We discuss the BRST quantization of General Relativity (GR) with a cosmological constant in the unimodular gauge. We show how to gauge fix the transverse part of the diffeomorphism 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 transverse diffeomorphism. We show that the physical spectrum is the same as GR in the usual covariant gauge fixing. We then suggest to define the quantum theory of Unimodular Gravity (UG) by making Fourier transform of GR in the unimodular gauge with respect to the cosmological constant and slightly generalizing it. This suggests that the quantum theory of UG may describe the same theory as GR but the spacetime volume is fixed. We also discuss problems left in this formulation of UG.
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