# Kinematics and Selection Rules for Light-by-Light Scattering in a Strong Magnetic Field

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## Abstract

**:**

## 1. Introduction

## 2. Kinematics of Scattering a Photon by a Photon

#### 2.1. Eigen-Value Problem for the Polarization Tensor, Polarization, and the Dispersion of Eigen-Modes

#### 2.1.1. Polarization of Eigen-Modes

#### 2.1.2. Dispersion of Eigen-Modes in the Heisenberg–Euler Approximation

#### 2.2. Conservation Laws and Selection Rules

#### 2.2.1. Perpendicular Incidence

#### 2.2.2. Parallel Incidence

#### 2.3. Quantitative Side of the Wave-Length Shifts

#### 2.3.1. Perpendicular Incidence

#### 2.3.2. Parallel Incidence

## 3. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1. | except, perhaps, the Schwinger effect of the pair production from the vacuum by a strong constant electric field. |

2. | It is interesting to note that, when the nonlinearity is determined by the Heisenberg–Euler Lagrangian, a traveling-wave solution of nonlinear Maxwell equations is provided by a dispersion law that involves a dependence on the amplitudes of the traveling wave [19]. This is, certainly, not our case, since we deal with the propagation of small-amplitude waves governed by equations linearized near an external field. |

3. | |

4. | These convexity properties are certainly respected by the HE approximation resulting in Equations (15) and (16). The third condition following from the causality principle, $1-{\mathfrak{L}}_{\mathfrak{F}}\u2a7e0$ can be violated for unrealistic exponentially strong fields $\sim exp\left(3\pi /\alpha \right)$ (in accord with Equation (30) below) as a manifestation of the intrinsic trouble of QED known as the lack of asymptotic freedom. |

5. | More generally, the dispersion curve for Mode 3 goes higher than that for Mode 2 ${\omega}^{\left(3\right)}({k}_{i}^{2},{({k}_{i}\xb7B)}^{2})>{\omega}^{\left(2\right)}({k}_{i}^{2},{({k}_{i}\xb7B)}^{2}),$ because the lowest energy threshold, where the photon may create an electron-positron pair, is for Mode 3 higher: one particle out of the two created by the photon of this mode belongs to an excited Landau level, whereas the Mode 2 photon may create a pair of particles, both in the ground Landau states. The point is that the polarization tensor is infinite at each of the thresholds; hence, each threshold strongly affects the dispersion law; see [21,22] for further explanations. |

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Shabad, A.
Kinematics and Selection Rules for Light-by-Light Scattering in a Strong Magnetic Field. *Universe* **2020**, *6*, 211.
https://doi.org/10.3390/universe6110211

**AMA Style**

Shabad A.
Kinematics and Selection Rules for Light-by-Light Scattering in a Strong Magnetic Field. *Universe*. 2020; 6(11):211.
https://doi.org/10.3390/universe6110211

**Chicago/Turabian Style**

Shabad, Anatoly.
2020. "Kinematics and Selection Rules for Light-by-Light Scattering in a Strong Magnetic Field" *Universe* 6, no. 11: 211.
https://doi.org/10.3390/universe6110211