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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:

Zbl 1168.81014
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References:

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