13 - Kinematical geometrical operators

Summary

In this chapter we will describe the so-called kinematical geometrical operators of Loop Quantum Gravity. These are gauge-invariant operators which measure the length, area and volume respectively of coordinate curves, surfaces and volumes for D = 3. The area and volume operators were first considered by Smolin in [660] and then formalised by Rovelli and Smolin in the loop representation [425]. In [575] Loll discovered that the volume operator vanishes on gauge-invariant states with at most trivalent vertices and used area and volume operators in her lattice theoretic framework [661–663]. Ashtekar and Lewandowski [427] used the connection representation defined in previous chapters and could derive the full spectrum of the area operator, while their volume operator differs from that of Rovelli and Smolin on graphs with vertices of valence higher than three, which can be seen as the result of using different diffeomorphism classes of regularisations. In [664] de Pietri and Rovelli computed the matrix elements of the RS volume operator in the loop representation and de Pietri created a computer code for the actual case-by-case evaluation of the eigenvalues. In [559] the connection representation was used in order to obtain the complete set of matrix elements of the AL volume operator.

Area and volume operators could be quantised using only the known quantisations of the electric flux of Section 6.3 but the construction of the length operator [424] required the new quantisation technique of using Poisson brackets with the volume operator, which was first employed for the Hamiltonian constraint, see Chapter 10. To the same category of operators also belong the ADM energy surface integral [442], angle operators [429, 430] and other similar operators that test components of the three-metric tensors.