Linear response, validity of semiclassical gravity, and the stability of flat space

Both spectral functions are positive for both transverse tensor and scalar gravitational perturbations of flat space.

“both spectral functions are positive for both transverse tensor and scalar gravitational perturbations of flat space.”

The tensor dispersion relation is always satisfied by k2=0.

“Let us examine first the tensor dispersion relation. It clearly is always satisfied by k2 = 0.”

For the free scalar field, the spectral functions have support only when s > 4m2, corresponding to the two particle threshold for timelike gravitational fluctuations.

“For this case, the spectral functions have support only when s > 4m2, which corresponds to the two particle threshold for timelike gravitational fluctuations.”

The spectral representation of the vacuum polarization function is computed in n dimensional Minkowski spacetime.

“The spectral representation of the vacuum polarization function is computed in n dimensional Minkowski spacetime”

The equations (3.4) are covariant in form and therefore are non-unique up to linearized coordinate (gauge) transformations (3.11).

“The equations (3.4) are covariant in form and therefore are non-unique up to linearized coordinate (gauge) transformations”

From the explicit form of F(T) for a scalar field, Re[F(T)(k2/m2)] → 1/(960π2) ln(|k2|/m2) as |k2| → ∞.

“Re [ F (T ) ( k2 m2 )] → 1 960π2 ln ( |k2| m2 ) as |k2| → ∞ ,”

Since the projections onto scalar and tensor modes are linearly independent (orthogonal), the coefficients of the two projection operators must satisfy the linear response relation separately.

“Since the projections onto scalar and tensor modes are linearly independent (in fact, orthogonal), the coefficients of the two projection operators must satisfy the linear response relation separately.”

The coefficient of k2 at k2=0 is unchanged from the classical value by the quantum parameter α and vacuum polarization corrections.

“The coefficient of k2 at k2 = 0 is unchanged from the classical value by the quantum parameter α and vacuum polarization corrections.”

There are no tensor mode solutions of the linear response eq. (3.4) on length scales much larger than the Planck length for a massive field theory around flat space, other than the usual linearized gravitational waves of the classical theory.

“Thus, there are no tensor mode solutions of the linear response eq. (3.4) on length scales much larger than the Planck length for a massive field theory around flat space, other than the usual linearized gravitational waves of the classical theory.”

By making use only of the positivity of ρ(T), and assuming that α is non-negative, the bracket is strictly positive for k2 > 0, provided 16παG_N k2 + 1 > 0.

“Thus, by making use only of the positivity of ρ(T ), and assuming that α is non-negative we observe that the bracket is strictly positive for k2 > 0, provided”

• AssumptionThe authors state this as an assumption: α is non-negative.

The flat space solution to the semi-classical Einstein equations for n = 4 is stable to all perturbations on distance scales much larger than the Planck length.

“used to show that the flat space solution to the semi-classical Einstein equations for n = 4 is stable to all perturbations on distance scales much larger than the Planck length.”

The equations of motion (2.9) are fourth order in derivatives of the metric.

“The equations of motion (2.9), which are the original eqs. (1.1) modified by the additional terms required by the UV renormalization of 〈Tab〉, are fourth order in derivatives of the metric.”

In the scalar case the explicit form of the response function (B45) yields Re[F(S)(k2/m2)] → 1/(96π2)(1−6ξ)2 log(|k2|/m2) as |k2| → ∞.

“In the scalar case the explicit form of the response function (B45) yields”

• Also stated in: § IV. STABILITY OF FLAT SPACETIME · plain

  “Re [ F (S) ( k2 m2 )] → 1 96π2 (1− 6ξ)2 log ( |k2| m2 ) as |k2| → ∞ .”

When s = -k^2 > 4m^2, f develops an imaginary part.

“Finally when s = −k2 > 4m2, f develops an imaginary part,”

In the case that the curvature coupling takes its conformal value, ξ = 1/6, the scalar spectral function does not have terms proportional to s^2 or to m^2 s.

“In the case that the curvature coupling takes its conformal value, ξ = 1/6, the scalar spectral function does not have terms proportional to s2 or to m2s, and becomes”

One obtains then the same conclusion as for the tensor case, namely that there are no new solutions of the linear response equations, stable or unstable, far from the Planck regime, despite the opposite sign of the classical Einstein term in the scalar sector.

“One obtains then the same conclusion as for the tensor case, namely that there are no new solutions of the linear response equations, stable or unstable, far from the Planck regime, despite the opposite sign of the classical Einstein term in the scalar sector.”

The large N approximation (2.7) is equivalent to the UV renormalization group improved one-loop approximation.

“The large N approximation (2.7) is also invariant under changes in the ultraviolet renormalization scale (by definition of the UV cutoff dependence of the local counterterms which cancel against those of the matter action), and is equivalent to the UV renormalization group (RG) improved one-loop approximation.”

No state-dependent divergences occur in the linear response framework.

“This ensures that no state-dependent divergences occur.”

The renormalization of the effective action (2.7) suffices in principle to renormalize the equations of motion (2.9) and all of its higher variations.

“Thus, the renormalization of the effective action (2.7) suffices in principle to renormalize the equations of motion (2.9) and all of its higher variations”

The perturbations are manifestly causal.

“Hence the perturbations are manifestly causal.”

The three subtractions in (4.4) correspond physically to renormalizing the coefficients of the cosmological constant (Λ), Newton's constant (1/G_N), and the coefficients of the fourth order terms (α and β).

“The three subtractions in (4.4) correspond physically to renormalizing the coefficients of the cosmological constant (Λ), Newton's constant (1/G N ), and the coefficients of the fourth order terms (α and β).”

Solutions with k2 = −(k0)2 + |~k|2 > 0 correspond to unstable modes with imaginary frequencies.

“Solutions with k2 = −(k0)2 + |~k|2 > 0 correspond to unstable modes with imaginary frequencies, since we can always consider these modes in a frame where ~k = 0.”

In the regime l ≫ lPl, the effects of the higher order local terms in eq. (2.9) are suppressed by at least two powers of lPl/l, provided the solutions remain regular in the limit of vanishing renormalized coefficients α and β.

“In this regime, the effects of the higher order local terms in eq. (2.9) are suppressed by at least two powers of ℓPl/ℓ, provided the solutions remain regular in the limit of vanishing renormalized coefficients α and β of the higher order terms.”

The correlation function can be computed using the Closed Time Path (CTP) effective action.

“In this case, the correlation function can be computed using the Closed Time Path (CTP) effective action [43].”

• cites [43] J. Schwinger, J. Math. Phys. 2, 407 (1961);Not matched

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

The k2=0 solution corresponds to physical, transverse linearized gravitational waves propagating in a flat space background.

“This solution corresponds to the physical, transverse linearized gravitational waves propagating in a flat space background.”

Flat spacetime is a solution of the semi-classical Einstein equations for vanishing expectation value of Tab and cosmological term, with the quantum matter field in its Lorentz invariant vacuum ground state.

“Flat spacetime is a solution of the semi-classical Einstein equations for vanishing expectation value of Tab and cosmological term, with the quantum matter field in its Lorentz invariant vacuum ground state.”

The positivity of the spectral representation is sufficient to demonstrate that there are no unstable modes of the linearized semi-classical equations around flat space at distance scales far larger than the Planck scale.

“The positivity of the spectral representation is sufficient to demonstrate that there are no unstable modes of the linearized semi-classical equations around flat space at distance scales far larger than the Planck scale”

There is no experimental basis for taking the Planck-scale solutions seriously because they predict that even empty flat space is unstable to arbitrarily short length and time scale perturbations.

“There is clearly no experimental basis for taking these solutions seriously (since they would predict that even empty flat space is unstable to arbitrarily short length and time scale perturbations) [32,33,46,49].”

• cites [32] G. T. Horowitz, Phys. Rev. D 21, 1445 (1980).Not matched

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

• cites [33] R. D. Jordan, Phys. Rev. D 36, 3593 (1987).Not matched

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

• cites [46] W.-M. Suen, Phys. Rev. Lett. 62, 2217 (1989); Phys. Rev. D 40, 315 (1989).Not matched

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

• cites [49] E. E. Flanagan and R. M. Wald, Phys. Rev. D 54, 6233 (1996).Not matched

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

Hence the no-unstable-tensor-perturbations conclusion is valid much more generally than for the specific scalar field example.

“Hence it is valid much more generally than for the specific scalar field example.”

Therefore, these linearized gravitational waves carry the same energy density in the semi-classical approximation as they do in the classical Einstein theory.

“Therefore, these linearized gravitational waves carry the same energy density in the semi-classical approximation as they do in the classical Einstein theory.”

The flat space example shows explicitly that the semi-classical approximation does give mathematically meaningful and physically sensible results, when properly restricted to its range of validity at space and distance scales very much larger than the Planck scale.

“However, the flat space example treated in some detail in this paper shows explicitly that the semi-classical approximation does give mathematically meaningful and physically sensible results, when properly restricted to its range of validity at space and distance scales very much larger than the Planck scale.”

The divergences in 〈Tab〉 are in one-to-one correspondence with the local counterterms in the action Seff[g].

“The divergences in 〈Tab〉 are in one-to-one correspondence with the local counterterms in the action Seff [g], whose variations with respect to gab produce, in addition to the terms in the classical Einstein equations, the fourth order tensors,”

Flat spacetime is completely infrared stable in semi-classical gravity.

“and hence, that flat spacetime is completely infrared stable in semi-classical gravity.”

A necessary condition for the validity of the large N semi-classical equations of motion (2.9) is that the linear response equations (3.4) should have no solutions with finite non-singular initial data for which any linearized gauge invariant scalar quantity grows without bound.

“A necessary condition for the validity of the large N semi-classical equations of motion (2.9) is that the linear response equations (3.4) should have no solutions with finite non-singular initial data for which any linearized gauge invariant scalar quantity grows without bound.”

General coordinate invariance is assured, provided only that the matter effective action is regularized and renormalized in a manner which respects that invariance.

“General coordinate invariance is assured, provided only that the matter effective action is regularized and renormalized in a manner which respects that invariance [7].”

• cites [7] B. S. DeWitt, in Les Houches 1985, Proceedings, Architecture Fundamental Interactions at Short Distances, Vol. 2, 1023-Not checked yet

  Not checked in the free run.

The UV divergences found in the unrenormalized linear response equations are exactly those required to renormalize the semi-classical theory itself.

“Moreover, the UV divergences found in the unrenormalized linear response equations are exactly those required to renormalize the semi-classical theory itself.”

The exact solution of the matter Green's function equation without perturbative re-expansion and the resulting self-consistent solution of (2.9) for the metric constitute the principal non-perturbative RG improved feature of the large N limit.

“It is the exact solution of this equation without any perturbative re-expansion of G[g], and the resulting self-consistent solution of eq. (2.9) for the metric gab, that constitutes the principal non-perturbative RG improved feature of the large N limit.”

If the order of the equations is changed by adding higher derivative terms, the solutions of the modified equations fall into two classes: those that approach the solutions of the lower order equations as α, β → 0, and those that become singular in that limit.

“if the order of the equations is changed by adding higher derivative terms, the solutions of the modified equations fall into two classes, viz., those that approach the solutions of the lower order equations as the new parameters α, β → 0, and those which become singular in that limit.”

Since this conclusion relies only on the positivity of the spectral function ρ(T)(s), it requires only causality, a bounded Hamiltonian, and a well-defined positive Hilbert space norm for the quantum matter theory.

“Since this conclusion relies only on the positivity of the spectral function ρ(T )(s), it requires only causality, a bounded Hamiltonian, and a well-defined positive Hilbert space norm for the quantum matter theory.”

• AssumptionRests on an assumption: The quantum matter theory has causality, a bounded Hamiltonian, and a well-defined positive Hilbert space norm.

The asymmetric treatment of matter as quantum and the metric as classical can be justified formally by replicating the number of matter fields N times and taking the large N limit of the quantum effective action for the matter fields in an arbitrary background metric.

“This asymmetric treatment can be justified formally by replicating the number of matter fields N times and taking the large N limit of the quantum effective action for the matter fields in an arbitrary background metric gab [10].”

• cites [10] E. Tomboulis, Phys. Lett. B 70, 361 (1977).Not matched

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

When the matter effective action is regularized and renormalized in a manner respecting general coordinate invariance, the quantum expectation value of the matter energy-momentum tensor is necessarily conserved.

“In that case the quantum expectation value of the matter energy-momentum tensor 〈Tab〉 is necessarily conserved.”

Since the action principle is fundamental to the present approach, any transformation of the form (3.11) for which the action (2.7) is not invariant due to boundary or surface terms is not a true invariance and should be excluded from allowable gauge transformations of the linear response equations (3.4).

“Since the action principle is fundamental to the present approach, any transformation of the form (3.11), for which the action (2.7) is not invariant (due to boundary or surface terms), is not a true invariance and should be excluded from the set of allowable gauge transformations of the linear response equations (3.4).”

• AssumptionThe authors state this as an assumption: The action principle is fundamental to the present approach.

The appearance of m2 in the lower limit of the logarithm is a result of the definition of renormalized α parameter at k2=0, which allows no other scale to appear in the logarithm.

“However the appearance of m2 in the lower limit of the logarithm is a result of our definition of renormalized α parameter at k2 = 0, which allows no other scale to appear in the logarithm.”

The existence of any solutions to the linear response equations with unbounded growth in time that cannot be removed by an allowed linearized gauge transformation (3.11) implies that the influence of the growing gravitational fluctuations on the semi-classical background geometry is large and must be taken into account in the evolution of the background itself.

“The existence of any solutions to the linear response equations with unbounded growth in time, that cannot be removed by an allowed linearized gauge transformation (3.11), implies that the influence of the growing gravitational fluctuations on the semi-classical background geometry are large, and must be taken into account in the evolution of the background itself.”

These divergences, as well as the local term M, can be handled by the same renormalization procedure needed to define the expectation value of the dimension four operator Tab in (1.1), namely by subtraction of the allowed covariant counterterms up to dimension four.

“Thus, these divergences, as well as the local term M cd ab , can be handled by the same renormalization procedure needed to define the expectation value of the dimension four operator Tab in (1.1), namely by subtraction of the allowed covariant counterterms up to dimension four.”

The existence of these Planck-scale instabilities precludes the validity of the semi-classical large N approximation at Planck time or distance scales.

“Their existence clearly precludes the validity of the semi-classical large N approximation at Planck time or distance scales.”

Hence the relation (4.7) cannot be satisfied except at k2 approaching G_N^{-1}, provided m2 > 0.

“Hence the relation (4.7) cannot be satisfied except at k2 approaching G−1 N , provided m2 > 0.”

The UV renormalization of the energy-momentum tensor and the covariant form of the equations of motion (2.9) are justified by formal appeal to an underlying covariant action principle (2.7), whose variation they are.

“It is worth emphasizing that the UV renormalization of the energy-momentum tensor and the covariant form of the equations of motion (2.9) are justified by formal appeal to an underlying covariant action principle (2.7), whose variation they are.”

The renormalization of the expectation value of Tab requires the introduction of fourth order counterterms in the effective action that modify the geometric terms on the left hand side of eq. (1.1).

“The renormalization of its expectation value requires the introduction of fourth order counterterms in the effective action, that modify the geometric terms on the left hand side of eq. (1.1) [3].”

• cites [3] N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space, Cambridge University Press (Cambridge), 1982, and sourceNot checked — no full text of the source

If these polarization effects are significant, the semi-classical approximation has certainly broken down, at least in the form specified by eq. (1.1), where all fluctuations of the metric have been ignored.

“Hence, if these polarization effects are significant, the semi-classical approximation has certainly broken down, at least in the form specified by eq. (1.1), where all fluctuations of the metric have been ignored.”

The large N approximation preserves the covariance properties of the theory, since it can be derived from an invariant action functional (2.7).

“The large N approximation also preserves the covariance properties of the theory, since it can be derived from an invariant action functional (2.7).”

If the gravitational fluctuations around the background grow, even if initially small, then the leading order semi-classical equations (2.9), which neglect these fluctuations, must eventually break down.

“if the gravitational fluctuations around the background grow, even if they were initially small, then the leading order semi-classical equations (2.9), which neglect these fluctuations, must eventually break down.”

If, for a given state and background geometry that solves the semi-classical equations, one or more solutions to the linear response equations experience unbounded growth in a gauge invariant sense, then the semi-classical approximation is not valid for that particular geometry, at least not for that particular state.

“If, for a given state and background geometry that solves the semi-classical equations, one or more solutions to the linear response equations experience unbounded growth in a gauge invariant sense, then the semi-classical approximation is not valid for that particular geometry, at least not for that particular state.”

The matter energy-momentum tensor correlator by itself suffers from possible ambiguities at coincident points.

“Although the matter energy-momentum tensor correlator by itself suffers from possible ambiguities at coincident points”

The exact finite number of order unity on the right hand side of the inequality 4πG_N |k^2| ≪ O(1) determines the values of k^2 for which new Planckian solutions and instabilities will appear.

“The exact finite number of order unity on the right hand side of this inequality determines the values of k2 for which new Planckian solutions and instabilities will appear.”

Also as in the tensor case the lower limit of the logarithm can be made finite in the m2 → 0 limit by redefining the renormalized β coefficient at a finite μ2 ≠ 0 analogously to eq. (4.10).

“Also as in the tensor case the lower limit of the logarithm can be made finite in the m2 → 0 limit by redefining the renormalized β coefficient at a finite µ2 6= 0 analogously to eq. (4.10).”

The linear response criterion provides a natural and well-defined way for the two-point correlation function to enter into the determination of the validity of the semiclassical approximation.

“The linear response criterion provides a natural and well-defined way for this two-point correlation function to enter into the determination of the validity of the semi-classical approximation.”

The net effect of either the covariant or canonical analysis is to remove the overall k2=0 propagating solution from the scalar sector, as it is not dynamically allowed by the constraints.

“The net effect of either the covariant or canonical analysis is to remove the overall k2 = 0 propagating solution from the scalar sector, as it is not dynamically allowed by the constraints.”

Linear response involves quantities that lie entirely within the semiclassical approximation itself, since the polarization tensor is computed on the semiclassical background geometry.

“Further, linear response involves quantities that lie entirely within the semi-classical approximation itself, since the polarization tensor is computed on the semi-classical background geometry.”

This corresponds to only a finite renormalization of the βR2 term in the effective action.

“This corresponds to only a finite renormalization of the βR2 term in the effective action.”

Physically the reason is that the conformal factor is constrained by the diffeomorphism invariance.

“Physically the reason is that the conformal factor is constrained by the diffeomorphism invariance.”

The ambiguities at coincident points in the matter energy-momentum tensor correlator are removed by a proper covariant regularization and renormalization procedure, which ties these divergences in the correlator at x = x′ to counterterms in the purely local gravitational effective action.

“these are removed by a proper covariant regularization and renormalization procedure, which ties these divergences in the correlator at x = x′ to counterterms in the purely local gravitational effective action.”

The resulting combination of all terms in eq. (3.4) becomes well-defined.

“The resulting combination of all terms in eq. (3.4) then becomes well-defined.”

In the covariant effective action formulation, the UV renormalization counterterms are the same as those needed to define the semiclassical approximation itself.

“In the covariant effective action formulation, it is clear that the UV renormalization counterterms are the same as those needed to define the semi-classical approximation itself”

The existence of growing modes that violate the validity criterion proposed here informs us not that flat spacetime is unstable, but only that the quantum fluctuations of the geometry should be included in some consistent way at short scales.

“The existence of such growing modes, which violate the validity criterion proposed here, informs us not that flat spacetime is unstable, but only that the quantum fluctuations of the geometry should be included in some consistent way at short scales.”

Working out the flat space case in detail provides an important clue as to how the validity criterion may fail to be satisfied in more interesting cases.

“working out this case in detail also provides an important clue as to how the validity criterion may fail to be satisfied in more interesting cases.”

However, analysis of the fully subtracted response function behavior in (4.11) above leads to the same result.

“However, analysis of the fully subtracted response function behavior in (4.11) above leads to the same result.”

In the covariant effective action formulation, there are no state-dependent divergences.

“and that there are no state-dependent divergences.”

In the Hartle-Hawking state one can construct a static solution to the semiclassical equations (2.9) that is quite close to the classical one near the horizon.

“In the Hartle-Hawking state [57] one can construct a static solution to the semi-classical equations (2.9) that is quite close to the classical one near the horizon [58–60].”

• cites [57] J. B. Hartle and S. W. Hawking, Phys. Rev. D 13, 2188 (1976).Not matched

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

• cites [58] J. W. York, Jr., Phys. Rev. D 31, 775 (1985).Not matched

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

• cites [59] D. Hochberg, T. W. Kephart, and J.W. York, Jr., Phys. Rev. D 48, 479 (1993).Not matched

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

• cites [60] P. R. Anderson, W. A. Hiscock, J. Whitesell, and J. W. York Jr., Phys. Rev. D 50, 6427 (1994).Not matched

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

The value of the exact finite number of order unity depends on both the matter theory and the values of the renormalized coefficients α and β of the fourth order terms.

“Its value depends on both the matter theory and the values of the renormalized coefficients α and β of the fourth order terms.”

What is required for the validity criterion to fail is that the polarization tensor of the matter fluctuations become singular, i.e., large and unbounded, in some region.

“What is required is simply that the polarization tensor of the matter fluctuations become singular, i.e., large and unbounded, in some region.”

From several different analyses [53,54] it is known that there are no physical wavelike scalar excitations of flat space in either the classical or semi-classical theory.

“From several different analyses [53,54] it is known that there are no physical wavelike scalar excitations of flat space in either the classical or semi-classical theory.”

• cites [53]Not matched

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

• cites [54] P. O. Mazur and E. Mottola, Nucl. Phys. B 341, 187 (1990).Not matched

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

By conservation of Tab the correlator must be transverse, with zero projection onto the vector subspace.

“By conservation of Tab this correlator must be transverse, with zero projection onto the vector subspace, i.e., [62]”

• cites [62] D. G. Boulware and S. Deser, Jour. Math. Phys. 8, 1468 (1967);Not matched

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

The validity criterion proposed in this paper provides a clear dynamical principle for the stability or instability of the self-consistent solutions in both the black hole and de Sitter cases.

“The validity criterion proposed in this paper provides a clear dynamical principle for the stability or instability of the self-consistent solutions in both the black hole and de Sitter cases.”

• AssumptionRests on an assumption: The stability of the Hartle-Hawking self-consistent solution has not been investigated in a dynamical approach.

The conclusions about the absence of solutions to the transverse linear response equation (4.5a) at length scales much greater than the Planck length remain unchanged.

“The conclusions about the absence of solutions to the transverse linear response equation (4.5a) at length scales much greater than the Planck length remains unchanged.”

The semiclassical approximation which does not incorporate the effects of these quantum fluctuations on the mean geometry is not valid in the Planck regime.

“The semi-classical approximation which does not incorporate the effects of these quantum fluctuations on the mean geometry is certainly not valid in the Planck regime.”

Because they are conserved tensors derived from invariant action functionals, all the local tensors on the left hand side of the linear response eq. (3.4) must be expressible in terms of only the scalar and tensor components of the metric fluctuations.

“Because they are conserved tensors derived from invariant action functionals, all the local tensors on the left hand side of the linear response eq. (3.4) must be expressible in terms of only the scalar and tensor components of the metric fluctuations.”

This is a necessary, though perhaps not a sufficient condition for the validity of the approximation.

“Clearly this is a necessary, though perhaps not a sufficient condition for the validity of the approximation.”

The linear response eq. cannot be satisfied for either sign of k2 unless G_N |k2| becomes of order unity.

“which shows that the linear response eq. cannot be satisfied for either sign of k2 unless G N |k2| becomes of order unity.”

The large N method, augmented by the causal CTP formulation of the effective action, provides a well-defined framework for applying the validity criterion, which is equivalent to a stability criterion for the semiclassical solution.

“The large N method, augmented by the causal CTP formulation of the effective action, provides a well-defined framework for applying the validity criterion, which is equivalent to a stability criterion for the semi-classical solution.”