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Foundations of Classical Electrodynamics
and
Optical Experiments to Measure
the Parameters of the PPM (Parametrized
Post-Maxewellian) Electrodynamics
經典電動力學的基礎(arXiv 1109.5501)
W.-T. Ni 倪維斗
Department of Physics,
National Tsing Hua University
[email protected]
2011.12.12. NTHU
Foundations of Classical Electrodynamics and PPM Framework
W-T Ni
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Outline
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Introduction – Maxwell equations and Lorentz force law
Photon mass constraints
Quantum corrections – quantum corrections
Parametrized Post-Maxwell (PPM) electrodynamics
Electromagnetic wave propagation
Measuring the parameters of the PPM electrodynamics
Electrodynamics in curved spacetime and EEP
Empirical tests of electromagnetism and the χ-g framework
Pseudoscalar-photon interaction and the cosmic pol. rotation
Discussion and outlook
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Introduction – Maxwell Equations
and Lorentz Force Law (I)
馬克士威方程式與羅倫茲力
(Jackson)
(法拉第定律)
(無磁單極)
(庫倫定律)
(安培-馬克士威定律)
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Introduction – Maxwell Equations
and Lorentz Force Law (II)
馬克士威方程式與羅倫茲力
(連續方程式-電荷守恆)
(羅倫茲力)
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Charges (and currents)  produce E and B fields  influence the Motion of charges
Maxwell Equations
Lorentz Force Law
電荷和電流產生電場和磁場；電場和磁場影響電荷的運動
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Introduction – Maxwell Equations
and Lorentz Force Law (III)
馬克士威方程式與羅倫茲力
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法拉第定律和無磁單極定律相當於電場和磁場可以由
標量勢和向量勢A表示：
4-vector potential A  (, A)
Second-rank, antisymmetric field-strength tensor
F = ∂A - ∂A
Electric field E [≡ (E1, E2, E3) ≡ (F01, F02, F03)] and magnetic induct
ion B [≡ (B1, B2, B3) ≡ (F32, F13, F21)]
2
2
Electromagnetic field Lagrangian density LEM = (1/8π)[E -B ].
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Lagrangian density LEMS
for a system of charged particles
in Gaussian units
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LEMS=LEM+LEM-P+LP
=-(1/(16π))[(1/2)ηikηjl-(1/2)ηilηkj]FijFkl
-Akjk-ΣI mI[(dsI)/(dt)]δ(x-xI),
LEMS 整个電荷系統拉格朗日
LEM 電磁場拉格朗日
LEM-P 電磁場-粒子相互作用拉格朗日
LP 粒子拉格朗日
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Test of Coulomb’s Law 1/r2+
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Cavendish 1772
||  0.02
4MHz 10kV p
Maxwell 1879
||  5  10-5
Plimpton and Lawton
||  2  10-9
Williams, Faller, and Hill
1971
 = (2.7  3.1)10-16
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1.5 m 
12.1 in 
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Proca (1936-8) Lagrangian density
and mass of photon
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LProca = (mphoton2c2/8πħ2)(AkAk)
the Coulomb law is modified to have the
electric potential A0 = q(e-μr/r)
where q is the charge of the source particle, r is
the distance to the source particle, and μ
(≡mphotonc/ħ) gives the inverse range of the
interaction
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Constraints on
the mass of photon
mphoton ≤ 10-14 eV
μ-1 ≥ 2 × 107 m
mphoton ≤ 4 × 10-16 eV
μ-1 ≥ 5 × 108 m
Ryutov (2007) Solar
wind magnetic field
mphoton ≤ 10-18 eV
μ-1 ≥ 2 × 1011 m
Chibisov (1976)
galactic sized fields
mphoton ≤ 2 × 10-27 eV
μ-1 ≥ 1020 m
Williams, Faller &
Hill (1971) Lab Test
Davis, Goldhaber &
Nieto (1975) Pioneer
10 Jupitor flyby
(= 2 ×
(= 7 ×
(= 2 ×
(= 4 ×
10-47
10-49
10-51
10-60
g)
g)
g)
g)
A good Reference: Goldhaber, A. S. & Nieto, M. M. (2010). Photon and Graviton Mass
Limits. Review of Modern Physics, Vol.82, No.1, (January-March 2010), pp. 939-979
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Quantum corrections to
classical electrodynamics
Heisenberg-Euler Lagrangian
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Born-Infeld Electrodynamics
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Parametrized Post-Maxwell (PPM)
Lagrangian density
(4 parameters: ξ, η1, η2, η3)

LPPM = (1/8π){(E2-B2)+ξΦ(E∙B)
+Bc-2[η1(E2-B2)2 +4η2(E∙B)2+2η3(E2-B2)(E∙B)]}
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LPPM = (1/(32π)){-2FklFkl -ξΦF*klFkl
+Bc-2 [η1(FklFkl)2+η2(F*klFkl)2+η3(FklFkl)(F*ijFij)]}
(manifestly Lorentz invariant form)
Dual electomagnetic field F*ij ≡ (1/2)eijkl Fkl
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Unified theory of nonlinear
electrodynamics and gravity
A. Torres-Gomez, K. Krasnov, & C. Scarinci PRD 83, 025023 (2011)
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A class of unified theories of
electromagnetism and
gravity with Lagrangian of
the BF type (F: Curvature of
the connection 1-form A
(), with a potential for the
B () field (Lie-algebra
valued 2-form), the gauge
group is U(2)
(complexified).
Given a choice of the
potential function the theory
is a deformation of
(complex) general relativity
and electromagnetism.
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Equations for nonlinear
electrodynamics (1)
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Equations for nonlinear
electrodynamics (2)
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Electromagnetic wave propagation
in PPM electrodynamics
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Birefringence
or no Birefringnce
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Measuring the parameters of
the PPM electrodynamics
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Δn = n║ - n┴ = 4.0 x 10-24 (Bext/1T)2
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Measuring the parameters of
the PPM electrodynamics
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Let’s choose z-axis to be in the propagation direction, x-axis in the Eext
direction and y-axis in the Bext direction, i.e., k = (0, 0, k), Eext = (E, 0, 0)
and Bext = (0, B, 0).
n± = 1 + (η1+η2)(E2+B2-EB)Bc-2± [(η1-η2)2(E2+B2-EB)2+η32(E2-B2)]1/2 Bc-2.
(i) E=B as in the strong microwave cavity, the indices of refraction for
light is
n± = 1 + (η1+η2)B2Bc-2±(η1-η2)B2Bc-2,
with birefringence Δn given by
Δn = 2(η1-η2)B2Bc-2;
(ii) E=0, B≠0, the indices of refraction for light is
n± = 1 + (η1+η2)B2Bc-2±[(η1-η2)2+η32]1/2B2Bc-2,
Δn = 2[(η1-η2)2+η32]1/2B2Bc-2.
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Measuring the parameters of
the PPM electrodynamics
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Lab Experiment:
Principle of Experiment
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Apparatus and
Finesse Measurement
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Suspension and
Analyzer’s Extinction ratio
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Injection Optical Bench
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Ni
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Vacuum Chamber and Magnet
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目前出光狀態與Finesse
(將耗散/投影損失視為反射率較差的表現)
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Current Optical Experiments
LNL
Ferrara
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PVLAS
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Rotation and ellipticity sensitivity
comparisons using ellipsometers with
optical path multipliers
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PVLAS Ferrara
on 3 competing experiments (I)
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Q & A in Taiwan: They have the mirrors of the FP installed in two
distant vacuum chambers suspended with attenators of ambient
vibrations of the type developed for interferometric gravitational
wave detectors. The separation of the two optical halves seems to
limit the sensitivity of their apparatus. The finesse is below 10^5
OSQAR at CERN: uses a LHC dipole magnet 15 m long that can
reach a 9 T field. The ellipsometer will exploit a FP to maximize the
number of reflections and a novel optical techique to modulate the
MBV effect. Since it is not feasible to set the LHC magnet in rotation
and a modulation of the LHC magnet field intensity could be
achieved only at very low frequency, the experiment foresees to
modulate the polarization of the light entering the ellipsometer by
setting in rotation the polarization plane.
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PVLAS Ferrara
on 3 competing experiments (II)
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According to our experience in this set-up the
rotation of the polarization will generate a very
large signal due to the intrinsic birefringence of the
FP mirrors.
BMV in Toulouse: homodyne, high intensity
magnetic field, employing pulsed magnets (few
ms), 40 cm long, that have already reached peak
intensities in excess of B = 15 T, L = 0.5 m total
length, 5 shots/hour. Finesse 10^5, 10^(-7) per
square root Hertz ellipticity sensitivity. For ten times
improvement in sensitivity, it takes 650 years of
continuous datataking.
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Comparisons on the
N2 magnetic birefringence measurement
PVLAS 2004
Q&A 2009
BMV2011
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(Pseudo)scalar field: WEP & EEP in EM field
(Pseudo)scalar-Photon
Interaction
Modified Maxwell Equations  Polarization Rotation in EM Propagaton
(Classical effect)
Constraints from CMB polarization observation  This talk
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Galileo’s experiment on inclined plane
(Contemporary painting of Giuseppe Bezzuoli)
Galileo Equivalence Principle:
Universality of free-fall trajectories
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GP-B and Rotational EP
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Einstein Equivalence Principle
EEP:(Einstein Elevator)
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Local physics is that of Special relativity
Study the relationship of Galileo
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Equivalence Principle and EEP in a
Relativistic Framework:   g framework
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Electromagnetism：
Charged particles and photons
Special Relativity

1
dsI
ikjl jl
k
1/ 2
LI  (
)η η Fij Fkl  Ak j ( g )   I mI
 ( x  xI )
16
dt
 g framework
1
dsI
ijkl
k
1/ 2
LI  (
)  Fij Fkl  Ak j ( g )   I mI
 ( x  xI )
16
dt
Galileo EP constrains  to：
ijkl
1 / 2 1 ik jl 1 il kj
  ( g ) [ g g  g g  ψ  ijkl ]
2
2
(Pseudo)scalar-Photon
Interaction
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Various terms in the Lagrangian
(W-T Ni, Reports on Progress in Physics, next month
/also in arXiv)
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Empirical Constraints: No Birefringence
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Empirical Constraints from Unpolarized EP
Experiment: constraint on Dilaton for EM:
φ = 1 ± 10^(-10)
Cho and Kim, Hierarchy Problem, Dilatonic Fifth, and Origin of Mass, ArXiv0708.2590v1
(4+3)-dim unification with G=SU(2), L<44 μm (Kapner et al., PRL 2007)
L<10 μm (Li, Ni, and Pulido Paton, ArXiv0708.2590v1
Lamb shift in Hydrogen and Muonium gr-qc
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Emprirical constraints: H  g
(One Metric)
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Constraint on axion: φ < 0.1
Solar-system 1973 (φ < 10^10)
Metric Theories of Gravity
 General Relativity
 Einstein Equivalence Principle
recovered
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For a recent exposition, see Hehl &
Obukhov ArXiv:0705.3422v1
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Change of Polarization due
to Cosmic Propagation
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The effect of φ is to change the phase of two different
circular polarizations of electromagnetic-wave propagation
in gravitation field and gives polarization rotation for
linearly polarized light.[6-8]
Polarization observations of radio galaxies put a limit of Δφ
≤ 1 over cosmological distance.[9-14]
Further observations to test and measure Δφ to 10-6 is
promising.
The natural coupling strength φ is of order 1. However, the
isotropy of our observable universe to 10-5 may leads to a
change (ξ)Δφ of φ over cosmological distance scale 10-5
smaller. Hence, observations to test and measure Δφ to 106 are needed.
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The angle between the direction of linear
polarization in the UV and the direction of the
UV axis for RG at z > 2. The angle predicted by
the scattering model is 90^o
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The advantage of the test using
the optical/UV polarization over
that using the radio one is that
it is based on a physical
prediction of the orientation of
the polarization due to
scattering, which is lacking in
the radio case,
and that it does not require a
correction for the Faraday
rotation, which is considerable
in the radio but negligible in the
optical/UV.
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Constraints on cosmic polarization rotation
from CMB polarization observations
[See Ni, RPP 73, 056901 (2010) for detailed references]
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CMB Polarization Observation
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In 2002, DASI microwave interferometer observed
the polarization of the cosmic background.
With the pseudoscalar-photon interaction , the
polarization anisotropy is shifted relative to the
temperature anisotropy.
In 2003, WMAP found that the polarization and
temperature are correlated to 10σ. This gives a
constraint of 10-1 rad or 6 degrees of the cosmic
polarization rotation angle Δφ.
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CMB Polarization Observation
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In 2005, the DASI results were extended (Leitch
et al.) and observed by CBI (Readhead et al.) and
CAPMAP (Barkats et al.)
In 2006, BOOMERANG CMB Polarization
DASI, CBI, and BOOMERANG detections of
Temperature-polarization cross correllation
QuaD
Planck Surveyor was launched last year with
better polarization-temperature measurement
sensitivity. Sensitivity to cosmic polarization
rotation Δφ of 10-2-10-3 expected.
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References
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Space contribution to the local polarization rotation
angle -- [μΣ13φ,μΔxμ] = |▽φ| cos θ Δx0. The time
contribution is φ,0 Δx0. The total contribution is
(|▽φ| cos θ + φ,0) Δx0. (Δx0 > 0)
Intergrated:
φ(2) - φ(1)
1: a point at the
decoupling epoch
2: observation
point
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Variations and Fluctuations
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Rotation φ(2) - φ(1)
δφ(2) - δφ(1): δφ(2) variations and
fluctuations at the last scattering surface
of the decoupling epoch; δφ(1), at
present observation point, fixed
<[δφ(2) - δφ(1)]^2> variance of
fluctuation ~ [couplingξ × 10^(-5)]^2
The coupling depends on various
cosmological models
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COSMOLOGICAL MODELS
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PSEUDO-SCALAR COSMOLOGY, e.g., BransDicke theory with pseudoscalar-photon
coupling
NEUTRINO NUMBER ASYMMETRY
BARYON ASYMMETRY
SOME other kind of CURRENT
LORENTZ INVARIANCE VIOLATION
CPT VIOLATION
DARK ENERGY (PSEUDO)SCALAR COUPLING
OTHER MODELS
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Constraints on
cosmic polarization rotation from CMB
Newest : Brown et al.
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11.2±8.7 ±8.7 mrad
QuaD 2009
All consistent with null detection and
with one another at 2 σ level
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Outlook
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Precision tests of Classical Electrodynamics
will continue to serve physics community in
frontier research, in the quantum regime, in
gravitation and in cosmology
Thank you for your attention
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