Quantum Gravitational Effects in De Sitter Space

We calculate the first quantum gravitational correction term to the trace anomaly in De Sitter space from the Wheeler-DeWitt equation. This is obtained through an expansion of the full wave functional for gravity and a conformally coupled scalar field in powers of the Planck mass. We also discuss a quantum gravity induced violation of unitarity and comment on its possible relevance for inflation. Invited Contribution to New Frontiers in Gravitation, edited by G. Sardanashvily and R. Santilli (Hadronic Press, 1995). A central role in the study of quantum field theory on a given classical background spacetime is played by the semiclassical Einstein equations Rμν − 1 2 gμνR = −8πG〈Tμν〉, (1) in which the renormalised expectation value of the energy-momentum tensor acts as a “back reaction” on the metric of some classical spacetime. The quantum matter state with respect to which this expectation value is taken is assumed to obey, in the Schrödinger picture, a functional Schrödinger equation, where the time evolution is generated by the matter Hamiltonian. A prominent example is the case of a conformally coupled scalar field in De Sitter spacetime. If one assumes this field to be in the Bunch-Davies vacuum state (which is the unique De Sitter invariant vacuum state [1]), one finds for the expectation value of the energy-momentum tensor the result (see e.g. [2]) 〈Tμν〉 = H 0 h̄ 960π2 gμν , (2) where H0 is the (constant) Hubble parameter of De Sitter space. Since the trace of this expression is non-vanishing, it leads to the so-called trace anomaly because conformal invariance would lead to a vanishing trace at the classical level. Since one expects that the gravitational field is fundamentally described by quantum theory, (1) can at best hold approximately. There have been many discussions in the literature which have investigated the range of validity of the semiclassical Einstein equations. Ford [3], e.g., has compared the emission of classical gravitational waves in the semiclassical theory with graviton emission in linear quantum gravity and found that this can be drastically different except if the mean deviation of Tμν is small. This is fulfilled, for example, if the quantum state approximately evolves adiabatically. A number of papers have strengthened this result by the attempt to derive (1) from the Wheeler-DeWitt equation, the central equation of canonical quantum gravity, in a semiclassical approximation and to study the emergence of the back reaction term with the help of Wigner’s function (see, e.g., [4]). A similar discussion was made for quantum electrodynamics [5]. If it is possible to derive the limit of quantum field theory in a classical background from the Wheeler-DeWitt equation [6], one should also be able to go beyond this limit and study quantum gravitational corrections. This has been achieved through an expansion with respect to the gravitational constant [6, 7], although in a formal sense only, since regularisation issues have not been adressed. More precisely, one obtains correction terms to the functional Schrödinger equation for matter fields on a given spacetime – the first quantum gravity induced “post-Schrödinger approximation”. In the present paper these correction terms are explicitly calculated for a conformally coupled scalar field in De Sitter space. Since the general discussion