Quantum field theory in curved spacetime

In theoretical , field in curved (QFTCS) is an extension of field from to a general curved . This treats as a fixed, classical background, while giving a -mechanical description of the matter and propagating through that . A general prediction of this is that particles can be created by time-dependent gravitational fields (multigraviton pair ), or by time-independent gravitational fields that contain horizons. The most famous example of the latter is the phenomenon of Hawking radiation emitted by holes.


Ordinary field theories, which form the basis of standard model, are defined in flat space, which is an excellent approximation when it comes to describing the behavior of microscopic particles in gravitational fields like those found on . In order to describe situations in which is strong enough to influence () matter, yet not strong enough to require quantization itself, physicists have formulated field theories in curved . These theories rely on general relativity to describe a curved background , and define a generalized field to describe the behavior of matter within that .

For non- cosmological constants, on curved spacetimes fields lose their interpretation as asymptotic particles. Only in certain situations, such as in asymptotically flat spacetimes ( 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 , the asymptotic particle interpretation depends on the observer (i.e., different observers may measure different numbers of asymptotic particles on a given ).

Another observation is that unless the background metric tensor has a global timelike Killing vector, there is no way to define a or ground canonically. The of a is not invariant under diffeomorphisms. This is because a mode decomposition of a field into and modes is not invariant under diffeomorphisms. If t′(t) is a diffeomorphism, in general, the Fourier transform of exp[ikt′(t)] will contain frequencies even if k > 0. Creation operators correspond to frequencies, while annihilation operators correspond to frequencies. This is why a which looks like a to one observer cannot look like a to another observer; it could even appear as a heat bath under suitable hypotheses.

Since the end of the 1980s, the local field approach due to Rudolf Haag and Daniel Kastler has been implemented in order to include an algebraic version of field in curved . Indeed, the viewpoint of local is suitable to generalize the renormalization procedure to the of fields developed on curved backgrounds. Several rigorous results concerning QFT in the presence of a hole have been obtained. In particular the algebraic approach allows one to deal with the problems mentioned above arising from the absence of a preferred reference , the absence of a notion of particle and the appearance of unitarily inequivalent representations of the of observables.


Using perturbation in field in curved geometry is known as the semiclassical approach to . This approach studies the interaction of fields in a fixed classical and among other thing predicts the creation of particles by time-varying spacetimes and Hawking radiation. The latter can be understood as a manifestation of the Unruh effect where an accelerating observer observes radiation. Other prediction of fields in curved spaces include, for example, the radiation emitted by a particle moving along a geodesic and the interaction of Hawking radiation with particles outside holes.

This formalism is also used to predict the primordial density perturbation arising in different models of cosmic inflation. These predictions are calculated using the Bunch–Davies or modifications thereto.

Approximation to

The of field in curved may be considered as an intermediate step towards . QFT in curved is expected to be a viable approximation to the of when curvature is not significant on the Planck scale. However, the fact that the true of remains unknown means that the precise criteria for when QFT on curved is a good approximation are also unknown.

is not renormalizable in QFT, so merely formulating QFT in curved is not a of .

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