# Application of the Phase-Space Path Integral to Strong-Laser-Field-Assisted Electron-Ion Radiative Recombination: A Gauge-Covariant Formulation

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## Abstract

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## 1. Introduction

## 2. Momentum-Space Matrix Element of the Time-Evolution Operator in Velocity Gauge

## 3. A Gauge-Covariant Transition Amplitude for Laser-Assisted Electron-Ion Radiative Recombination

## 4. Equivalence of the Transition Amplitudes in Velocity and Length Gauges and Expansion in Powers of the Atomic Potential

## 5. Conclusions and Comments

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Length-Gauge Momentum-Space Matrix Element of the Time-Evolution Operator

## Appendix B. Gauge-Covariant Energy Operator

## Appendix C. Conventional Velocity-Gauge SFA

## References

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**Figure 1.**Schematic diagram of laser-assisted radiative recombination (LAR) (left-hand part), LAR with rescattering (middle part—SLAR), and LAR with double scattering (right-hand part—DSLAR). For the DSLAR process the corresponding Feynman diagram is also presented.

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**MDPI and ACS Style**

Esquembre Kučukalić, A.; Becker, W.; Milošević, D.B.
Application of the Phase-Space Path Integral to Strong-Laser-Field-Assisted Electron-Ion Radiative Recombination: A Gauge-Covariant Formulation. *Symmetry* **2020**, *12*, 1606.
https://doi.org/10.3390/sym12101606

**AMA Style**

Esquembre Kučukalić A, Becker W, Milošević DB.
Application of the Phase-Space Path Integral to Strong-Laser-Field-Assisted Electron-Ion Radiative Recombination: A Gauge-Covariant Formulation. *Symmetry*. 2020; 12(10):1606.
https://doi.org/10.3390/sym12101606

**Chicago/Turabian Style**

Esquembre Kučukalić, Ali, Wilhelm Becker, and Dejan B. Milošević.
2020. "Application of the Phase-Space Path Integral to Strong-Laser-Field-Assisted Electron-Ion Radiative Recombination: A Gauge-Covariant Formulation" *Symmetry* 12, no. 10: 1606.
https://doi.org/10.3390/sym12101606