Quantum field theory in curved spacetime

In particle physics, 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.


Interesting new phenomena occur; owing 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, 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 tensor 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 t(t) is a diffeomorphism, in general, the Fourier transform of exp[ikt(t)] will contain negative frequencies even if k > 0. 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 cannot look like a vacuum state to another observer; it could even appear as a heat bath under suitable hypotheses.

Since the end of the eighties, the local quantum field theory approach due to Rudolf Haag and Daniel Kastler has been implemented in order to include an algebraic version of quantum field theory in curved spacetime. Indeed, the viewpoint of local quantum physics is suitable to generalize the renormalization procedure to the theory of quantum fields developed on curved backgrounds. Several rigorous results concerning QFT in the presence of a black hole have been obtained. In particular the algebraic approach allows one to deal with the problems, above mentioned, arising from the absence of a preferred reference vacuum state, the absence of a natural notion of particle and the appearance of unitarily inequivalent representations of the algebra of observables. (See these lecture notes [1] for an elementary introduction to these approaches and the more advanced review [2])


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 Dirac equation can be formulated in curved spacetime, see Dirac equation in curved spacetime for details.

Approximation to quantum gravity

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). However gravity is not renormalizable in QFT,[3] so merely formulating QFT in curved spacetime is not a theory of quantum gravity.

See also


  1. C. J. Fewster (2008). "Lectures on quantum field theory in curved spacetime (Lecture Note 39/2008 Max Planck Institute for Mathematics in the Natural Sciences (2008))" (PDF). York, UK.
  2. I. Khavkine and V. Moretti (2015). "Algebraic QFT in Curved Spacetime and quasifree Hadamard states: an introduction)" (PDF). Trento, Italy. arXiv:1412.5945. Bibcode:2014arXiv1412.5945K.
  3. A. Shomer (2007). "A pedagogical explanation for the non-renormalizability of gravity". arXiv:0709.3555.

Further reading

  • N.D. Birrell & P.C.W. Davies. Quantum fields in curved space. CUP (1982).
  • S.A. Fulling. Aspects of quantum field theory in curved space-time. CUP (1989).
  • R.M. Wald. Quantum field theory in curved space-time and black hole thermodynamics. Chicago U. (1995).
  • V. Mukhanov and S. Winitzki. Introduction to Quantum Effects in Gravity. CUP (2007).
  • L. Parker & D. Toms. Quantum Field Theory in Curved Spacetime. (2009).
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