Apart from characterizing geometric properties of lattice quantum states, it is needed in the construction of the quantum Hamiltonian of the real connection approach.
Loll  showed that a lattice Wilson loop state has to have intersections of valence at least 4 in order not to be annihilated by the volume operator . This result is independent of the choice of gauge group ( or ). The volume operator has discrete eigenvalues, and part of its non-vanishing spectrum, for the simplest case of four-valent intersections, was first calculated by Loll . These spectral calculations were confirmed by De Pietri and Rovelli , who derived a formula for matrix elements of the volume operator on intersections of general valence (as did Thiemann ).
This still leaves questions about the spectrum itself unanswered, since the eigenspaces of grow rapidly, and diagonalization of the matrix representations becomes a technical problem. Nevertheless, one can achieve a better understanding of some general spectral properties of the lattice volume operator, using symmetry properties. It was observed in  that all non-vanishing eigenvalues of come in pairs of opposite sign. Loll subsequently proved that this is always the case . A related observation concerns the need for imposing an operator condition on physical states in non-perturbative quantum gravity , a condition which distinguishes its state space from that of a gauge theory already at a kinematical level.
The symmetry group of the cubic three-dimensional lattice is the discrete octagonal group , leaving the classical local volume function invariant. Consequently, one can find a set of operators that commute among themselves and with the action of the volume operator, and simplify its spectral analysis by decomposing the Hilbert space into the corresponding irreducible representations . This method is most powerful when applied to states which are themselves maximally symmetric under the action of , in which case it leads to a dramatic reduction of the dimension of the eigenspaces of .
In addition to the volume operator, one may define geometric lattice operators measuring areas and lengths [150, 152]. They are based on (non-unique) discretizations of the continuum spatial integrals of the square root of the determinant of the metric induced on subspaces of dimension 2 and 1. For the case of the length operator, operator-ordering problems arise in the quantization.
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