Let us assume a quantum state formed by an isolated system, which consists of a superposition with
equal amplitude of one configuration of mass with the center of mass at
and another
configuration of the same mass with the center of mass at
. 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
. 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
centered at
and half of the time to the field centered at
. The two predictions are clearly different; note that the fluctuation in the position of the
center of masses is on the order of
. Although this example raises the issue of 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 [392
, 393
] 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 with 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 [181] where it
was proposed that one view the backreaction problem in the framework of an open quantum system: the
quantum fields acting 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 in which
(semiclassical) Einstein–Langevin equations were derived [52
, 58
, 73
, 74
, 192
, 206
, 248
]. The key
technical factor to most of these results was the use of the influence-functional method of Feynman
and Vernon [108
], 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 [43, 99, 100, 101, 126, 136, 144
, 145, 146
, 149, 209, 210, 211, 212, 228, 229, 230, 275, 276, 277, 278, 279, 280, 299, 350],
for the existence of a semiclassical regime for the dynamics of the system, one has 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 in the system
through backreaction, thus inducing a semiclassical stochastic dynamic in the system. As shown by different
authors [46
, 127
, 131
, 221
, 352
, 389
, 390
, 391
, 392
, 393
], 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 only be made formally [256
] or
under certain assumptions, such as adopting the Born–Oppenheimer approximation in quantum
cosmology [297
, 298
].
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 [257]. It was shown that the same equation could be derived, in this general
case, from the influence functional of Feynman and Vernon [258
]. 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 stochastic-gravity theory, is
the physical observable that measures the stress-energy fluctuations, namely, the expectation
value of the symmetrized bitensor 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 [317
].
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