Now, using GUP correction ( 3) and ( 5) take the form
#AREA IN CURVED SPACE FREE#
Where is the rest mass of the particle, is the Dirac spinor, and are Dirac matrices, are momentum operators,, is the Schwarzschild radius of massive body, related to its mass by, is the gravitational constant, and is the speed of light in free space. GUP Dirac Equations in Schwarzschild Metricĭirac equation in Schwarzschild metric without (GUP) can be written as follows : In this paper we study Dirac equation in Schwarzschild metric using GUP and show that we arrive at the quantization of space. In, the above results were extended to a relativistic particle in two and three dimensions. This corrected Hamiltonian implies not only the usual quantization of energy, but also that the box length is quantized. In, it was shown that any nonrelativistic Hamiltonian of the form can be written as using ( 3). With and satisfying the canonical commutation relations, such that. The following definitions are proposed in and used in : It is natural to take for more details see. GUP-induced terms become important near the Planck scale (for earlier version of GUP motivated by string theory, black hole physics, and DSR, see, and for some phenomenological implications, see ).Įquation ( 1) implies the following minimum measurable length and maximum measurable momentum :
Where, = Planck mass, m = Planck length, and = Planck energy GeV. In, the following proposed GUP is consistent with Doubly special relativity or DSR theories and black hole physics which ensure that : This implies a modification of the commutation relations between position coordinates and momentum. Introductionĭiverse approaches to quantum gravity expect a minimum measurable length and a modification of the Heisenberg uncertainty principle to a so-called generalized uncertainty principle or GUP. We showed that we arrived at the quantization of space by solving Dirac equation with GUP in this metric. In this paper, we extend the above results to the case of curved spacetime (Schwarzschild metric). By solving the GUP corrected equations, the authors arrived at quantization not only of energy but also of box length, area, and volume.
It was shown by some authors that the GUP gives rise to corrections to the Schrodinger, Klein-Gordon, and Dirac equations. Diverse theories of quantum gravity expect modifications of the Heisenberg's uncertainty principle near the Planck scale to a so-called Generalized uncertainty principle (GUP).
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