Quantum deformation of quantum cosmology: A framework to discuss the cosmological constant problem

Abstract

We endorse the context that the cosmological constant problem is a quantum cosmology issue. Therefore, in this paper we investigate the q-deformed Wheeler-DeWitt equation of a spatially closed homogeneous and isotropic Universe in the presence of a conformally coupled scalar field. Specifically, the quantum deformed Universe is a quantized minisuperspace model constructed from quantum Heisenberg-Weyl U q (h 4 ) and U q (su(1, 1)) groups. These intrinsic mathematical features allow to establish that (i) the scale factor, the scalar field and corresponding momenta are quantized and (ii) the phase space has a non-equidistance lattice structure. On the other hand, such quantum group structure provides us a new framework to discuss the cosmological constant problem. Subsequently, we show that a ultraviolet cutoff can be obtained at 10 −3 eV , i.e., at a scale much larger than the expected Planck scale. In addition, an infrared cutoff, at the size of the observed Universe, emerges from within such quantum deformation of Universe. In other words, the spectrum of the scale factor is upper bounded. Moreover, we show that the emerged cosmological horizon is a quantum sphere S 2 q or, alternatively, a fuzzy sphere S 2 F which explicitly exhibits features of the holographic principle. The corresponding number of fundamental cells equals the dimension of the Hilbert space and hence, the cosmological constant can be presented as a consequence of the quantum deformation of the FLRW minisuperspace.

Remarks on SU_q(2) fermions

“…As remarked in the introduction, SU q (N ) bosons are strictly related to GUP [42]. This fact represents a suggestive hint about the basic role played by quantum groups in the fundamental descriptions of nature (see also [3,4,5,6,7,8]).…”

“…Quantum groups, originally introduced in connection with Quantum Inverse Scattering Method and Yang-Baxter equation [1], nowadays represent a widespread mathematical framework: their applications cover various areas of physics, including condensed matter physics [2] and quantum gravity [3,4,5,6,7,8].…”

From Fractional Quantum Mechanics to Quantum Cosmology: An Overture

“…Regarding that the parameter γ has dimension of inverse of length (as pointed out in [25]), then γ would be a new fundamental constant of theory. However, as addressed in [26], the origin of this unwanted new constant is the effective matter field Lagrangian in action (28). If we use a "real" matter field, for example a scalar field or the Maxwell's field Lagrangian instead of ρ in (28), then the value of γ will be fixed to only following two acceptable values…”

Quantum Groups and Polymer Quantum Mechanics

“…On a similar line of reasoning, q-deformed formulations of Loop Quantum Gravity (LQG) were proposed along the years [8][9][10][11], which naturally incorporate the cosmological constant. This relationship was even used in different models [12]. It has been also shown that the quantum group structure of the κ-Poincaré is the natural symmetry for effective field theories, emerging from discrete gravity theories coupled to matter [13], and for noncommutative description of the spacetime [14][15][16].…”