Quantum Field Theory in Curved Spacetime

Quantum field theory in curved spacetime is an extension of standard, Minkowski-space quantum field theory to curved spacetime. A general prediction of this theory is that particles can be created by time dependent gravitational fields (multigraviton pair production), or by time independent gravitational fields that contain horizons.

Thanks to the equivalence principle the quantization procedure locally resembles that of normal coordinates where the affine connection at the origin is set to zero and a nonzero Riemann tensor in general once the proper (covariant) formalism is chosen; however, interesting new phenomena occur. Even in flat spacetime quantum field theory, the number of particles is not well-defined locally. For non-zero cosmological constants, on curved spacetimes quantum fields lose their interpretation as asymptotic particles. Only in certain situations, such as in asymptotically flat spacetimes (zero cosmological curvature), can the notion of incoming and outgoing particle be recovered, thus enabling one to define an S-matrix. Even then, as in flat spacetime, the asymptotic particle interpretation depends on the observer (i.e., different observers may measure different numbers of asymptotic particles on a given spacetime).

Another observation is that unless the background metric has a global timelike Killing vector, there is no way to define a vacuum or ground state canonically. The concept of a vacuum is not invariant under diffeomorphisms. This is because a mode decomposition of a field into positive and negative frequency modes is not invariant under diffeomorphisms. If is a diffeomorphism, in general, the Fourier transform of will contain negative frequencies even if . Creation operators correspond to positive frequencies, while annihilation operators correspond to negative frequencies. This is why a state which looks like a vacuum to one observer can look like a heat bath to another accelerating with respect to the former observer.

The most striking application of the theory is Hawking's prediction that Schwarzschild black holes radiate with a thermal spectrum. A related prediction is the Unruh effect: accelerated observers in the vacuum measure a thermal bath of particles.

This formalism is also used to predict the primordial density perturbation spectrum arising from cosmic inflation, i.e. the Bunch–Davies vacuum. Since this spectrum is measured by a variety of cosmological measurements—such as the CMB -- if inflation is correct this particular prediction of the theory has already been verified.

The theory of quantum field theory in curved spacetime can be considered as a first approximation to quantum gravity. A second step towards that theory would be semiclassical gravity, which would include the influence of particles created by a strong gravitational field on the spacetime (which is still considered classical and the equivalence principle still holds).

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