Induced quantum metric fluctuations and the validity of semiclassical gravity

May combine two claims.

Flat space is stable and a description based on SCG is a valid approximation in that case.

“We illustrate our criterion by showing explicitly that flat space is stable and a description based on SCG is a valid approximation in that case.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Eq. (18) corresponds to adding the stochastic source to the linearized semiclassical Einstein equation (7).

“which simply corresponds to adding the stochastic source to the linearized semiclassical Einstein equation ͑7͒, where F ␣␤ (p) was given by Eq. ͑8͒.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The Einstein-Langevin equation can be entirely written in terms of the linearized Einstein tensor G~(1)(p) as Eq. (18).

“Following Ref. ͓54͔, the Einstein-Langevin equation can be entirely written in terms of the linearized Einstein tensor G ˜ (1) (p) as follows:”

• cites [54]Not matched

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May combine two claims.

In addition to solutions with G_ab^(1)(x)=0, there are other solutions that in Fourier space take the form G~(1)(p) ∝ δ(p^2 - p0^2) for some particular values of p0^2, but they all exhibit exponential instabilities with Planckian characteristic timescales.

“As described there, in addition to the solutions with G ab (1) (x) ϭ0, there are other solutions that in Fourier space take the form G ˜ (1) (p)ϰ␦(p 2 Ϫp 0 2 ) for some particular values of p 0 2 , but they all exhibit exponential instabilities with Planckian characteristic timescales.”

• cites [3]Not matched

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• cites [21]Not matched

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May combine two claims.

The expectation value of the anticommutator is given by Eq. (14) and is the sum of intrinsic and induced fluctuations.

“On the other hand, the expectation value of the anticommutator is given by Eq. ͑14͒ and is the sum of two separate contributions: intrinsic and induced fluctuations.”

May combine two claims.

This propagator is no longer the retarded one since it exhibits causality violations at Planckian scales.

“However, it should be emphasized that this propagator is no longer the retarded one since it exhibits causality violations at Planckian scales.”

Classified as probably not a substantial claim. May combine two claims.

The second term on the right-hand side of Eq. (14) corresponds to the fluctuations of the metric perturbations induced by the fluctuations of the quantum matter fields and is given by Eq. (17).

“The second term on the right-hand side of Eq. ͑14͒ corresponds to the fluctuations of the metric perturbations induced by the fluctuations of the quantum matter fields and is given by”

May combine two claims.

Following the Hawking et al. procedure, for a massless conformally coupled scalar field with ᾱ=0 and β̄>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 which renders Minkowski spacetime stable but plays a crucial role in providing a graceful exit for infla

“Following that procedure we get, for example, that for a massless conformally coupled scalar field with ␣ ¯ϭ0 6 and ␤ ¯Ͼ0 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 which renders Minkowski spacetime stable but plays a crucial role in providing a graceful”

• cites [64]Not matched

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• cites [65]Not matched

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Probably not a substantial claim: may not pass S3 (load-bearing). The quote only partly states this claim.

The expression for the massless case can be obtained by adding a term ln(m2/μ2) to Eq. (B5) and subtracting the stated terms from 2ᾱ and 2β̄ in Eqs. (B2) and (B6) before taking the limit m2→0.

“Nevertheless, the expression for the massless case can still be obtained by adding a term ln(m 2 / 2 ) to Eq. ͑B5͒ and subtracting (1920 2 ) Ϫ1 ln( 2 /m 2 ) and (Ϫ1/6) 2 (96 2 ) Ϫ1 ln( 2 /m 2 ) respectively from 2␣ ¯and 2␤ ¯ in Eqs.͑B2͒ and ͑B6͒ before taking the limit m 2 →0.”

May combine two claims.

The noise kernel has in general a singular coincidence limit x→x' and is still singular for x≠x' when the two points are connected by a null geodesic, which translates into an ultraviolet divergence when integrating over momenta in Fourier space.

“The noise kernel defined by Eq. ͑12͒ has in general a singular coincidence limit x→xЈ ͑in fact, it is still singular even for x xЈ when the two points are connected by a null geodesic͒, which translates into an ultraviolet divergence when integrating over momenta in Fourier space, as can be seen from Eq. ͑B7͒.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The expression for the noise kernel in Fourier space is given by Eq. (B7).

“Finally, the expression for the noise kernel in Fourier space is given by”

May combine two claims.

Even though the noise kernel is not well defined as a tensor-valued function, it is well defined as a tensor-valued distribution and yields finite results when integrated with suitable test functions.

“In fact, even though the noise kernel is not well defined as a tensor-valued function,foot_6 it is well defined as a tensor-valued distribution and yields finite results when integrated with suitable test functions.”

May combine two claims.

Boundary terms at finite times t1 and t2 are finite, at least when t1≠t2; boundary terms corresponding to t1' and t2' both equal to ti are divergent.

“The contributions from the boundary terms at the finite times t 1 and t 2 are finite, at least when t 1 t 2 ͓the fact that t 1 t 2 may be required to get a finite result is simply indicating that expression ͑D2͒ is also a distribution͔.On the other hand, the boundary terms that correspond to taking t 1 Ј and t 2 Ј both equal to t i are divergent.”

May combine two claims.

The local kernel Mren is proportional to the Einstein tensor and can be absorbed in a finite renormalization of the gravitational coupling constant.

“In the case under consideration the local kernel M ren ␣␤( xϪy) is proportional to the Einstein tensor and can be absorbed in a finite renormalization of the gravitational coupling constant.”

May combine two claims.

Even if the massless case is excluded, the result would be finite, but it would become larger and larger as one chose a positive but arbitrarily small mass.

“In fact, even if we exclude the massless case, the result would be finite, but it would become larger and larger as we chose a positive but arbitrarily small mass.”

May combine two claims.

There are two separate contributions to the two-point correlation function: the first is related to the dispersion of the initial state for the metric perturbations, and the second is proportional to the noise kernel and accounts for fluctuations induced by interaction with the environment; these are called intrinsic and induced fluctuations.

“One can see that there are two separate contributions to the two-point correlation function: the first one is related to the dispersion of the initial state for the metric perturbations, whereas the second one is proportional to the noise kernel and accounts for the fluctuations induced by their interaction with the environment ͑in this case, the quantum matter fields͒.We refer to these two”

• AssumptionThe authors state this as an assumption: The initial states of the matter fields and the metric perturbations are initially uncorrelated.

May combine two claims.

These runaway solutions only exist in spacetime coordinates because their Fourier transform is not well defined, and for p ~ l_p^-1 they grow exponentially on time scales comparable to the Planck 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 ϳl p Ϫ1 they grow exponentially in time scales comparable to the Planck time.”

May combine two claims.

For epsilon < 0, the potential term of the negative-kinetic harmonic oscillator is also negative and the classical solutions do not exhibit instabilities, but configurations of that oscillator can have negative energies arbitrarily large in absolute value.

“For Ͻ0 the potential term of the harmonic oscillator with the negative kinetic term is also negative and the classical solutions do not exhibit instabilities.However, in any case the configurations for the harmonic oscillator with the negative kinetic term can have negative energies arbitrarily large in absolute value.”

May combine two claims. The quote only partly states this claim.

For p < crit, the two zeros on the upper half of the complex plane correspond to solutions exponentially growing in time, and the two on the lower half correspond to solutions exponentially decreasing in time.

“The two zeros on the upper half of the complex plane correspond to solutions in spacetime coordinates exponentially growing in time, whereas the two on the lower half correspond to solutions exponentially decreasing in time.”

May combine two claims. The quote only partly states this claim.

In the order reduction prescription for the semiclassical Einstein equation, the perturbative parameter is hbar or equivalently l_p^2; differentiating twice with respect to the background covariant derivative shows second-order derivatives of the Einstein tensor are of order hbar, and substituting back gives Eq. (E7) up to O(hbar^2) with no vacuum polarization effects left.

“In the case of the semiclassical Einstein equation, the perturbative parameter employed is ប or, equivalently, the square of the Planck length l p 2 ϭ/8.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

For the quantum Brownian motion model, substituting the solution X(t)=X0(t)+(G_ret • xi)(t) into the correlation function gives Eq. (A2), where the first term is intrinsic fluctuations from initial-state dispersion and the term proportional to the noise kernel reflects fluctuations induced by interaction with the environment.

“where the first contribution corresponds to the intrinsic fluctuations connected to the dispersion of the initial state of the system, and the term proportional to the noise kernel reflects the fluctuations induced by the system's interaction with the environment.”

May combine two claims.

For beta < 0, the scalar component solutions correspond to a tachyonic field with m^2 = -(12|beta|)^-1, exhibiting exponential growth or decrease in time for wavelengths larger than 4(3|beta|)^1/2 and oscillatory behavior for smaller wavelengths.

“for ␤ ¯Ͻ0 the solutions correspond to a tachyonic field with m 2 ϭϪ(12͉␤ ¯͉) Ϫ1 : in spacetime coordinates they exhibit an exponential behavior in timegrowing or decreasing-for wavelengths larger than 4(3͉␤ ¯͉) 1/2 and an oscillatory behavior for wavelengths smaller than 4(3͉␤ ¯͉) 1/2”

May combine two claims.

The frequency of the negative-kinetic oscillator is proportional to epsilon^-1/2 and diverges as epsilon -> 0; at the quantum level the theory gives negative Hamiltonian eigenvalues arbitrarily large in absolute value but can be formulated with negative-norm ghost states.

“Moreover, the frequency for that oscillator is proportional to Ϫ1/2 and diverges as →0.At the quantum level, such a theory also gives rise to negative eigenvalues of the Hamiltonian arbitrarily large in absolute value, but can be alternatively formulated in terms of a Hamiltonian without negative energies by introducing states with negative norm ͑commonly referred to as ghosts͒ ͓65͔.”

• cites [65]Not matched

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May combine two claims.

Imposing final boundary conditions that discard unboundedly growing solutions is equivalent to taking a different integration contour; the resulting propagator has no exponential instabilities but violates causality with characteristic timescale of order sqrt(N) l_p.

“The resulting propagator does not exhibit exponential instabilities, but gives rise to causality violations since D ␣␤ (xϪy) 0 for t x Ͻt y ͑the characteristic timescale of these causality violations is of order ͱNl p ).”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

For an ohmic environment, the kernel H_ren becomes local with H_ren(t,t')=M gamma delta'(t-t') and the homogeneous solution X0(t) takes the form of Eq. (A3).

“Then the kernel H ren becomes local with H ren (t,tЈ)ϭM ␥␦Ј(t ϪtЈ) and the homogeneous solution X 0 (t) takes on the following simple form:”

May combine two claims.

Effective action estimates: the Einstein-Hilbert term is order l_p^-2 R, vacuum polarization terms involving matter loops are order R^2, and higher loop corrections involving internal graviton propagators are order l_p^2 R^3 or higher.

“The Einstein-Hilbert term is of order l p Ϫ2 R ͑the characteristic curvature R is simply given by L Ϫ2 , where L is the characteristic length scale of our problem͒, the vacuum polarization terms involving loops of matter fields are of order R 2 , and higher loop corrections involving internal graviton propaga-tors are of order l p 2 R 3 or higher.”

May combine two claims.

Runaway solutions still exist in stochastic gravity and some method to deal with them is required.

“Runaway solutions, however, still exist and some method to deal with them is required.”

May combine two claims.

In the linearized semiclassical Einstein equation around Minkowski spacetime, the vectorial part vanishes, while the scalar and tensorial contributions satisfy Eqs. (E1) and (E2).

“Taking Eq. ͑7͒ and using a decomposition for the linearized Einstein tensor analogous to that introduced in Sec.IV C for the metric perturbation, the vectorial part is found to vanish,foot_7 whereas the scalar and tensorial contributions satisfy the equations”

May combine two claims.

For induced fluctuations, the retarded propagator D_ab(x-y)=0 for t_x < t_y, but for t_x > t_y it increases exponentially in time due to the two poles on the upper half of the complex plane.

“It exhibits the appropriate causal behavior: D ␣␤ (xϪy)ϭ0 for t x Ͻt y , as can be seen by closing the integration contour on the upper half of the complex plane.However, for t x Ͼt y it increases exponentially in time due to the contributions from the two poles on the upper half of the complex plane when closing the path on the lower half.”

May combine two claims.

With a large number N of fields, vacuum polarization terms involving matter loops become order N R^2, and higher loop corrections involving internal graviton propagators are order N l_p^2 R^3 or higher.

“The vacuum polarization terms involving loops of matter become of order NR 2 and, similarly, the higher loop corrections involving internal graviton propagators are of order Nl p 2 R 3 or higher”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Stochastic gravity in the broad sense entails the whole hierarchy of correlation functions, which would imply going beyond order 1/N in the generating functional.

“Thus, stochastic gravity in this broad sense entails the whole hierarchy of correlation functions, which would imply going beyond order 1/N in the generating functional.”

Probably not a substantial claim: may not pass S4 (based).

When metric perturbations are quantized, the semiclassical equation can be interpreted as the equation governing the evolution of the expectation value of the operator for the metric perturbations.

“When the metric perturbations are quantized, the semiclassical equation can be interpreted as the equation governing the evolution of the expectation value of the operator for the metric perturbations.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The actual implementations of stochastic gravity so far have been restricted to linear metric perturbations around the background geometry and a Gaussian stochastic source.

“Although technically the actual implementations of stochastic gravity so far ͑to which our present discussion ap-plies͒ have been restricted to linear metric perturbations around the background geometry and a Gaussian stochastic source, the theoretical construct of the stochastic gravity program has a much broader meaning beyond these limitations.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Following the order reduction approaches, Minkowski spacetime is shown to be a stable solution of SCG with respect to small metric perturbations.

“Following these approaches Minkowski spacetime is shown to be a stable solution of SCG with respect to small metric perturbations.”

• cites [21]Not matched

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Probably not a substantial claim: may not pass S3 (load-bearing).

Such a massive scalar field would not be in conflict with present observations because, for the range of parameters usually considered, the mass would be far too large to have observational consequences.

“͑such a massive scalar field would not be in conflict with present observations because, for the range of parameters usually considered, the mass would be far too large to have observational consequences ͓64͔͒.”

• cites [64]Not matched

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May combine two claims.

Information on the stability of the intrinsic fluctuations can be retrieved from an analysis of the solutions of the perturbed semiclassical Einstein equation, whereas the effect of the induced fluctuations is properly accounted for only in the stochastic semiclassical gravity framework based on the Einstein-Langevin equation.

“Whereas information on the stability of the intrinsic fluctuations can be retrieved from an analysis of the solutions of the perturbed semiclassical Einstein equation, the effect of the induced fluctuations is properly accounted for only in the stochastic semiclassical gravity framework based on the Einstein-Langevin equation.”

Classified as probably not a substantial claim.

A necessary condition for stability of a solution of SCG is that no gauge invariant scalar quantity constructed just from the linearized metric perturbation h_ab and its derivatives grows without bound.

“According to Ref. ͓3͔, a necessary condition for the stability of a solution of SCG requires that no gauge invariant scalar quantity constructed just from the linearized metric perturbation h ab ͑which satisfies the semiclassical Einstein equation linearized around the semiclassical solution under consider-ation͒ and its derivatives grows without bound.”

• cites [3]Not matched

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May combine two claims.

The solutions of Eq. (7) include linear gravitational waves propagating in Minkowski spacetime and additional solutions with characteristic time scales comparable to the Planck scale, which are usually regarded as unphysical.

“There is an obvious solution for G ˜ab (1) (p)ϭ0, which corresponds to linear gravitational waves propagating in Minkowski spacetime.In addition, there are solutions of the form G ˜ (1) ϰ␦(p 2 Ϫ p 0 2 ) for particular values of p 0 2 ͑positive or negative͒ comparable to m p 2 .Since they exhibit characteristic time scales or length scales comparable to the Planck scale, where semiclassical gravity”

• cites [16]Not matched

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Probably not a substantial claim: may not pass S3 (load-bearing).

The previous result for the expectation value of the stress tensor when small metric perturbations around a background solution of SCG are considered has been obtained in two ways: by linear response theory and by the influence functional formalism.

“The previous result for the expectation value of the stress tensor when small metric perturbations around a background solution of SCG are considered has been obtained in two ways: ͑1͒ by applying the usual techniques of linear response theory ͑see, for instance, ͓46͔͒ to SCG ͓3,47,48͔ and ͑2͒ by applying the influence functional formalism ͓49,50͔ to linearized metric perturbations regarded as an”

• cites [3]Not matched

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• cites [47]Not matched

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• cites [48]Not matched

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• cites [49]Not matched

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• cites [51]Not matched

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• cites [52]Not matched

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May combine two claims.

The closed-time-path formalism naturally yields true expectation values and causal evolution equations for their dynamics.

“The closed-time-path ͑CTP͒ formalism, which naturally yields true expectation values and causal evolution equations for their dynamics ͓35-39͔, should be used instead.”

• cites [35]Not matched

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• cites [39]Not matched

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Classified as probably not a substantial claim.

QFTCST is only an approximation in that the matter fields are treated as test fields evolving on a given spacetime.

“QFTCST is only an approximation in that the matter fields are treated as test field evolving on a given spacetime.”

• cites [32]Not matched

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• cites [33]Not matched

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Classified as probably not a substantial claim.

Horowitz found unstable solutions of the equations describing metric perturbations that grow exponentially with characteristic time scales comparable to the Planck time.

“Horowitz was the first one to analyze the equations describing those perturbations, which involved higher order derivatives ͑up to fourth order͒, and found unstable solutions that grow exponentially with characteristic time scales comparable to the Planck time ͓16,17͔.”

• cites [16]Not matched

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• cites [17]Not matched

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Probably not a substantial claim: may not pass S3 (load-bearing).

Stochastic semiclassical gravity can be understood as the Gaussian approximation to stochastic gravity.

“Stochastic semiclassical gravity can be understood as the Gaussian approximation to stochastic gravity.”

• cites [1]Not matched

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May combine two claims.

Although the expectation values of the commutator and anticommutator are related by a fluctuation-dissipation relation in some particular cases, that is not true in general, and the induced fluctuations need to be explicitly analyzed.

“Furthermore, although the expectation values of the commutator and anticommutator are related by a fluctuation-dissipation relation in some particular cases ͓51,54͔, that is not true in general and the induced fluctuations need to be explicitly analyzed.”

• cites [51]Not matched

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• cites [54]Not matched

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Probably not a substantial claim: may not pass S3 (load-bearing).

The Einstein-Langevin equation was previously derived using a formal analogy with open quantum systems and the influence functional formalism.

“The Einstein-Langevin equation had been previously derived making use of a formal analogy with open quantum systems and employing the influence functional formalism ͓49,50͔.”

• cites [49]Not matched

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Probably not a substantial claim: may not pass S3 (load-bearing).

The method employed in Ref. [3], although slightly different, yields the same result.

“The method employed in Ref. ͓3͔, although slightly different, yields the same result.”

• cites [3]Not matched

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Probably not a substantial claim: may not pass S3 (load-bearing).

Jordan reanalyzed the equations and reached similar conclusions about unstable solutions.

“This was later reanalyzed by Jordan with similar conclusions ͓18͔.”

• cites [18]Not matched

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May combine two claims.

Induced fluctuations are proportional to the noise kernel and are induced by the quantum fluctuations of the matter field stress tensor operator.

“The former are connected to the dispersion of the initial state of the metric perturbations, whereas the latter are proportional to the noise kernel and are induced by the quantum fluctuations of the matter field stress tensor operator.”

Classified as probably not a substantial claim. May combine two claims.

The noise kernel needed for the discussion is for a Minkowski background spacetime with a scalar field in the Minkowski vacuum, and it was obtained in Ref. [54] and is given by Eq. (B7).

“The noise kernel that we need for our discussions is for the particular case of a Minkowski background spacetime with a scalar field in the Minkowski vacuum.It was obtained in Ref. ͓54͔ and is given by Eq. ͑B7͒ in Appendix B.”

• cites [54]Not matched

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May combine two claims.

In reality N is finite and the semiclassical Einstein equation can only be understood as the lowest order contribution in a 1/N expansion.

“Of course, in reality N is finite and the semiclassical Einstein equation can only be understood as the lowest order contribution in a 1/N expansion.”

Probably not a substantial claim: may not pass S4 (based).

Einstein's theory requires the spacetime dynamics to determine and be determined by the matter fields.

“Einstein's theory requires the spacetime dynamics to determine and be determined by the matter fields.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Ford and collaborators have considered both intrinsic and induced fluctuations, usually referred to as active and passive fluctuations respectively.

“They have considered both intrinsic ͓6,9-11͔ and induced fluctuations ͓4 -8͔, which they usually refer to as active and passive fluctuations respectively.”

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Probably not a substantial claim: may not pass S3 (load-bearing).

The analogy between SCG and the equation for the expectation value of the electromagnetic field to leading order in a large N expansion for N charged quantum fields has been discussed by a number of authors.

“In fact, the analogy between SCG and the equation for the expectation value of the electromagnetic field to leading order ͑order 1͒ in a large N expansion for N charged quantum fields has been discussed by a number of authors ͓17,18,34͔.”

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Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

The Lorentz gauge condition does not fix completely the gauge freedom under local diffeomorphisms, but it can be completely fixed by imposing additional gauge fixing conditions on the state of the metric perturbations at the initial time.

“The Lorentz gauge condition does not fix completely the gauge freedom under local diffeomorphisms.However, it can be completely fixed by imposing additional gauge fixing conditions on the state of the metric perturbations at the initial time.”

May combine two claims.

The stability of the intrinsic contribution is closely related to the stability of linearized perturbations of the semiclassical Einstein equation, whereas the induced contribution is equivalent to the stochastic correlation functions in stochastic semiclassical gravity obtained from solutions of the Einstein-Langevin equation.

“The stability of the first contribution turns out to be closely related to the stability of linearized perturbations of the semiclassical Einstein equation, whereas the second contribution is equivalent to the stochastic correlation functions in stochastic semiclassical gravity obtained from solutions of the Einstein-Langevin equation.”

May combine two claims.

The leading order contribution to WCTP is entirely given by WCTP(LO), which is of order 1/N and from which the leading order contribution to all quantum correlation functions with an even number of points can be obtained.

“Therefore, one can conclude that the leading order contribution to W CTP is entirely given by W CTP (LO) , which is of order 1/N and from which the leading order contribution to all the quantum correlation functions with an even number of points can be obtained.”

May combine two claims.

In the gravitational case, the dissipation kernel will not damp the intrinsic fluctuations, but the induced ones will still play an important role.

“In a more general context, such as the gravitational case, the dissipation kernel will not damp the intrinsic fluctuations, but the induced ones will still play an important role.”

May combine two claims.

To preserve conservation of the source in the Einstein-Langevin equation, the interaction has to be turned on adiabatically and asymptotically past initial conditions are required.

“However, in order to preserve the conservation of the source in the Einstein-Langevin equation, which guarantees the integrability of the equation through the Bianchi identity, the interaction has to be turned on adiabatically and asymptotically past initial conditions are required.”

May combine two claims.

That neglect is indeed the case when considering a Minkowski background if the order reduction prescription is employed, but it is not clear whether it remains true under more general conditions.

“That is indeed the case when considering a Minkowski background if the order reduction prescription is employed, but it is not clear whether it remains true under more general conditions.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Tomboulis considered fermionic rather than scalar fields, but that just amounts to a change in one coefficient.

“͑he actually considered fermionic rather than scalar fields, but that just amounts to a change in one coefficient͒.”

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Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Ford and collaborators have stressed the importance of the metric fluctuations and investigated some of their physical implications.

“As we pointed out in the Introduction, Ford and collaborators have stressed the importance of the metric fluctuations and investigated some of their physical implications ͓4 -11͔.”

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• cites [5]Not matched

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• cites [8]Not matched

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• cites [9]Not matched

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• cites [10]Not matched

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Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

The main result can be generalized to nonminimally coupled scalar fields or even vectorial and fermionic fields.

“In particular we will consider N minimally coupled free scalar fields, but the main result can be generalized to nonminimally coupled scalar fields or even vectorial and fermionic fields.”

May combine two claims.

In the gravitational case, the appropriate way to deal with the singular coincidence limit of the noise kernel is by integrating over some smearing function; smearing in time is needed to get a finite result.

“There are some situations, such as the effect of stress tensor fluctuations on the propagation of null geodesics, in which the appropriate way to deal with the singular coincidence limit of the noise kernel is by integrating over some smearing function ͓8͔ ͑in general smearing just along the spatial directions is not enough: smearing in time is needed to get a finite result͒.”

• cites [8]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

Tomboulis's propagator is equivalent to the time-ordered CTP propagator when asymptotic initial conditions are considered because in Minkowski spacetime there is no real particle creation and the in and out vacua are equivalent up to a phase.

“Nevertheless, his propagator is equivalent to the time-ordered CTP propagator when asymptotic initial conditions are considered because in Minkowski spacetime there is no real particle creation and the in and out vacua are equivalent ͑up to some phase which is absorbed in the usual normalization of the in-out propagator͒.”

• cites [67]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

One gets an infrared divergence for the massless case when computing the inverse Fourier transform for the induced contribution to the correlation function of the metric, but that seems to be purely a gauge effect.

“In fact, one gets an infrared divergence for the massless case when computing the inverse Fourier transform for the induced contribution to the correlation function of the metric, but that seems to be purely a gauge effect, as argued in footnote 7.”

Classified as probably not a substantial claim.

Ford and collaborators justify neglecting the nonlocal term by arguing that those terms would be of higher order in a perturbative expansion.

“Their justification is by arguing that those terms would be of higher order in a perturbative expansion.”

• cites [4]Not matched

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Probably not a substantial claim: may not pass S3 (load-bearing).

In Ref. [59] it was shown that, even in the absence of decoherence, a stochastic description based on a Langevin type equation contains nontrivial information on fully quantum properties of simple linear open quantum systems.

“Later, in Ref. ͓59͔ it was shown that, even in the absence of decoherence, a stochastic description based on a Langevin type equation contains nontrivial information on fully quantum properties of simple linear open quantum systems.”

• cites [59]Not matched

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Probably not a substantial claim: may not pass S3 (load-bearing).

The two-point quantum correlation functions for the electromagnetic field to leading order in 1/N are completely analogous to the quantum correlation functions for the metric perturbations considered here.

“The two-point quantum correlation functions ͑the CTP propagators͒ for the electromagnetic field to leading order in 1/N ͑order 1/N), which play a crucial role there, are completely analogous to the quantum correlation functions for the metric perturbations considered here.”

Probably not a substantial claim: may not pass S3 (load-bearing).

In order to deal with the singular coincidence limit of the noise kernel, in Ref. [5] Kuo and Ford opted to subtract a number of terms including the fluctuations for the Minkowski vacuum.

“Finally, in order to deal with the singular coincidence limit of the noise kernel, in Ref. ͓5͔ Kuo and Ford opted to subtract a number of terms including the fluctuations for the Minkowski vacuum.”

• cites [5]Not matched

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May combine two claims.

Equation (C15) can be regarded as a stochastic Langevin equation and coincides with the Einstein-Langevin equation expressed in the Lorentz gauge.

“Equation ͑C15͒ can then be regarded as a stochastic Langevin equation and coincides with the Einstein-Langevin equation expressed in the Lorentz gauge”

May combine two claims.

Assuming a Gaussian initial state with vanishing expectation value for the metric perturbations, the generating functional in Eq. (C17) becomes Gaussian and any other n-point quantum correlation function has a simple expression in terms of the two-point functions.

“Moreover, assuming a Gaussian initial state with vanishing expectation value for the metric perturbations, the expression for the generating functional in Eq. ͑C17͒ becomes Gaussian and any other n-point quantum correlation function has a simple expression in terms of the two-point functions.”

• AssumptionThe authors state this as an assumption: The initial state for the metric perturbations is Gaussian with vanishing expectation value.

May combine two claims.

ZCTP(LO) is also the generating functional for the stochastic correlation functions for the solutions of the Einstein-Langevin equation, and therefore the stochastic correlation functions are equivalent to quantum correlation functions for the metric perturbations.

“Furthermore, Z CTP (LO) ͓J ab ⌺ ϭ0,J cd ⌬ ͔ is also the generating functional for the stochastic correlation functions for the solutions of the Einstein-Langevin equation and, therefore, the stochastic correlation functions are actually equivalent to quantum correlation functions for the metric perturbations.”

• AssumptionThe authors state this as an assumption: The initial states of the matter fields and the metric perturbations are initially uncorrelated.

May combine two claims.

Higher order corrections not included in semiclassical gravity are negligible provided L >> l_p; in that regime vacuum polarization terms only yield a small correction to the Einstein-Hilbert term and any classical gravitational source.

“Thus, we see that the higher order corrections not included in SCG are negligible provided that Lӷl p .In that regime, however, the vacuum polarization terms only yield a small correction to the Einstein-Hilbert term and any classical gravitational source which were present.”

May combine two claims.

There is a regime where vacuum polarization of matter fields and the Einstein-Hilbert term are comparable when L ~ sqrt(N) l_p, while higher loop corrections remain much smaller if L >> l_p; both conditions are compatible if N is very large.

“There is then a regime in which the vacuum polarization of the matter fields and the Einstein-Hilbert term are comparable when L ϳͱNl p .On the other hand, the higher loop corrections will still be much smaller if Lӷl p .Both conditions are compatible provided that the number of fields, N, is very large.”

Probably not a substantial claim: may not pass S3 (load-bearing).

If the counterterms quadratic in curvature are taken into account, the generalized canonical formalism and corresponding Wigner functional should be used.

“If the counterterms quadratic in the curvature, which give rise to higher order derivative terms, are also taken into account, the generalized canonical formalism referred to above and the corresponding Wigner functional should be used.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Hawking et al. argued that well-behaved results can be obtained by imposing boundary conditions that discard solutions growing unboundedly in time; these conditions can be implemented by a suitable integration contour, but causality is violated at small time scales.

“Hawking et al. have argued that well-behaved results can still be obtained by imposing boundary conditions which discard solutions which grow unboundedly in time ͓64,65͔.Those conditions can be implemented by a suitable choice of the integration contour on the complex plane when computing the inverse Fourier transform of the propagator, but causality is violated at small time scales”

• cites [64]Not matched

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• cites [65]Not matched

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Probably not a substantial claim: may not pass S1 (asserted), S3 (load-bearing).

It was suggested that introducing a finite 3-momentum cutoff and a running coupling constant small enough at low energies could circumvent the Landau pole problem at least practically.

“In fact, it was suggested in Ref. ͓45͔ that by introducing a finite 3-momentum cutoff and considering a running coupling constant small enough at low energies, the problem with the Landau pole could be circumvented ͑at least from a practical point of view͒.”

• cites [45]Not matched

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May combine two claims. The quote only partly states this claim.

Applying Hawking et al.'s final-boundary-condition method to intrinsic fluctuations: tensorial growing solutions are discarded, decreasing ones vanish with regular initial conditions at asymptotic initial time; for the scalar part, beta < 0 is analogous, while beta > 0 oscillatory solutions are not discarded.

“The exponentially growing solutions are discarded when the final boundary condition is imposed, and the contributions from the exponentially decreasing ones also vanish if regular initial conditions are specified at an asymptotic initial time.”

May combine two claims. The quote only partly states this claim.

The absence of the dissipation kernel allows specifying initial conditions at a finite initial time for the homogeneous solutions and the retarded propagator, but asymptotic initial conditions are still forced for induced fluctuations due to the singular coincidence limit of the noise kernel.

“Furthermore, the absence of the dissipation kernel also allows the possibility of specifying initial conditions at a finite initial time as far as the homogeneous solutions ͑relevant for the computation of the intrinsic fluc-tuations͒ and the retarded propagator are concerned.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

In Ref. [59], a stochastic description based on a Langevin-type equation could be used to gain information on quantum properties of the open system; the symmetrized two-point quantum correlation function for the system is equivalent to the correlation function obtained in the stochastic description.

“In Ref. ͓59͔ it was shown that a stochastic description based on a Langevin type equation could be used to gain information on the quantum properties of the open system.”

• cites [59]Not matched

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May combine two claims.

Under order reduction, the Einstein-Langevin equation becomes Eq. (E8), the stochastic source is unaffected, so induced fluctuations still have a nontrivial contribution from quantum matter polarization but no contribution from the dissipation kernel.

“Therefore, in contrast to the intrinsic fluctuations, there will still be a nontrivial contribution to the induced fluctuations due to the polarization of the quantum matter fields, but no contribution from the dissipation kernel is left in the Einstein-Langevin equation.”

May combine two claims.

Since the linearized semiclassical Einstein equation coincides with the homogeneous part of the Einstein-Langevin equation, Eq. (E7) governs the contribution of intrinsic fluctuations to the quantum correlation function, which coincides with that of free gravitons.

“Since the linearized semiclassical Einstein equation coincides with the homogeneous part of the Einstein-Langevin equation, Eq. ͑E7͒ governs the contribution of the intrinsic fluctuations to the quantum correlation function, which coincides with that of free gravitons.”

May combine two claims.

It seems unlikely that a similar cutoff procedure could work for the gravitational case due to the existence of higher derivatives; introducing a 3-momentum cutoff would break general covariance and pose serious difficulties for a consistent renormalization scheme in general curved spacetimes.

“Yet it seems unlikely that a similar procedure could work for the gravitational case due to the existence of higher derivatives.Moreover, introducing a 3-momentum cutoff would break general covariance and that would pose serious difficulties when implementing a consistent and natural renormalization scheme in general curved spacetimes.”

May combine two claims.

This propagator is the only one with a well-defined Fourier transform; it was employed in Ref. [54], where it was argued any other propagator should yield an equivalent result for the correlation function.

“This propagator is the only one which has a well-defined Fourier transform.It was employed in Ref. ͓54͔, where it was argued that any other propagator should yield an equivalent result for the correlation function obtained by solving the Einstein-Langevin equation.”

• cites [54]Not matched

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May combine two claims.

The order reduction prescription would completely discard the effect from vacuum polarization of matter fields even though it is comparable to the Einstein-Hilbert term, whereas Hawking et al.'s procedure keeps the contribution from matter 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.”

May combine two claims.

Hawking et al.'s method, though applied to quantum propagators, can also be used when solving the semiclassical Einstein equation and the Einstein-Langevin equation; the stochastic correlation functions obtained are equivalent to the quantum correlation functions (CTP propagators) from the prescription.

“In fact, Hawking et al. applied their method to quantum propagators, but, as we have described, it can also be used when solving the semiclassical Einstein equation and the Einstein-Langevin equation.The stochastic correlation functions obtained are then equivalent to the quantum correlation functions ͑CTP propagators͒ which would result from the application of the prescription.”