**7. Pseudoscalar-photon interaction**

In this section, we discuss the modified electromagnetism in gravity with the pseudoscalarphoton interaction which was reached in the last section, i.e., the theory with the constitutive tensor density (96). Its Lagrangian density is as follows

$$\mathbf{L}\_{l} = -\left(\mathbf{1}/(16\,\mathrm{m})\right)(-\mathbf{g})^{l/2}[(1/2)\mathbf{g}^{\mathrm{ik}}\mathbf{g}^{\mathrm{il}}\mathbf{\hat{r}}(1/2)\mathbf{g}^{\mathrm{il}}\mathbf{g}^{\mathrm{ik}} + \mathbf{g}^{\mathrm{cj}\bar{\mathbf{g}}}[\mathbf{F}\_{\bar{\mathbf{g}}}\mathbf{F}\_{\mathrm{il}} - \mathbf{A}\_{\bar{\mathbf{k}}}\mathbf{j}](-\mathbf{g})^{\mathrm{(1/2)}} - \Sigma\_{l}\,\mathbf{m}\_{l}\mathrm{(ds)}/\{\mathrm{(dt)}\delta(\mathbf{x}-\mathbf{x})\}. \tag{97}$$

In the constitutive tensor density and the Lagrangian density, ϕ is a scalar or pseudoscalar function of relevant variables. If we assume that the ϕ-term is local CPT invariant, then ϕ

Foundations of Electromagnetism, Equivalence Principles and Cosmic Interactions 61

 *Fik,k* + *eikml Fkm φ*,*l* = 0. (100) Analyzing the wave into Fourier components, imposing the radiation gauge condition, and solving the dispersion eigenvalue problem, we obtain *k* = *ω* + (*nμφ,*<sup>μ</sup> + *φ,0*) for right circularly polarized wave and *k* = *ω* – (*nμφ,*<sup>μ</sup> + *φ,0*) for left circularly polarized wave in the eikonal approximation (Ni 1973). Here *nμ* is the unit 3-vector in the propagation direction. The

 *v*g = ∂*ω*/∂*k* = 1, (101) which is independent of polarization. There is no birefringence. For the right circularly

*α* = *φ*(P2) - *φ*(P1) to the wave; for left circularly polarized light, the added phase will be opposite in sign (Ni 1973). Linearly polarized electromagnetic wave is a superposition of circularly polarized waves. Its polarization vector will then rotate by an angle *α*. Locally, the

The rotation angle in (102) consists of 2 parts -- *φ,0*Δx0 and [Σμ=13*φ,*μΔxμ]. For light in a local inertial frame, |Δxμ| = |Δx0|. In Fig. 4, space part of the rotation angle is shown. The amplitude of the space part depends on the direction of the propagation with the tip of magnitude on upper/lower sphere of diameter |Δxμ| × |*φ,μ*|. The time part is equal to Δx0 *φ,*0. (∇*φ* ≡ [*φ,μ*]) When we integrate along light (wave) trajectory in a global situation, the total polarization rotation (relative to no *φ*-interaction) is again Δ*φ = φ*<sup>2</sup> – *φ*1 for *φ* is a scalar field where *φ*1 and *φ*2 are the values of the scalar field at the beginning and end of the wave. When the propagation distance is over a large part of our observed universe, we call this

In the CMB polarization observations, there are variations and fluctuations. The variations and fluctuations due to scalar-modified propagation can be expressed as δ*φ*(2) - δ*φ*(1), where 2 denotes a point at the last scattering surface in the decoupling epoch and 1 observation point. δ*φ*(2) is the variation/fluctuation at the last scattering surface. δ*φ*(1) at the present observation point is zero or fixed. Therefore the covariance of fluctuation <[δ*φ*(2) - δ*φ*(1)]2> gives the covariance of δ*φ*2(2) at the last scattering surface. Since our Universe is isotropic to ~ 10-5, this covariance is ~ (ς× 10-5)2 where the parameter ς depends

Birefringence, also called double refraction, refers to the two different directions of propagation that a given incident ray can take in a medium, depending on the direction of

Dichroic materials have the property that their absorption constant varies with polarization. When polarized light goes through dichroic material, its polarization is rotated due to difference in absorption in two principal directions of the material for the two polarization

polarization. The index of refraction depends on the direction of polarization.

components. This phenomenon or property of the medium is called dichroism.

)] =Σi=03 [*φ,i*Δx*<sup>i</sup>*


} = {x(1)0; x(1)μ} = {x(1)0,

} = {x(2)0; x(2)μ} = {x(2)0, x(2)1, x(2)2, x(2)3} adds a phase of

] = - (½) *V*0Δx0 – (½) [Σμ=13*V*μΔxμ] (102)

] = *φ,*0Δx0 + [Σμ=13*φ,*μΔxμ]

polarized electromagnetic wave, the propagation from a point P1 = {x(1)*<sup>i</sup>*

group velocity is

x(1)1, x(1)2, x(1)3} to another point P2 = {x(2)*<sup>i</sup>*

polarization rotation angle can be approximated by

= - (½) Σi=03 [*Vi*Δxi

phenomenon cosmic polarization rotation (Ni, 2008, 2009a, 2010).

on various cosmological models. (Ni, 2008, 2009a, 2010)

Now we must say something about nomenclature.

*α* = *φ*(P2)-*φ*(P1) =Σi=03 [*φ,i* ×(x(2)*<sup>i</sup>*

should be a pseudoscalar (function) since *εijkl* is a pseudotensor. The pseudoscalar(scalar) photon interaction part (or the nonmetric part) of the Lagrangian density of this theory is

$$L^{\langle \text{p} \rangle \text{\textquotedblleft}} = L^{\langle \text{NM} \rangle} = \text{- (1/16 \text{m})} \text{ } q \, e^{\bar{q} \cdot \text{l}} F\_{\bar{q}} F\_{\text{il}} = \text{- (1/4 \text{m})} \, q \, \_i e^{\bar{q} \cdot \text{l}} A\_{\bar{l}} A\_{\bar{l}} \text{ (mod } \text{div)}, \tag{98}$$

where 'mod div' means that the two Lagrangian densities are related by integration by parts in the action integral. This term gives pseudoscalar-photon-photon interaction in the quantum regime and can be denoted by *L*(φγγ). This term is also the *ξ*-term in the PPM Lagrangian density *LPPM* with the *φ* ≡ (1/4)*ξΦ* correspondence. The Maxwell equations (Ni 1973, 1977) from Eq. (97) become

$$F^{ik}\_{\;\;\;k} + \varepsilon^{ikm\!\!\;} F\_{km} \varphi\_{\;\!\!\!/ \!} = \text{-} 4\text{II}\text{j}\text{'},\tag{99}$$

where the derivation ; is with respect to the Christoffel connection of the metric. The Lorentz force law is the same as in metric theories of gravity or general relativity. Gauge invariance and charge conservation are guaranteed. For discussions on the tests of charge conservation, and on the limits of differences in active and passive charges, please see Lämmerzahl et al. (2005, 2007). The modified Maxwell equations (99) are also conformally invariant.

The rightest term in equation (99) is reminiscent of Chern-Simons (1974) term *eαβγAαFβγ*. There are two differences: (i) Chern-Simons term is in 3 dimensional space; (ii) Chern-Simons term in the integral is a total divergence (Table 1). However, it is interesting to notice that the cosmological time may be defined through the Chern-Simons invariant (Smolin and Soo, 1995).


Table 1. Various terms in the Lagrangian and their meaning.

A term similar to the one in equation (98), axion-gluon interaction term, occurs in QCD in an effort to solve the strong CP problem (Peccei & Quinn, 1977; Weinberg, 1978; Wilczek, 1978). Carroll, Field and Jackiw (1990) proposed a modification of electrodynamics with an additional *eijkl Vi Aj Fkl* term with *Vi* a constant vector (See also Jackiw, 2007). This term is a special case of the term *eijkl φ Fij Fkl* (mod div) with *φ,i =* - ½*Vi*.

Various terms in the Lagrangians discussed in this subsection are listed in Table 1. Empirical tests of the pseudoscalar-photon interaction (98) will be discussed in next section.
