**2. Equations for nonlinear electrodynamics**

In analogue with the nonlinear electrodynamics of continuous media, we can define the electric displacement **D** and the magnetic field **H** as follows:

$$\mathbf{D} \equiv 4\boldsymbol{\pi}(\partial\mathcal{L}\_{\mathrm{PDM}}\mathcal{\partial}\mathbf{E}) = [1 + 2\boldsymbol{\eta}\_{1}(\mathbf{E}\mathbf{-B}^{2})\mathcal{B}\_{\mathrm{c}}\cdot^{2} + 2\boldsymbol{\eta}\_{\overline{\mathcal{D}}}(\mathbf{E}\cdot\mathbf{B})\mathcal{B}\_{\mathrm{c}}\cdot^{2}]\mathbf{E} + [\mathcal{O} + 4\boldsymbol{\eta}\_{\overline{\mathcal{D}}}(\mathbf{E}\cdot\mathbf{B})\mathcal{B}\_{\mathrm{c}}\cdot^{2} + \boldsymbol{\eta}\_{\overline{\mathcal{D}}}(\mathbf{E}\cdot\mathbf{B}^{2})\mathcal{B}\_{\mathrm{c}}\cdot^{2}]\mathbf{B},\tag{21}$$

$$\mathbf{H} \equiv 4\mathbf{m}(\partial \mathbf{L}\_{\mathrm{PPM}} \partial \mathbf{B}) = [1 + 2\eta\_1(\mathbf{E} \mathbf{\cdot} \mathbf{B}^2) \mathbf{B}\_{\varepsilon} \, \varepsilon + 2\eta\_3(\mathbf{E} \cdot \mathbf{B}) \mathbf{B}\_{\varepsilon} \, \varepsilon] \mathbf{B} \cdot [\mathcal{O} + 4\eta\_2(\mathbf{E} \cdot \mathbf{B}) \mathbf{B}\_{\varepsilon} \, \varepsilon + \eta\_3(\mathbf{E} \cdot \mathbf{B}^2) \mathbf{B}\_{\varepsilon} \, \varepsilon] \mathbf{E} \,, \quad (22)$$

From **D** & **H**, we can define a second-rank *Gij* tensor, just like from **E** & **B** to define *Fij* tensor. With these definitions and following the standard procedure in electrodynamics [see, e.g., Jackson (1999), p. 599], the nonlinear equations of the electromagnetic field are

$$\operatorname{curl} \mathbf{H} = \left( \mathbf{1}/\mathbf{c} \right) \partial \mathbf{D} / \partial t + 4 \operatorname{tr} \mathbf{J} \tag{23}$$

$$\operatorname{div} \mathbf{D} = 4 \text{tr} \,\rho,\tag{24}$$

48 Trends in Electromagnetism – From Fundamentals to Applications

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

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

 *eijkl* ≡ 1 if (*ijkl*) is an even permutation of (0123); -1 if odd; 0 otherwise. (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.

In analogue with the nonlinear electrodynamics of continuous media, we can define the

*-2*]**E+**[*Φ+*4*η*2(**E**·**B**)*Bc*

*-2*]**B-**[*Φ+*4*η*2(**E**·**B**)*Bc*

*-2*+*η*3(**E**2-**B**2)*Bc*

*-2*+*η*3(**E**2-**B**2)*Bc*

*-2*]**B**, (21)

*-2*]**E.** (22)

*-2*+2*η*3(**E**·**B**)*Bc*

*-2*+2*η*3(**E**·**B**)*Bc*

Jackson (1999), p. 599], the nonlinear equations of the electromagnetic field are

From **D** & **H**, we can define a second-rank *Gij* tensor, just like from **E** & **B** to define *Fij* tensor. With these definitions and following the standard procedure in electrodynamics [see, e.g.,

curl **H** = (1/c) ∂**D**/∂*t +* 4π **J**, (23)

div **D** = 4π *ρ*, (24)

*-2*[*η*1(**E**2-**B**2)2+4*η*2(**E**·**B**)2+2*η*3(**E**2-**B**2)(**E**·**B**)]}. (17)

*-2*[*η*1(*FklFkl*)2+*η*2(*F*\**klFkl*)2+*η*3(*FklFkl*)(*F*\**ijFij*)]}, (18)

*F\*ij* ≡ (1/2)*eijkl Fkl*, (19)

various corrections and modifications to be tested by experiments and observations,

and Kravtsov (2004). The manifestly Lorentz covariant form of Eq. (17) is

 *LPPM* = (1/8π){(**E**2-**B**2)+*ξΦ*(**E**·**B**)+*Bc*

 *LPPM =* (*1/*(32π)){-2*FklFkl* -*ξΦF*\**klFkl*+*Bc*

**2. Equations for nonlinear electrodynamics** 

 **D***≡*4π(∂*LPPM/∂***E**)=[1+2*η*1(**E**2-**B**2)*Bc*

 **H***≡*-4π(∂*LPPM/∂***B**)=[1+2*η*1(**E**2-**B**2)*Bc*

electric displacement **D** and the magnetic field **H** as follows:

where

with *eijkl* defined as

$$\text{curl } \mathbf{E} = \text{-}(1/c) \,\partial \mathbf{B}/\partial t,\tag{25}$$

$$\text{div } \mathbf{B} = 0. \tag{26}$$

We notice that it has the same form as in macroscopic electrodynamics. The Lorentz force law remains the same as in classical electrodynamics:

$$d[(\mathbf{1} \cdot \mathbf{v})^2 / \mathbf{c}^2]^{1/2} m |\mathbf{v}| / \det = q[\mathbf{E} + (\mathbf{1}/\mathbf{c})\mathbf{v} \times \mathbf{B}] \tag{27}$$

for the *I*-th particle with charge *qI* and velocity **v***I* in the system. The source of *Φ* in this system is (**E**·**B**) and the field equation for *Φ* is

$$
\partial \mathbf{J}^{\perp} \!/ \partial (\partial \!/ \otimes \mathbf{0}) \text{ - } \partial \!\mathbf{J}^{\perp} \!/ \partial \!\!/ \partial \!\!/ \mathbf{0} \text{= } \mathbf{E} \cdot \mathbf{B} \,, \tag{28}
$$

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