Abstract
There is a famous criterion for objective wave function reduction which is derived by using the Shrödinger–Newton equation [L Diosi, Phys. Lett. A 105(4–5), 199 (1984)]. In this regard, a critical mass for the transition from quantum world to the classical world is determined for a particle or an object. In this paper, we shall derive that criterion by using the concept of Bohmian trajectories. This study has two consequences. The first is, it provides a geometric framework for the problem of wave function reduction. The second is, it represents the role of quantum and gravitational forces in the reduction process.
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The Bohmian quantum potential for the relativistic electron has been derived recently by Hiley and co-workers through complicated methods of Clifford algebra \(C^{0}_{3}\,[33]\). There, the quantum potential is considered as additional term in the Hamilton–Jacobi equation of the electron and is obtained by comparing with the original equation \(p_\mu p^\mu =m^2\). In other words, in that case too the relation \(p_\mu p^\mu ={\cal{M}}^2\) holds, where \(\cal{M}^2=m^2+Q_{\text{Dirac}}\)
The polar form substitution of the wave function into the Klein–Gordon equation is not the only way to get the Hamilton–Jacobi equation (11)
B J Hiley and R E Callaghan, Found. Phys. 42, 192 (2012)
F Rahmani and M Golshani, Int. J. Theor. Phys. 56, 3096 (2017)
F Rahmani, M Golshani and Gh Jafari, Int. J. Mod. Phys. A 33(22) (2018), https://doi.org/10.1142/S0217751X18501294
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Rahmani, F., Golshani, M. & Jafari, G. A geometric look at the objective gravitational wave function reduction. Pramana - J Phys 94, 163 (2020). https://doi.org/10.1007/s12043-020-02032-6
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DOI: https://doi.org/10.1007/s12043-020-02032-6
Keywords
- Gravitational reduction of the wave function
- Bohmian quantum potential
- Bohmian geodesic deviation equation
- Bohmian trajectories