**1. Introduction**

24 Electromagnetic Theory

44 Trends in Electromagnetism – From Fundamentals to Applications

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Standard electromagnetism is based on Maxwell equations and Lorentz force law. It can be derived by a least action with the following Lagrangian density for a system of charged particles in Gaussian units (e.g., Jackson, 1999),

$$L\_{EM} = L\_{EM} + L\_{EM \cdot P} + L\_P = -(1/(16 \,\text{n}))[(1/2)\eta^{\text{n}}\eta^{\text{n}} - (1/2)\eta^{\text{n}}\eta^{\text{n}}]F\_{\bar{\eta}}F\_{\bar{\mathbf{k}} \text{T}}A\_{\bar{\mathbf{k}}}\bar{\mathbf{k}} - \Sigma\_{\bar{\mathbf{t}}}m\_{\bar{\mathbf{t}}}[\text{d}\mathbf{s}\_{\mathbf{l}}]/\langle\text{d}\mathbf{t}\rangle\langle\text{d}\mathbf{t}\rangle[\delta(\mathbf{x}-\mathbf{x}\_{\mathbf{l}})] \quad \text{(1)}$$

where *Fij ≡ Aj,i - Ai,j* is the electromagnetic field strength tensor with *Ai* the electromagnetic 4 potential and comma denoting partial derivation, *ηij* is the Minkowskii metric with signature (+, -, -, -), *mI* the mass of the *I*th charged particle*, sI* its 4-line element, and *jk* the charge 4 current density. Here, we use Einstein summation convention, i.e., summation over repeated indices. There are three terms in the Lagrangian density *LEMS* – (i) *LEM* for the electromagnetic field, (ii) *LEM-P* for the interaction of electromagnetic field and charged particles and (iii) *LP* for charged particles.

The electromagnetic field Lagrangian density can be written in terms of the electric field **E** [*≡* (*E1*, *E2*, *E3*) *≡* (*F01*, *F02*, *F03*)] and the magnetic induction **B** [*≡* (*B1*, *B2*, *B3*) *≡* (*F32*, *F13*, *F21*)] as

$$L\_{\rm EM} = (1/8 \text{m})[\mathbf{E}2\mathbf{-B}2].\tag{2}$$

This classical Lagrangian density is based on the photon having zero mass. To include the effects of nonvanishing photon mass *mphoton*, a mass term *LProca*,

$$L\_{Prava} = (m\_{photon}2c2\beta \text{nft}^2)(A\_kA^k),\tag{3}$$

needs to be added (Proca, 1936a, 1936b, 1936c, 1937, 1938). We use *ηij* and its inverse *ηij* to raise and lower indices. With this term, the Coulomb law is modified to have the electric potential *A0*,

$$A\_0 = q(e^{\mu r}/r),\tag{4}$$

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.

Foundations of Electromagnetism, Equivalence Principles and Cosmic Interactions 47

 *B*<sup>c</sup> ≡ *E*<sup>c</sup> ≡ *m2c3/eħ* =4.4x10l3 G=4.4x109 T=4.4x10l3 statvolt/cm=1.3x10l8 V/m, (11)

For time varying and space varying effects of external fields, and higher order corrections in quantum electrodynamics, please see Dittrich and Reuter (1985) and Kim (2011a, 2011b) and

Fig. 1. On the left is the basic diagram for light-light scattering and for nonlinear

proposed the following Lagrangian density for the electromagnetic field

wave) propagation in strong electric and/or magnetic field.

small compared with *b*, (14) can be expanded into

The next order corrections are of the form of Eq. (12) with

electrodynamics; on the right is the basic diagram for the nonlinear light (electromagnetic-

Before Heisenberg & Euler (1936), Born and Infeld (Born, 1934; Born & Infeld, 1934)

 *LBorn-Infeld* = -(*b*2/4π) [1 - (**E**2-**B**2)/*b*2 - (**E**·**B**)2/*b*4]1/2, (14) where *b* is a constant which gives the maximum electric field strength. For field strength

 *LBorn-Infeld* = (1/8π) [(**E**2-**B**2) + (**E**2-**B**2)2/*b*2 + (**E**·**B**)2/*b*2 + O(*b*-4)]. (15) The lowest order of Born-Infeld electrodynamics agrees with the classical electrodynamics.

*η*1 = *η*2 = *Bc*

In the Born-Infeld electrodynamics, *b* is the maximum electric field. Electric fields at the edge of heavy nuclei are of the order of 1021 V/m. If we take *b* to be 1021 V/m, then, *η*1 = *η*2 =

For formulating a phenomenological framework for testing corrections to Maxwell-Lorentz classical electrodynamics, we notice that (**E**2-**B**2) and (**E**·**B**) are the only Lorentz invariants second order in the field strength, and (**E**2-**B**2)2, (**E**·**B**)2 and (**E**2-**B**2) (**E**·**B**) are the only Lorentz invariants fourth order in the field strength. However, (**E**·**B**) is a total divergence and, by itself in the Lagrangian density, does not contribute to the equation of motion (field

*η*1 = α/(45π) = 5.1x10-5 and *η*2 = 7α/(180π) = 9.0 x10-5. (13)

*-2* [*η*1(**E**2-**B**2)2 + 4*η*2(**E**·**B**)2], (12)

*<sup>2</sup>*/*b*2. (16)

this Lagrangian density can be written as

references therein.

5.9 x 10-6.

 *LHeisenberg-Euler* = (1/8π) *Bc*

Experimental test on Coulomb's law (Williams, Faller & Hill, 1971) gives a constraint of the photon mass as

$$m\_{photon} \le 10^{14} \text{ eV (= } 2 \times 10^{47} \text{ g)},\tag{5}$$

on the interaction range *μ*-1 as

$$
\mu^{\cdot 1} \ge 2 \times 10^{\prime} \,\mathrm{m}.\tag{6}
$$

Photon mass affects the structure and the attenuation of magnetic field and therefore can be constrained by measuring the magnetic field of Earth, Sun or an astronomical body (Schrödinger, 1943; Bass & Schrödinger, 1955). From the magnetic field measurement of Jupiter during Pioneer 10 flyby, constraints are set as (Davis, Goldhaber & Nieto, 1975)

$$m\_{\text{phdown}} \le 4 \times 10^{\text{-16 eV}} \text{ eV (= 7 \times 10^{49} \text{ g}); } \mu \cdot 1 \ge 5 \times 10^{8} \text{ m.} \tag{7}$$

Using the plasma and magnetic field data of the solar wind, constraints on the photon mass are set recently as (Ryutov, 2007)

$$m\_{\text{photon}} \le 10^{\cdot 18} \text{ eV} \left(= 2 \times 10^{\cdot 51} \text{ g} \right); \mu^{\cdot 1} \ge 2 \times 10^{11} \text{ m}. \tag{8}$$

Large-scale magnetic fields in vacuum would be direct evidence for a limit on their exponential attenuation with distance, and hence a limit on photon mass. Using observations on galactic sized fields, Chibisov limit is obtained (Chibisov, 1976)

$$m\_{\text{photon}} \le 2 \times 10^{\circ 27} \,\text{eV} \, (= 4 \times 10^{\circ 60} \,\text{g}); \,\mu^{\circ 1} \ge 10^{20} \,\text{m}.\tag{9}$$

For a more detailed discussion of this work and for a comprehensive review on the photon mass, please see Goldhaber and Nieto (2010).

As larger scale magnetic field discovered and measured, the constraints on photon mass and on the interaction range may become more stringent. If cosmic scale magnetic field is discovered, the constraint on the interaction range may become bigger or comparable to Hubble distance (of the order of radius of curvature of our observable universe). If this happens, the concept of photon mass may lose significance amid gravity coupling or curvature coupling of photons.

Now we turn to quantum corrections to classical electrodynamics. In classical electrodynamics, the Maxwell equations are linear in the electric field **E** and magnetic field **B**, and we have the principle of superposition of electromagnetic field in vacuum. However, in the electrodynamics of continuous matter, media are usually nonlinear and the principle of superposition of electromagnetic field is not valid. In quantum electrodynamics, due to loop diagrams like the one in Fig. 1, photon can scatter off photon in vacuum. This is the origin of invalidity of the principle of superposition and makes vacuum a nonlinear medium also. The leading order of this effect in slowly varying electric and magnetic field is derived in Heisenberg and Euler (1936) and can be incorporated in the Heisenberg-Euler Lagrangian density

$$L\_{\text{Alleisembry-Iialer}} = \left[2\,\text{n}^2\text{/}\,^2 \text{45(4\,\text{n})}\,^2 \text{m}^4\,\text{c}^4\right] \left[(\text{E}\cdot\text{B}^2)^2 + 7(\text{E}\cdot\text{B})^2\right] \tag{10}$$

where α is the fine structure constant and *m* the electron mass. In terms of critical field strength *B*c defined as

$$B\_{\mathbb{C}} \equiv E\_{\mathbb{C}} \equiv m^2 c^3 / e\hbar = 4.4 \times 10^3 \text{ G} = 4.4 \times 10^9 \text{ T} = 4.4 \times 10^3 \text{ statvolt/cm} = 1.3 \times 10^8 \text{ V/m},\tag{11}$$

this Lagrangian density can be written as

46 Trends in Electromagnetism – From Fundamentals to Applications

Experimental test on Coulomb's law (Williams, Faller & Hill, 1971) gives a constraint of the

 *mphoton* ≤ 10-14 eV (= 2 × 10-47 g), (5)

Photon mass affects the structure and the attenuation of magnetic field and therefore can be constrained by measuring the magnetic field of Earth, Sun or an astronomical body (Schrödinger, 1943; Bass & Schrödinger, 1955). From the magnetic field measurement of Jupiter during Pioneer 10 flyby, constraints are set as (Davis, Goldhaber & Nieto, 1975)

 *mphoton* ≤ 4 × 10-16 eV (= 7 × 10-49 g); *μ*-1 ≥ 5 × 108 m. (7) Using the plasma and magnetic field data of the solar wind, constraints on the photon mass

 *mphoton* ≤ 10-18 eV (= 2 × 10-51 g); *μ*-1 ≥ 2 × 1011 m. (8) Large-scale magnetic fields in vacuum would be direct evidence for a limit on their exponential attenuation with distance, and hence a limit on photon mass. Using

 *mphoton* ≤ 2 × 10-27 eV (= 4 × 10-60 g); *μ*-1 ≥ 1020 m. (9) For a more detailed discussion of this work and for a comprehensive review on the photon

As larger scale magnetic field discovered and measured, the constraints on photon mass and on the interaction range may become more stringent. If cosmic scale magnetic field is discovered, the constraint on the interaction range may become bigger or comparable to Hubble distance (of the order of radius of curvature of our observable universe). If this happens, the concept of photon mass may lose significance amid gravity coupling or

Now we turn to quantum corrections to classical electrodynamics. In classical electrodynamics, the Maxwell equations are linear in the electric field **E** and magnetic field **B**, and we have the principle of superposition of electromagnetic field in vacuum. However, in the electrodynamics of continuous matter, media are usually nonlinear and the principle of superposition of electromagnetic field is not valid. In quantum electrodynamics, due to loop diagrams like the one in Fig. 1, photon can scatter off photon in vacuum. This is the origin of invalidity of the principle of superposition and makes vacuum a nonlinear medium also. The leading order of this effect in slowly varying electric and magnetic field is derived in Heisenberg and Euler (1936) and can be incorporated in the Heisenberg-Euler

where α is the fine structure constant and *m* the electron mass. In terms of critical field

observations on galactic sized fields, Chibisov limit is obtained (Chibisov, 1976)

*μ*-1 ≥ 2 × 107 m. (6)

/45(4π)2*m*4*c*6][(**E**2-**B**2)2 + 7(**E**·**B**)2], (10)

photon mass as

on the interaction range *μ*-1 as

are set recently as (Ryutov, 2007)

curvature coupling of photons.

Lagrangian density

strength *B*c defined as

 *LHeisenberg-Euler* = [2α2*ħ*<sup>2</sup>

mass, please see Goldhaber and Nieto (2010).

$$L\_{\text{Hieenberg}\cdot\text{Euler}} = \left(1/8\,\text{n}\right)B\_c\,^2\left[\eta\_1(\mathbf{E}\cdot\mathbf{B}^2)^2 + 4\eta\_2(\mathbf{E}\cdot\mathbf{B})^2\right] \tag{12}$$

$$
\eta\_1 = \mathbf{a} / (45\,\text{n}) = 5.1 \times 10^5 \,\text{and}\,\eta\_2 = 7\mathbf{a} / (180\,\text{n}) = 9.0 \times 10^5.\tag{13}
$$

For time varying and space varying effects of external fields, and higher order corrections in quantum electrodynamics, please see Dittrich and Reuter (1985) and Kim (2011a, 2011b) and references therein.

Fig. 1. On the left is the basic diagram for light-light scattering and for nonlinear electrodynamics; on the right is the basic diagram for the nonlinear light (electromagneticwave) propagation in strong electric and/or magnetic field.

Before Heisenberg & Euler (1936), Born and Infeld (Born, 1934; Born & Infeld, 1934) proposed the following Lagrangian density for the electromagnetic field

$$L\_{\text{Born-lufel}} = \text{-}(b^2/4\text{m})\left[1 - (\mathbf{E}^2 \cdot \mathbf{B}^2)/b^2 \cdot (\mathbf{E} \cdot \mathbf{B})^2/b^4\right]^{1/2},\tag{14}$$

where *b* is a constant which gives the maximum electric field strength. For field strength small compared with *b*, (14) can be expanded into

$$L\_{\text{Born-hydfeld}} = (1/8\,\text{m})\left[ (\text{E-B2}) + (\text{E-B2})2/b2 + (\text{E-B2})2/b2 + \text{O}(b\cdot 4) \right]. \tag{15}$$

The lowest order of Born-Infeld electrodynamics agrees with the classical electrodynamics. The next order corrections are of the form of Eq. (12) with

$$
\eta\_1 = \eta\_2 = B\_c \mathbb{Z}/b^2. \tag{16}
$$

In the Born-Infeld electrodynamics, *b* is the maximum electric field. Electric fields at the edge of heavy nuclei are of the order of 1021 V/m. If we take *b* to be 1021 V/m, then, *η*1 = *η*2 = 5.9 x 10-6.

For formulating a phenomenological framework for testing corrections to Maxwell-Lorentz classical electrodynamics, we notice that (**E**2-**B**2) and (**E**·**B**) are the only Lorentz invariants second order in the field strength, and (**E**2-**B**2)2, (**E**·**B**)2 and (**E**2-**B**2) (**E**·**B**) are the only Lorentz invariants fourth order in the field strength. However, (**E**·**B**) is a total divergence and, by itself in the Lagrangian density, does not contribute to the equation of motion (field

Foundations of Electromagnetism, Equivalence Principles and Cosmic Interactions 49

curl **E** = -(1/*c*) ∂**B**/∂*t*, (25)

 div **B** = 0. (26) We notice that it has the same form as in macroscopic electrodynamics. The Lorentz force

for the *I*-th particle with charge *qI* and velocity **v***I* in the system. The source of *Φ* in this

Here we follow the previous method (Ni et al., 1991; Ni, 1998), and separate the electric field and the magnetic induction field into the wave part (small compared to external part) and

 **E** = **E**wave + **E**ext, (29)

 **B** = **B**wave + **B**ext. (30) We use the following expressions to calculate the displacement field **D**wave [= (*D*wave*α*) = (*D*wave*1*, *D*wave*2*, *D*wave*3*)] and the magnetic field **H**wave [= (*H*wave*α*) = (*H*wave*1*, *H*wave*2*, *H*wave*3*)] of

 *D*wave*<sup>α</sup>* = *Dα – D*ext*α* = (4π*)*[(∂*L*PPM/∂E*α*)**E**&**B** - (∂*L*PPM/∂E*α*)ext], (31)

 *H*wave*<sup>α</sup>* = *Hα – H*ext*α* = - (4π*)*[(∂*L*PPM/∂B*α*) **E**&**B** - (∂*L*PPM/∂B*α*)ext], (32) where (…) **E**&**B** means that the quantity inside paranthesis is evaluated at the total field values **E** & **B** and (…)ext means that the quantity inside paranthesis is evaluated at the

Since both the total field and the external field satisfy Eqs. (23)-(26), the wave part also

curl **Hwave** = (1/c) ∂**Dwave**/∂*t*, (33)

div **Dwave** = 0, (34)

curl **Ewave** = -(1/c) ∂**Bwave**/∂*t*, (35)

 div **Bwave** = 0. (36) After calculating *D*wave*α* and *H*wave*α* from Eqs. (31) & (32), we express them in the following

satisfy the same form of Eqs. (23)-(26) with the source terms subtracted:

2/c2)-1/2*mI***v***I*]/dt = *qI*[**E** + (1/c)**v***<sup>I</sup>* × **B**] (27)

*Φ) -* ∂*LΦ*/∂*Φ*= **E**·**B**, (28)

law remains the same as in classical electrodynamics:

∂*i LΦ*/∂(∂*<sup>i</sup>*

**3. Electromagnetic wave propagation in PPM electrodynamics** 

where *LΦ* is the Lagrangian density of the pseudoscalar field *Φ*.

system is (**E**·**B**) and the field equation for *Φ* is

 *d*[(1-**v***<sup>I</sup>*

external part as follows:

the electromagnetic waves:

external field values **E**ext & **B**ext.

form:

equation). Multiplying (**E**·**B**) by a pseudoscalar field *Φ*, the term *Φ*(**E**·**B**) is the Lagrangian density for the pseudoscalar-photon (axion-photon) interaction. When this term is included together with the fourth-order invariants, we have the following phenomenological Lagrangian density for our Parametrized Post-Maxwell (PPM) Lagrangian density including various corrections and modifications to be tested by experiments and observations,

$$L\_{\rm PPM} = (1/8\,\text{n})[(\mathbf{E}\cdot\mathbf{B}^2) + \xi\otimes(\mathbf{E}\cdot\mathbf{B}) + \mathcal{B}\_c\cdot\mathbf{j}\,\text{[}\eta\_1(\mathbf{E}\cdot\mathbf{B}^2) + 4\eta\_2(\mathbf{E}\cdot\mathbf{B})\mathbf{2} + 2\eta\_3(\mathbf{E}\cdot\mathbf{B}^2)(\mathbf{E}\cdot\mathbf{B})]).\tag{17}$$

This PPM Lagrangian density contains 4 parameters *ξ, η*1, *η*2 & *η*3, and is an extension of the two-parameteer (*η*1 and *η*2) post-Maxwellian Lagrangian density of Denisov, Krivchenkov and Kravtsov (2004). The manifestly Lorentz covariant form of Eq. (17) is

$$L\_{\rm PPM} = \left(1/\text{(52m)}\right) \left\{-2F^{\text{il}}F\_{\text{il}} - \xi \mathcal{O}F^{\text{st}}F\_{\text{il}} + B\_{\text{c}}\mathcal{I} \left[\eta\_1 (F^{\text{il}}F\_{\text{il}})^2 + \eta\_2 (F^{\text{st}}F\_{\text{il}})^2 + \eta\_3 (F^{\text{il}}F\_{\text{il}}) (F^{\text{st}}F\_{\text{il}})\right] \right\},\tag{18}$$

where

$$F^{\*ij} \equiv (\mathbf{1}/\mathbf{2})e^{ijkl} \text{ F}\_{kl} \tag{19}$$

with *eijkl* defined as

$$e^{j\text{jil}} \equiv 1 \text{ if (ijkl) is an even permutation of (0123): -1 if odd;} 0 \text{ otherwise.} \tag{20}$$

In section 2, we derive the PPM nonlinear electrodynamic equations, and in section 3, we use them to derive the light propagation equation in PPM nonlinear electrodynamics. In section 4, we discuss ultra-high precision laser interferometry experiments to measure the parameters of PPM electrodynamics. In section 5, we treat electromagnetism in curved spacetime using Einstein Equivalence Principle, and discuss redshift as an application with examples from astrophysics and navigation. In section 6, we discuss empirical tests of electromagnetism in gravity and the χ-g framework and find pseudoscalar-photon interaction uniquely standing out. In section 7, we discuss the pseudoscalar-photon interaction and its relation to other approaches. In section 8, we use Cosmic Microwave Background (CMB) observations to constrain the cosmic polarization rotation and discuss radio galaxy observations. In section 9, we present a summary and an outlook briefly.
