Abstract
Unimodular gravity (UG) is an important theory which may explain the smallness of the cosmological constant. To get insight into the covariant quantization of UG, we discuss the BRST quantization of General Relativity (GR) with a cosmological constant in the unimodular gauge. We develop a novel way to gauge fix the transverse diffeomorphism (TDiff) and then further to fulfill the unimodular gauge. This process requires the introduction of an additional pair of BRST doublets which decouple from the physical sector together with the other three pairs of BRST doublets for the TDiff. We show that the physical spectrum is the same as GR in the usual covariant gauge fixing. We then study a theory derived by making "Fourier transform" of GR in the unimodular gauge with respect to the cosmological constant as a candidate of "quantum UG." We clarify the difference from GR and point out problems in this theory.
Highlights
General relativity (GR) well describes gravitational lowenergy phenomena
We develop a novel way to gauge fix the transverse diffeomorphism (TDiff) and further to fulfill the unimodular gauge
We show that the physical spectrum is the same as general relativity (GR) in the usual covariant gauge fixing
Summary
General relativity (GR) well describes gravitational lowenergy phenomena. The action is given by. Baulieu [21] has recently proposed an interesting way of gauge fixing which seems to realize this unimodular gauge in a manifestly covariant manner He introduces a new set of “BRST quartet” fields consisting of an additional scalar field together with ghosts and a scalar. We propose a novel and general way of decomposing a d-vector condition into a scalar plus d − 1 (i.e., transverse-vector) conditions in a manifestly covariant and local manner, which naturally leads to the introduction of a scalar field and the emergence of a new gauge symmetry This reproduces essentially the same unimodular gaugefixed Lagrangian as Baulieu’s one. We introduce an additional longitudinal mode and there appears additional invariance, which removes the remaining degrees of freedom Since this is the quantization of GR, the cosmological constant is present in the field egqauuagteiown itehvepn ffi−ffitffiffihgffiffio1⁄4ugωh. We find that such a formulation has problem with unitarity, but we hope that our discussions clarify the problems in the covariant quantization of UG
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