Dilaton cosmology, noncommutativity, and generalized uncertainty principle

Abstract

The effects of noncommutativity and of the existence of a minimal length on the phase space of a dilatonic cosmological model are investigated. The existence of a minimum length, results in the Generalized Uncertainty Principle (GUP), which is a deformed Heisenberg algebra between the minisuperspace variables and their momenta operators. We extend these deformed commutating relations to the corresponding deformed Poisson algebra. For an exponential dilaton potential, the exact classical and quantum solutions in the commutative and noncommutative cases, and some approximate analytical solutions in the case of GUP, are presented and compared.

A cosmological viewpoint on the correspondence between deformed phase-space and canonical quantization

“…For example, if in a model field theory the fields are taken as noncommutative, as has been done in [35,36], the resulting effective theory predicts the same Lorentz violation as a field theory in which the coordinates are considered as noncommutative [37][38][39]. As a further example, it is well known that string theory can be used to suggest a modification to the bracket structure of coordinates, also known as GUP [34] which is used to modify the phase-space structure [40][41][42][43][44][45]. Over the years, a large number of works on noncommutative fields [25][26][27][28][29] have been inspired by noncommutative geometry model theories [31][32][33].…”

Dilaton cosmology and the modified uncertainty principle

“…Its non-singular boundary is the line a = 0 with |φ| < ∞, while at singular boundaries at least one of the two variables is infinite [23]. In terms of the new variables x and y the minisuperspace is recovered by x > 0, x > |y| and the non singular boundary may be represented by x = y = 0 [15].…”

“…2 In general, the main reason for applying the transformation like (15) is that the minisuperspace is curved in terms of the original coordinates (see (14)) and the application of some deformed commutation relation like GUP is not an easy task. However, we know that any 2-D Riemannian space is conformally flat and if the Ricci scalar is zero this space will be Minkowskian.…”

“…In our case the 2-D minisuperspace (spanned by a and φ) is a Riemannian space with vanishing Ricci scalar. Therefore one can find a coordinate transformation (15) in terms of which the minisupermetric takes the form of a Minkowskian space (see (19)). We also note that we did not introduce a canonical transformation and therefore after introducing the new coordinates all of the quantum analysis should be done based on them and one should not make an inverse transformation in the wave functions.…”

“…Here we use an approximation method (as used in [15]) for the solution. As we are now studying the quantum model so we require the solutions in the limit x → 0 and y → 0.…”