Linearized gravity in “closed” universes with continuous symmetries

Abstract

It has been pointed out that all the states in linearized quantum gravity in spacetime with compact Cauchy surfaces and with continuous symmetries are required to be invariant under those symmetries [1]. For de Sitter spacetime this requirement is rather paradoxical because the vacuum is the only state that is invariant under the de Sitter group SO(4, 1) in the Fock space of linearized gravity. It has been shown that there are infinite-norm invariant states and that they can be used to construct a new Hilbert space of invariant states by factoring out infinity from the norm of these states [2]. Here, I will review how this requirement arises and argue that it is a natural consequence of the underlying diffeomorphism invariance of the full quantnm gravity and explain briefly the results of Ref. [2]. It is well known that some solutions to linearized gravity in spacetime with continuous symmetries and with compact Cauchy surfaces are spurious in the sense that they do not correspond to any exact solutions [3]. This fact can be seen as follows. In the Hamiltonian formulation [4] one parametrizes the line element as ds 2 = ( N 2 -NaNa)dt2+ 2N~dxadt +g~dxadz b, where indices are raised and lowered by gab. The lapse function N and the shift vector N~ have no conjugate momenta. Hence, there are corresponding constraints