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 of
the order of
. 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 [149], 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 [43
, 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
].