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 those solutions of semiclassical gravity, such as the Minkowski metric, for which the expectation value of the stress-energy operator can always be chosen to be zero, the fluctuations of this operator are non-vanishing. This fact leads us to consider the stochastic metric perturbations induced by these fluctuations.
Here we derive the Einstein-Langevin equation for the metric perturbations in a Minkowski background. We solve this equation for the linearized Einstein tensor and compute the associated two-point correlation functions, as well as the two-point correlation functions for the metric perturbations. Even though in this case we expect to have negligibly small values for these correlation functions for points separated by lengths larger than the Planck length, there are several reasons why it is worth carrying out this calculation.
On the one hand, these are the first backreaction solutions of the full Einstein-Langevin equation. There are analogous solutions to a “reduced” version of this equation inspired in a “minisuperspace” model [52, 74], and there is also a previous attempt to obtain a solution to the Einstein-Langevin equation in [73], but there the non-local terms in the Einstein-Langevin equation are neglected.
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, but also predict that the fluctuations are strongly suppressed on small scales, can be considered a first test of stochastic semiclassical gravity. These results also reveal an important connection between stochastic gravity and the large N expansion of quantum gravity. In addition, they are used in Section 6.5 to study the stability of the Minkowski metric as a solution of semiclassical gravity, which constitutes an application of the validity criterion introduced in Section 3.3. This calculation also requires a discussion of the problems posed by the so-called runaway solutions, which arise in the backreaction equations of semiclassical and stochastic gravity, and some of the methods to deal with them. As a result we conclude that Minkowski spacetime is a stable and valid solution of semiclassical gravity.
We advise the reader that Section 6 is rather technical since it deals with an explicit non trivial backreaction computation in stochastic gravity. We tried to make it reasonable self-contained and detailed, however a more detailed exposition can be found in [259].
Perturbations around Minkowski spacetime
The Minkowski metric ηab in a manifold \({\mathcal M}\), which is topologically ℝ4, together with the usual Minkowski vacuum, denoted as |0⟩, is the simplest solution to the semiclassical Einstein equation (8), the so-called trivial solution of semiclassical gravity [110]. It constitutes the ground state of semiclassical gravity. In fact, we can always choose a renormalization scheme in which the renormalized expectation value \(\langle 0\vert \hat T_{\rm{R}}^{ab}[\eta ]\vert 0\rangle = 0\). Thus, the Minkowski spacetime (ℝ4,ηab) plus the vacuum state |0⟩ is a solution 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 [359], and it may be understood as a renormalization convention [121, 135]. Note that other possible solutions of semiclassical gravity with zero vacuum expectation value of the stress-energy tensor are the exact gravitational plane waves, since they are known to be vacuum solutions of Einstein equations, which induce neither particle creation nor vacuum polarization [91, 125, 128].
As we have already mentioned, the vacuum |0⟩ is an eigenstate of the total four-momentum operator in Minkowski spacetime, but not an eigenstate of \(\hat T_{ab}^R[\eta ]\). Hence, even in the Minkowski background there are quantum fluctuations in the stress-energy tensor and, as a result, the noise kernel does not vanish. This fact leads us to consider stochastic corrections to this class of trivial solutions of semiclassical gravity. Since in this case the Wightman and Feynman functions (44), their values in the two-point coincidence limit and the products of derivatives of two of such functions appearing in expressions (45) and (46) are known in dimensional regularization, we can compute the Einstein-Langevin equation using the methods outlined in Sections 3 and 4.
To perform explicit 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 simplyημν = diag(−1, 1,…, 1). In Minkowski spacetime, the components of the classical stress-energy tensor (4) reduce to
$${T^{\mu \nu}}[\eta, \phi ] = {\partial ^\mu}\phi {\partial ^\nu}\phi - {1 \over 2}{\eta ^{\mu \nu}}{\partial ^\rho}\phi {\partial _\rho}\phi - {1 \over 2}{\eta ^{\mu \nu}}{m^2}{\phi ^2} + \xi ({\eta ^{\mu \nu}}\square - {\partial ^\mu}{\partial ^\nu}){\phi ^2},$$
(85)
where □ = ∂μ∂μ and the formal expression for the components of the corresponding “operator” in dimensional regularization, see Equation (5), is
$$\hat T_n^{\mu \nu}[\eta ] = {1 \over 2}\left\{{{\partial ^\mu}{{\hat \phi}_n},{\partial ^\nu}{{\hat \phi}_n}} \right\} + {\mathcal{D}^{\mu \nu}}\hat \phi _n^2,$$
(86)
where \({{\mathcal D}^{\mu \nu}}\) is the differential operator (6), with gμν = ημν, Rμν = 0, and ∇μ = ∂μ. The field \({{\hat \phi}_n}(x)\) is the field operator in the Heisenberg representation in an n-dimensional Minkowski spacetime, which satisfies the Klein-Gordon equation (2). We use here a stress-energy tensor, which differs from the canonical one that corresponds to ξ = 0; both tensors, however, define the same total momentum.
The Wightman and Feynman functions (44) for gμν = ημν are well known:
$$G_n^ + (x,y) = i\Delta _n^ + (x - y),\quad {G_{{F_n}}}(x,y) = {\Delta _{{F_n}}}(x - y),$$
(87)
with
$$\begin{array}{*{20}c}{\Delta _n^ + (x) = - 2\pi i\int {{{{d^n}k} \over {{{(2\pi)}^n}}}{e^{ikx}}\delta ({k^2} + {m^2})\theta ({k^0}),} \quad \;\;}\\{{\Delta _{{F_n}}}(x) = - \int {{{{d^n}k} \over {{{(2\pi)}^n}}}} {{{e^{ikx}}} \over {{k^2} + {m^2} - i\epsilon}}\quad \quad {\rm{for}}\;\epsilon \rightarrow {0^ +},}\end{array}$$
(88)
where k2 ≡ ημνkμkν and kx ≡ ημνkμkν. Note that the derivatives of these functions satisfy \(\partial _\mu ^x\Delta _n^ + (x - y) = {\partial _\mu}\Delta _n^ + (x - y)\) and \(\partial _\mu ^y\Delta _n^ + (x - y) = {\partial _\mu}\Delta _n^ + (x - y)\), and similarly for the Feynman propagator \({\Delta _{{F_n}}}(x - y)\).
To write down the semiclassical Einstein equation (8) in n dimensions for this case, we need to compute the vacuum expectation value of the stress-energy operator components (86). Since, from (87), we have that \(\langle 0\vert \hat \phi _n^2(x)\vert 0\rangle = i{\Delta _{{F_n}}}(0) = i\Delta _n^ + (0)\), which is a constant (independent of x), we have simply
$$\left\langle {0\left\vert {\hat T_n^{\mu \nu}[\eta ]} \right\vert 0} \right\rangle = - i\;\int {{{{d^n}k} \over {{{(2\pi)}^n}}}{{{k^\mu}{k^\nu}} \over {{k^2} + {m^2} - i\epsilon}}} = {{{\eta ^{\mu \nu}}} \over 2}{\left({{{{m^2}} \over {4\pi}}} \right)^{n/2}}\Gamma \left({- {n \over 2}} \right),$$
(89)
where the integrals in dimensional regularization have been computed in the standard way [259], and where Γ(z) is Euler’s gamma function. The semiclassical Einstein equation (8) in n dimensions before renormalization reduces now to
$${{{\Lambda _{\rm{B}}}} \over {8\pi {G_{\rm{B}}}}}{\eta ^{\mu \nu}} = {\mu ^{- (n - 4)}}\left\langle {0\left\vert {\hat T_n^{\mu \nu}[\eta ]} \right\vert 0} \right\rangle {.}$$
(90)
Thus, this equation simply sets the value of the bare coupling constant ΛB/GB. Note from Equation (89) that in order to have \(\langle 0\vert \hat T_{\rm{R}}^{\mu \nu}\vert 0\rangle [\eta ] = 0\), the renormalized and regularized stress-energy tensor “operator” for a scalar field in Minkowski spacetime, see Equation (7), has to be defined as
$$\hat T_{\rm{R}}^{\mu \nu}[\eta ] = {\mu ^{- (n - 4)}}\hat T_n^{\mu \nu}[\eta ] - {{{\eta ^{\mu \nu}}} \over 2}{{{m^4}} \over {{{(4\pi)}^2}}}{\left({{{{m^2}} \over {4\pi {\mu ^2}}}} \right)^{{{n - 4} \over 2}}}\Gamma \left({- {n \over 2}} \right),$$
(91)
which corresponds to a renormalization of the cosmological constant
$${{{\Lambda _{\rm{B}}}} \over {{G_{\rm{B}}}}} = {\Lambda \over G} - {2 \over \pi}{{{m^4}} \over {n(n - 2)}}{\kappa _n} + \mathcal{O}(n - 4),$$
(92)
where
$${\kappa _n} \equiv {1 \over {n - 4}}{\left({{{{e^\gamma}{m^2}} \over {4\pi {\mu ^2}}}} \right)^{{{n - 4} \over 2}}} = {1 \over {n - 4}} + {1 \over 2}\ln \left({{{{e^\gamma}{m^2}} \over {4\pi {\mu ^2}}}} \right) + \mathcal{O}(n - 4),$$
(93)
with γ being 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 Equation (90), we can take the limit n → 4. We find that, for (ℝ4, ηab, |0⟩) to satisfy the semiclassical Einstein equation, we must take Λ = 0.
We can now write down the Einstein-Langevin equations for the components hμν of the stochastic metric perturbation in dimensional regularization. In our case, using \(\langle 0\vert \hat \phi _n^2(x)\vert 0\rangle = i{\Delta _{{F_n}}}(0)\) and the explicit expression of Equation (41), we obtain
$$\begin{array}{*{20}c}{{1 \over {8\pi {G_{\rm{B}}}}}\left[ {{G^{(1)\mu \nu}} + {\Lambda _{\rm{B}}}\left({{h^{\mu \nu}} - {1 \over 2}{\eta ^{\mu \nu}}h} \right)} \right](x) - {4 \over 3}{\alpha _{\rm{B}}}{D^{(1)\mu \nu}}(x) - 2{\beta _{\rm{B}}}{B^{(1)\mu \nu}}(x)\quad \quad \quad \quad \quad \quad}\\{- \xi {G^{(1)\mu \nu}}(x){\mu ^{- (n - 4)}}i{\Delta _{{F_n}}}(0) + {1 \over 2}\int {{d^n}} y{\mu ^{- (n - 4)}}H_n^{\mu \nu \alpha \beta}(x,y){h_{\alpha \beta}}(y) = {\xi ^{\mu \nu}}(x){.}}\end{array}$$
(94)
The indices in hμν are raised with the Minkowski metric and \(h \equiv h_\rho ^\rho\); here a superindex (1) denotes the components of a tensor linearized around the flat metric. Note that in n dimensions the two-point correlation functions for the field ξμν is written as
$${\langle {\xi ^{\mu \nu}}(x){\xi ^{\alpha \beta}}(y)\rangle _{\rm{s}}} = {\mu ^{- 2(n - 4)}}N_n^{\mu \nu \alpha \beta}(x,y){.}$$
(95)
Explicit expressions for D(1)μν and B(1)μν are given by
$${D^{(1)\mu \nu}}(x) = {1 \over 2}\mathcal{F}_x^{\mu \nu \alpha \beta}{h_{\alpha \beta}}(x),\quad \quad {B^{(1)\mu \nu}}(x) = 2\mathcal{F}_x^{\mu \nu}\mathcal{F}_x^{\alpha \beta}{h_{\alpha \beta}}(x),$$
(96)
with the differential operators \({\mathcal F}_x^{\mu \nu} \equiv {\eta ^{\mu \nu}}\Box_x - \partial _x^\mu \partial _x^\nu\) and \({\mathcal F}_x^{\mu \nu \alpha \beta} \equiv 3{\mathcal F}_x^{\mu (\alpha}{\mathcal F}_x^{\beta)\nu} - {\mathcal F}_x^{\mu \nu}{\mathcal F}_x^{\alpha \beta}\).
The kernels in the Minkowski background
Since the two kernels (43) are free of ultraviolet divergences in the limit n → 4, we can deal directly with the \({F^{\mu \nu \alpha \beta}}(x - y) \equiv {\lim\nolimits _{n \rightarrow 4}}{\mu ^{- 2(n - 4)}}F_n^{\mu \nu \alpha \beta}\) in Equation (42). The kernels Nμναβ(x, y) = Re Fμναβ(x − y) and \(H_{\rm{A}}^{\mu \nu \alpha \beta}(x,y) = {\rm{Im}}\,{F^{\mu \nu \alpha \beta}}(x - y)\) are actually the components of the “physical” noise and dissipation kernels that will appear in the Einstein-Langevin equations once the renormalization procedure has been carried out. The bitensor Fμναβ be expressed in terms of the Wightman function in four spacetime dimensions, according to Equation (45). The different terms in this kernel can be easily computed using the integrals
$$I(p) \equiv \int {{{{d^4}k} \over {{{(2\pi)}^4}}}\delta ({k^2} + {m^2})\theta (- {k^0})\delta [{{(k - p)}^2} + {m^2}]\theta ({k^0} - {p^0})}$$
(97)
and \({I^{{\mu _1} \ldots {\mu _r}}}(p)\), which are defined as in Equation (97) by inserting the momenta \({k^{{\mu _1}}} \ldots {k^{{\mu _r}}}\) with r = 1,…, 4 into the integrand. All these integrals can be expressed in terms of I(p); see [259] for the explicit expressions. It is convenient to separate I(p) into its even and odd parts with respect to the variables pμ as
$$I(p) = {I_{\rm{S}}}(p) + {I_{\rm{A}}}(p),$$
(98)
where IS(−p) = IS(p) and IA(−p) = −IA(p). These two functions are explicitly given by
$$\begin{array}{*{20}c}{{I_{\rm{S}}}(p) = {1 \over {8{{(2\pi)}^3}}}\theta (- {p^2} - 4{m^2})\sqrt {1 + 4{{{m^2}} \over {{p^2}}}}, \quad \quad \quad \;}\\{{I_{\rm{A}}}(p) = {{- 1} \over {8{{(2\pi)}^3}}}{\rm{sign}}\;({p^0})\;\theta (- {p^2} - 4{m^2})\sqrt {1 + 4{{{m^2}} \over {{p^2}}}} {.}}\end{array}$$
(99)
After some manipulations, we find
$$\begin{array}{*{20}c}{{F^{\mu \nu \alpha \beta}}(x) = {{{\pi ^2}} \over {45}}\mathcal{F}_x^{\mu \nu \alpha \beta}{{\int {{{{d^4}p} \over {{{(2\pi)}^4}}}{e^{- ipx}}\left({1 + 4{{{m^2}} \over {{p^2}}}} \right)}}^2}I(p)\quad \quad \quad \quad \quad \quad \quad}\\{+ {{8{\pi ^2}} \over 9}\mathcal{F}_x^{\mu \nu}\mathcal{F}_x^{\alpha \beta}\int {{{{d^4}p} \over {{{(2\pi)}^4}}}{e^{- ipx}}{{\left({3\Delta \xi + {{{m^2}} \over {{p^2}}}} \right)}^2}I(p),}}\end{array}$$
(100)
where \(\Delta \xi \equiv \xi - {1 \over 6}\). The real and imaginary parts of the last expression, which yield the noise and dissipation kernels, are easily recognized as the terms containing IS(p) and IA(p), respectively. To write them explicitly, it is useful to introduce the new kernels
$$\begin{array}{*{20}c}{{N_{\rm{A}}}(x;{m^2}) \equiv {1 \over {480\pi}}\int {{{{d^4}p} \over {{{(2\pi)}^4}}}{e^{ipx}}\theta (- {p^2} - 4{m^2})\sqrt {1 + 4{{{m^2}} \over {{p^2}}}} \left({1 + 4{{{m^2}} \over {{p^2}}}} \right),\quad \quad} \quad \quad}\\{{N_{\rm{B}}}(x;{m^2},\Delta \xi) \equiv {1 \over {72\pi}}\int {{{{d^4}p} \over {{{(2\pi)}^4}}}{e^{ipx}}\theta (- {p^2} - 4{m^2})\sqrt {1 + 4{{{m^2}} \over {{p^2}}}} {{\left({3\Delta \xi + {{{m^2}} \over {{p^2}}}} \right)}^2},\quad \quad \quad \quad \quad}}\\{{D_{\rm{A}}}(x;{m^2}) \equiv {{- i} \over {480\pi}}\int {{{{d^4}p} \over {{{(2\pi)}^4}}}{e^{ipx}}{\rm{sign}}} \;({p^0})\theta (- {p^2} - 4{m^2})\sqrt {1 + 4{{{m^2}} \over {{p^2}}}} {{\left({1 + 4{{{m^2}} \over {{p^2}}}} \right)}^2},}\\{{D_{\rm{B}}}(x;{m^2},\Delta \xi) \equiv {{- i} \over {72\pi}}\int {{{{d^4}p} \over {{{(2\pi)}^4}}}{e^{ipx}}{\rm{sign}}} ({p^0})\theta (- {p^2} - 4{m^2})\sqrt {1 + 4{{{m^2}} \over {{p^2}}}} {{\left({3\Delta \xi + {{{m^2}} \over {{p^2}}}} \right)}^2}{.}\quad \;\;}\end{array}$$
(101)
Finally, we get
$$\begin{array}{*{20}c}{{N^{\mu \nu \alpha \beta}}(x,y) = {1 \over 6}\mathcal{F}_x^{\mu \nu \alpha \beta}{N_{\rm{A}}}(x - y;{m^2}) + \mathcal{F}_x^{\mu \nu}\mathcal{F}_x^{\alpha \beta}{N_{\rm{B}}}(x - y;{m^2},\Delta \xi),}\\{H_{\rm{A}}^{\mu \nu \alpha \beta}(x,y) = {1 \over 6}\mathcal{F}_x^{\mu \nu \alpha \beta}{D_{\rm{A}}}(x - y;{m^2}) + \mathcal{F}_x^{\mu \nu}\mathcal{F}_x^{\alpha \beta}{D_{\rm{B}}}(x - y;{m^2},\Delta \xi){.}\;}\end{array}$$
(102)
Notice that the noise and dissipation kernels defined in Equation (101) are actually real because, for the noise kernels, only the cospx terms of the exponentials eipx contribute to the integrals, and, for the dissipation kernels, the only contribution of such exponentials comes from the isinpx terms.
The evaluation of the kernel \(H_{{{\rm{S}}_n}}^{\mu \nu \alpha \beta}(x,y)\) is a more involved task. Since this kernel contains divergences in the limit n → 4, we use dimensional regularization. Using Equation (46), this kernel can be written in terms of the Feynman propagator (88) as
$${\mu ^{- (n - 4)}}H_{{{\rm{S}}_n}}^{\mu \nu \alpha \beta}(x,y) = {{\rm Im}}\; {K^{\mu \nu \alpha \beta}}(x - y),$$
(103)
where
$$\begin{array}{*{20}c}{{K^{\mu \nu \alpha \beta}}(x) \equiv - {\mu ^{- (n - 4)}}\left\{{2{\partial ^\mu}{\partial ^{(\alpha}}{\Delta _{{F_n}}}(x){\partial ^{\beta)}}{\partial ^\nu}{\Delta _{{F_n}}}(x) + 2{\mathcal{D}^{\mu \nu}}({\partial ^\alpha}{\Delta _{{F_n}}}(x){\partial ^\beta}{\Delta _{{F_n}}}(x))\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \right.}\\{+ 2{\mathcal{D}^{\alpha \beta}}({\partial ^\mu}{\Delta _{{F_n}}}(x){\partial ^\nu}{\Delta _{{F_n}}}(x)) + 2{\mathcal{D}^{\mu \nu}}{\mathcal{D}^{\alpha \beta}}(\Delta _{{F_n}}^2(x))\quad \quad \quad \quad \quad \quad \quad}\\{+ \left[ {{\eta ^{\mu \nu}}{\partial ^{(\alpha}}{\Delta _{{F_n}}}(x){\partial ^{\beta)}} + {\eta ^{\alpha \beta}}{\partial ^{(\mu}}{\Delta _{{F_n}}}(x){\partial ^{\nu)}} + {\Delta _{{F_n}}}(0) ({\eta ^{\mu \nu}}{\mathcal{D}^{\alpha \beta}} + {\eta ^{\alpha \beta}}{\mathcal{D}^{\mu \nu}})} \right.}\\{\left. {\left. {+ {1 \over 4}{\eta ^{\mu \nu}}{\eta ^{\alpha \beta}}({\Delta _{{F_n}}}(x)\square - {m^2}{\Delta _{{F_n}}}(0))} \right]{\delta ^n}(x)} \right\}{.}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad}\end{array}$$
(104)
Let us define the integrals
$${J_n}(p) \equiv {\mu ^{- (n - 4)}}\int {{{{d^n}k} \over {{{(2\pi)}^n}}}} {1 \over {({k^2} + {m^2} - i\epsilon)\;[{{(k - p)}^2} + {m^2} - i\epsilon ]}},$$
(105)
and \(J_n^{{\mu _1} \ldots {\mu _r}}(p)\) obtained by inserting the momenta \({k^{{\mu _1}}} \ldots {k^{{\mu _r}}}\) into Equation (105), together with
$${I_{{0_n}}} \equiv {\mu ^{- (n - 4)}}\int {{{{d^n}k} \over {{{(2\pi)}^n}}}{1 \over {({k^2} + {m^2} - i\epsilon)}}},$$
(106)
and \(I_{{0_n}}^{{\mu _1} \ldots {\mu _r}}\), which are also obtained by inserting momenta into the integrand. Then the different terms in Equation (104) can be computed; these integrals are explicitly given in [259]. It is found that \(I_{{0_n}}^\mu = 0\), and the remaining integrals can be written in terms of \({I_{{0_n}}}\) and Jn(p). It is useful to introduce the projector Pμν orthogonal to pμ and the tensor Pμναβ as
$${p^2}{P^{\mu \nu}} \equiv {\eta ^{\mu \nu}}{p^2} - {p^\mu}{p^\nu},\quad \quad {P^{\mu \nu \alpha \beta}} \equiv 3{P^{\mu (\alpha}}{P^{\beta)\nu}} - {P^{\mu \nu}}{P^{\alpha \beta}}{.}$$
(107)
then the action of the operator \({\mathcal F}_x^{\mu \nu}\) is simply written as \({\mathcal F}_x^{\mu \nu}\int {{d^n}p\,{e^{ipx}}f(p) = - \int {{d^n}p\,{e^{ipx}}} f(p){p^2}{P^{\mu \nu}}}\), where f(p) is an arbitrary function of pμ.
Finally, after a rather long but straightforward calculation, and after expanding around n = 4, we get
$$\begin{array}{*{20}c}{{K^{\mu \nu \alpha \beta}}(x) = {i \over {{{(4\pi)}^2}}}\left\{{{\kappa _n}\left[ {{1 \over {90}}\mathcal{F}_x^{\mu \nu \alpha \beta}{\delta ^n}(x) + 4\Delta {\xi ^2}\mathcal{F}_x^{\mu \nu}\mathcal{F}_x^{\alpha \beta}{\delta ^n}(x)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \right.} \right.\quad \quad \quad \quad}\\{+ {2 \over 3}{{{m^2}} \over {(n - 2)}}\left({{\eta ^{\mu \nu}}{\eta ^{\alpha \beta}}\square _\alpha - {\eta ^{\mu (\alpha}}{\eta ^{\beta)\nu}}\square _x + {\eta ^{\mu (\alpha}}\partial _x^{\beta)}\partial _x^\nu}\right. \quad \quad \quad \quad\quad}\\\left. {+ {\eta ^{\nu (\alpha}}\partial _x^{\beta)}\partial _x^\mu - {\eta ^{\mu \nu}}\partial _x^\alpha \partial _x^\beta - {\eta ^{\alpha \beta}}\partial _x^\mu \partial _x^\nu} \right){\delta ^n}(x)\\{\left. {+ {{4{m^4}} \over {n(n - 2)}}(2{\eta ^{\mu (\alpha}}{\eta ^{\beta)\nu}} - {\eta ^{\mu \nu}}{\eta ^{\alpha \beta}}){\delta ^n}(x)} \right]\quad \quad \quad \quad \quad \quad \quad \quad \quad \;\;}\\{+ {1 \over {180}}\mathcal{F}_x^{\mu \nu \alpha \beta}\int {{{{d^n}p} \over {{{(2\pi)}^n}}}} {e^{ipx}}{{\left({1 + 4{{{m^2}} \over {{p^2}}}} \right)}^2}\bar \phi ({p^2})\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad}\\{+ {2 \over 9}\mathcal{F}_x^{\mu \nu}\mathcal{F}_x^{\alpha \beta}\int {{{{d^n}p} \over {{{(2\pi)}^n}}}{e^{ipx}}{{\left({3\Delta \xi + {{{m^2}} \over {{p^2}}}} \right)}^2}\bar \phi ({p^2})} \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad}\\{- \left[ {{4 \over {675}}\mathcal{F}_x^{\mu \nu \alpha \beta} + {1 \over {270}}(60\xi - 11)F_x^{\mu \nu}F_x^{\alpha \beta}} \right]{\delta ^n}(x)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad}\\{\left. {- {m^2}\left[ {{2 \over {135}}\mathcal{F}_x^{\mu \nu \alpha \beta} + {1 \over {27}}\mathcal{F}_x^{\mu \nu}\mathcal{F}_x^{\alpha \beta}} \right]{\Delta _n}(x)} \right\} + \mathcal{O}(n - 4),\quad \quad \quad \quad \quad \quad \quad \quad}\end{array}$$
(108)
where κn has been defined in Equation (93), and \(\bar \phi ({p^2})\) and Δn(x) are given by
$$\bar \phi ({p^2}) \equiv \int\nolimits_0^1 {d\alpha} \ln \left({1 + {{{p^2}} \over {{m^2}}}\alpha (1 - \alpha) - i\epsilon} \right) = - i\pi \theta (- {p^2} - 4{m^2})\sqrt {1 + 4{{{m^2}} \over {{p^2}}}} + \varphi ({p^2}),$$
(109)
$${\Delta _n}(x) \equiv \int {{{{d^n}p} \over {{{(2\pi)}^n}}}\;} {e^{ipx}}{1 \over {{p^2}}},$$
(110)
where
$$\varphi ({p^2}) \equiv \int\nolimits_0^1 {d\alpha} \ln \left\vert {1 + {{{p^2}} \over {{m^2}}}\alpha (1 - \alpha)} \right\vert {.}$$
The imaginary part of Equation (108) gives the kernel components \({\mu ^{- (n - 4)}}H_{{{\rm{S}}_n}}^{\mu \nu \alpha \beta}(x,y)\), according to Equation (103). It can be easily obtained by multiplying this expression by −i and retaining only the real part φ(p2) of the function \(\bar \phi ({p^2})\).
The Einstein-Langevin equation
With the previous results for the kernels we can now write the n-dimensional Einstein-Langevin equation (94), previous to the renormalization. Also taking into account Equations (89) and (90), we may finally write:
$$\begin{array}{*{20}c}{{1 \over {8\pi {G_{\rm{B}}}}}{G^{(1)\mu \nu}}(x) - {4 \over 3}{\alpha _{\rm{B}}}{D^{(1)\mu \nu}}(x) - 2{\beta _{\rm{B}}}{B^{(1)\mu \nu}}(x)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad}\\{+ {{{\kappa _n}} \over {{{(4\pi)}^2}}}\left[ {- 4\Delta \xi {{{m^2}} \over {(n - 2)}}{G^{(1)\mu \nu}} + {1 \over {90}}{D^{(1)\mu \nu}}\Delta {\xi ^2}{B^{(1)\mu \nu}}} \right]\;(x)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad}\\{+ {1 \over {2880{\pi ^2}}}\left\{{- {{16} \over {15}}{D^{(1)\mu \nu}}(x) + \left({{1 \over 6} - 10\Delta \xi} \right)\;{B^{(1)\mu \nu}}(x)} \right.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad}\\{+ \int {{d^n}y} \int {{{{d^n}p} \over {{{(2\pi)}^n}}}{e^{ip(x - y)}}\varphi ({p^2})\left[ {{{\left({1 + 4{{{m^2}} \over {{p^2}}}} \right)}^2}{D^{(1)\mu \nu}}(y) + 10{{\left({3\Delta \xi + {{{m^2}} \over {{p^2}}}} \right)}^2}{B^{(1)\mu \nu}}(y)} \right]}}\\{\left. {- {{{m^2}} \over 3}\int {{d^n}y{\Delta _n}(x - y)\;(8{D^{(1)\mu \nu}} + 5{B^{(1)\mu \nu}})\;(y)}} \right\}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\;}\\{+ {1 \over 2}\int {{d^n}y{\mu ^{- (n - 4)}}H_{{{\rm{A}}_n}}^{\mu \nu \alpha \beta}(x,y)\;{h_{\alpha \beta}}(y) + \mathcal{O}(n - 4) = {\xi ^{\mu \nu}}(x){.}} \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad}\end{array}$$
(111)
Notice that the terms containing the bare cosmological constant have cancelled. These equations can now be renormalized; that is, we can now write the bare coupling constants as renormalized coupling constants plus some suitably-chosen counterterms, and take the limit as n → 4. In order to carry out such a procedure, it is convenient to distinguish between massive and massless scalar fields. The details of the calculation can be found in [259].
It is convenient to introduce the two new kernels
$$\begin{array}{*{20}c}{{H_{\rm{A}}}(x;{m^2}) \equiv {1 \over {480{\pi ^2}}}\int {{{{d^n}p} \over {{{(2\pi)}^4}}}{e^{ipx}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\;}}\\{\times \left\{{{{\left({1 + 4{{{m^2}} \over {{p^2}}}} \right)}^2}\left[ {- i\pi \;{\rm{sign}}\;({p^0})\theta (- {p^2} - 4{m^2})\sqrt {1 + 4{{{m^2}} \over {{p^2}}}} + \varphi ({p^2})} \right] - {8 \over 3}{{{m^2}} \over {{p^2}}}} \right\},}\\{{H_{\rm{B}}}(x;{m^2},\Delta \xi) \equiv {1 \over {72{\pi ^2}}}\int {{{{d^4}p} \over {{{(2\pi)}^4}}}{e^{ipx}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad}}\\{\times \left\{{{{\left({3\Delta \xi + {{{m^2}} \over {{p^2}}}} \right)}^2}\left[ {- i\pi \;{\rm{sign}}\;({p^0})\theta (- {p^2} - 4{m^2})\sqrt {1 + 4{{{m^2}} \over {{p^2}}}} + \varphi ({p^2})} \right] - {1 \over 6}{{{m^2}} \over {{p^2}}}} \right\},}\end{array}$$
(112)
where φ(p2) is given by the restriction to n = 4 of expression (109). The renormalized coupling constants 1/G, α and β are easily computed as in Equation (92). Substituting their expressions into Equation (111), we can take the limit as n → 4. Using the fact that, for n = 4, \({D^{(1)\mu \nu}}(x) = {3 \over 2}{A^{(1)\mu \nu}}(x)\), we obtain the corresponding semiclassical Einstein-Langevin equation.
For the massless case one needs the limit as m → 0 of Equation (111). In this case it is convenient to separate κn in Equation (93) as \({\kappa _n} = {{\tilde \kappa}_n} + {1 \over 2}{\rm{In(}}{{\rm{m}}^2}/{\mu ^2}) + {\mathcal O}(n - 4)\), where
$${\bar \kappa _n} \equiv {1 \over {n - 4}}{\left({{{{e^\gamma}} \over {4\pi}}} \right)^{{{n - 4} \over 2}}} = {1 \over {n - 4}} + {1 \over 2}\ln \left({{{{e^\gamma}} \over {4\pi}}} \right) + \mathcal{O}(n - 4),$$
(113)
and use that, from Equation (109), we have
$$\lim \limits_{{m^2} \rightarrow 0} \left[ {\varphi ({p^2}) + \ln {{{m^2}} \over {{\mu ^2}}}} \right] = - 2 + \ln \left\vert {{{{p^2}} \over {{\mu ^2}}}} \right\vert {.}$$
(114)
The coupling constants are then easily renormalized. We note that in the massless limit, the Newtonian gravitational constant is not renormalized and, in the conformal coupling case, Δξ = 0, we have that βB = β. Note also that, by making m = 0 in Equation (101), the noise and dissipation kernels can be written as
$$\begin{array}{*{20}c}{{N_{\rm{A}}}(x;{m^2} = 0) = N(x),\quad \quad {N_{\rm{B}}}(x;{m^2} = 0,\Delta \xi) = 60\Delta {\xi ^2}N(x),}\\{{D_{\rm{A}}}(x;{m^2} = 0) = D(x),\quad \quad \;{D_{\rm{B}}}(x;{m^2} = 0,\Delta \xi) = 60\Delta {\xi ^2}D(x),}\end{array}$$
(115)
where
$$N(x) \equiv {1 \over {480\pi}}\int {{{{d^4}p} \over {{{(2\pi)}^4}}}} {e^{ipx}}\theta (- {p^2}),\quad \quad D(x) \equiv {{- i} \over {480\pi}}\int {{{{d^4}p} \over {{{(2\pi)}^4}}}} {e^{ipx}}\;{\rm{sign}}\;({p^0})\theta (- {p^2}){.}$$
(116)
It is also convenient to introduce the new kernel
$$\begin{array}{*{20}c}{H(x;{\mu ^2}) \equiv {1 \over {480{\pi ^2}}}\int {{{{d^4}p} \over {{{(2\pi)}^4}}}} {e^{ipx}}\left[ {\ln \left\vert {{{{p^2}} \over {{\mu ^2}}}} \right\vert - i\pi \;{\rm{sign}}\;({p^0})\theta (- {p^2})} \right]\quad \quad \quad}\\{= {1 \over {480{\pi ^2}}}\lim \limits_{\epsilon \rightarrow {0^ +}} \int {{{{d^4}p} \over {{{(2\pi)}^4}}}} {e^{ipx}}\ln \left({{{- {{({p^0} + i\epsilon)}^2} + {p^i}{p_i}} \over {{\mu ^2}}}} \right){.}}\end{array}$$
(117)
This kernel is real and can be written as the sum of an even part and an odd part in the variables xμ, where the odd part is the dissipation kernel D(x). The Fourier transforms (116) and (117) can actually be computed and, thus, in this case, we have explicit expressions for the kernels in position space; see, for instance, [71, 169, 220].
Finally, the Einstein-Langevin equation for the physical stochastic perturbations hμν can be written in both cases, for m ≠ 0 and for m = 0, as
$$\begin{array}{*{20}c}{{1 \over {8\pi G}}{G^{(1)\mu \nu}}(x) - 2(\bar \alpha {A^{(1)\mu \nu}}(x) + \bar \beta {B^{(1)\mu \nu}}(x))}\\{+ {1 \over 4}\int {{d^4}y} [{H_{\rm{A}}}(x - y){A^{(1)\mu \nu}}(y) + {H_{\rm{B}}}(x - y){B^{(1)\mu \nu}}(y)] = {\xi ^{\mu \nu}}(x),}\end{array}$$
(118)
where, in terms of the renormalized constants α and β, the new constants are \(\bar \alpha = \alpha + {(3600{\pi ^2})^{- 1}}\) and \(\bar \beta = \beta - ({1 \over {12}} - 5\Delta \xi)\,{(2880{\pi ^2})^{- 1}}\). The kernels HA(x) and HB(x) are given by Equations (112) when m ≠ 0, and by HA(x) = H(x; μ2) and HB(x) = 60Δξ2H(x; μ2) when m = 0. In the massless case, we can use the arbitrariness of the mass scale p to eliminate one of the parameters \({\bar \alpha}\) or \({\bar \beta}\). The components of the Gaussian stochastic source ξμν have zero mean value, and their two-point correlation functions are given by ⟨ξμν(x)ξαβ(y)⟩s = Nμναβ(x, y), where the noise kernel is given in Equation (102), which in the massless case reduces to Equation (115).
It is interesting to consider the massless conformally-coupled scalar field, i.e., the case Δξ = 0, which is of particular interest because of its similarities with the electromagnetic field, and also because of its interest to cosmology; massive fields become conformally invariant when their masses are negligible compared to the spacetime curvature. We have already mentioned that, for a conformally coupled field, the stochastic source tensor must be traceless (up to first order perturbations around semiclassical gravity), in the sense that the stochastic variable \(\xi _\mu ^\mu \equiv {\eta _{\mu \nu}}{\xi ^{\mu \nu}}\) behaves deterministically as a vanishing scalar field. This can be directly checked by noticing from Equations (102) and (115) that when Δξ = 0, one has \({\langle \xi _\mu ^\mu (x){\xi ^{\alpha \beta}}(y)\rangle _{\rm{S}}} = 0\), since \({\mathcal F}_\mu ^\mu = 3\square\) and \({{\mathcal F}^{\mu \alpha}}{\mathcal F}_\mu ^\beta = \square{{\mathcal F}^{\alpha \beta}}\). The Einstein-Langevin equations for this particular case (and generalized to a spatially-flat Robertson-Walker background) were first obtained in [73], where the coupling constant β was fixed to be zero. See also [208] for a discussion of this result and its connection to the problem of structure formation in the trace anomaly driven inflation [162, 339, 355].
Note that the expectation value of the renormalized stress-energy tensor for a scalar field can be obtained by comparing Equation (118) with the Einstein-Langevin equation (15); its explicit expression is given in [259]. The results agree with the general form found by Horowitz [169, 170] using an axiomatic approach and coincide with that given in [110]. The particular cases of conformal coupling, Δξ = 0, and minimal coupling, Δξ = −1/6, are also in agreement with the results for these cases given in [72, 169, 170, 223, 340], modulo local terms proportional to A(1)μν and B(1)μν due to different choices of the renormalization scheme. For the case of a massive minimally-coupled scalar field, \(\Delta \xi = - {1 \over 6}\), our result is equivalent to that of [347].
Correlation functions for gravitational perturbations
Here we solve the Einstein-Langevin equations (118) for the components G(1)μν of the linearized Einstein tensor. Then we use these solutions to compute the corresponding two-point correlation functions, which give a measure of the gravitational fluctuations predicted by the stochastic semi-classical theory of gravity in the present case. Since the linearized Einstein tensor is invariant under gauge transformations of the metric perturbations, these two-point correlation functions are also gauge invariant. Once we have computed the two-point correlation functions for the linearized Einstein tensor, we find the solutions for the metric perturbations and compute the associated two-point correlation functions. The procedure used to solve the Einstein-Langevin equation is similar to the one used by Horowitz [169] (see also [110]) to analyze the stability of Minkowski spacetime in semiclassical gravity.
We first note that the tensors A(1)μν and B(1)μν can be written in terms of G(1)μν as
$${A^{(1)\mu \nu}} = {2 \over 3}({\mathcal{F}^{\mu \nu}}G_{\quad \;\alpha}^{(1)\alpha} - \mathcal{F}_\alpha ^\alpha {G^{(1)\mu \nu}}),\quad \quad {B^{(1)\mu \nu}} = 2{\mathcal{F}^{\mu \nu}}G_{\quad \;\alpha}^{(1)\alpha},$$
(119)
where we have used \(3 \square = {\mathcal F}_\alpha ^\alpha\). Therefore, the Einstein-Langevin equation (118) can be seen as a linear integro-differential stochastic equation for the components G(1)μν. In order to find solutions to Equation (118), it is convenient to Fourier transform it. With the convention \(\tilde f(p) = \int {{d^4}x{e^{- ipx}}} f(x)\) for a given field f(x), one finds, from Equation (119),
$$\begin{array}{*{20}c} {{{\tilde A}^{(1)\mu \nu}}(p) = 2{p^2}{{\tilde G}^{(1)\mu \nu}}(p) - {2 \over 3}{p^2}{P^{\mu \nu}}\tilde G_{\quad \;\alpha}^{(1)\alpha}(p),} \\ {{{\tilde B}^{(1)\mu \nu}}(p) = - 2{p^2}{P^{\mu \nu}}\tilde G_{\quad \;\alpha}^{(1)\alpha}(p).\quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(120)
The Fourier transform of the Einstein-Langevin Equation (118) now reads
$${F^{\mu \nu}}_{\alpha \beta}(p){\tilde G^{(1)\alpha \beta}}(p) = 8\pi G{\tilde \xi ^{\mu \nu}}(p),$$
(121)
where
$${F^{\mu \nu}}{}_{\alpha \beta}(p) \equiv {F_1}(p)\delta _{(\alpha}^\mu \delta _{\beta)}^\nu + {F_2}(p){p^2}{P^{\mu \nu}}{\eta _{\alpha \beta}},$$
(122)
with
$$\begin{array}{*{20}c} {{F_1}(p) \equiv 1 + 16\pi G{p^2}\left[ {{1 \over 4}{{\tilde H}_{\rm{A}}}(p) - 2\bar \alpha} \right],\quad \quad \quad \quad \quad} \\ {{F_2}(p) \equiv - {{16} \over 3}\pi G\left[ {{1 \over 4}{{\tilde H}_{\rm{A}}}(p) + {3 \over 4}{{\tilde H}_{\rm{B}}}(p) - 2\bar \alpha - 6\bar \beta} \right].} \\ \end{array}$$
(123)
In the Fourier transformed Einstein-Langevin Equation (121), \({{\tilde \xi}^{\mu \nu}}(p)\), the Fourier transform of ξμν(x), is a Gaussian stochastic source of zero average, and
$${\left\langle {{{\tilde \xi}^{\mu \nu}}(p){{\tilde \xi}^{\alpha \beta}}({p{\prime}})} \right\rangle _{\rm{s}}} = {(2\pi)^4}{\delta ^4}(p + {p{\prime}}){\tilde N^{\mu \nu \alpha \beta}}(p),$$
(124)
where we have introduced the Fourier transform of the noise kernel. The explicit expression for Ñμναβ(p) is found from Equations (101) and (102) to be
$${\tilde N^{\mu \nu \alpha \beta}}(p) = {{\theta (- {p^2} - 4{m^2})} \over {720\pi}}\sqrt {1 + 4{{{m^2}} \over {{p^2}}}} \left[ {{1 \over 4}{{\left({1 + 4{{{m^2}} \over {{p^2}}}} \right)}^2}{{({p^2})}^2}{P^{\mu \nu \alpha \beta}} + 10{{\left({3\Delta \xi + {{{m^2}} \over {{p^2}}}} \right)}^2}{{({p^2})}^2}{P^{\mu \nu}}{P^{\alpha \beta}}} \right],$$
(125)
which in the massless case reduces to
$$\lim \limits_{m \rightarrow 0} {\tilde N^{\mu \nu \alpha \beta}}(p) = {1 \over {480\pi}}\theta (- {p^2})\left[ {{1 \over 6}{{({p^2})}^2}{P^{\mu \nu \alpha \beta}} + 60\Delta {\xi ^2}{{({p^2})}^2}{P^{\mu \nu}}{P^{\alpha \beta}}} \right].$$
(126)
Correlation functions for the linearized Einstein tensor
In general we can write \({G^{(1)\mu \nu}} = {\langle {G^{(1)\mu \nu}}\rangle _{\rm{S}}} + G_{\rm{f}}^{(1)\mu \nu}\), where \(G_{\rm{f}}^{(1)\mu \nu}\) is a solution to Equations (118) with zero average, or Equation (121) in the Fourier transformed version. The averages ⟨G(1)μν⟩s must be a solution of the linearized semiclassical Einstein equations obtained by averaging Equations (118) or (121). Solutions to these equations (especially in the massless case, m = 0) have been studied by several authors [110, 152, 169, 170, 174, 223, 309, 310, 330, 344, 345], particularly in connection with the problem of the stability of the ground state of semiclassical gravity. The two-point correlation functions for the linearized Einstein tensor are defined by
$$\begin{array}{*{20}c} {{\mathcal{G}^{\mu \nu \alpha \beta}}(x,{x{\prime}}) \equiv {{\left\langle {{G^{(1)\mu \nu}}(x){G^{(1)\alpha \beta}}({x{\prime}})} \right\rangle}_{\rm{s}}} - {{\left\langle {{G^{(1)\mu \nu}}(x)} \right\rangle}_{\rm{s}}}{{\left\langle {{G^{(1)\alpha \beta}}({x{\prime}})} \right\rangle}_{\rm{s}}}} \\ {= {{\left\langle {G_{\rm{f}}^{(1)\mu \nu}(x)G_{\rm{f}}^{(1)\alpha \beta}({x{\prime}})} \right\rangle}_{\rm{s}}}.\quad \quad \quad \quad \quad \;\;} \\ \end{array}$$
(127)
Now we shall seek the family of solutions to the Einstein-Langevin equation, which can be written as a linear functional of the stochastic source, and whose Fourier transform \({{\tilde G}^{(1)\mu \nu}}(p)\) depends locally on \({{\tilde \xi}^{\alpha \beta}}(p)\). Each of these such solutions is a Gaussian stochastic field and thus can be completely characterized by the averages ⟨G(1)μν⟩s and the two-point correlation functions (127). For such a family of solutions, \(\tilde G_{\rm{f}}^{(1)\mu \nu}(p)\) is the most general solution to Equation (121), which is linear, homogeneous, and local in \({{\tilde \xi}^{\alpha \beta}}(p)\). It can be written as
$$\tilde G_{\rm{f}}^{(1)\mu \nu}(p) = 8\pi G\;{D^{\mu \nu}}{}_{\alpha \beta}(p){\tilde \xi ^{\alpha \beta}}(p),$$
(128)
where Dμναβ(p) are the components of a Lorentz-invariant tensor-field distribution in Minkowski spacetime (by “Lorentz-invariant” we mean invariant under transformations of the orthochronous Lorentz subgroup; see [169] for more details on the definition and properties of these tensor distributions). This tensor is symmetric under the interchanges of α ↔ β and μ↔ν, and is the most general solution of
$${F^{\mu \nu}}{}_{\rho \sigma}(p){D^{\rho \sigma}}{}_{\alpha \beta}(p) = \delta _{(\alpha}^\mu \delta _{\beta)}^\nu.$$
(129)
In addition, we must impose the conservation condition, \({p_\nu}\tilde G_{\rm{f}}^{(1)\mu \nu}(p) = 0\), where this zero must be understood as a stochastic variable, which behaves deterministically as a zero vector field. We can write \({D^{\mu \nu}}_{\alpha \beta}(p) = D_{\rm{p}}^{\mu \nu}{}_{\alpha \beta}(p) + D_{\rm{h}}^{\mu \nu}{}_{\alpha \beta}(p)\), where \({D{_P^{\mu \nu }}_{\alpha \beta }}(p)\) is a particular solution to Equation (129) and \({D{_{\rm{h}}^{\mu \nu}}_{\alpha \beta}}(p)\) is the most general solution to the homogeneous equation. Consequently, see Equation (128), we can write \(\tilde G_{\rm{f}}^{(1)\mu \nu}(p) = \tilde G_{\rm{p}}^{(1)\mu \nu}(p) + \tilde G_{\rm{h}}^{(1)\mu \nu}(p)\). To find the particular solution, we try an ansatz of the form
$$D_{\rm{p}}^{\mu \nu}{}_{\alpha \beta}(p) = {d_1}(p)\delta _{(\alpha}^\mu \delta _{\beta)}^\nu + {d_2}(p){p^2}{P^{\mu \nu}}{\eta _{\alpha \beta}}.$$
(130)
Substituting this ansatz into Equations (129), it is easy to see that it solves these equations if we take
$${d_1}(p) = {\left[ {{1 \over {{F_1}(p)}}} \right]_{\rm{r}}},\quad \quad {d_2}(p) = - {\left[ {{{{F_2}(p)} \over {{F_1}(p){F_3}(p)}}} \right]_{\rm{r}}},$$
(131)
with
$${F_3}(p) \equiv {F_1}(p) + 3{p^2}{F_2}(p) = 1 - 48\pi G{p^2}\left[ {{1 \over 4}{{\tilde H}_{\rm{B}}}(p) - 2\bar \beta} \right],$$
(132)
and where the notation [ ]r means that the zeros of the denominators are regulated with appropriate prescriptions in such a way that d1(p) and d2(p) are well-defined Lorentz-invariant scalar distributions. This yields a particular solution to the Einstein-Langevin equations,
$$\tilde G_{\rm{p}}^{(1)\mu \nu}(p) = 8\pi G\;D_{\rm{p}}^{\mu \nu}{}_{\alpha \beta}(p){\tilde \xi ^{\alpha \beta}}(p),$$
(133)
which, since the stochastic source is conserved, satisfies the conservation condition. Note that, in the case of a massless scalar field (m = 0), the above solution has a functional form analogous to that of the solutions of linearized semiclassical gravity found in the appendix of [110]. Notice also that, for a massless conformally-coupled field (m = 0 and Δξ = 0), the second term on the right-hand side of Equation (130) will not contribute in the correlation functions (127), since in this case the stochastic source is traceless.
A detailed analysis given in [259] concludes that the homogeneous solution \(\tilde G_{\rm{h}}^{(1)\mu \nu}(p)\) gives no contribution to the correlation functions (127). Consequently \({{\mathcal G}^{\mu \nu \alpha \beta}}(x,{x{\prime}}) = {\langle G_{\rm{p}}^{(1)\mu \nu}(x)G_{\rm{p}}^{(1)\alpha \beta}({x{\prime}})\rangle _{\rm{s}}}\), where \(G_{\rm{p}}^{(1)\mu \nu}(x)\) is the inverse Fourier transform of Equation (133), and the correlation functions (127) are
$${\left\langle {\tilde G_{\rm{p}}^{(1)\mu \nu}(p)\tilde G_{\rm{p}}^{(1)\alpha \beta}({p{\prime}})} \right\rangle _{\rm{s}}} = 64{(2\pi)^6}{G^2}{\delta ^4}(p + {p{\prime}})D_{\rm{p}}^{\mu \nu}{}_{\rho \sigma}(p)D_{\rm{p}}^{\alpha \beta}{}_{\lambda \gamma}(- p){\tilde N^{\rho \sigma \lambda \gamma}}(p).$$
(134)
It is easy to see from the above analysis that the prescriptions [ ]r in the factors Dp are irrelevant in the last expression and thus can be suppressed. Taking into account that \({F_l}(- p) = F_l^{\ast}(p)\), with l = 1, 2, 3, we get from Equations (130) and (131)
$$\begin{array}{*{20}c} {{{\left\langle {\tilde G_{\rm{p}}^{(1)\mu \nu}(p)\tilde G_{\rm{p}}^{(1)\alpha \beta}({p{\prime}})} \right\rangle}_{\rm{s}}} = 64{{(2\pi)}^6}{G^2}{{{\delta ^4}(p + {p{\prime}})} \over {\vert {F_1}(p){\vert ^2}}}\left[ {{{\tilde N}^{\mu \nu \alpha \beta}}(p) - {{{F_2}(p)} \over {{F_3}(p)}}{p^2}{P^{\mu \nu}}{{\tilde N}^{\alpha \beta \rho}}{}_\rho (p)} \right.\quad \quad \quad \quad \quad} \\ {\quad \quad \quad \quad \quad \quad \quad \quad \quad \left. {- {{F_2^{\ast}(p)} \over {F_3^{\ast}(p)}}{p^2}{P^{\alpha \beta}}{{\tilde N}^{\mu \nu \rho}}{}_\rho (p) + {{\vert {F_2}(p){\vert ^2}} \over {\vert {F_3}(p){\vert ^2}}}{p^2}{P^{\mu \nu}}{p^2}{P^{\alpha \beta}}\tilde N_{\;\;\rho \;\;\sigma}^{\rho \;\;\sigma}(p)} \right].} \\ \end{array}$$
(135)
This last expression is well-defined as a bi-distribution and can be easily evaluated using Equation (125). The final explicit result for the Fourier-transformed correlation function for the Einstein tensor is thus
$$\begin{array}{*{20}c} {{{\left\langle {\tilde G_{\rm{p}}^{(1)\mu \nu}(p)\tilde G_{\rm{p}}^{(1)\alpha \beta}({p{\prime}})} \right\rangle}_{\rm{s}}} = {2 \over {45}}{{(2\pi)}^5}{G^2}{{{\delta ^4}(p + {p{\prime}})} \over {\vert {F_1}(p){\vert ^2}}}\theta (- {p^2} - 4{m^2})\sqrt {1 + 4{{{m^2}} \over {{p^2}}}} \quad \quad \quad \quad \quad \quad} \\ {\quad \quad \times \left[ {{1 \over 4}{{\left({1 + 4{{{m^2}} \over {{p^2}}}} \right)}^2}{{({p^2})}^2}{P^{\mu \nu \alpha \beta}}} \right.\quad \quad \quad \quad \quad} \\ {\quad \quad \quad \quad \quad \quad \quad \quad \left. {+ 10{{\left({3\Delta \xi + {{{m^2}} \over {{p^2}}}} \right)}^2}{{({p^2})}^2}{P^{\mu \nu}}{P^{\alpha \beta}}{{\left\vert {1 - 3{p^2}{{{F_2}(p)} \over {{F_3}(p)}}} \right\vert}^2}} \right].} \\ \end{array}$$
(136)
To obtain the correlation functions in coordinate space, Equation (127), we take the inverse Fourier transform. The final result is
$${\mathcal{G}^{\mu \nu \alpha \beta}}(x,{x{\prime}}) = {\pi \over {45}}{G^2}\mathcal{F}_x^{\mu \nu \alpha \beta}{\mathcal{G}_{\rm{A}}}(x - {x{\prime}}) + {{8\pi} \over 9}{G^2}\mathcal{F}_x^{\mu \nu}\mathcal{F}_x^{\alpha \beta}{\mathcal{G}_{\rm{B}}}(x - {x{\prime}}),$$
(137)
with
$$\begin{array}{*{20}c} {{{\tilde {\mathcal{G}}}_{\rm{A}}}(p) \equiv \theta (- {p^2} - 4{m^2})\sqrt {1 + 4{{{m^2}} \over {{p^2}}}} {{\left({1 + 4{{{m^2}} \over {{p^2}}}} \right)}^2}{1 \over {\vert {F_1}(p){\vert ^2}}},\quad \quad \quad \quad \quad \quad \quad} \\ {{{\tilde {\mathcal{G}}}_{\rm{B}}}(p) \equiv \theta (- {p^2} - 4{m^2})\sqrt {1 + 4{{{m^2}} \over {{p^2}}}} {{\left({3\Delta \xi + {{{m^2}} \over {{p^2}}}} \right)}^2}{1 \over {\vert {F_1}(p){\vert ^2}}}{{\left\vert {1 - 3{p^2}{{{F_2}(p)} \over {{F_3}(p)}}} \right\vert}^2},} \\ \end{array}$$
(138)
where Fl(p), l = 1, 2, 3, are given in Equations (123) and (132). Notice that, for a massless field (m = 0), we have
$$\begin{array}{*{20}c} {{F_1}(p) = 1 + 4\pi G{p^2}\tilde H(p;{{\bar \mu}^2}),\quad \quad \quad \quad \quad \quad \quad \quad \;} \\ {{F_2}(p) = - {{16} \over 3}\pi G\left[ {(1 + 180\Delta {\xi ^2}){1 \over 4}\tilde H(p;{{\bar \mu}^2}) - 6\Upsilon} \right],} \\ {{F_3}(p) = 1 - 48\pi G{p^2}\left[ {15\Delta {\xi ^2}\tilde H(p;{{\bar \mu}^2}) - 2\Upsilon} \right],\quad \quad \;} \\ \end{array}$$
(139)
with \(\bar \mu \equiv \mu \exp (1920{\pi ^2}\bar \alpha)\) and \(\Upsilon \equiv \bar \beta - 60\Delta {\xi ^2}\bar \alpha\), and where \(\tilde H(p;{\mu ^2})\) is the Fourier transform of H(x; μ2) given in Equation (117).
Correlation functions for the metric perturbations
Starting from the solutions found for the linearized Einstein tensor, which are characterized by the two-point correlation functions (137) (or, in terms of Fourier transforms, Equation (136)), we can now solve the equations for the metric perturbations. Working in the harmonic gauge, \({\partial _\nu}{{\bar h}^{\mu \nu}} = 0\) (this zero must be understood in a statistical sense), where \({{\bar h}_{\mu \nu}} \equiv {h_{\mu \nu}} - {1 \over 2}{\eta _{\mu \nu}}h_\alpha ^\alpha\), the equations for the metric perturbations in terms of the Einstein tensor are
$$\square{\bar h^{\mu \nu}}(x) = - 2{G^{(1)\mu \nu}}(x),$$
(140)
or, in terms of Fourier transforms, \({p^2}{{\tilde \bar h}^{\mu \nu}}(p) = 2{{\tilde G}^{(1)\mu \nu}}(p)\). Similarly to the analysis of the equation for the Einstein tensor, we can write \({{\bar h}^{\mu \nu}} = {\langle {{\bar h}^{\mu \nu}}\rangle _{\rm{S}}} + \bar h_{\rm{f}}^{\mu \nu}\), where \(\bar h_{\rm{f}}^{\mu \nu}\) is a solution to these equations with zero average, and the two-point correlation functions are defined by
$$\begin{array}{*{20}c} {{\mathcal{H}^{\mu \nu \alpha \beta}}(x,{x{\prime}}) \equiv {{\left\langle {{{\bar h}^{\mu \nu}}(x){{\bar h}^{\alpha \beta}}({x{\prime}})} \right\rangle}_{\rm{s}}} - {{\left\langle {{{\bar h}^{\mu \nu}}(x)} \right\rangle}_{\rm{s}}}{{\left\langle {{{\bar h}^{\alpha \beta}}({x{\prime}})} \right\rangle}_{\rm{s}}}} \\ {= {{\left\langle {\bar h_{\rm{f}}^{\mu \nu}(x)\bar h_{\rm{f}}^{\alpha \beta}({x{\prime}})} \right\rangle}_{\rm{s}}}.\quad \quad \quad \;\;\;} \\ \end{array}$$
(141)
We can now seek solutions of the Fourier transform of Equation (140) of the form \(\tilde \bar h_{\rm{f}}^{\mu \nu}(p) = 2D(p)\tilde G_{\rm{f}}^{(1)\mu \nu}(p)\), where D(p) is a Lorentz-invariant scalar distribution in Minkowski spacetime, which is the most general solution of p2D(p) = 1. Note that, since the linearized Einstein tensor is conserved, solutions of this form automatically satisfy the harmonic gauge condition. As in Section 6.4.1 we can write D(p) = [1/p2]r + Dh(p), where Dh(p) is the most general solution to the associated homogeneous equation and, correspondingly, we have \(\tilde \bar h_{\rm{f}}^{\mu \nu}(p) = \tilde \bar h_{\rm{p}}^{\mu \nu}(p) + \tilde \bar h_{\rm{h}}^{\mu \nu}(p)\). However, since Dh(p) has support on the set of points for which p2 = 0, it is easy to see from Equation (136) (from the factor θ(−p2 − 4m2)) that \({\langle \tilde \bar h_{\rm{h}}^{\mu \nu}(p)\tilde G_{\rm{f}}^{(1)\alpha \beta}({p{\prime}})\rangle _{\rm{s}}} = 0\) and, thus, the two-point correlation functions (141) can be computed from \({\langle \tilde \bar h_{\rm{f}}^{\mu \nu}(p)\tilde \bar h_{\rm{f}}^{\alpha \beta}({p{\prime}})\rangle _{\rm{s}}} = {\langle \tilde \bar h_{\rm{p}}^{\mu \nu}(p)\tilde \bar h_{\rm{p}}^{\alpha \beta}({p{\prime}})\rangle _{\rm{s}}}\). From Equation (136), and due to the factor θ(−p2 − 4m2), it is also easy to see that the prescription [ ]r is irrelevant in this correlation function, and we obtain
$${\left\langle {\tilde \bar h_{\rm{p}}^{\mu \nu}(p)\tilde \bar h_{\rm{p}}^{\alpha \beta}({p{\prime}})} \right\rangle _{\rm{s}}} = {4 \over {{{({p^2})}^2}}}{\left\langle {\tilde G_{\rm{p}}^{(1)\mu \nu}(p)\tilde G_{\rm{p}}^{(1)\alpha \beta}({p{\prime}})} \right\rangle _{\rm{s}}},$$
(142)
where \({\langle \tilde G_{\rm{P}}^{(1)\mu \nu}(p)\tilde G_{\rm{P}}^{(1)\alpha \beta}({p{\prime}})\rangle _{\rm{S}}}\) is given by Equation (136). The right-hand side of this equation is a well-defined bi-distribution, at least for m ≠ 0 (the θ function provides the suitable cutoff). In the massless field case, since the noise kernel is obtained as the limit m → 0 of the noise kernel for a massive field, it seems that the natural prescription to avoid divergences on the lightcone p2 =0 is a Hadamard finite part (see [322, 388] for its definition). Taking this prescription, we also get a well-defined bi-distribution for the massless limit of the last expression.
The final result for the two-point correlation function for the field \({{\bar h}^{\mu \nu}}\) is
$${\mathcal{H}^{\mu \nu \alpha \beta}}(x,{x\prime}) = {{4\pi} \over {45}}{G^2}\mathcal{F}_x^{\mu \nu \alpha \beta}{\mathcal{H}_{\rm{A}}}(x - {x{\prime}}) + {{32\pi} \over 9}{G^2}\mathcal{F}_x^{\mu \nu}\mathcal{F}_x^{\alpha \beta}{\mathcal{H}_{\rm{B}}}(x - {x{\prime}}),$$
(143)
where \({{\tilde {\mathcal H}}_{\rm{A}}}(p) \equiv [1/{({p^2})^2}]{{\tilde {\mathcal G}}_{\rm{A}}}(p)\) and \({{\tilde {\mathcal H}}_{\rm{B}}}(p) \equiv [1/{({p^2})^2}]{{\tilde {\mathcal G}}_{\rm{B}}}(p)\), with \({{\tilde {\mathcal G}}_{\rm{A}}}(p)\) and \({{\tilde {\mathcal G}}_{\rm{B}}}(p)\) given by Equation (138). The two-point correlation functions for the metric perturbations can be easily obtained using \({h_{\mu \nu}} = {\bar h_{\mu \nu}} - {1 \over 2}{\eta _{\mu \nu}}\bar h_\alpha ^\alpha\).
Conformally-coupled field
For a conformally coupled field, i.e., when m = 0 and Δξ = 0, the previous correlation functions are greatly simplified and can be approximated explicitly in terms of analytic functions. The detailed results are given in [259]; here we outline the main features.
When m = 0 and Δξ = 0, we have \({{\mathcal G}_{\rm{B}}}(x)\) and \({{\tilde {\mathcal G}}_{\rm{A}}}(p) = \theta (- {p^2})\vert {F_1}(p){\vert ^{- 2}}\). Thus the two-point correlations functions for the Einstein tensor are written
$${\mathcal{G}^{\mu \nu \alpha \beta}}(x,{x{\prime}}) = {\pi \over {45}}{G^2}\mathcal{F}_x^{\mu \nu \alpha \beta}\int {{{{d^4}p} \over {{{(2\pi)}^4}}}{{{e^{ip(x - {x{\prime}})}}\theta (- {p^2})} \over {{{\left\vert {1 + 4\pi G{p^2}\tilde H(p;{{\bar \mu}^2})} \right\vert}^2}}},}$$
(144)
where \(\tilde H(p;{{\bar \mu}^2}) = {(480{\pi ^2})^{- 1}}\ln [ - ({({p^0} + i\varepsilon)^2} + {p^i}{p_i})/{\mu ^2}]\); (see Equation (117)).
To estimate this integral, let us consider spacelike separated points (x − x′)μ = (0, x − x′), and define y = x − x′. We may now formally change the momentum variable pμ by the dimensionless vector sμ, pμ = sμ/|y|. Then the previous integral denominator is \(\vert 1 + 16\pi {({L_{\rm{P}}}/\vert y\vert)^2}{s^2}\tilde H(s){\vert ^2}\), where we have introduced the Planck length \({L_{\rm{P}}} = \sqrt G\). It is clear that we can consider two regimes: (a) when LP ≪ |y|, and (b) when |y| ∼ LP. In case (a) the correlation function, for the 0000 component, say, will be of the order
$$\mathcal{G}^{0000}({\bf{y}})\sim {{L_{\rm{P}}^4} \over {\vert {\bf{y}}{\vert ^8}}}.$$
In case (b), when the denominator has zeros, a detailed calculation carried out in [259] shows that
$${\mathcal{G}^{0000}}({\bf{y}})\sim {e^{- \vert {\bf{y}}\vert/{L_{\rm{P}}}}}\left({{{{L_{\rm{P}}}} \over {\vert {\bf{y}}{\vert ^5}}} + \ldots + {1 \over {L_{\rm{P}}^2\vert {\bf{y}}{\vert ^2}}}} \right),$$
which indicates an exponential decay at distances around the Planck scale. Thus short scale fluctuations are strongly suppressed.
For the two-point metric correlation the results are similar. In this case we have
$${\mathcal{H}^{\mu \nu \alpha \beta}}(x,{x{\prime}}) = {{4\pi} \over {45}}{G^2}\mathcal{F}_x^{\mu \nu \alpha \beta}\int {{{{d^4}p} \over {{{(2\pi)}^4}}}{{{e^{ip(x - {x{\prime}})}}\theta (- {p^2})} \over {{{({p^2})}^2}{{\left\vert {1 + 4\pi G{p^2}\tilde H(p;{{\bar \mu}^2})} \right\vert}^2}}}.}$$
(145)
The integrand has the same behavior as the correlation function of Equation (144), thus matter fields tends to suppress the short-scale metric perturbations. In this case we find, as for the correlation of the Einstein tensor, that for case (a) above we have
$${\mathcal{H}^{0000}}({\bf{y}})\sim {{L_{\rm{P}}^4} \over {\vert {\bf{y}}{\vert ^4}}},$$
and for case (b) we have
$${\mathcal{H}^{0000}}({\bf{y}})\sim {e^{- \vert {\bf{y}}\vert/{L_{\rm{P}}}}}\left({{{{L_{\rm{P}}}} \over {\vert {\bf{y}}\vert}} + \ldots} \right).$$
It is interesting to write expression (145) in an alternative way. If we use the dimensionless tensor Pμναβ introduced in Equation (107), which accounts for the effect of the operator \({\mathcal F}_x^{\mu \nu \alpha \beta}\), we can write
$${\mathcal{H}^{\mu \nu \alpha \beta}}(x,{x{\prime}}) = {{4\pi} \over {45}}{G^2}\int {{{{d^4}p} \over {{{(2\pi)}^4}}}{{{e^{ip(x - {x{\prime}})}}{P^{\mu \nu \alpha \beta}}\theta (- {p^2})} \over {{{\left\vert {1 + 4\pi G{p^2}\tilde H(p;{{\bar \mu}^2})} \right\vert}^2}}}.}$$
(146)
This expression allows a direct comparison with the graviton propagator for linearized quantum gravity in the 1/N expansion found by Tomboulis [348]. One can see that the imaginary part of the graviton propagator leads, in fact, to Equation (146). In [317] it is shown that the two-point correlation functions for the metric perturbations derived from the Einstein-Langevin equation are equivalent to the symmetrized quantum two-point correlation functions for the metric fluctuations in the large N expansion of quantum gravity interacting with N matter fields.
Stability of Minkowski spacetime
In this section we apply the validity criterion for semiclassical gravity introduced in Section 3.3 to flat spacetime. The Minkowski metric is a particularly simple and interesting solution of semiclassical gravity. In fact, as we have seen in Section 6.1, when the quantum fields are in the Minkowski vacuum state, one may take the renormalized expectation value of the stress tensor as \(\left\langle {\hat T_{ab}^R[\eta ]} \right\rangle = 0\); this is equivalent to assuming that the cosmological constant is zero. Then the Minkowski metric ηab is a solution of the semiclassical Einstein equation (8). Thus, we can look for the stability of Minkowski spacetime against quantum matter fields. According to the criteria we have established, we have to look for the behavior of the two-point quantum correlations for the metric perturbations hab(x) over the Minkowski background, which are given by Equations (16) and (17). As we have emphasized before, these metric fluctuations separate in two parts: the first term on the right-hand side of Equation (17), which corresponds to the intrinsic metric fluctuations, and the second term, which corresponds to the induced metric fluctuations.
Intrinsic metric fluctuations
Let us first consider the intrinsic metric fluctuations,
$${\langle {{h_{ab}}(x){h_{cd}}(y)} \rangle _{{{\rm int}}}} = {\langle {h_{ab}^0(x)h_{cd}^0(y)} \rangle _s},$$
(147)
where \(h_{ab}^0\) are the homogeneous solutions of the Einstein-Langevin equation (15), or equivalently the linearly-perturbed semiclassical equation, and where the statistical average is taken with respect to the Wigner distribution that describes the initial quantum state of the metric perturbations. Since these solutions are described by the linearized semiclassical equation around flat spacetime, we can make use of the results derived in [10, 11, 110, 169]. The solutions for the case of a massless scalar field were first discussed in [169] and an exhaustive description can be found in Appendix A of [110]. It is convenient to decompose the perturbation around Minkowski spacetime into scalar, vectorial and tensorial parts, as
$${h_{ab}} = \bar \phi {\eta _{ab}} + ({\nabla _{(a}}{\nabla _{b)}} - {\eta _{ab}}{\nabla _c}{\nabla ^c})\psi + 2{\nabla _{(a}}{v_{b)}} + h_{ab}^{{\rm{TT}}},$$
(148)
where va is a transverse vector and \(h_{ab}^{{\rm{TT}}}\) is a transverse and traceless symmetric tensor, i.e., \({\nabla _a}{\upsilon ^a} = 0,\,{\nabla ^a}h_{ab}^{{\rm{TT}}} = 0\) and \(({h^{{\rm{TT}}}})_a^a = 0\). A vector field ζa characterizes the gauge freedom due to infinitesimal diffeomorphisms as hab → hab + Δaζb + Δbζa. We may use this freedom to choose a gauge; a convenient election is the Lorentz or harmonic gauge defined as
$${\nabla ^a}\left({{h_{ab}} - {1 \over 2}{\eta _{ab}}h_c^c} \right) = 0.$$
(149)
When this gauge is imposed we have the following conditions on the metric perturbations ∇a ∇avb = 0 and \({\nabla _b}\bar \phi = 0\), which implies \(\bar \phi = {\rm{const}}\). A remaining gauge freedom compatible with the Lorentz gauge is still possible provided the vector field ζa satisfies the condition ∇a∇aζb = 0. One can easily see [203] that the vectorial and scalar part \({\bar \phi}\) can be eliminated, as well as the contribution of the scalar part ψ, which corresponds to Fourier modes \(\tilde \psi (p)\) with p2 =0. Thus, we will assume that we impose the Lorentz gauge with additional gauge transformations, which leave only the tensorial component and the modes of the scalar component ψ with p2 ≠ 0 in Fourier space.
Using the metric decomposition (148) we may compute the linearized Einstein tensor \(G_{ab}^{(1)}\). It is found that the vectorial part of the metric perturbation gives no contribution to this tensor, and the scalar and tensorial components give rise, respectively, to scalar and tensorial components: \(G_{ab}^{(1)\,(S)}\) and \(G_{ab}^{(1)\,(T)}\). Thus, let us now write the Fourier transform of the homogeneous Einstein-Langevin equation (121), which is equivalent to the linearized semiclassical Einstein equation,
$${F^{\mu \nu}}_{\alpha \beta}(p)\;{\tilde G^{(1)\alpha \beta}}(p) = 0.$$
(150)
Using the previous decomposition of the Einstein tensor this equation can be re-written in terms of its scalar and tensorial parts as
$$[{F_1}(p) + 3{p^2}{F_2}(p)]\tilde G_{\mu \nu}^{(1)\;({\rm{S}})}(p) = 0,$$
(151)
$${F_1}(p)\tilde G_{\mu \nu}^{(1)\;({\rm{T}})}(p) = 0.$$
(152)
where F1(p) and F2(p) are given by Equations (123), and \(\tilde G_{\mu \nu}^{(1)\,({\rm{S}})}\) and \(\tilde G_{\mu \nu}^{(1)\,({\rm{T}})}\) denote, respectively, the Fourier-transformed scalar and tensorial parts of the linearized Einstein tensor. To simplify the problem and to illustrate, in particular, how the runaway solutions arise, we will consider the case of a massless and conformally coupled field (see [110] for the massless case with arbitrary coupling and [10, 259] for the general massive case). Thus substituting m = 0 and ξ = 1/6 into the functions F1(p) and F2(p), and using Equation (117), the above equations become
$$(1 + 12\kappa \bar \beta {p^2})\tilde G_{\mu \nu}^{(1)\;({\rm{S}})}(p) = 0,$$
(153)
$$\lim\limits_{\epsilon \rightarrow {0^ +}} \left[ {1 + {{\kappa {p^2}} \over {960{\pi ^2}}}\ln \left({{{- {{({p^0} + i\epsilon)}^2} + {{\vec p}^2}} \over {{\mu ^2}}}} \right)} \right]\tilde G_{\mu \nu}^{(1)\;({\rm{T}})}(p) = 0,$$
(154)
where κ = 8πG. Let us consider these two equations separately.
For the scalar component when \(\bar \beta = 0\) the only solution is \(\tilde G_{\mu \nu}^{(1)\,({\rm{S}})}(p) = 0\). When \(\bar \beta > 0\) the solutions for the scalar component exhibit an oscillatory behavior in spacetime coordinates, which corresponds to a massive scalar field with \({m^2} = {(12\kappa \vert \bar \beta \vert)^{- 1}}\); for \(\bar \beta < 0\) the solutions correspond to a tachyonic field with \({m^2} = - {(12\kappa \vert \bar \beta \vert)^{- 1}}\). In spacetime coordinates they exhibit an exponential behavior in time, growing or decreasing, for wavelengths larger than \(4\pi {(3\kappa \vert \bar \beta \vert)^{1/2}}\) and an oscillatory behavior for wavelengths smaller than \(4\pi {(3\kappa \vert \bar \beta \vert)^{1/2}}\). On the other hand, the solution \(\tilde G_{\mu \nu}^{(1)\,({\rm{S}})}(p) = 0\) is completely trivial since any scalar metric perturbation \({{\tilde h}_{\mu \nu}}(p)\) giving rise to a vanishing linearized Einstein tensor can be eliminated by a gauge transformation.
For the tensorial component, when \(\mu \leq {\mu _{{\rm{crit}}}} = l_p^{- 1}{(120\pi)^{1/2}}{e^\gamma}\), where lp is the Planck length \((l_p^2 \equiv \kappa/8\pi)\), the first factor in Equation (154) vanishes for four complex values of p0 of the form ±ω and ±ω*, where ω is some complex value. This means that, in the corresponding propagator, there are two poles on the upper half-plane of the complex p0 plane and two poles in the lower half-plane. We will consider here the case in which μ < μcrit; a detailed description of the situation for μ ≥ μcrit can be found in Appendix A of [110]. The two zeros on the upper half of the complex plane correspond to solutions in spacetime coordinates, which exponentially grow in time, whereas the two on the lower half correspond to solutions exponentially decreasing in time. Strictly speaking, these solutions only exist in spacetime coordinates, since their Fourier transform is not well-defined. They are commonly referred to as runaway solutions and for \(\mu \sim l_p^{- 1}\) they grow exponentially in time scales comparable to the Planck time.
Consequently, in addition to the solutions with \(G_{ab}^{(1)}(x) = 0\), there are other solutions that in Fourier space take the form \(\tilde G_{ab}^{(1)}(p) \propto \delta ({p^2} - p_0^2)\) for some particular values of p0, but all of them exhibit exponential instabilities with characteristic Planckian time scales. In order to deal with those unstable solutions, one possibility is to make use of the order-reduction prescription [295], which we will briefly summarize in Section 6.5.3. Note that the p2 terms in Equations (153) and (154) come from two spacetime derivatives of the Einstein tensor, moreover, the p2 ln p2 term comes from the nonlocal term of the expectation value of the stress tensor. The order-reduction prescription amounts here to neglecting these higher derivative terms. Thus, neglecting the terms proportional to p2 in Equations (153) and (154), we are left with only the solutions, which satisfy \(\tilde G_{ab}^{(1)}(p) = 0\). The result for the metric perturbation in the gauge introduced above can be obtained by solving for the Einstein tensor, which in the Lorentz gauge of Equation (149) reads:
$$\tilde G_{ab}^{(1)}(p) = {1 \over 2}{p^2}\left({{{\tilde h}_{ab}}(p) - {1 \over 2}{\eta _{ab}}\tilde h_c^c(p)} \right).$$
(155)
These solutions for \({{\tilde h}_{ab}}(p)\) simply correspond to free linear gravitational waves propagating in Minkowski spacetime expressed in the transverse and traceless (TT) gauge. When substituting back into Equation (147) and averaging over the initial conditions we simply get the symmetrized quantum correlation function for free gravitons in the TT gauge for the state given by the Wigner distribution. As far as the intrinsic fluctuations are concerned, it seems that the order-reduction prescription is too drastic, at least in the case of Minkowski spacetime, since no effects due to the interaction with the quantum matter fields are left.
A second possibility, proposed by Hawking et al. [161, 162], is to impose boundary conditions, which discard the runaway solutions that grow unbounded in time. These boundary conditions correspond to a special prescription for the integration contour when Fourier transforming back to spacetime coordinates. As we will discuss in more detail in Section 6.5.2, this prescription reduces here to integrating along the real axis in the p0 complex plane. Following that procedure we get, for example, that for a massless conformally-coupled matter field with \(\bar \beta > 0\) the intrinsic contribution to the symmetrized quantum correlation function coincides with that of free gravitons plus an extra contribution for the scalar part of the metric perturbations. This extra-massive scalar renders Minkowski spacetime stable, but also plays a crucial role in providing a graceful exit in inflationary models driven by the vacuum polarization of a large number of conformal fields. Such a massive scalar field would not be in conflict with present observations because, for the range of parameters considered, the mass would be far too large to have observational consequences [162].
Induced metric fluctuations
Induced metric fluctuations are described by the second term in Equation (17). They are dependent on the noise kernel that describes the stress-tensor fluctuations of the matter fields,
$${\langle {{h_{ab}}(x){h_{cd}}(y)} \rangle _{{\rm{ind}}}} = {{{{\bar \kappa}^2}} \over N}\int {{d^4}{x{\prime}}{d^4}{y{\prime}}\sqrt {g({x{\prime}})g({y{\prime}})} G_{abef}^{{\rm{ret}}}(x,{x{\prime}}){N^{efgh}}({x{\prime}},{y{\prime}})G_{cdgh}^{{\rm{ret}}}(y,{y{\prime}}),}$$
(156)
where here we have written the expression in the large N limit, so that \(\bar \kappa = N\kappa\), where κ = 8πG and N is the number of independent free scalar fields. The contribution corresponding to the induced quantum fluctuations is equivalent to the stochastic correlation function obtained by considering just the inhomogeneous part of the solution to the Einstein-Langevin equation. We can make use of the results for the metric correlations obtained in Sections 6.3 and 6.4 for solving the Einstein-Langevin equation. In fact, one should simply take N = 1 to transform our expressions here to those of Sections 6.3 and 6.4 or, more precisely, one should multiply the noise kernel in these expressions by N in order to use those expressions here, as follows from the fact that we now have N independent matter fields.
As we have seen in Section 6.4, following [259], the Einstein-Langevin equation can be entirely written in terms of the linearized Einstein tensor. The equation involves second spacetime derivatives of that tensor and, in terms of its Fourier components, is given in Equation (121) as
$${F^{\mu \nu}}{}_{\alpha \beta}(p){\tilde G^{(1)\alpha \beta}}(p) = \bar \kappa {\tilde \xi ^{\mu \nu}}(p),$$
(157)
where we have now used the rescaled coupling \({\bar \kappa}\). The solution for the linearized Einstein tensor is given in Equation (133) in terms of the retarded propagator Dμνρσ(p) defined in Equation (129). Now this propagator, which is written in Equation (130), exhibits two poles in the upper half complex p0 plane and two poles in the lower half-plane, as we have seen analyzing the zeros in Equations (153) and (154) for the massless and conformally coupled case. The retarded propagator in spacetime coordinates is obtained, as usual, by taking the appropriate integration contour in the p0 plane. It is convenient in this case to deform the integration path along the real p0 axis so as to leave the two poles of the upper half-plane below that path. In this way, when closing the contour by an upper half-circle, in order to compute the anti-causal part of the propagator, there will be no contribution. The problem now is that when closing the contour on the lower half-plane, in order to compute the causal part, the contribution of the upper half-plane poles gives an unbounded solution, a runaway instability. If we adopt the Hawking et al. [161, 162] criterion of imposing final boundary conditions, which discard solutions growing unboundedly in time, this implies that we just need to take the integral along the real axis, as was done in Section 6.4.2. But now that the propagator is no longer strictly retarded, there are causality violations in time scales on the order of \(\sqrt N {l_p}\), which should have no observable consequences. This propagator, however, has a well-defined Fourier transform.
Following the steps after Equation (133), the Fourier transform of the two-point correlation for the linearized Einstein tensor can be written in our case as,
$${\langle \tilde G_{\mu \nu}^{(1)}(p)\tilde G_{\alpha \beta}^{(1)}({p\prime})\rangle _{{\rm{ind}}}} = {{{{\bar \kappa}^2}} \over N}{(2\pi)^4}{\delta ^4}(p + {p\prime}){D_{\mu \nu \rho \sigma}}(p){D_{\alpha \beta \lambda \gamma}}(- p){\tilde N^{\rho \sigma \lambda \gamma}}(p),$$
(158)
where the noise kernel Ñρσλγ(p) is given by Equation (125). Note that these correlation functions are invariant under gauge transformations of the metric perturbations because the linearized Einstein tensor is invariant under those transformations.
We may also use the order-reduction prescription, which amounts in this case to neglecting terms in the propagator, which are proportional to p2, corresponding to two spacetime derivatives of the Einstein tensor. The propagator then becomes a constant, and we have
$${\langle {\tilde G_{\mu \nu}^{(1)}(p)\tilde G_{\alpha \beta}^{(1)}({p{\prime}})} \rangle _{{\rm{ind}}}} = {{{{\bar \kappa}^2}} \over N}{(2\pi)^4}{\delta ^4}(p + {p{\prime}}){\tilde N_{\mu \nu \alpha \beta}}(p).$$
(159)
Finally, we may derive the correlations for the metric perturbations from Equations (158) or (159). In the Lorentz or harmonic gauge the linearized Einstein tensor takes the particularly simple form of Equation (155) in terms of the metric perturbation. One may derive the correlation functions for \({{\tilde h}_{\mu \nu}}(p)\) as it was done in Section 6.4.2 to get
$${\langle {{{\tilde \bar h}_{\mu \nu}}(p){{\tilde \bar h}_{\alpha \beta}}({p{\prime}})} \rangle _{{\rm{ind}}}} = {4 \over {{{({p^2})}^2}}}{\langle {{{\tilde G}_{\mu \nu}}(p){{\tilde G}_{\alpha \beta}}({p{\prime}})} \rangle _{{\rm{ind}}}}.$$
(160)
There will be one possible expression for the two-point metric correlation, which corresponds to the Einstein-tensor correlation of Equation (158), and another expression corresponding to Equation (158), when the order-reduction prescription is used. We should note that, contrary to the correlation functions for the Einstein tensor, the two-point metric correlation is not gauge invariant (it is given in the Lorentz gauge). Moreover, when taking the Fourier transform to get the correlations in spacetime coordinates, there is an apparent infrared divergence when p2 = 0 in the massless case. This can be seen from the expression for the noise kernel Ñμναβ(p) defined in Equation (125). For the massive case no such divergence due to the factor θ(− p2 − 4m2) exists, but as one takes the limit m → 0 it will show up. This infrared divergence, however, is a gauge artifact that has been enforced by the use of the Lorentz gauge. A gauge different from the Lorentz gauge should be used in the massless case; see [203] for a more detailed discussion of this point.
Let us now write the two-point metric correlation function in spacetime coordinates for the massless and conformally coupled fields. In order to avoid runaway solutions we use the prescription that the propagator should have a well-defined Fourier transform by integrating along the real axis in the complex p0 plane. This was, in fact, done in Section 6.4.3 and we may now write Equation (146) as
$${\langle {{{\tilde \bar h}_{\mu \nu}}(x){{\tilde \bar h}_{\alpha \beta}}(y)} \rangle _{{\rm{ind}}}} = {{{{\bar \kappa}^2}} \over {720\pi N}}\int {{{{d^4}p} \over {{{(2\pi)}^4}}}{{{e^{ip(x - y)}}{P_{\mu \nu \alpha \beta}}\theta (- {p^2})} \over {\vert 1 + (\bar \kappa/2){p^2}\tilde H(p;{{\bar \mu}^2}){\vert ^2}}},}$$
(161)
where the projector Pμναβ is defined in Equation (107). This correlation function for the metric perturbations is in agreement with the real part of the graviton propagator obtained by Tomboulis in [348] using a large N expansion with Fermion fields. Note that when the order-reduction prescription is used the terms in the denominator of Equation (161) that are proportional to p2 are neglected. Thus, in contrast to the intrinsic metric fluctuations, there is still a nontrivial contribution to the induced metric fluctuations due to the quantum matter fields in this case.
To estimate the above integral let us follow Section 6.4.3 and consider spacelike separated points x − y = (0,r) and introduce the Planck length lp. For space separations ∣r∣ ≫ lp we have that the two-point correlation (161) goes as \(\sim Nl_p^4\vert r{\vert ^4}\), and for \(\vert r\vert \sim \sqrt N {l_p}\) we have that it goes as \(\sim \exp (- \vert r\vert/\sqrt N lp){l_p}/\vert r\vert\). Since these metric fluctuations are induced by the matter stress fluctuations we infer that the effect of the matter fields is to suppress metric fluctuations at small scales. On the other hand, at large scales the induced metric fluctuations are small compared to the free graviton propagator, which goes like \(l_p^2/\vert r{\vert ^2}\).
We thus conclude that, once the instabilities giving rise to the unphysical runaway solutions have been discarded, the fluctuations of the metric perturbations around the Minkowski spacetime induced by the interaction with quantum scalar fields are indeed stable (instabilities lead to divergent results when Fourier transforming back to spacetime coordinates). We have found that, indeed, both the intrinsic and the induced contributions to the quantum correlation functions of metric perturbations are stable, and consequently Minkowski spacetime is stable.
Order-reduction prescription and large N
Runaway solutions are a typical feature of equations describing backreaction effects, such as in classical electrodynamics, and are due to higher than two time derivatives in the dynamical equations. Here we will give a qualitative analysis of this problem in semiclassical gravity. In a very schematic way the semiclassical Einstein equations have the form
$${G_h} + l_p^2{\ddot G_h} = 0,$$
(162)
where, say, Gh stands for the linearized Einstein tensor over the Minkowski background and we have simplified the equation as much as possible. The second term of the equation is due to the vacuum polarization of matter fields and contains four time derivatives of the metric perturbation. Some specific examples of such an equation are, in momentum space, Equations (153) and (154). The order-reduction procedure is based on treating perturbatively the terms involving higher-order derivatives, differentiating the equation under consideration, and substituting back the higher derivative terms in the original equation, keeping only terms up to the required order in the perturbative parameter. In the case of the semiclassical Einstein equation, the perturbative parameter is \(l_p^2\). If we differentiate Equation (162) with respect to time twice, it is clear that the second-order derivatives of the Einstein tensor are of order \(l_p^2\). Substituting back into the original equation, we get the following equation up to order \(l_p^2:\,{G_h} = 0 + O(l_p^4)\). Now there are certainly no runaway solutions but also no effect due to the vacuum polarization of matter fields. Note that the result is not so trivial when there is an inhomogeneous term on the right-hand side of Equation (162), this is what happens with the induced fluctuations predicted by the Einstein-Langevin equation.
Semiclassical gravity is expected to provide reliable results as long as the characteristic length scales under consideration, say L, satisfy that L ≫ lp [110]. This can be qualitatively argued by estimating the magnitude of the different contributions to the effective action for the gravitational field, considering the relevant Feynman diagrams and using dimensional arguments. Let us write the effective gravitational action, again in a very schematic way, as
$${S_{{\rm{eff}}}} = \int {{d^4}x\sqrt {- g} \left({{1 \over {l_p^2}}R + \alpha {R^2} + l_p^2{R^3} + \ldots} \right)},$$
(163)
where R is the Ricci scalar. The first term is the usual classical Einstein-Hilbert term. The second stands for terms quadratic in the curvature (square of Ricci and Weyl tensors). These terms appear as radiative corrections due to vacuum polarization of matter fields. Here α is a dimensionless parameter presumably of order 1 and the R3 terms are higher-order corrections, which appear, for instance, when one considers internal graviton propagators inside matter loops. Let us assume that R ∼ L−2; then the different terms in the action are on the order of R2 ∼ L−4 and \(l_p^2{R^3} \sim l_p^2{L^{- 6}}\). Consequently, when \(L \gg l_p^2\) the term due to matter loops is a small correction to the Einstein-Hilbert term \((1/l_p^2)R \gg {R^2}\) and this term can be treated as a perturbation. The justification for the order-reduction prescription is actually based on this fact. Therefore, significant effects from the vacuum polarization of the matter fields are only expected when their small corrections accumulate in time, as would be the case for an evaporating macroscopic black hole all the way before reaching Planckian scales (see Section 8.3).
However, if we have a large number N of matter fields, the estimates for the different terms change in a remarkable way. This is interesting because the large N expansion seems, as we have argued in Section 3.3.1, the best justification for semiclassical gravity. In fact, now the N vacuum-polarization terms involving loops of matter are of order NR2 ∼ NL−4. For this reason, the contribution of the graviton loops, which is just of order R2 as is any other loop of matter, can be neglected in front of the matter loops; this justifies the semiclassical limit. Similarly, higher-order corrections are of order \(Nl_p^2{R^3} \sim Nl_p^2{L^{- 6}}\). Now there is a regime, when \(L \sim \sqrt {N{l_p}}\), where the Einstein-Hilbert term is comparable to the vacuum polarization of matter fields, \((1/l_p^2)R \sim N{R^2}\), and yet the higher correction terms can be neglected because we still have L ≫ lp, provided N ≫ 1. This is the kind of situation considered in trace anomaly driven inflationary models [162], such as that originally proposed by Starobinsky [339], see also [355], where exponential inflation is driven by a large number of massless conformal fields. The order-reduction prescription would completely discard the effect from the vacuum polarization of the matter fields even though it is comparable to the Einstein-Hilbert term. In contrast, the procedure proposed by Hawking et al. keeps the contribution from the matter fields. Note that here the actual physical Planck length lp is considered, not the rescaled one, \(\bar l_p^2 = \bar \kappa/8\pi\), which is related to lp by \(l_p^2 = \kappa/8\pi = \bar l_p^2/N\).
Summary
An analysis of the stability of any solution of semiclassical gravity with respect to small quantum perturbations should include not only the evolution of the expectation value of the metric perturbations around that solution, but also their fluctuations encoded in the quantum correlation functions. Making use of the equivalence (to leading order in 1/N, where N is the number of matter fields) between the stochastic correlation functions obtained in stochastic semiclassical gravity and the quantum correlation functions for metric perturbations around a solution of semiclassical gravity, the symmetrized two-point quantum correlation function for the metric perturbations can be decomposed into two different parts: the intrinsic metric fluctuations due to the fluctuations of the initial state of the metric perturbations itself, and the fluctuations induced by their interaction with the matter fields. From the linearized perturbations of the semiclassical Einstein equation, information on the intrinsic metric fluctuations can be retrieved. On the other hand, the information on the induced metric fluctuations naturally follows from the solutions of the Einstein-Langevin equation.
We have analyzed the symmetrized two-point quantum correlation function for the metric perturbations around the Minkowski spacetime interacting with N scalar fields initially in the Minkowski vacuum state. Once the instabilities that arise in semiclassical gravity, which are commonly regarded as unphysical, have been properly dealt with by using the order-reduction prescription or the procedure proposed by Hawking et al. [161, 162], both the intrinsic and the induced contributions to the quantum correlation function for the metric perturbations are found to be stable [203]. Thus, we conclude that Minkowski spacetime is a valid solution of semiclassical gravity.