The importance of quantum fluctuations

2.1 The importance of quantum fluctuations

For a free quantum field, semiclassical gravity is fairly well understood. The theory is in some sense unique, since the only reasonable c-number stress-energy tensor that one may construct [282

, 283

] with the stress-energy operator is a renormalized expectation value. However, the scope and limitations of the theory are not so well understood. It is expected that the semiclassical theory would break down at the Planck scale. One can conceivably assume that it would also break down when the fluctuations of the stress-energy operator are large [92

, 194

]. Calculations of the fluctuations of the energy density for Minkowski, Casimir and hot flat spaces as well as Einstein and de Sitter universes are available [194

, 241

, 163

, 242

, 243

, 245

, 244

, 208

, 209

, 251 , 254

, 227 , 69

]. It is less clear, however, how to quantify what a large fluctuation is, and different criteria have been proposed [194

, 93

, 95

, 163

, 242

, 10

, 9

]. The issue of the validity of the semiclassical gravity viewed in the light of quantum fluctuations is summarized in our Erice lectures [168

]. One can see the essence of the problem by the following example inspired by Ford [92

Let us assume a quantum state formed by an isolated system which consists of a superposition with equal amplitude of one configuration of mass $M$ with the center of mass at $X1$ , and another configuration of the same mass with the center of mass at $X2$ . The semiclassical theory as described by the semiclassical Einstein equation predicts that the center of mass of the gravitational field of the system is centered at $1 2(X1 + X2 )$ . However, one would expect that if we send a succession of test particles to probe the gravitational field of the above system, half of the time they would react to a gravitational field of mass $M$ centered at $X1$ and half of the time to the field centered at $X2$ . The two predictions are clearly different; note that the fluctuation in the position of the center of masses is of the order of $(X − X )2 1 2$ . Although this example raises the issue of how to place the importance of fluctuations to the mean, a word of caution should be added to the effect that it should not be taken too literally. In fact, if the previous masses are macroscopic, the quantum system decoheres very quickly [306 , 307 ] and instead of being described by a pure quantum state it is described by a density matrix which diagonalizes in a certain pointer basis. For observables associated to such a pointer basis, the density matrix description is equivalent to that provided by a statistical ensemble. The results will differ, in any case, from the semiclassical prediction.

In other words, one would expect that a stochastic source that describes the quantum fluctuations should enter into the semiclassical equations. A significant step in this direction was made in [148 ], where it was proposed to view the backreaction problem in the framework of an open quantum system: the quantum fields seen as the “environment” and the gravitational field as the “system”. Following this proposal a systematic study of the connection between semiclassical gravity and open quantum systems resulted in the development of a new conceptual and technical framework where (semiclassical) Einstein–Langevin equations were derived [44 , 157 , 167 , 58 , 59 , 38 , 202 ]. The key technical factor to most of these results was the use of the influence functional method of Feynman and Vernon [89 ], when only the coarse-grained effect of the environment on the system is of interest. Note that the word semiclassical put in parentheses refers to the fact that the noise source in the Einstein–Langevin equation arises from the quantum field, while the background spacetime is classical; generally we will not carry this word since there is no confusion that the source which contributes to the stochastic features of this theory comes from quantum fields.

In the language of the consistent histories formulation of quantum mechanics [114 , 221 , 222 , 223 , 224 , 225 , 226 , 105 , 125 , 83 , 120 , 122 , 30 , 239 , 278 , 170 , 171 , 172 , 121 , 81 , 82 , 185 , 186 , 187 , 173 ] for the existence of a semiclassical regime for the dynamics of the system, one needs two requirements: The first is decoherence, which guarantees that probabilities can be consistently assigned to histories describing the evolution of the system, and the second is that these probabilities should peak near histories which correspond to solutions of classical equations of motion. The effect of the environment is crucial, on the one hand, to provide decoherence and, on the other hand, to produce both dissipation and noise to the system through backreaction, thus inducing a semiclassical stochastic dynamics on the system. As shown by different authors [106 , 303 , 304 , 305 , 306 , 180 , 33 , 279 , 307 , 109 ], indeed over a long history predating the current revival of decoherence, stochastic semiclassical equations are obtained in an open quantum system after a coarse graining of the environmental degrees of freedom and a further coarse graining in the system variables. It is expected but has not yet been shown that this mechanism could also work for decoherence and classicalization of the metric field. Thus far, the analogy could be made formally [206 ] or under certain assumptions, such as adopting the Born–Oppenheimer approximation in quantum cosmology [237 , 238 ].

An alternative axiomatic approach to the Einstein–Langevin equation without invoking the open system paradigm was later suggested, based on the formulation of a self-consistent dynamical equation for a perturbative extension of semiclassical gravity able to account for the lowest order stress-energy fluctuations of matter fields [207 ]. It was shown that the same equation could be derived, in this general case, from the influence functional of Feynman and Vernon [208 ]. The field equation is deduced via an effective action which is computed assuming that the gravitational field is a c-number. The important new element in the derivation of the Einstein–Langevin equation, and of the stochastic gravity theory, is the physical observable that measures the stress-energy fluctuations, namely, the expectation value of the symmetrized bi-tensor constructed with the stress-energy tensor operator: the noise kernel. It is interesting to note that the Einstein–Langevin equation can also be understood as a useful intermediary tool to compute symmetrized two-point correlations of the quantum metric perturbations on the semiclassical background, independent of a suitable classicalization mechanism [255 ].