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The quantum theory of the gravitational field; also, the study of quantum fields in a curved space-time. In classical general relativity, the gravitational field is represented by the metric tensor gμ&ngr; of space-time. This tensor satisfies Einstein's field equation, with the energy-momentum tensor of matter and radiation as a source. However, the equations of motion for the matter and radiation fields also depend on the metric.
Classical field theories such as Maxwell's electromagnetism or the classical description of particle dynamics are approximations valid only at the level of large-scale macroscopic observations. At a fundamental level, elementary interactions of particles and fields must be described by relativistic quantum mechanics, in terms of quantum fields. Because the geometry of space-time in general relativity is inextricably connected to the dynamics of matter and radiation, a consistent theory of the metric in interaction with quantum fields is possible only if the metric itself is quantized. See Maxwell's equations, Quantum field theory, Relativistic quantum theory, Relativity
Under ordinary laboratory conditions the curvature of space-time is so extremely small that in most quantum experiments gravitational effects are completely negligible. Quantization in Minkowski space is then justified. Gravity is expected to play a significant role in quantum physics only at rather extreme conditions of strongly time-dependent fields, near or inside very dense matter. The scale of energies at which quantization of the metric itself becomes essential is given by (ℏc5/G)1/2 ≈ 1019 GeV, where G is the gravitational constant, ℏ is Planck's constant divided by 2&pgr;, and c is the velocity of light. Energies that can be reached in the laboratory or found in cosmic radiation are far below this order of magnitude. Only in the very early stages of the universe, within a proper time of the order of (Gℏ/c5)1/2 ≈ 10-43 s after the big bang, would such energies have been produced.
In most physical systems the metric is quasistationary over macroscopic distances so that its fluctuations can be ignored. A quantum description of fields in a curved space-time can then be given by treating the metric as a classical external field in interaction with the quantum fields.
Quantum effects of black holes
The most striking quantum effect in curved space-time is the emission of radiation by black holes. A black hole is an object that has undergone gravitational collapse. Classically this means that it becomes confined to a space-time region in which the metric has a singularity (the curvature becomes infinite). This region is bounded by a surface, called the horizon, such that any matter or radiation falling inside becomes trapped. Therefore, classically the mass of a black hole can only increase. However, this is no longer the case if quantum effects are taken into account. When, because of fluctuations of the quantum field, particle-antiparticle or photon pairs are created near the horizon of a black hole, one of the particles carrying negative energy may move toward the hole, being absorbed by it, while the other moves out with positive energy.
It is found that the total rate of emission is inversely proportional to the square of the mass. For stellar black holes whose masses are of the order of a solar mass, the emission rate is negligibly small and unobservable. Only primordial black holes, of mass less than 1013 kg, formed very early in the quantum era of the universe, would have been small enough to produce quantum effects that could play any significant role in astrophysics or in cosmology.
Quantization of the metric
There are basically two approaches to the quantization of the metric, the canonical and the covariant quantization. A third method, which can be derived from the first and is now most widely used, is based on the Feynman path integral representation for the vacuum-to-vacuum amplitude, which is the generator of Green's functions for the quantum theory. One important feature of this method is that since the topology of the manifold is not specified at the outset it is possible to include a sum over paths in different topologies. The outcome of this idea is that the vacuum would, at the level of the Planck length, (Gℏ/c3)1/2 ≈ 10-35 m, acquire a foamlike structure. See Feynman integral, Green's function
At present a complete, consistent theory of quantum gravity is still lacking. The formal theory fails to satisfy the power-counting criterion for renormalizability. In every order of the perturbation expansion, new divergences appear which could only be canceled by counterterms that do not exist in the original lagrangian. This may not be just a technical problem but the reflection of a conceptual difficulty stemming from the dual role, geometric and dynamic, played by the metric. See Renormalization
Supergravity and superstrings
Supergravity is a geometric extension of general relativity which incorporates the principle of supersymmetry. Supersymmetry is a kind of symmetry, discovered in the 1970s, that allows for the transformation of fermions and bosons into each other. (Fermions carry half-integer spin while bosons carry integer spin; they also obey different statistics.) Supergravity can be formulated in space-time manifolds with a total of D = d + 1 dimensions, where d, the number of space dimensions, can be as large as 10. They constitute truly unified theories of all interactions including gravity.
In the early 1980s, some encouraging results were found with a theory based on the idea that the basic objects of nature are not pointlike but actually one-dimensional objects like strings, which can be open or closed. Incorporating supersymmetry into the theory leads to a critical dimension D = 10.
In the approximation of neglecting string excitations, certain superstring models may be described in terms of local fields as a D = 10 supergravity theory. At present, these are the only theories that both include gravity and can be consistently quantized. Although a superstring theory may eventually become the ultimate theory of all the interactions, there is still a very long way to go in making the connection between its fundamental fields and the fields representing the particles and their interactions as observed at low energies. See Fundamental interactions, Gravitation, Supergravity, Superstring theory, Supersymmetry