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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 [282Jump To The Next Citation Point283Jump To The Next Citation Point] 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 [92Jump To The Next Citation Point194Jump To The Next Citation Point]. 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 [194Jump To The Next Citation Point241Jump To The Next Citation Point163Jump To The Next Citation Point242Jump To The Next Citation Point243Jump To The Next Citation Point245Jump To The Next Citation Point244Jump To The Next Citation Point208Jump To The Next Citation Point209Jump To The Next Citation Point251254Jump To The Next Citation Point22769Jump To The Next Citation Point]. It is less clear, however, how to quantify what a large fluctuation is, and different criteria have been proposed [194Jump To The Next Citation Point93Jump To The Next Citation Point95Jump To The Next Citation Point163Jump To The Next Citation Point242Jump To The Next Citation Point10Jump To The Next Citation Point9Jump To The Next Citation Point]. The issue of the validity of the semiclassical gravity viewed in the light of quantum fluctuations is summarized in our Erice lectures [168Jump To The Next Citation Point]. One can see the essence of the problem by the following example inspired by Ford [92Jump To The Next Citation Point].

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 [306Jump To The Next Citation Point307Jump To The Next Citation Point] 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 [148Jump To The Next Citation Point], 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 [44Jump To The Next Citation Point157Jump To The Next Citation Point167Jump To The Next Citation Point58Jump To The Next Citation Point59Jump To The Next Citation Point38Jump To The Next Citation Point202Jump To The Next Citation Point]. The key technical factor to most of these results was the use of the influence functional method of Feynman and Vernon [89Jump To The Next Citation Point], 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 [11422122222322422522610512583Jump To The Next Citation Point120Jump To The Next Citation Point122Jump To The Next Citation Point302392781701711721218182185186187173] 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 [106Jump To The Next Citation Point303Jump To The Next Citation Point304Jump To The Next Citation Point305Jump To The Next Citation Point306Jump To The Next Citation Point180Jump To The Next Citation Point33Jump To The Next Citation Point279Jump To The Next Citation Point307Jump To The Next Citation Point109Jump To The Next Citation Point], 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 [206Jump To The Next Citation Point] or under certain assumptions, such as adopting the Born–Oppenheimer approximation in quantum cosmology [237Jump To The Next Citation Point238Jump To The Next Citation Point].

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 [207Jump To The Next Citation Point]. It was shown that the same equation could be derived, in this general case, from the influence functional of Feynman and Vernon [208Jump To The Next Citation Point]. 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 [255Jump To The Next Citation Point].

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