General relativity as an effective field theory: The leading quantum corrections

The low-energy fluctuations are very weakly coupled, with a typical strength κ^2 q^2 ∼ 10^-40 [10^-70] for q^2 ∼ 1/(1 fm)^2 [q^2 ∼ 1/(1 m)^2].

“However, the low energy fluctua- tions are very weakly coupled, with a typical strength κ2q2 ∼ 10−40[10−70] for q2 ∼ 1/(1fm)2 [q2 ∼ 1/(1m)2].”

The full matrix element has the form M_full = A q^2 (1 + α κ^2 q^2 + β κ^2 q^2 ln(-q^2) + ...).

“Mfull = A q2 ( 1 + ακ2q2 + βκ2q2 ln (−q2) + . . . )”

• AssumptionThe authors state this as an assumption: Gravitons are the only low-energy particles which are remnants of the full gravity theory.

The diagrams of Fig. 4d, 4e, and 4f do not have any nonanalytic terms of the form considered here because the matter fields are massive or the loop integrals are independent of q2.

“The diagrams of Fig. 4d,e,f do not have any non-analytic terms of the form considered here because the matter fields are massive or the loop integrals are independent of q2.”

The nonanalytic terms lead to power law behavior, as shown by the Fourier transforms in Eq. 60.

“The non analytic terms however lead to power law behavior since ∫ d3q (2π)3 e−i~q·~r 1 q = 1 2π2r2 ∫ d3q (2π)3 e−i~q·~rln~q2 = −1 2π2r3 (60)”

The resulting nonanalytic contributions to F1 and F2 are given by Eq. 68.

“The resulting non- analytic contributions to F1, F2 are then F1(q 2) = 1 + κ2 32π2 q2 [ −3 4 ln(−q2) + 1 16 π2m√ −q2 ] F2(q 2) = κ2m2 32π2 [ −4 3 ln(−q2) + 7 8 π2m√ −q2 ] (68)”

In the static limit, the interaction takes the approximate form given in Eq. 58.

“≈ κ2m2 1m 2 2 2 { 1 q2 + 2 (d1 − 2d2 − 2d3) +κ2 ( (2ℓ1 − ℓ4) + (2ℓ2 − ℓ5)ln ( −q2 µ2 ) + (2ℓ3 − ℓ6) √ m2 −q2 )} (58)”

Any factor of k2 or (k−q)2 in the numerator automatically removes any nonanalytic behavior.

“For example, any factor of k2 or (k−q)2 in the numerator automatically removes any non-analytic behavior.”

The gravitational logs in the diagram are given by Eq. 71.

“This is therefore equivalent to the gravitational logs in the diagram being given by Παβ,γδ = − κ2 32π2 ln(−q2) { 21 120 q4Iαβγδ + 23 120 q4ηαβηγδ − 23 120 (ηαβqγqδ + ηγδqαqβ) + 21 240 (qαqδηβγ + qαqγηβδ + qβqγηαδ + qβqδηαγ) + 11 30 qαqβqγqδ } + (nonlogs) (71)”

Unless an infinite number of accidents occur, at least some of the higher-order coefficients will be nonzero.

“Unless an infinite number of”

For loops involving low-order terms in the energy expansion, the renormalization involves the coefficients which appear at higher order.

“Moreover, it can be shown that for loops involving low order terms in the energy expansion, that the renormalization involves the coefficients which appear at higher order.”

Small quantum fluctuations at ordinary energies behave normally in perturbation theory.

“Since small quantum fluctuations at or- dinary energies behave normally in perturbation theory, it is natural to try to separate these quantum corrections from the problematic (and most likely incorrect) high energy fluctuations.”

The class of leading quantum corrections is generally dominant at large distances over the other one-loop gravitational corrections.

“Moreover this class, the”

• AssumptionThe authors state this as an assumption: Gravitons are the only low-energy particles which are remnants of the full gravity theory.

The involvement of higher powers of R at higher loops is a consequence of the structure of the energy expansion in a massless theory.

“This is a consequence of the structure of the energy expansion in a massless theory.”

Low-energy particles, whether massless or with mass comparable to or less than the external energy scale, cannot be integrated out of the theory but must be included explicitly in the quantum calculations.

“The low energy particles (massless ones or ones whose mass is comparable to or less than the external energy scale) cannot be integrated out of the theory but must be included explicitly in the quantum calculations.”

The quantum effects of massless particles cannot all be accounted for by local Lagrangian shifts.

“On the other hand, the quantum effects of massless particles cannot all be accounted for in this way.”

Reasonable values of c1 and c2 do not influence physics at low energies.

“We can instead view the gravitational action as being organized in an energy expansion, and then reasonable values of c1, c2 do not influence physics at low energies.”

Nonanalytic effects can be dominant in magnitude over analytic corrections.

“Moreover, the nonanalytic effects can be dominant in magnitude over analytic corrections.”

• AssumptionThe authors state this as an assumption: Gravitons are the only low-energy particles which are remnants of the full gravity theory.

The divergences can be absorbed into renormalized values for the unknown coefficients which appear in the general Lagrangian.

“They can then be absorbed into renormalized values for the unknown coefficients which appear in this general Lagrangian.”

Quantum corrections involving the known elementary particles will generate corrections to c1, c2, etc.

“For example, quantum corrections involving the known elementary particles (whether or not gravity itself is quantized) will generate corrections to c1, c2 etc.”

The low-energy propagation of massless particles leads to long-distance quantum corrections which are distinct from the effects of the local effective Lagrangian.

“However the low energy propagation of massless particles leads to long distance quantum corrections which are distinct from the effects of the local effective Lagrangian.”

• AssumptionThe authors state this as an assumption: Gravitons are the only low-energy particles which are remnants of the full gravity theory.

On distance scales much larger than 1/M_H, the effects of virtual heavy particles look local, as if they were described by a local Lagrangian.

“On distance scales much larger than this, their effects will look local, as if they were described by a local Lagrangian.”

Unless c1R or c1R^λ_;λ/R ≥ 10^70 m^-2, the influence of the c1R^2 terms is negligible.

“Unless c1R or c1R λ ;λ /R ≥ 1070m−2 the influence of the c1R 2 terms is negli- gible.”

A graviton loop will have a logarithmic nonanalytic modification around flat space.

“As we will see in the next section, a graviton loop will have a logarithmic nonanalytic modification around flat space.”

• AssumptionThe authors state this as an assumption: Gravitons are the only low-energy particles which are remnants of the full gravity theory.

There is a class of quantum predictions that are parameter-free other than Newton's constant G and that dominates over other quantum predictions in the low-energy limit.

“In the course of our study we will find a class of quantum predictions which are parameter free (other than Newton's constant G) and which domi- nates over other quantum predictions in the low energy limit.”

• AssumptionThe authors state this as an assumption: Gravitons are the only low-energy particles which are remnants of the full gravity theory.

Classes II and III corrections, if present, are most often larger than the gravitational leading quantum corrections (Class I) because their form need not be expansions in the small quantity GN.

“Classes II and III corrections (if present) are most often larger than the gravitational leading quantum corrections (Class I) because their form need not be expansions in the small quantity GN .”

At two loops, the divergence of pure gravity is given by the standard two-loop pure-gravity divergence expression.

“At two loops, the divergence of pure gravity is[16]”

• cites [16] [16] M. Goroff and A. Sagnotti, Nucl. Phys. B266, 799 (1986).Not matched

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

At one loop, the divergences due to gravitons have the form given by the standard one-loop gravity divergence expression.

“At one loop, the divergences due to gravitons have the form [3]”

• cites [3]Not matched

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

Within the effective field theory framework, gravity is a well-behaved quantum field theory at ordinary energies.

“Within this framework, gravity is a well behaved quantum field theory at ordi- nary energies.”

• Also stated in: § 1 Introduction · plain

  “We will explore quantum gravity as an effective field theory and find no obstacle to its successful implementa- tion.”

J. Simon has shown that the problems with R+R^2 theories do not appear when the theory is treated as an effective field theory.

“However J. Simon[14] has shown that these problems do not appear when the theory is treated as an effective field theory.”

• cites [14] [14] J. Simon, Phys. Rev. D41, 3720 (1990); 43, 3308 (1991).Not matched

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

Class III quantum predictions may occur for analytic terms in the energy expansion that are accompanied by a parameter with dimension 1/(mass)^n, n ≥ 1.

“In addition there is a district type of quantum predictions (Class III) which may also be predictions of the low energy theory once we allow interactions in addition to gravity. This occurs for analytic terms in the energy expansion which are accompanied by a parameter with dimension 1/(mass)n n ≥ 1.”

The quantum effects of virtual heavy particles appear as shifts in the coefficients of the most general possible local Lagrangian.

“The quantum effects of virtual heavy particles then appear as shifts in the coefficients of most general possible local Lagrangian.”

It is very unlikely that in fact c1, c2 = 0.

“However it is very un- likely that in fact c1, c2 = 0.”

An exception is the gravitational potential at large distance, where analytic corrections have no effect on the 1/r3 term.

“[An exception is the gravitational potential at large distance, where analytic corrections have no effect on the 1/r3 term.]”

At low enough energies, terms of order p^4 are small compared to those of order p^2.

“The graviton energy can be arbitrarily small and at low enough energies terms of order p4 are small compared to those of order p2.”

In distinction to the analytic contributions, which depend on unknown parameters c1, c2, ..., the leading quantum corrections are parameter free.

“So in distinction to the analytic contributions, which depend on the unknown parameters c1, c2 . . ., the leading quantum corrections are parameter free.”

• AssumptionRests on an assumption: The massless particles at low energy are the gravitons, photons, and maybe neutrinos.
• AssumptionRests on an assumption: The low energy couplings are contained in the Einstein action and depend only on the gravitational constant GN.

Experiment tells us that the cosmological constant is very small.

“For example, the constant Λ is proportional to the cosmological con- stant (λ = −8πGΛ), which experiment tells us is very small.[11]”

• cites [11] [11] Review of Particle Properties, Phys. Rev. D45, I.1 (1992).Not matched

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

Higher-order Lagrangians have little effect at low energies compared to the Einstein term R.

“The higher order Lagrangians will have little effect at low energies compared to the Einstein term, R.”

The key feature is that higher powers of R are involved at higher loops.

“The key feature is that higher powers of R are involved at higher loops.”

The dominant low-energy quantum corrections at large distance are due to the propagation of massless particles, including gravitons, and are manifested in nonlocal/nonanalytic contributions to vertex functions and propagators.

“In studying the class of quantum corrections at low en- ergy, the dominant effects at large distance can be isolated, as these are due to the propagation of the massless particles (including gravitons) of the theory and are manifested in the nonlocal/nonanalytic contri- butions to vertex functions and propagators.”

• AssumptionThe authors state this as an assumption: Gravitons are the only low-energy particles which are remnants of the full gravity theory.

Only the lowest energy couplings are needed, since higher order effects at the vertices introduce more powers of q2.

“Only the lowest energy couplings are needed, since higher order effects at the vertices introduce more powers of q2.”

There is a distinct class of quantum corrections, uncovered in the above procedure, which are independent of the unknown coefficients.

“There is a distinct class of quantum corrections, uncovered in the above procedure, which are independent of the unknown coefficients.”

At the lowest energy, only κ^2 is important.

“At the lowest energy, only κ2 is important.”

The low-energy manifestations of massless particles are not all local, as the particles can propagate for long distances.

“However the low energy manifestations of massless particles are not all local, as the particles can propagate for long distances.”

In the case of quantum gravity, effective field theory allows one to separate the effects of the unknown high energy theory from the known degrees of freedom at low energy.

“In the case of quantum gravity it allows one to separate the effects of the unknown high energy theory from the known degrees of freedom at low energy.”

Therefore, the long distance corrections to the gravitational interactions come uniquely from the nonanalytic terms in the loop diagrams.

“Therefore, the long distance corrections to the gravitational interactions come uniquely from the non analytic terms in the loop diagrams.”

Despite the description of quantizing, gauge fixing, etc., the Newtonian potential result is purely classical.

“Of course, despite the description of quantizing, gauge fixing etc., this is purely a classical result.”

The divergent parts of the vacuum polarization have been calculated by 't Hooft and Veltman using the same gauge fixing scheme as used in the present paper, using dimensional regularization.

“The divergent parts of the vacuum polarization have been calculated, by 't Hooft and Veltman[3] employing the same gauge fixing scheme as used in the present paper, using dimensional regularization.”

• cites [3]Not matched

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

We expect a modification to the potential of the form V(r) = −Gm1m2/r [1 + a G(m1+m2)/(rc2) + b Ghbar/(r2c3) + ...].

“We then expect a modification to the potential of the form V (r) = −Gm1m2 r [ 1 + a G(m1 + m2) rc2 + b Gh̄ r2c3 + . . . ] (41)”

The set of intermediate states is a known consequence of the low energy theory.

“The set of intermediate states is a known consequence of the low energy theory.”

There is no such thing as a purely classical source for gravity.

“One interesting consequence of the quantum correction is that it appears that there is no such thing as a purely classical source for gravity.”

For timelike q2, ln(−q2) and √(m2/−q2) pick up imaginary parts corresponding to physical on-shell intermediate states as described by unitarity.

“For q2 time-like, ln(−q2) and √ m2 −q2 pick up imaginary parts which correspond to the physical (on shell) intermediate states as described by unitarity.”

In this range, quantum predictions can be extracted from generally covariant theories in the same way as done in other effective field theories.

“In this range quantum predictions can be extracted from generally covariant theories in the same way as done in other effective field theories.”

The exact assumptions of the singularity theorems apply to essentially all situations we care about, although simple cases such as Minkowski space or a single stable star are exceptions.

“[The exact assumptions are more precisely stated in the original works, but apply to essentially all situations that we care about although simple cases such as Minkowski space or a single stable star are exceptions.]”

• cites [21] [21] S.W. Hawking and R. Penrose, Proc. R. Soc. London A314, 529 (1970). S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space- Time, (Cambridge Univ. Press, Cambridge, 1973). Figure Captions Fig. 1. a) A tree level vertex of order κ2p2, b) a one loop vertex of order κ4p4. Fig. 2. One graviton exchange for the Newtonian potential. The matter fields are represented by solid lines and wavy lines represent gravitons. Fig. 3. The set of corrections included in the one particle reduceable poten- tial. Fig. 4. The diagrams involved in the vertex correction. Fig. 5. The diagrams in the grNot checked yet

  Not checked in the free run.

There can be no corrections in F1(q2) of the form κ2m2 because of the normalization condition F1(0) = 1.

“There can be no corrections in F1(q 2) of the form κ2m2 because of the normalization condition F1(0) = 1.”

The class of leading quantum corrections is not optional.

“The class of leading quantum corrections is not optional.”

These nonanalytic terms come uniquely from the long distance propagation of massless particles.

“These nonanalytic terms come uniquely from the long distance propagation of the massless particles.”

When only massless particles appear in the diagram, the ln(−q2) terms can be read off of the coefficients of 1/(d−4) using a relatively well known trick.

“When only massless particles appear in the diagram, the ln(−q2) terms can be read off of the coefficients of 1/(d − 4) using a relatively well known trick.”

The logarithm will always share the coefficient of 1/(d−4) in the combination given in Eq. 70(b).

“the logarithm will always share the coefficient of 1 d−4 in the combination given in Eq. 70(b).”

For gravitational effective field theory, the content of the singularity theorems is that it is difficult to specify a spacetime where the curvature is everywhere small.

“For gravitational effective field theory, the content of the singularity theorems is that it is difficult to specify a space-time where the curvature is everywhere small.”

The leading quantum corrections will also be found in a full quantum gravity theory as long as our universe is reproduced in the low energy limit.

“It will also be found in a full quantum gravity theory as long as our universe is reproduced in the low energy limit.”

't Hooft and Veltman find that the divergent part of the graviton plus ghost vacuum polarization diagrams is equivalent to the Lagrangian in Eq. 30.

“'t Hooft and Veltman find that the divergent part of the graviton plus ghost vacuum polarization diagrams is equivalent to the Lagrangian in Eq. 30.”

• cites [3]Not matched

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

The nonanalytic terms vanish for any integrands that would vanish on shell gravitons.

“Thus the non-analytic terms vanish for any integrands which would vanish on shell gravitons.”