Jinsong Yang

A new alternative volume operator is constructed for loop quantum gravity by using the so-called cotriad operators as building blocks. It is shown that the new volume operator shares the same qualitative properties with the standard volume operator. Moreover, a new alternative inverse volume operator is also constructed in the light of the construction of the alternative volume operator, which is possessed of the same qualitative properties as those of the alternative volume operator. The new inverse volume operator can be employed to construct the Hamiltonian operator of matter fields, which may lead to an anomaly-free on-shell quantum constraint algebra without any special restriction on the regularization procedure for gravity coupled to matter fields.

A new alternative volume operator is constructed for loop quantum gravity by using the so-called cotriad operators as building blocks. It is shown that the new volume operator shares the same qualitative properties with the standard volume operator. Moreover, a new alternative inverse volume operator is also constructed in the light of the construction of the alternative volume operator, which is possessed of the same qualitative properties as those of the alternative volume operator. The new inverse volume operator can be employed to construct the Hamiltonian operator of matter fields, which may lead to an anomaly-free on-shell quantum constraint algebra without any special restriction on the regularization procedure for gravity coupled to matter fields.

This is the second paper in the series to introduce a graphical method to loop quantum gravity. We employ the graphical method as a powerful tool to calculate the actions of the Euclidean Hamiltonian constraint operator and the so-called inverse volume operator on spin network states with trivalent vertices. Both of the operators involve the co-triad operator which contains holonomies by construction. The non-ambiguous, concise and visual characters of our graphical method ensure the rigour for our calculations. Our results indicate some corrections to the existing results in literatures.

To adopt a practical method to calculate the action of geometrical operators on quantum states is a crucial task in loop quantum gravity. In the series of papers, we will introduce a graphical method, developed by Yutsis and Brink, to loop quantum gravity along the line of previous works. The graphical method provides a very powerful technique for simplifying complicated calculations. In this first paper, the closed formula of volume operator is derived via the graphical method. By employing suitable and non-ambiguous graphs to represent the acting of operators as well as the spin network states, we use the simple rules of transforming graphs to yield the resulting formula. Comparing with the complicated algebraic derivation in some literatures, our procedure is more concise, intuitive and visual. The resulting matrix elements of volume operator is compact and uniform, fitting for both gauge-invariant and gauge-variant spin network states.

A new symmetric Hamiltonian constraint operator is proposed for loop quantum gravity, which is well defined in the Hilbert space of diffeomorphism invariant states up to non-planar vertices with valence higher than three. It inherits the advantage of the original regularization method, so that its regulated version in the kinematical Hilbert space is diffeomorphism covariant and creates new vertices to the spin networks. The quantum algebra of this Hamiltonian is anomaly-free on shell, and there is less ambiguity in its construction in comparison with the original method. The regularization procedure for this Hamiltonian constraint operator can also be applied to the symmetric model of loop quantum cosmology, which leads to a new quantum dynamics of the cosmological model.

Both projectable and non-projectable versions of Horava-Lifshitz gravity face serious challenges. In the non-projectable version, the constraint algebra is seemingly inconsistent. The projectable version lacks a local Hamiltonian constraint, thus allowing for an extra scalar mode which can be problematic. A new formulation of non-projectable Horava-Lifshitz gravity, naturally realized as a representation of the master constraint algebra studied by loop quantum gravity researchers, is presented. This yields a consistent canonical theory with first class constraints. It captures the essence of Horava-Lifshitz gravity in retaining only spatial diffeomorphisms (instead of full space-time covariance) as the physically relevant non-trivial gauge symmetry; at the same time the local Hamiltonian constraint needed to eliminate the extra mode is equivalently enforced by the master constraint.

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.

Although there is no known meaningful notion of the energy density of the gravitational field in general relativity, a few notions of quasi-local energy of gravity associated to extended but finite domains have been proposed. In this paper, the notions of quasi-local energy are studied in the framework of loop quantum gravity, in order to see whether these notions can be carried out at quantum level. Two basic quasi-local geometric quantities are quantized, which lead to well-defined operators in the kinematical Hilbert space of loop quantum gravity. We then use them as basic building blocks to construct different versions of quasi-local energy operators. The operators corresponding to Brown-York energy, Liu-Yau energy, Hawking energy, and Geroch energy are obtained respectively. The virtue of the Geroch energy operator is beneficial for us to derive a rather general entropy-area relation and thus a holographic principle from loop quantum gravity.