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
. This result is independent of the choice of gauge group (SU
(2) 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 [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
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
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
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
, 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
[154]. 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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Discrete Approaches to Quantum Gravity in Four Dimensions
Renate Loll http://www.livingreviews.org/lrr-1998-13 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |