Composite system in deformed space with minimal length

Abstract

For composite systems made of $N$ different particles living in a space characterized by the same deformed Heisenberg algebra, but with different deformation parameters, we define the total momentum and the center-of-mass position to first order in the deformation parameters. Such operators satisfy the deformed algebra with new effective deformation parameters. As a consequence, a two-particle system can be reduced to a one-particle problem for the internal motion. As an example, the correction to the hydrogen atom $n$S energy levels is re-evaluated. Comparison with high-precision experimental data leads to an upper bound of the minimal length for the electron equal to $3.3\times 10^{-18} {\rm m}$. The effective Hamiltonian describing the center-of-mass motion of a macroscopic body in an external potential is also found. For such a motion, the effective deformation parameter is substantially reduced due to a factor $1/N^2$. This explains the strangely small result previously obtained for the minimal length from a comparison with the observed precession of the perihelion of Mercury. From our study, an upper bound of the minimal length for quarks equal to $2.4\times 10^{-17}{\rm m}$ is deduced, which appears close to that obtained for electrons.Comment: 22 pages, no figure; small additions in Secs. I, III and V

Physics of the so q (4) hydrogen atom

“…3 The q-deformed so (4) We choose to deform the so(4) = su(2) ⊕ su(2) dynamical symmetry of the hydrogen atom by a parameter q by means of the well-established theory of the quantum group su q (2) [20,21,22,23], where the commutation relations are written as…”

“…quantum field theory on noncommutative spaces [1,2]. They might have been initially investigated in the context of the quantum Yang-Baxter equation as a way of generating its solutions, but they are also the mathematical tool used to investigate the idea of a minimal length [3,4]. For this reason, much attention has been devoted to looking into deformed symmetries in physical systems, such as the deformed rotational symmetry su q (2) of the harmonic oscillator [5].…”

Hydrogen atom in momentum space with a minimal length

“…[4][5][6][7][8][9][10][11][12][13]. The idea of modifying the standard Heisenberg uncertainty relation in such a way that it includes a minimal length has first been proposed in the context of quantum gravity and string theory [14,15].…”

Some Details of Statistical Mechanics of Many-Body Systems in the Presence of a Measurable Minimal Length

“…It is important to emphasize also that the idea of minimal length has its origin in quantum gravity and here we have presented a toy model to see how this minimal length scale affects statistical mechanics of many-body systems at high temperature. At this point we note that the issue of composite system in a deformed space with minimal length has been studied by [26]. Following this seminal work, here we have studied some other statistical features of these composite systems.…”

“…Naturally our results could be consistent with their results and this is actually the case. We refer also to [27] for another study on minimal length physics.…”