decomposing the
electromagnetic field with respect to dx i ∧ dx j , one should do it with
respect to the coframe ϑα = ei α dx i of a noninertial observer:
Z
1
Hαβ (τ, ξ) =
dτ ′ Kαβ γδ (τ, τ ′ ) Fγδ (τ ′ , ξ) .
2
A priori, a metric is not needed, but a linear connection Γα β = Γi α β dx i is
required. A suitable kernel was proposed, namely
1
Kαβ γδ (τ, τ ′ ) = ǫαβ µ[δ δµγ] δ(τ − τ ′ ) − u⌋Γµ γ] (τ ′ ) ,
2
but Mashhoon argued that this kernel cannot be applied to Fαβ , but at
most for the corresponding formula for the potential A(x) = Aα ϑα ...
Possible nonlocality in QFT is discussed by Grinstein, arXiv:1102.4011.
Suitable non-local models could still be Lorentz invariant,
renormalizable...
[A nonlocal gravitational theory generalizing GR has been propsed by
Mashhoon and H.]
13. Local nonlinear
constitutive law
ij
LocalpLorentz metric g is a prerequisit, since the premetric Lagrangian
L ∼ det Fij = 81 ǫ̂klmn Fkl Fmn can only describe the axion.
• Initial value problem can be formulated and solved under such
circumstances, see Volker Perlick, ask him!
• The Plebański nonlinear ED has the constitutive relation
H = U(I1 , I2 ) ⋆ F + V (I1 , I2 ) F ,
where U and V are functions of the two invariants
1
1
I1 := ⋆ (F ∧ ⋆ F )
and
I2 := ⋆ (F ∧ F ) ,
2
2
• The Born–Infeld ED U and V depend on both invariants:
r
⋆
F + 2f1 2 ⋆ (F ∧ F ) F
ε0
e
q
H=
, f0 ∼ Ee /c ≈ 6.4×1011 T .
µ0 1 − 12 ⋆ (F ∧ ⋆ F ) − 1 4 [⋆ (F ∧ F )]2
f
4f
e
e
1
Fij )]1/2 + b 2 , see Zwiebach
b
• in the Heisenberg–Euler case we have UHE (I1 ) and VHE (I2 ):
r ⋆
ε0
8 αf ⋆
14 αf ⋆
⋆
H=
1+
F+
F∧ F
F ∧ F F , Bk ≈ 4.4×109 T.
µ0
45 Bk2
45 Bk2
LBI ∼ −b 2 [− det(gij +
Only HE ED is a valid phys. theory: ML ED with 1st order qu. corrects.
14. Local and linear constitutive law
Local and linear spacetime relation H = κ(F ) with 36 functions:
Hij (x) =
1 kl
κij (x) Fkl (x) ,
2
κij kl = κ[ij] kl = κij [kl] .
Can be applied to vacuum, matter, and metamatter (“metamaterials”).
Decompose κij kl (“algebraic RC-curvature tensor”) into irreduc. pieces:
κi k := κil kl ,
κij kl
=
κ := κk k ,
l]
κij kl + 26 κ[i [k δj] + κ δ[ik δj]l /6 .
| {z }
| {z } | {z }
(1)
20, principal
kl
6 κi k := κi k − κ δik /4 ,
15,skewon
1,axion
κij is sold in string theory in any dimensions as area metric (Schuller et
al. 2006): “Spacetime is an area metric manifold”.
Convenient notation for dilaton λ, skewon, and axion fields:
√
(1)
6 Si j = − 6 κi j /2,
α = κ/12 .
κij kl = 4!λkij kl ,
Spacetime relation explicitly (kij kl kkl ij = −1, kij kl is tracefree):
Hij =
1
λkij kl Fkl + 26 S [i k Fj]k + α Fij .
2
(cf. the 3-dim. 6 × 6 Hookean tensor of elasticity and the Cauchy rels.)
1. χijkl algebraic Riemann or Riemann-Cartan tensor, classific. à la
Petrov, see Schuller et al., multipicities of the roots of the corr.
quartic polynomial. For alg. Riemann (skewon=axion= 0)
Petrov types I [1111], II [211], D 22, III [31], N [4]
2. Study electromagnetic shock waves in vacuum with the local linear
law, see lecture of Yakov Itin =⇒ Tamm-Rubilar 4th rank totally
symmetric tensor density G ijkl , generalized Fresnel equation. Forbid
birefringence (Lämmerzahl, Obukhov, H): the light cone emerges,
the light cone as an electromagnetic ‘animal’
3. Classification of matter according to the wave propagation
properties, see Bergamin, Favaro, Lindell, lecture of Alberto Favaro
4. Theory of propag. skewons in vacuum PRD’04, with gravity PLA’05
5. Theory of propag. axions in vacuum =⇒ Wilczek’s axion ED 1987
6. Proof of the existence of an axion piece in chromium sesquioxide
Cr2 O3 (H, Obukhov, Rivera, Schmid, PRA 2008), is T odd and P
odd, “Post constraint” of Post and Lakhtakia ruled out
7. Relation of the latter to the gyrator (Tellegen), PEMC (Lindell &
Sihvola)
To item 2. Electromagnetic shock waves in vacuum
“Light” propagation studied à la Hadamard. Sourcefree Maxwell eqs.
dH = 0 ,
dF = 0 .
Jumps [ ] in dH and dF investigated, wave covector q = qα ϑα := dΦ.
Derivation in Yakov Itin’s lecture. Find (extended) 4D Fresnel equation:
G ijkl (e
χ ) qi qj qk ql = 0 ,
with 4th-order Tamm–Rubilar (TR) tensor density of weight +1,
G ijkl (χ) :=
1
ǫ̂mnpq ǫ̂rstu χmnr (i χj|ps|k χl)qtu .
4!
It is totally symmetric G ijkl (χ) = G (ijkl) (χ) . with 35 independent
components. Explicitly,
G ijkl (χ) = G ijkl ((1) χ) + (1) χ m(i |n|j6 Smk 6 Sn l) ,
that is, the axion piece drops out. Wave covectors q lie on a quartic
Fresnel wave surface. We have shown that vanishing birefringence of the
vacuum yields the lightcone. Lightcone was derived from eldyn.!
Dilaton λ(x), metric g ij (x), axion α(x)
Then, for vanishing birefingence, the spacetime relation reads
√
Hij = [ λ(x) −g g ik (x) g jl (x) + α(x) ǫijkl ] Fkl ,
|{z}
|{z}
axion
dilaton
or, in exterior calculus,
H = λ(x) ⋆ F + α(x) F .
Axion not found so far: α = 0. Moreover, the dilaton seems to be a
constant field, i.e., λ(x) = λ0 −→ admittance of free space of 1/(377 Ω).
Accordingly,
√
Hij = λ0 −g g ik (x) g jl (x) Fkl
or
H = λ0 ⋆ F .
Hypothesis: dilaton and axion are fundamental fields with own
propagation degrees of freedom. Add corresponding kinetic terms to the
Lagrangian, and moreover, Born-Infeld ED =⇒ the theory of Burton,
Dereli, and Tucker, see Robin Tucker’s lecture.
Fresnel-Kummer surfaces (Clemens Schaefer):
Ray surface of a simple anisotropic crystal
[Algebraic geometry: Fresnel → Kummer → K3 → Calabi-Yau]
−1
εab = diag(39.7, 15.4, 2.3), µ−1
ab = µ0 diag(1, 1, 1), skewon piece
vanishes. Dimensionless variables x := cq1 /q0 , y := cq2 /q0 , z := cq3 /q0
[Sergei Tertychniy].
6-4
y
4
2
0
5
z
0
-5
x
-2
0
2
4
−1
Permittivity εab = ε0 diag(1, 1, 1), impermeability µ−1
ab = µ0 diag(1, 1, 1),
skewon field of electric Faraday type 6 S 3 0 = 3.1λ0 (all other components
vanish) [Sergei Tertychniy].
1
1
z
0
2
-1
0
-2
0
-2
x
2
y
−1
Permittivity εab = ε0 diag(1, 1, 1), impermeability µ−1
ab = µ0 diag(1, 1, 1),
skewon field of the magnetoelectric optical activity type
6 S1 2 =6 S2 1 = 0.8 λ0 (all other components vanish) [Sergei Tertychniy].
1
0.5
z
0
-0.5
1
-1
0
-1
-1
0
x
1
y
Permittivity εab = diag(2.4, 14.8, 54), impermeability
−1
µ−1
ab = µ0 diag(1, 1, 1), spatially isotropic skewon field
S6 1 1 =6S2 2 =6S3 3 = − 31 S6 0 0 = 0.25 λ0 (all other components vanish) [Sergei
Tertychniy].
-5
y
0
5
5
z
0
-5
-2
0
2
x
15. Metamaterials and the four families of
(1 + 3)-dimensional Maxwell equations
From the abstract of Itin & Friedman AdP 2007 (my emphasis):
“...the derivation of four different possible sets of electrodynamic
equations which may occur in different types of isotropic electromagnetic
media. The wave propagation in each of these media is described by the
Minkowskian optical metrics. Moreover, the electric and magnetic
energies are nonnegative in all cases. We also show that the correct
directions of the Lorentz force (as a consequence of the Dufay and the
Lenz rules) hold true for all these cases. However, the differences
between these four types of media must have a physical meaning. In
particular, the signs of the three electromagnetic invariants are different.”
In other words, out of one set of the 4d Mawell equations they derive via
(1+3) decomposition four sets. Make four out of one!
———–
——–
Addendum: A check: The TR-tensor for the conventional
Maxwell-Lorentz spacetime relation
The constitutive tensor density for vacuum reads
√
χijkl = λ0 −g g ik g jl − g il g jk .
3
We calculate the corresponding TR-tensor density, f := (λ30 /3!)(−g ) 2 :
G ijkl = f b
ǫmnpq b
ǫrstu g mr g ni (g js g pk − g jk g ps )g lt g qu (ijkl)
ikab jl
=f b
ǫ
b
ǫ ab + b
ǫ iabc b
ǫ l abc g jk (ijkl)
= f g (ij g kl) ,
G ijkl =
3
λ30
(−g ) 2 g ij g kl + g kj g il + g lj g ik .
18
This yields, indeed, via G ijkl qi qj qk ql = 0, the light cone (twice):
(g ij qi qj )(g kl qk ql ) = 0
⇒
light cone, q.e.d.