New length operator for loop quantum gravity

Abstract

An alternative expression for the length operator in loop quantum gravity is presented. The operator is background independent, symmetric, positive semidefinite, and well defined on the kinematical Hilbert space. The expression for the regularized length operator can moreover be understood both from a simple geometrical perspective as the average of a formula relating the length to area, volume and flux operators, and also consistently as the result of direct substitution of the densitized triad operator with the functional derivative operator into the regularized expression of the length. Both these derivations are discussed, and the origin of an undetermined overall factor in each case is also elucidated.Comment: 10 pages, 2 figures; version published in PR

Loop quantumf(R)theories

“…Note that the spatial geometric operator of LQG, such as the area [18] , the volume [19] and the length [20,21] operators, are still valid in H gr kin , though their properties in the physical Hilbert space still need to be clarified [22,23].…”

Curvature operator for loop quantum gravity

“…(16) and the dihedral angle eq. (24). Therefore, the operator is naturally defined in the space of gauge invariant cylindrical functions.…”

“…In LQG we have three proposals for length operator [22][23][24]. Since our approach to construct a scalar curvature operator is using the dual picture, we choose Bianchi's operator [23] which is constructed based on the same dual picture of quantum geometry.…”

“…lengths and angles. Operators of length [22][23][24] and angle [25][26][27] are available in LQG and Spinfoams and by using a suitable regularization procedure for the classical quantities we can implement directly the integral of the Ricci scalar as an operator acting on spin-network states and in this way settle the first step to bypass many of the complications appearing in the Lorentzian constraint. The considerations due to the change of regularization procedure bring us close to the perspective of [10] and make our proposal intriguing also for the spinfoam formalism.…”

Loop quantum modified gravity and its cosmological application

“…The basic operators of quantum STT are the quantum analogue of holonomies h e (A) = P exp e A a of a connection along edges e ⊂ Σ, densitized triads E(S , f ) := S ǫ abc E a i f i smeared over 2-surfaces, point holonomies U λ = exp(iλφ(x)) [31], and scalar momenta π(R) := R d 3 xπ(x) smeared on 3-dimensional regions. It is worth noting that the spatial geometric operator, such as the area [32] , the volume [33] and the length [34,35] operators, are still valid in H gr kin of quantum STT. As in LQG, it is natural to promote the Gaussian constraint G(Λ) as a well-defined operator [2,4].…”