2.16 The real dynamics2 Gauge-theoretic discretizations of gravity2.14 The role of diffeomorphisms

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 tex2html_wrap_inline2389 . The continuum dreibein determinant tex2html_wrap_inline2391 (with tex2html_wrap_inline2393), has a natural lattice analogue, given by

  equation227

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 tex2html_wrap_inline2395 . This result is independent of the choice of gauge group (SU (2) or tex2html_wrap_inline2365). 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 In The Article]. 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 tex2html_wrap_inline2401 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 tex2html_wrap_inline2401 come in pairs of opposite sign. Loll subsequently proved that this is always the case [151Jump To The Next Citation Point In The Article]. A related observation concerns the need for imposing an operator condition tex2html_wrap_inline2405 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 tex2html_wrap_inline2407, leaving the classical local volume function tex2html_wrap_inline2409 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 tex2html_wrap_inline2407, in which case it leads to a dramatic reduction of the dimension of the eigenspaces of tex2html_wrap_inline2401 .

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.



2.16 The real dynamics2 Gauge-theoretic discretizations of gravity2.14 The role of diffeomorphisms

image Discrete Approaches to Quantum Gravity in Four Dimensions
Renate Loll
http://www.livingreviews.org/lrr-1998-13
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