

Various properties of gauge gravity models were analyzed by
Menotti and Pelissetto in a series of papers in the mid-eighties.
They first studied a discrete
O
(4)-version of the Lagrangian (1
) [165], including the
O
(4)-Haar measure and a general local
O
(4)-invariant measure for the vierbein fields, and showed that
reflection-positivity holds only for a restricted class of
functions. Furthermore, expanding about flat space, and after
appropriate gauge-fixing, they discovered a doubling phenomenon
similar to that found for chiral fermions in lattice gauge
theory, a behaviour that also persists for different
gauge-fixings. One finds the same mode doubling also for a
flat-background expansion of conformal gravity [167
]. In the same paper, they gave a unified treatment of Poincaré,
de Sitter and conformal gravity, and showed that reflection
positivity for
O
(4)-gravity (as well as for the two other gauge groups) holds
exactly and for general functions only provided a signature
factor sign
is included in the Lagrangian.
To ensure the convergence of the functional integration, one
has to introduce a damping factor for the vierbeins in the
measure, both for Poincaré and conformal gravity [167
]. An extension of the results of [167] to supergravity with the super-Poincaré group is also possible
[62]. One can prove reflection positivity and finds a matching
gravitino doubling in the perturbative expansion.


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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
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livrev@aei-potsdam.mpg.de
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