Kepler Problem in Space with Deformed Lorentz-Covariant Poisson Brackets
- PDF / 1,042,386 Bytes
- 18 Pages / 439.37 x 666.142 pts Page_size
- 54 Downloads / 190 Views
Kepler Problem in Space with Deformed Lorentz‑Covariant Poisson Brackets M. I. Samar1 · V. M. Tkachuk1 Received: 15 July 2019 / Accepted: 8 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We propose a Lorentz-covariant deformed algebra describing a (3 + 1)-dimensional quantized spacetime, which in the nonrelativistic limit leads to undeformed one. The deformed Poincaré transformations leaving the algebra invariant are identified. In the classical limit the Lorentz-covariant deformed algebra yields the deformed Lorentz-covariant Poisson brackets. Kepler problem with the deformed Lorentz-covariant Poisson brackets is studied. We obtain that the precession angle of an orbit of the relativistic particle in the gravitational field depends on the mass of the particle, i.e. equivalence principle is violated. We propose a condition for the recovery of the equivalence principle in the space with the deformed Poisson brackets. Comparing our analytical result with the experimental data for the precession angle of Mercury’s orbit we provide an estimation of minimal length. Keywords Kepler problem · Precession of an orbit · Lorentz-covariant deformed algebra · Minimal length
1 Introduction Deformed commutation relations with minimal length and their classical limit ( ℏ → 0 ) have been intensively studied lately. For the first time this idea was considered by Snyder as a way of the regularization of UV divergences of quantum field theory [1]. The deformation proposed by Snyder preserves the Lorentz invariance breaking the Poincaré one. However, his paper had not attracted much attention for many years. Motivated by the studies in string theory and quantum gravity [2–4], the interest in the minimal length hypothesis was renewed after several decades. These studies led to the generalized uncertainty principle (GUP Generalized Uncertainty Principle (GUP) * M. I. Samar [email protected] 1
Department for Theoretical Physics, Ivan Franko National University of Lviv, 12 Drahomanov St., Lviv, UA 79005, Ukraine
13
Vol.:(0123456789)
Foundations of Physics
ΔX ≥
( ) ℏ 1 + 𝛽ΔP 2 ΔP
(1)
√ and suggested the existence of the fundamental minimal length ΔXmin = ℏ 𝛽 , which is supposed to be of the order of the Planck length. Kempf et al. showed that the effect of minimal length as minimal uncertainty for position operators can be achieved in quantum mechanics by modifying usual canonical commutation relations [5–8]. According to Kempf the deformed commutaion relation in one-dimensional space may read ̂ P] ̂ = i�(1 + 𝛽 P̂ 2 ), [X,
(2)
In cases of higher dimensions deformed algebra (3) can be generalized as
[X̂ i , P̂ j ] =i�[(1 + 𝛽 P̂ 2 )𝛿ij + 𝛽 � P̂ i P̂ j ], 2𝛽 − 𝛽 � + (2𝛽 + 𝛽 � )𝛽 P̂ 2 ̂ ̂ [X̂ i , X̂ j ] =i� (Pi Xj − P̂ j X̂ i ), 1 + 𝛽 P̂ 2 [P̂ i , P̂ j ] =0,
(3)
with 𝛽 and 𝛽 ′ being two small nonnegative parameters. Deformed commutation relations suggested by Kempf are not Lorentz-covariant. The Lorentz-covariant version of that kind of commutation relations was proposed in
Data Loading...
