The volume operator

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

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 [149 ]. 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 [151 ]. 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 [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.