Quantum gravitational corrections to the real Klein-Gordon field in the presence of a minimal length. (English) Zbl 1202.83112
Summary: The \((D+1)\)-dimensional \((\beta,\beta')\)-two-parameter Lorentz-covariant deformed algebra introduced by C. Quesne and V. M. Tkachuk [J. Phys. A, Math. Gen. 39, No. 34, 10909–10922 (2006; Zbl 1168.81014)], leads to a nonzero minimal uncertainty in position (minimal length). The Klein-Gordon equation in a (3+1)-dimensional space-time described by Quesne-Tkachuk Lorentz-covariant deformed algebra is studied in the case where \(\beta'=2\beta \) up to first order over deformation parameter \(\beta \). It is shown that the modified Klein-Gordon equation which contains fourth-order derivative of the wave function describes two massive particles with different masses. We have shown that physically acceptable mass states can only exist for \(\beta<\frac{1}{8m^2c^2}\) which leads to an isotropic minimal length in the interval \(10^{-17}\,\text{m}<(\Delta X^i)_0< 10^{-15}\,\text{m}\). Finally, we have shown that the above estimation of minimal length is in good agreement with the results obtained in previous investigations.
MSC:
| 83E15 | Kaluza-Klein and other higher-dimensional theories |
| 83C45 | Quantization of the gravitational field |
| 83D05 | Relativistic gravitational theories other than Einstein’s, including asymmetric field theories |
| 35L05 | Wave equation |
| 81S10 | Geometry and quantization, symplectic methods |
| 53D55 | Deformation quantization, star products |
Citations:
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