*2.4. Field Equations*

Neglecting the fermion sector, let us derive the field equations for the Lagrangian (18). Technically, we need to make variations with respect to the axial torsion field *α<sup>i</sup>* and the

electromagnetic field potential *A<sup>i</sup>* . The corresponding Euler–Lagrange equation *δL*/*δα<sup>i</sup>* = 0 for the torsion reads:

$$-\left.\partial\_{\dot{j}}f^{i\dot{j}} + \mu^2 a^i + \lambda \partial^i(\partial a) + \chi \sqrt{\frac{\varepsilon\_0}{\mu\_0}} \eta^{i\dot{j}kl} A\_{\dot{j}} \partial\_k A\_l = 0. \tag{22}$$

Introducing the "electric" and "magnetic" components of the torsion variable by means of

$$\mathcal{E}\_a = f\_{a0\prime} \qquad \mathcal{B}^a = \frac{1}{2} \epsilon^{abc} f\_{bc\prime} \qquad a, b = 1, 2, 3,\tag{23}$$

along with the usual definitions of the electric and magnetic fields

$$E\_{a} = F\_{a0\prime} \qquad B^{a} = \frac{1}{2} \epsilon^{abc} F\_{bc\prime} \qquad a, b = 1, 2, 3,\tag{24}$$

we recast (22) into the three-dimensional form:

$$
\nabla \cdot \mathbf{E} - \mu^2 \varrho - \lambda \partial\_t (\partial \alpha) = -\frac{\chi}{\mu\_0} \mathbf{A} \cdot \mathbf{B} \,\tag{25}
$$

$$
\nabla \times \mathbf{B} - \frac{1}{c^2} \partial\_l \mathcal{E} + \mu^2 \mathbf{a} + \lambda \nabla(\partial \mathbf{a}) = -\chi \varepsilon\_0 (\phi \mathbf{B} + \mathbf{E} \times \mathbf{A}).\tag{26}
$$

In a similar way, we derived the modified Maxwell equations as the Euler–Lagrange equation *δL*/*δA<sup>i</sup>* = 0:

$$-\left.\partial\_{\dot{f}}F^{ij} + \frac{\chi}{2}\eta^{ijkl}A\_{j}f\_{kl} - \chi\eta^{ijkl}a\_{j}F\_{kl} = 0.\tag{27}$$

An immediate observation is in order. Since *∂i∂jF ij* = 0 identically (symmetric lower indices contracted with the antisymmetric upper indices), by taking the divergence *∂<sup>i</sup>* of the field Equation (22) (in the special class of stable and unitary models under consideration with *λ* = 0 and *µ* <sup>2</sup> = 0) and (27), we derive

$$
\eta^{ijkl} F\_{\rm ij} F\_{kl} = 0, \qquad \eta^{ijkl} F\_{\rm ij} f\_{kl} = 0,\tag{28}
$$

respectively. Making use of (23) and (24), we find that only crossed-field configurations are actually allowed:

$$
\mathbf{E} \cdot \mathbf{B} = 0, \qquad \mathbf{E} \cdot \mathbf{B} + \mathbf{E} \cdot \mathbf{B} = 0. \tag{29}
$$

In particular, this includes wave configurations.

By making use of (20), (23), and (24), we rewrite the inhomogeneous Maxwell Equation (27) in the three-dimensional form:

$$\nabla \cdot \mathbf{E} = 2\chi \mathbf{c} \,\mathbf{a} \cdot \mathbf{B} - \chi \mathbf{c} \,\mathbf{A} \cdot \mathbf{B} \,\prime \tag{30}$$

$$
\nabla \times \mathbf{B} - \frac{1}{c^2} \partial\_l \mathbf{E} = \frac{2\chi}{c} (\phi \mathbf{B} + \mathbf{E} \times \mathbf{a}) - \frac{\chi}{c} (\phi \mathbf{B} + \mathbf{E} \times \mathbf{A}).\tag{31}
$$

As usual, we have to add the homogeneous Maxwell system,

$$
\nabla \cdot \mathbf{B} = 0,\tag{32}
$$

$$
\nabla \times \mathbf{E} + \partial\_l \mathbf{B} = 0.\tag{33}
$$

### **3. Influence of Axial Torsion on Electromagnetic Wave**

Here, we focus on the analysis of the dynamics of the electromagnetic field under the action of the background axial torsion, whereas the full coupled system will be considered elsewhere. Among the possible background configurations, of special interest are the cases of the uniform homogeneous field and the wave configurations. The uniform background field may arise on macroscopic scales, mimicking a distinguished cosmic frame violating the Lorentz symmetry, similar to the mechanisms discussed in [63–66]. It seems natural to turn to the case of the uniform axial torsion background that was extensively considered in earlier literature [28,60,61]

### *3.1. The Case of the Uniform External Axial Torsion Field*

Assuming the uniform axial torsion field, when the components *α<sup>i</sup>* = {− *ϕ*, *α*} are constant in time and do not change in space, we can solve Maxwell's equations for the plane wave ansatz

$$\mathbf{E} = \mathbf{E}\_0 e^{-i\omega t + i\mathbf{k}\cdot\mathbf{r}}, \qquad \mathbf{B} = \mathbf{B}\_0 e^{-i\omega t + i\mathbf{k}\cdot\mathbf{r}}.\tag{34}$$

Substituting this into the homogeneous system, (32) and (33), we derive *k* · *B* = 0, and

$$\mathbf{B} = \frac{\mathbf{k} \times \mathbf{E}}{\omega} \,, \tag{35}$$

and making use of this in (31), we obtain the algebraic equation

$$
\mathbf{k} \times (\mathbf{k} \times \mathbf{E}) + \frac{\omega^2}{c^2} \mathbf{E} + i \frac{2\chi}{c} (\varphi \mathbf{k} - \omega \mathbf{a}) \times \mathbf{E} = \mathbf{0}.\tag{36}
$$

Evaluating the determinant, we find the dispersion relation

$$
\left(\frac{\omega^2}{c^2} - k^2\right)^2 \frac{\omega^2}{c^2} - \left(\frac{\omega^2}{c^2} - k^2\right) u^2 - (ku)^2 = 0. \tag{37}
$$

Here we denoted

$$
\mathfrak{u} := \frac{2\chi}{c} (\mathfrak{q}\mathfrak{k} - \omega\mathfrak{a}),
\tag{38}
$$

whereas *k* <sup>2</sup> <sup>=</sup> *<sup>k</sup>* · *<sup>k</sup>*, *<sup>u</sup>* <sup>2</sup> <sup>=</sup> *<sup>u</sup>* · *<sup>u</sup>*, and (*ku*) = *<sup>k</sup>* · *<sup>u</sup>*. It is worthwhile to notice that, similar to the wave in a vacuum, the magnetic field is orthogonal to the electric field and the wave vector,

$$\mathbf{B} \cdot \mathbf{E} = 0, \qquad \mathbf{B} \cdot \mathbf{k} = 0,\tag{39}$$

which follows from (35). However, the electric field is not orthogonal to the wave vector, in general,

$$i\mathbf{k}\cdot\mathbf{E} = 2\chi\mathbf{c}\,\mathbf{a}\cdot\mathbf{B} = \frac{2\chi\mathbf{c}}{\omega}\,\mathbf{a}\cdot(\mathbf{k}\times\mathbf{E}),\tag{40}$$

which is derived by substituting the plane wave ansatz (34) into the inhomogeneous Equation (30); this is also a direct consequence of (36).

**Special case 1**. Assuming *ϕ* 6= 0, *α* = 0, from (37), we find a simpler dispersion relation

$$
\left(\frac{\omega^2}{c^2} - k^2\right)^2 - \left(\frac{2\chi}{c}\right)^2 \rho^2 k^2 = 0. \tag{41}
$$

This can be immediately recast into

$$\frac{\omega^2}{c^2} = \mathbf{k} \cdot \mathbf{k} \pm \frac{2\chi}{c} \,\varphi \sqrt{\mathbf{k} \cdot \mathbf{k}}.\tag{42}$$

The second term on the right-hand side describes a deformation of the light cone under the influence of the torsion component *ϕ*.

**Special case 2**. Assuming *ϕ* = 0, *α* 6= 0, the general result (37) reduces to

$$
\left[\left(\frac{\omega^2}{c^2} - k^2\right)^2 - \left(2\chi\right)^2\right] \left[\left(\frac{\omega^2}{c^2} - k^2\right)a^2 + \left(ka\right)^2\right] = 0. \tag{43}
$$

Here, *α* <sup>2</sup> <sup>=</sup> *<sup>α</sup>* · *<sup>α</sup>*, and (*kα*) = *<sup>k</sup>* · *<sup>α</sup>*. The resulting dispersion relation can be straightforwardly simplified into

$$\frac{\omega^2}{c^2} = \mathbf{k} \cdot \mathbf{k} + 2\chi^2 \mathbf{a} \cdot \mathbf{a} \pm 2\chi\sqrt{\chi^2(\mathbf{a} \cdot \mathbf{a})^2 + (\mathbf{k} \cdot \mathbf{a})^2} = 0.\tag{44}$$

### *3.2. Rotation of the Polarization Plane*

Without loss of generality, one can assume that the electromagnetic wave propagates along the *z*-axis, in other words, we take *k* = (0, 0, *kz*). It is more convenient to analyze the two special cases separately. In the first case (*ϕ* 6= 0, *α* = 0), the dispersion relation (42) yields two values for the wave vector,

$$k\_z = k\_\pm = \frac{\omega \pm \chi \varrho}{c},\tag{45}$$

in the leading order of the small coupling constant *χ*. Hence, there are two independent waves propagating along *z* with two different phase velocities. As a result, after extracting the corresponding amplitudes *E*<sup>±</sup> for the electric field from the algebraic Equation (36), we find the solution

$$E = E\_{+}e^{-i\omega t + i\mathbf{k}\_{+}z} \mathbf{e}\_{+} + E\_{-}e^{-i\omega t + i\mathbf{k}\_{-}z} \mathbf{e}\_{-} \tag{46}$$

where we denote the combinations of the basis vectors

$$e\_{\pm} = \frac{e\_{\times} \mp ie\_{y}}{2}.\tag{47}$$

Therefore, from the point of view of physics, the solution (46) describes the superposition of the right-hand (counter-clockwise) circularly polarized and the left-hand (clockwise) circularly polarized waves.

Recalling that the linearly polarized wave arises as the sum of the right-hand and left-hand circular waves with equal amplitudes, *E*<sup>+</sup> = *E*−, we then obtain the real solution

$$\mathbf{E} = E\_0 \cos[\omega(t - z/c)] \left\{ \cos(\chi \varrho z/c) \,\mathbf{e}\_x - \sin(\chi \varrho z/c) \,\mathbf{e}\_y \right\}. \tag{48}$$

Thus, we recover the *Faraday effect* when the polarization vector continuously rotates with the propagation of the plane wave. The polarization rotation angle *γ*, see Figure 1, from the initial point *z* = 0 until the point *z* = *h* is determined by

**Figure 1.** Faraday effect of polarization rotation under the action of uniform axial torsion.

For the second case of the axial torsion field configuration (*ϕ* = 0, *α* 6= 0), the form of the solution depends on the relative orientation of the *α* with respect to the wave vector of the electromagnetic wave *k*. Quite generally, given the direction of the wave propagation, we can decompose *<sup>α</sup>* = *<sup>α</sup>*|| + *<sup>α</sup>*<sup>⊥</sup> into the longitudinal and transversal projections on the wave vector *k* and the plane orthogonal to it. Accordingly, we derive the solution for the real electric field of the linearly polarized electromagnetic plane wave, to the first order in the interaction constant *χ*:

$$E = E\_0 \cos[\omega(t - z/c)] \left\{ \cos(\chi a\_{||} z) \,\mathbf{e}\_x - \sin(\chi a\_{||} z) \,\mathbf{e}\_y \right\}$$

$$- \frac{2\chi a\_\perp c}{\omega} \, E\_0 \sin[\omega(t - z/c)] \sin(\chi a\_{||} z + \phi\_0) \,\mathbf{e}\_z \,. \tag{50}$$

Since the wave propagates along the *<sup>z</sup>*-axis, we have *<sup>k</sup>* · *<sup>α</sup>* = *<sup>k</sup>zα*||, and

$$\mathfrak{a}\_{||} = \mathfrak{e}\_{\mathbb{Z}} \cdot \mathfrak{a} = \mathfrak{a} \cos \theta, \qquad \mathfrak{a}\_{\perp} = \mathfrak{a} \sin \theta,\tag{51}$$

and cos *<sup>φ</sup>*<sup>0</sup> = *<sup>e</sup><sup>x</sup>* · *<sup>α</sup>*<sup>⊥</sup> measures the angle between the projection *<sup>α</sup>*<sup>⊥</sup> and the basis vector *<sup>e</sup>x*. Obviously, by choosing the coordinate frame appropriately, we can always make *φ*<sup>0</sup> = 0. It is worthwhile to note that, in accordance with (40), the electric field has a nontrivial component along the wave vector and the third term in (50) vanishes only when the wave propagates along the axial torsion *α*. In that case, the polarization rotation angle

$$\gamma = \chi \mathfrak{a}\_{||} \mathfrak{h} = \chi \mathfrak{a} \cos \theta \mathfrak{h} \tag{52}$$

from the initial point *z* = 0 to the endpoint *z* = *h* is maximal. However, when the axial vector field is orthogonal to the direction of propagation of the electromagnetic wave *α* ⊥ *k*, there will be no rotation of the polarization plane. This generalized Faraday effect is supported by the solution of the dispersion Equation (43),

$$k\_z = k\_\pm = \frac{\omega}{c} \pm \chi a\_{||} = \frac{\omega}{c} \pm \chi a \cos \theta,\tag{53}$$

in the leading order of the small coupling constant *χ*, which gives rise to the two waves traveling with two different phase velocities.

### **4. Discussion and Conclusions**

The search for exotic new forces and interactions generated by the spin of matter particles, fields, and continuous media has a long history. Since the corresponding spin– torsion coupling constant *χ* is very small, the detection of such new forces becomes a challenging issue and requires high-precision measurements. On the other hand, the new fields should be truly highly penetrating.

Taking into account that optical experiments are among the most accurate ones, the analysis of possible optical effects of the axial vector torsion field appears to be quite promising. Here, the classical dynamics of the axial vector field were studied in the framework of the Yang–Mills type Poincaré gravity model with a focus on the interaction with the electromagnetic field. We derived the dynamical equations for the axial vector torsion field, (25) and (26), and identified the source of the "electric" component of the torsion variable with the helicity density of the electromagnetic field ∼ *A* · *B*, which characterizes non-trivial topological properties of the field configuration, whereas the "magnetic" component of the torsion variable is generated by the spin density of the electromagnetic field.

Continuing the earlier studies of the spin–torsion effects in the fermion sector [24–26], we here turn to the boson sector. Maxwell's equations are modified (30)–(33) in the presence of the axial torsion field. The analysis of the propagation of electromagnetic waves under the action of the axial vector torsion field reveals the Faraday effect of the rotation of the wave's polarization and the angle of rotation is determined by the coupling constant, the magnitude of the axial vector torsion field and the travel distance: *γ* = *χϕh c* or *<sup>γ</sup>* = *χα*||*h*. This is consistent with the similar effect arising due to the Lorentz symmetry violation [63,64] or due to the action of the pseudoscalar axion [65,66].

Compared to earlier literature that focused on the evaluation of the spin–spin interaction potential [20,30,34,42,43,48,49,51], the results obtained provide a qualitatively new approach to the search for possible manifestations of the spin–torsion coupling with the help of the optical polarization methods. When discussing the most general model of the interacting electromagnetic field and propagating axial torsion field [67,68], one has to pay

special attention to the acausal (i.e., superluminal propagation) anomalies. In agreement with the conclusions of [67,68], the class of models under consideration with *b*<sup>3</sup> = 2*b*<sup>1</sup> yields *λ* = 0 for the axial pseudovector field, and so the stability, unitarity, and causality issues are safely avoided.

Assuming the cosmic nature of the background axial torsion, we can apply the results obtained in the astrophysical situation, and analyze the distribution over the sky sphere of the polarization of radiation, coming from distant radio galaxies. Then taking the observational data collected in Table I of Reference [65], and repeating verbatim the computations by replacing the Lorentz-violating parameter with the axial torsion pseudovector, we establish the strong bounds on the torsion's magnitude: <sup>|</sup>*T*<sup>|</sup> . 8.7 <sup>×</sup> <sup>10</sup>−<sup>27</sup> <sup>m</sup>−<sup>1</sup> . This turns out to be significantly lower than the limits found from the analysis of the spin–torsion coupling effects in the fermion sector [12,24–29].

**Author Contributions:** All authors discussed the results and contributed to the final manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** The work of MIT was supported by the Russian Science Foundation under grant 22-72- 00036.

**Data Availability Statement:** No data is associated with this theoretical work; accordingly no data will be deposited.

**Acknowledgments:** We express our gratitude to I. A. Kudryashov (Nuclear Physics Research Institute of Moscow State University) for discussing the problem.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **Appendix A. Poincaré Gauge Gravity Dynamics**

In the literature, the quadratic Poincaré gravity theories are often formulated in terms of the standard tensor objects, which are not decomposed into irreducible parts. As usual, we introduce the *Ricci tensor* and the *co-Ricci tensor* as

$$R\_{ij} := R\_{kij}{}^k \,, \qquad \overline{R}^{ij} := \frac{1}{2} \ R\_{klm}{}^i \eta^{klmj} \, \,, \tag{A1}$$

from which the curvature scalar and pseudoscalar arise naturally as the traces

$$R = \mathbf{g}^{ij} R\_{ij} = R\_{ij}{}^{ji}{}, \qquad \overline{R} = \mathbf{g}\_{ij} \overline{R}^{ij} = \frac{1}{2} R\_{ijkl}{}^{jkl} \eta^{ijkl}. \tag{A2}$$

We consider the general quadratic model with the Yang–Mills type Lagrangian that contains all possible quadratic invariants of the torsion and the curvature:

$$\begin{split} L &= -\frac{1}{2\kappa c} \Big\{ a\_0 \boldsymbol{R} + \overline{a}\_0 \overline{\boldsymbol{R}} + 2\lambda\_0 \\ &\quad + a\_1 \operatorname{T}\_{kl}{}^i T^{kl}{}\_i + a\_2 \operatorname{T}\_i \, \boldsymbol{T}^i + a\_3 \operatorname{T}\_{kl}{}^i T\_i^{kl} \\ &\quad + \overline{a}\_1 \eta^{klmn} \operatorname{T}\_{kli}{}^i T\_{mn}{}^i + \overline{a}\_2 \eta^{klmn} \operatorname{T}\_{klm}{}^n \operatorname{T}\_n \\ &\quad + \ell\_p^2 \Big( b\_1 \operatorname{R}\_{ijkl}{}^i \boldsymbol{R}^{ijkl} + b\_2 \operatorname{R}\_{ijkl}{}^i \boldsymbol{R}^{klij} + b\_3 \operatorname{R}\_{ijkl}{}^i \boldsymbol{R}^{ikjl} \\ &\quad + b\_4 \operatorname{R}\_{ij} \boldsymbol{R}^{ij} + b\_5 \operatorname{R}\_{ij}{}^i \boldsymbol{R}^{jk} + b\_6 \operatorname{R}^2 \\ &\quad + \overline{b}\_1 \eta^{klmn} \operatorname{R}\_{klij}{}^n \operatorname{R}\_{mn}{}^{ij} + \overline{b}\_2 \eta^{klmn} \operatorname{R}\_{kl}{}^n \operatorname{R}\_{mn} \\ &\quad + \overline{b}\_3 \eta^{klmn} \operatorname{R}\_{klm}{}^i \operatorname{R}\_{ni} + \overline{b}\_4 \ \eta^{klmn} \operatorname{R}\_{klmn}{}^R \Big) \Big\}. \end{split} \tag{A3}$$

Here, *κ* = <sup>8</sup>*π<sup>G</sup> c* 4 is Einstein's gravitational constant with the dimension of [*κc*] =s kg−<sup>1</sup> . *<sup>G</sup>* <sup>=</sup> 6.67 <sup>×</sup> <sup>10</sup>−<sup>11</sup> <sup>m</sup><sup>3</sup> kg−<sup>1</sup> s −2 is Newton's gravitational constant. The speed of light: *<sup>c</sup>* <sup>=</sup> 2.9 <sup>×</sup> <sup>10</sup><sup>8</sup> m/s. For completeness, we include the cosmological term *<sup>λ</sup>*0.

Besides the linear "Hilbert type" part characterized by *a*<sup>0</sup> and *a*0, the Lagrangian (A3) contains several additional coupling constants, which fix the structure of the "Yang–Mills type" part: *a*1, *a*2, *a*3, *a*1, *a*2, *b*1, · · · , *b*6, *b*1, · · · , *b*4, and ℓ 2 *ρ* . The coupling constants *a<sup>I</sup>* , *a<sup>I</sup>* , *b<sup>I</sup>* , and *b<sup>I</sup>* are dimensionless, whereas the dimension [ℓ 2 *ρ* ] = [area] so that [ℓ 2 *<sup>ρ</sup>*/*κc*] = [*h*¯].

The analysis of the particle spectrum for the quadratic model (A3) reveals that the dynamics of gravitational modes in different *J P* (spin*parity*) sectors are determined by the following combinations of the coupling constants: 2± sector

$$
\Lambda\_1 = 4(b\_1 + b\_2) + 2b\_3 + b\_4 + b\_{5\prime} \qquad \Lambda\_2 = 4b\_1 + b\_{3\prime} \tag{A4}
$$

0 ± sector

$$
\Lambda\_3 = 4(b\_1 + b\_2) + 2b\_3 + 4(b\_4 + b\_5) + 12b\_{6\prime} \qquad \Lambda\_4 = 4b\_1 - 2b\_3 \tag{A5}
$$

and 1± sector

$$
\Lambda\_5 = 4(b\_1 - b\_2) + b\_4 - b\_{5\prime} \qquad \qquad \Lambda\_6 = 4b\_1 + b\_3 + 2b\_{4\prime} \tag{A6}
$$

$$
\overline{\Lambda}\_5 = 4(\overline{b}\_2 - 2\overline{b}\_3), \qquad \qquad \overline{\Lambda}\_6 = -4(\overline{b}\_2 + 3\overline{b}\_4). \tag{A7}
$$

whereas the mass terms are specified by

$$
\mu\_1 \quad = \quad -a\_0 + a\_1 - a\_2, \qquad \mu\_2 = -2a\_0 + \frac{2a\_1 + a\_2 + 3a\_3}{4}, \tag{A8}
$$

$$
\mu\_3 \quad = \quad -a\_0 - \frac{2a\_1 + a\_2}{4}, \quad \quad \overline{\mu}\_1 = \frac{8}{3}(4\overline{a}\_1 + 3\overline{a}\_2 - 2\overline{a}\_0). \tag{A9}
$$

### **References**


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