_{1}

Modifications of the Weyl-Heisenberg algebra
are proposed where the classical limit
corresponds to a metric in (curved) momentum spaces. In the simplest scenario, the
*2D* de Sitter metric of constant curvature in momentum space furnishes a hierarchy of modified uncertainty relations leading to a minimum value for the position uncertainty . The first uncertainty relation of this hierarchy has the same functional form as the
*stringy* modified uncertainty relation with a Planck scale minimum value for
at
. We proceed with a discussion of the most general curved phase space scenario (cotangent bundle of spacetime) and provide the noncommuting phase space coordinates algebra in terms of the symmetric
and nonsymmetric
metric components of a Hermitian complex metric
, such
. Yang’s noncommuting phase-space coordinates algebra, combined with the Schrodinger-Robertson inequalities involving angular momentum eigenstates, reveals how a quantized area operator in units of
emerges like it occurs in Loop Quantum Gravity (LQG). Some final comments are made about Fedosov deformation quantization, Noncommutative and Nonassociative gravity.

Recently, we studied the generalized gravitational field equations in curved phase spaces (the cotangent bundle of spacetime) [

Most of the work devoted to Quantum Gravity has been focused on the geometry of spacetime rather than phase space per se. The first indication that phase space should play a role in Quantum Gravity was raised by [

It is better understood now that the Planck-scale modifications of the particle dispersion relations can be encoded in the nontrivial geometrical properties of momentum space [

We shall focus in this work on two main points. Firstly, on solutions to the field equations in momentum space with the inclusion of the momentum analog of a cosmologically constant

the solutions to the above field equations will be used in the modified uncertainty relations. The momentum-space analog

Secondly, on the rotationally invariant commutator of the form [

one can see that under rotations

the left and right hand side of Equation (2) become

and the commutator relations remain invariant. Consequently, if one is to set

A close inspection reveals that the 4D momentum-space metric analog of the de Sitter metric in a 4D spacetime (written in static coordinates and using the momentum- space analog of the cosmological constant

does not have the required form indicated by Equation (5). To verify this one simply rewrites the de Sitter metric in Cartesian coordinates. One then finds that the rota- tionally invariant commutation relations, leading to the metrics (5), are not compatible with a spherically symmetric momentum space de Sitter metric (6).

One may insert the metric (5) into the field equations in momentum space in order to determine whether or not there exist actual functions

which is trivially rotational invariant.

momentum space when

to coincide with the Planck length

Inspired by the 2D de Sitter momentum space metric (7), and by promoting the classical momentum variable

One may notice that since

consistent with the cosmological momentum-horizon

Inserting the inequality of the equation below

into Equation (8), yields to leading order in

which has the same functional form as the stringy modified uncertainty relations [

(11).

The minimum value for the position uncertainty is

and which coincides with the location of the cosmological momentum horizon. If one equates the minimum value of the position uncertainty to the Planck scale length it gives

To sum up, to leading order in L, the de Sitter momentum space metric in 2D

furnishes: 1) a cosmological momentum-horizon

cutoff; 2) a Planck scale minimal length uncertainty for the position coordinate

The next-to-leading order term can be obtained after using the inequality

that simply follows from

after replacing

The minimum position uncertainty now turns out to be

The value of

on Equation (12a), Equation (12b), by a process of successive squaring, a hierarchy of modified uncertainty relations of the form are derived

The most salient feature of the modified uncertainty relations (11), (12c), (12d) is that there is a minimum value for the position uncertainty

In general one can postulate the following modification of the Weyl-Heisenberg algebra

combined with the additional commutation relations

with the provision that the above commutators obey the Jacobi identities [

The more general commutator than the one in Equation (13)

may be chosen such that the classical limit

An important remark is in order. By Hermitian metric one usually means

example if

the argument of the metric matrix is Hermitian. Similarly, by anti-Hermitian metric one usually means

Since the commutator of two Hermitian operators in anti-Hermitian, one may postulate the following commutators below (in a fully relativistic phase space) given in terms of of a real metric which has both symmetric

the right hand sides are anti-Hermitian due to

It is at this point where the following Schrodinger-Robertson inequalities for 2n observables

the covariance is defined as

uncorrelated variables have zero covariance. The uncertainty squared is

For the 2n phase space coordinates, the

Due to the nontrivial commutation relations (16)-(18), the Schrodinger-Robertson inequalities

Closely related to the nontrivial commutation relations (16)-(18) is Yang’s algebra in an 8D Noncommutative phase space [

Yang’s algebra can be obtained simply by replacing

and recurring to the angular momentum algebra in 6D. The Jacobi identities are satisfied because the angular momentum algebra in 6D obeys them. The noncommuting coordinates and momenta are just rotations/boosts involving the extra directions.

One may notice that Yang’s algebra and the algebra of Eqsuations (16)-(18) bears a certain resemblance if one were to set the numerical coefficient B to zero;

which bears a similarity to the results associated to the area operator obtained in Loop Quantum Gravity (LQG) and based on spin networks. The Planck area is the quantum of minimal area [

Symplectic geometry is the realm of phase spaces [

To conclude, we may add that non-geometric fluxes in string theory give rise to noncommutative/nonassociative structures. More recently, the differential geometry on the simplest nonassociative (phase) space arising for a constant non-geometric R-flux has been analyzed in [

We thank M. Bowers for very kind assistance.

Castro, C. (2016) Generalized Uncertainty Relations, Curved Phase-Spaces and Quantum Gravity. Journal of Applied Mathematics and Physics, 4, 1870-1878. http://dx.doi.org/10.4236/jamp.2016.410189