2.15 The volume operator

An important quantity in Hamiltonian lattice quantum gravity is the volume operator, the quantum analogue of the classical volume function ∫ ----- d3x √ detg. The continuum dreibein determinant detE (x) (with |det E (x )| = det g), has a natural lattice analogue, given by
1 detp (n ) := --𝜖abc𝜖ijkpi(n, ˆa)pj(n,ˆb)pk(n,ˆc). (10 ) 3!

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 [147] 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 ∘ -------- ∑ det ˆp(n) n. This result is independent of the choice of gauge group (SU (2) or SL (2,ℂ)). 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 [149Jump To The Next Citation Point]. These spectral calculations were confirmed by De Pietri and Rovelli [94], who derived a formula for matrix elements of the volume operator on intersections of general valence (as did Thiemann [188]).

This still leaves questions about the spectrum itself unanswered, since the eigenspaces of det ˆp(n) 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 [149] that all non-vanishing eigenvalues of ˆ detp(n ) come in pairs of opposite sign. Loll subsequently proved that this is always the case [151Jump To The Next Citation Point]. A related observation concerns the need for imposing an operator condition deˆt p > 0 on physical states in non-perturbative quantum gravity [151], 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 𝒪, leaving the classical local volume function det p(n) 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 [154]. This method is most powerful when applied to states which are themselves maximally symmetric under the action of 𝒪, in which case it leads to a dramatic reduction of the dimension of the eigenspaces of ˆ det p(n).

In addition to the volume operator, one may define geometric lattice operators measuring areas and lengths [150152]. 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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