Stochastic semiclassical fluctuations in Minkowski spacetime

Abstract

The semiclassical Einstein-Langevin equations which describe the dynamics of stochastic perturbations of the metric induced by quantum stress-energy fluctuations of matter fields in a given state are considered on the background of the ground state of semiclassical gravity, namely, Minkowski spacetime and a scalar field in its vacuum state. The relevant equations are explicitly derived for massless and massive fields arbitrarily coupled to the curvature. In doing so, some semiclassical results, such as the expectation value of the stress-energy tensor to linear order in the metric perturbations and particle creation effects, are obtained. We then solve the equations and compute the two-point correlation functions for the linearized Einstein tensor and for the metric perturbations. In the conformal field case, explicit results are obtained. These results hint that gravitational fluctuations in stochastic semiclassical gravity have a ``non-perturbative'' behavior in some characteristic correlation lengths. Comment: 28 pages, RevTeX, no figures

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arXiv:gr-qc/0001098v1 28 Jan 2000
Stochastic semiclassical fluctuations in Minkowski spacetime
Rosario Mart´ ın and Enric Verdaguer∗
Departament de F´ ısica Fonamental, Universitat de Barcelona, Av. Diagonal 647, 08028 Barcelona, Spain
(February 7, 2008)
The semiclassical Einstein-Langevin equations which describe the dynamics of stochastic pertur-
bations of the metric induced by quantum stress-energy fluctuations of matter fields in a given state
are considered on the background of the ground state of semiclassical gravity, namely, Minkowski
spacetime and a scalar field in its vacuum state. The relevant equations are explicitly derived for
massless and massive fields arbitrarily coupled to the curvature. In doing so, some semiclassical
results, such as the expectation value of the stress-energy tensor to linear order in the metric per-
turbations and particle creation effects, are obtained. We then solve the equations and compute the
two-point correlation functions for the linearized Einstein tensor and for the metric perturbations.
In the conformal field case, explicit results are obtained. These results hint that gravitational fluctu-
ations in stochastic semiclassical gravity have a “non-perturbative” behavior in some characteristic
correlation lengths.
04.62.+v, 05.40.+j
I. INTRODUCTION
It has been pointed out that the semiclassical theory of gravity [1–5] cannot provide a correct description of the
dynamics of the gravitational field in situations where the quantum stress-energy fluctuations are important [1,2,4,6–8].
In such situations, these fluctuations may have relevant back-reaction effects in the form of induced gravitational
fluctuations [6] which, in a certain regime, are expected to be described as classical stochastic fluctuations.
generalization of the semiclassical theory is thus necessary to account for these effects. In two previous papers,
Refs. [9] and [10], we have shown how a stochastic semiclassical theory of gravity can be formulated to improve the
description of the gravitational field when stress-energy fluctuations are relevant.
In Ref. [9], we adopted an axiomatic approach to construct a perturbative generalization of semiclassical gravity
which incorporates the back reaction of the lowest order stress-energyfluctuations in the form of a stochastic correction.
We started noting that, for a given solution of semiclassical gravity, the lowest order matter stress-energy fluctuations
can be associated to a classical stochastic tensor. We then sought a consistent equation in which this stochastic tensor
was the source of linear perturbations of the semiclassical metric. The equation obtained is the so-called semiclassical
Einstein-Langevin equation.
In Ref. [10], we followed the idea, first proposed by Hu [11] in the context of back reaction in semiclassical gravity, of
viewing the metric field as the “system” of interest and the matter fields (modeled in that paper by a single scalar field)
as being part of its “environment.” We then showed that the semiclassical Einstein-Langevin equation introduced in
Ref. [10] can be formally derived by a method based on the influence functional of Feynman and Vernon [12] (see also
Ref. [13]). That derivation shed light into the physical meaning of the semiclassical Langevin-type equations around
specific backgrounds previously obtained with the same functional approach [14–23], since the stochastic source term
was shown to be closely linked to the matter stress-energy fluctuations. We also developed a method to compute the
semiclassical Einstein-Langevin equation using dimensional regularization, which provides an alternative and more
direct way of computing this equation with respect to previous calculations.
This paper is intended to be a first application of the full stochastic semiclassical theory of gravity, where we
evaluate the stochastic gravitational fluctuations in a Minkowski background. In order to do so, we first use the
method developed in Ref. [10] to derive the semiclassical Einstein-Langevin equation around a class of trivial solutions
of semiclassical gravity consisting of Minkowski spacetime and a linear real scalar field in its vacuum state, which
may be considered the ground state of semiclassical gravity. Although the Minkowski vacuum is an eigenstate of the
total four-momentum operator of a field in Minkowski spacetime, it is not an eigenstate of the stress-energy operator.
Hence, even for these solutions of semiclassical gravity, for which the expectation value of the stress-energy operator
A
∗Institut de F´ ısica d’Altes Energies (IFAE)
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can always be chosen to be zero, the fluctuations of this operator are non-vanishing. This fact leads to consider the
stochastic corrections to these solutions described by the semiclassical Einstein-Langevin equation.
We then solve the Einstein-Langevin equation for the linearized Einstein tensor and compute the associated two-
point correlation functions. Even though, in this case, we expect to have negligibly small values for these correlation
functions at the domain of validity of the theory, i.e., for points separated by lengths larger than the Planck length,
there are several reasons why we think that it is worth carrying out this calculation.
On the one hand, these are, to our knowledge, the first solutions obtained to the full semiclassical Einstein-
Langevin equation. We are only aware of analogous solutions to a “reduced” version of this equation inspired in a
“mini-superspace” model [20]. There is also a previous attempt to obtain a solution to the Einstein-Langevin equation
in Ref. [17], but, there, the non-local terms in the Einstein-Langevin equation were neglected.
The Einstein-Langevin equations computed in this paper are simple enough to be explicitly solved and, at least for
the case of a conformal field, the expressions obtained for the correlation functions can be explicitly evaluated in terms
of elementary functions. Thus, our calculation can serve as a testing ground for the solutions of the Einstein-Langevin
equation in more complex situations of physical interest (for instance, for a Robertson-Walker background and a field
in a thermal state).
On the other hand, the results of this calculation, which confirm our expectations that gravitational fluctuations are
negligible at length scales larger than the Planck length, can be considered as a first check that stochastic semiclassical
gravity predicts reasonable results.
In addition, we can extract conclusions on the possible qualitative behavior of the solutions to the Einstein-Langevin
equation. Thus, it is interesting to note that the correlation functions are characterized by correlation lengths of the
order of the Planck length; furthermore, such correlation lengths enter in a non-analytic way in the correlation
functions. This kind of non-analytic behavior is actually quite common in the solutions to Langevin-type equations
with dissipative terms and hints at the possibility that correlationfunctions for other solutions to the Einstein-Langevin
equation are also non-analytic in their characteristic correlation lengths.
The plan of the paper is the following. In Sec. II, we give a brief overview of the method developed in Ref. [10]
to compute the semiclassical Einstein-Langevin equation. We then consider the background solutions of semiclassical
gravity consisting of a Minkowski spacetime and a real scalar field in the Minkowski vacuum. In Sec. III, we compute
the kernels which appear in the Einstein-Langevin equation. In Sec. IV, we derive the Einstein-Langevin equation
for metric perturbations around Minkowski spacetime. As a side result, we obtain some semiclassical results, which
include the expectation value of the stress-energy tensor of a scalar field with arbitrary mass and arbitrary coupling
parameter to linear order in the metric perturbations, and also some results concerning the production of particles
by metric perturbations: the probability of particle creation and the number and energy of created particles. In
Sec. V, we solve this equation for the components of the linearized Einstein tensor and compute the corresponding
two-point correlation functions. For the case of a conformal field and spacelike separated points, explicit calculations
show that the correlation functions are characterized by correlation lengths of the order of the Planck length. We
conclude in Sec. VI with a discussion of our results. We also include some appendices with technical details used in
the calculations.
Throughout this paper we use the (+ + +) sign conventions and the abstract index notation of Ref. [24], and we
work with units in which c = ¯ h = 1.
II. OVERVIEW
In this section, we give a very brief summary of the main results of Refs. [9] and [10] which are relevant for
the computations in the present paper. One starts with a solution of semiclassical gravity consisting of a globally
hyperbolic spacetime (M,gab), a linear real scalar field quantized on it and some physically reasonable state for this
field (we work in the Heisenberg picture). According to the stochastic semiclassical theory of gravity [9,10], quantum
fluctuations in the stress-energy tensor of matter induce stochastic linear perturbations habto the semiclassical metric
gab. The dynamics of these perturbations is described by a stochastic equation called the semiclassical Einstein-
Langevin equation.
Assuming that our semiclassical gravity solution allows the use of dimensional analytic continuation to define
regularized matrix elements of the stress-energy “operator,” we shall write the equations in dimensional regularization,
that is, assuming an arbitrary dimension n of the spacetime. Using this regularization method, we use a notation in
which a subindex n is attached to those quantities that have different physical dimensions from the corresponding
physical quantities. The n-dimensional spacetime (M,gab) has to be a solution of the semiclassical Einstein equation
in dimensional regularization
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1
8πGB
?Gab[g] + ΛBgab?−
?4
3αBDab+ 2βBBab
?
[g] = µ−(n−4)?ˆTab
n
?
[g], (2.1)
where GB, ΛB, αB and βB are bare coupling constants and Gab is the Einstein tensor. The tensors Daband Bab
are obtained by functional derivation with respect to the metric of the action terms corresponding to the Lagrangian
densities RabcdRabcd− RabRaband R2, respectively, where Rabcdis the Riemann tensor, Rabis the Ricci tensor and
R is the scalar curvature (see Ref. [10] for the explicit expressions for the tensors Daband Bab). In the last equation,
ˆTab
n
is the stress-energy “operator” in dimensional regularization and the expectation value is taken in some state
for the scalar field in the n-dimensional spacetime. Writing the bare coupling constants in Eq. (2.1) as renormalized
coupling constants plus some counterterms which absorb the ultraviolet divergencies of the right hand side, one can
take the limit n→4, which leads to the physical semiclassical Einstein equation.
Assuming that gabis a solution of Eq. (2.1), the semiclassical Einstein-Langevin equation can be similarly written
in dimensional regularization as
1
8πGB
?
Gab[g+h] + ΛB
?gab−hab??
−
?4
3αBDab+ 2βBBab
?
[g+h] = µ−(n−4)?ˆTab
n
?
[g+h]+ 2µ−(n−4)ξab
n,(2.2)
where hab is a linear stochastic perturbation to gab, and hab≡ gacgbdhcd. In this last equation, ξab
stochastic tensor characterized by the correlators
n is a Gaussian
?ξab
n(x)?c= 0,
n(y)}?[g], withˆtab
?ξab
n−?ˆTab
n(x)ξcd
n(y)?c= Nabcd
n
(x,y),(2.3)
where 8Nabcd
an anticommutator. As we pointed out in Ref. [10], the noise kernel Nabcd
the limit n→4. Therefore, in the semiclassical Einstein-Langevin equation (2.2), one can perform exactly the same
renormalization procedure as the one for the semiclassical Einstein equation (2.1), and Eq. (2.2) yields the physical
semiclassical Einstein-Langevin equation in four spacetime dimensions.
In Ref. [10], we used a method based on the CTP functional technique applied to a system-environment interaction,
more specifically, on the influence action formalism of Feynman and Vernon, to obtain an explicit expression for the
expansion of ?ˆTab
more explicit form. This expansion involves the kernel Habcd
n
n
(x,y) ≡ ?{ˆtab
n(x),ˆtcd
n≡ˆTab
n?; here, ? ?cmeans statistical average and { , } denotes
n
(x,y) is free of ultraviolet divergencies in
n?[g+h] up to first order in hcd. In this way, we can write the Einstein-Langevin equation (2.2) in a
(x,y) ≡ Habcd
(x,y) ≡1
where [ , ] means a commutator, and we use the symbol T∗to denote that we have to time order the field operators
ˆΦnfirst and then to apply the derivative operators which appear in each term of the product Tab(x)Tcd(y), where
Tabis the classical stress-energy tensor; see Ref. [10] for more details. In Eq. (2.2), all the ultraviolet divergencies in
the limit n→4, which shall be removed by renormalization of the coupling constants, are in some terms containing
?ˆΦ2
two last kernels can be related to the real and imaginary parts of?ˆtab
Nabcd
n
4Re?ˆtab
We now consider the case in which we start with a vacuum state |0? for the field quantized in spacetime (M,gab).
In this case, it was shown in Ref. [10] that all the expectation values entering the Einstein-Langevin equation (2.2)
can be written in terms of the Wightman and Feynman functions, defined as
Sn
(x,y) + Habcd
An
(x,y), with
Habcd
Sn
4Im
?
T∗?ˆTab
n(x)ˆTcd
n(y)
??
[g],Habcd
An(x,y) ≡ −i
8
??ˆTab
n(x),ˆTcd
n(y)
??
[g],(2.4)
n(x)? and in Habcd
Sn
(x,y), whereas the kernels Nabcd
n
(x,y) and Habcd
An
(x,y) are free of ultraviolet divergencies. These
n(x)ˆtcd
n(y)?by
4Im?ˆtab
(x,y) =1
n(x)ˆtcd
n(y)?,Habcd
An
(x,y) =1
n(x)ˆtcd
n(y)?.(2.5)
G+
n(x,y) ≡ ?0|ˆΦn(x)ˆΦn(y)|0?[g], iGFn(x,y) ≡ ?0|T
?ˆΦn(x)ˆΦn(y)
?
|0?[g].(2.6)
For instance, we can write ?ˆΦ2
our calculations, can be found in Appendix A.
n(x)? = iGFn(x,x) = G+
n(x,x). The expressions for the kernels, which shall be used in
A. Perturbations around Minkowski spacetime
An interesting case to be analyzed in the framework of the semiclassical stochastic theory of gravity is that of a
Minkowski spacetime solution of semiclassical gravity. The flat metric ηabin a manifold M≡IR4(topologically) and
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the usual Minkowski vacuum, denoted as |0?, give the class of simplest solutions to the semiclassical Einstein equation
[note that each possible value of the parameters (m2,ξ) leads to a different solution], the so called trivial solutions
of semiclassical gravity [25]. In fact, we can always choose a renormalization scheme in which the renormalized
expectation value ?0|ˆTab
to the semiclassical Einstein equation with renormalized cosmological constant Λ = 0. The fact that the vacuum
expectation value of the renormalized stress-energy operator in Minkowski spacetime should vanish was originally
proposed by Wald [2] and it may be understood as a renormalization convention [3,5]. There are other possible
renormalization prescriptions (see, for instance, Ref. [26]) in which such vacuum expectation value is proportional to
ηab, and this would determine the value of the cosmological constant Λ in the semiclassical equation. Of course, all
these renormalization schemes give physically equivalent results: the total effective cosmological constant, i.e., the
constant of proportionality in the sum of all the terms proportional to the metric in the semiclassical Einstein and
Einstein-Langevin equations, has to be zero.
Although the vacuum |0? is an eigenstate of the total four-momentum operator in Minkowski spacetime, this state
is not an eigenstate ofˆTR
ab[η]. Hence, even in these trivial solutions of semiclassical gravity, there are quantum
fluctuations in the stress-energy tensor of matter and, as a result, the noise kernel does not vanish. This fact leads
to consider the stochastic corrections to this class of trivial solutions of semiclassical gravity. Since, in this case,
the Wightman and Feynman functions (2.6), their values in the two-point coincidence limit, and the products of
derivatives of two of such functions appearing in expressions (A1) and (A3) (Appendix A) are known in dimensional
regularization, we can compute the semiclassical Einstein-Langevin equation using the method outlined above.
In order to perform the calculations, it is convenient to work in a global inertial coordinate system {xµ} and in
the associated basis, in which the components of the flat metric are simply ηµν = diag(−1,1,...,1). In Minkowski
spacetime, the components of the classical stress-energy tensor functional reduce to
R|0?[η] = 0. Thus, Minkowski spacetime (IR4,ηab) and the vacuum state |0? are a solution
Tµν[η,Φ] = ∂µΦ∂νΦ −1
2ηµν∂ρΦ∂ρΦ −1
2ηµνm2Φ2+ ξ (ηµν2 − ∂µ∂ν)Φ2,(2.7)
where 2 ≡ ∂µ∂µ, and the formal expression for the components of the corresponding “operator” in dimensional
regularization is
ˆTµν
n[η] =1
2
?
∂µˆΦn,∂νˆΦn
?
+ DµνˆΦ2
n,(2.8)
where Dµνare the differential operators Dµν
Heisenberg picture in an n-dimensional Minkowski spacetime, which satisfies the Klein-Gordon equation (2−m2)ˆΦn=
0.
Notice, from (2.8), that the stress-energy tensor depends on the coupling parameter ξ of the scalar field to the
scalar curvature even in the limit of a flat spacetime. Therefore, that tensor differs in general from the canonical
stress-energy tensor in flat spacetime, which corresponds to the value ξ =0. Nevertheless, it is easy to see [10] that
the n-momentum density componentsˆT0µ
n(ξ)[η] (we temporary use this notation to indicate the dependence on the
parameter ξ) andˆT0µ
n(ξ=0)[η] differ in a space divergence and, hence, dropping surface terms, they both yield the same
n-momentum operator:
x
≡ (ξ − 1/4)ηµν2x− ξ ∂µ
x∂ν
xandˆΦn(x) is the field operator in the
ˆPµ≡
?
dn−1x :ˆT0µ
n(ξ)[η]:=
?
dn−1x :ˆT0µ
n(ξ=0)[η]:,(2.9)
where the integration is on a hypersurface x0= constant (ˆPµis actually independent of the value of x0) and we use the
notation for coordinates xµ≡ (x0,x), i.e., x are space coordinates on each of the hypersurfaces x0= constant. The
symbol : : in Eq. (2.9) means normal ordering of the creation and annihilation operators on the Fock space built on
the Minkowski vacuum |0? (in n spacetime dimensions), which is an eigenstate with zero eigenvalue of the operators
(2.9).
The Wightman and Feynman functions (2.6) in Minkowski spacetime are well known:
G+
n(x,y) ≡ ?0|ˆΦn(x)ˆΦn(y)|0?[η] = i∆+
GFn(x,y) ≡ −i?0|T
n(x − y),
|0?[η] = ∆Fn(x − y),
?ˆΦn(x)ˆΦn(y)
?
(2.10)
with
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∆+
n(x) = −2πi
?
dnk
(2π)neikxδ(k2+ m2)θ(k0),
dnk
(2π)n
∆Fn(x) = −
?
eikx
k2+ m2− iǫ,
ǫ→0+, (2.11)
where k2≡ ηµνkµkνand kx ≡ ηµνkµxν. Note that the derivatives of these functions satisfy ∂x
and ∂y
n(x − y), and similarly for the Feynman propagator ∆Fn(x − y).
To write down the semiclassical Einstein equation (2.1) for this case, we need to compute the vacuum expectation
value of the stress-energy operator components (2.8). Since, from (2.10), we have that ?0|ˆΦ2
i∆+
n(0), which is a constant (independent of x), we have simply
µ∆+
n(x−y) = ∂µ∆+
n(x−y)
µ∆+
n(x − y) = −∂µ∆+
n(x)|0? = i∆Fn(0) =
?0|ˆTµν
n |0?[η] =1
2?0|
?
∂µˆΦn,∂νˆΦn
?
|0?[η] = −i(∂µ∂ν∆Fn)(0) = −i
?
dnk
(2π)n
kµkν
k2+ m2− iǫ=ηµν
2
?m2
4π
?n/2
Γ
?
−n
2
?
,
(2.12)
where the integrals in dimensional regularization have been computed in the standard way (see Appendix B) and
where Γ(z) is the Euler’s gamma function. The semiclassical Einstein equation (2.1), which now reduces to
ΛB
8πGB
ηµν= µ−(n−4)?0|ˆTµν
n|0?[η], (2.13)
simply sets the value of the bare coupling constant ΛB/GB. Note, from (2.12), that in order to have ?0|ˆTab
the renormalized (and regularized) stress-energy tensor “operator” for a scalar field in Minkowski spacetime has to
be defined as
m4
(4π)2
R|0?[η]=0,
ˆTab
R[η] = µ−(n−4)ˆTab
n[η] −ηab
2
?
m2
4πµ2
?n−4
2
Γ
?
−n
2
?
, (2.14)
which corresponds to a renormalization of the cosmological constant
ΛB
GB
=Λ
G−2
π
m4
n(n−2)κn+ O(n − 4),(2.15)
where
κn≡
1
(n−4)
?eγm2
4πµ2
?n−4
2
=
1
n−4+12ln
?eγm2
4πµ2
?
+ O(n − 4),(2.16)
being γ the Euler’s constant. In the case of a massless scalar field, m2=0, one simply has ΛB/GB= Λ/G. Introducing
this renormalized coupling constant into Eq. (2.13), we can take the limit n→4. We find again that, for (IR4,ηab,|0?)
to satisfy the semiclassical Einstein equation, we must take Λ=0.
We are now in the position to write down the Einstein-Langevin equations for the components hµνof the stochastic
metric perturbation in dimensional regularization. In our case, using ?0|ˆΦ2
for Eq. (2.2) found in Ref. [10], we obtain that this equation reduces to
n(x)|0? = i∆Fn(0) and the explicit expression
1
8πGB
?
G(1)µν+ ΛB
?
hµν−1
2ηµνh
??
(x) −4
3αBD(1)µν(x) − 2βBB(1)µν(x)
−ξ G(1)µν(x)µ−(n−4)i∆Fn(0) + 2
where ξµνare the components of a Gaussian stochastic tensor of zero average and
?
dny µ−(n−4)Hµναβ
n
(x,y)hαβ(y) = 2ξµν(x),(2.17)
?ξµν(x)ξαβ(y)?c= µ−2(n−4)Nµναβ
n
(x,y), (2.18)
and where indices are raised in hµνwith the flat metric and h ≡ hρ
of a tensor linearized around the flat metric. In the last expressions, Nµναβ
of the kernels defined above. In Eq. (2.17), we have made use of the explicit expression for G(1)µν. This expression
and those for D(1)µνand B(1)µνare given in Appendix E; the last two can also be written as
ρ. We use a superindex (1) to denote the components
(x,y) and Hµναβ
nn
(x,y) are the components
D(1)µν(x) =1
2(3Fµα
xFνβ
x − Fµν
xFαβ
≡ ηµν2x− ∂µ
x)hαβ(x),B(1)µν(x) = 2Fµν
xFαβ
xhαβ(x), (2.19)
where Fµν
x
is the differential operator Fµν
xx∂ν
x.
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