Bootstrapped Newtonian quantum gravity
aa r X i v : . [ g r- q c ] J un Bootstrapped Newtonian quantum gravity
Roberto Casadio ab ∗ and Iberê Kuntz ab † a Dipartimento di Fisica e Astronomia, Università di Bolognavia Irnerio 46, 40126 Bologna, Italy b I.N.F.N., Sezione di Bologna, I.S. FLAGviale B. Pichat 6/2, 40127 Bologna, Italy
June 16, 2020
Abstract
We compute quantum corrections for the gravitational potential obtained by including aderivative self-coupling in its classical dynamics as a toy model for analysing quantum gravityin the strong field regime. In particular, we focus on quantum corrections to the classicalsolutions in the vacuum outside localised matter sources. ∗ E-mail: [email protected] † E-mail: [email protected] Introduction
In quantum field theory, fundamental forces are associated with the exchange of (virtual) quanta ofthe interaction fields among matter fields and static potentials only emerge as approximate descrip-tions for particular configurations. For instance, the linear interaction among fermions in QuantumElectro-Dynamics (QED) is carried by the two polarisations of the massless vector field whosequanta are the photons. The Coulomb potential describing the force acting on a (strictly speaking)static charge then emerges in the non-relativistic limit of the tree-level transition amplitude forthe scattering between two charged fermions via the exchange of these (virtual) photons [1]. Forextended sources involving many charged particles, the calculation of amplitudes becomes imme-diately very cumbersome already at the tree level, and things only get worse when nonlinearitiesstemming from quantum (loop) corrections are included.The linear Newtonian interaction likewise emerges from a suitable limit for the exchange of spin 2(virtual) gravitons between two massive particles in the weak field regime. However, according toGeneral Relativity (GR), nonlinearities should already be present at the classical level, which makesexplicit quantum calculations for configurations in the strong field regime very difficult, if possibleat all. As an effective description of the gravitational force, say applied on a test particle by anextended matter source, one can instead consider the static potential as the mean field generatedby that extended source and quantise it canonically. This approach applied to QED leads to thedescription of a static electric field in terms of the coherent state of (virtual) longitudinal photons [2].It then appears straightforward that one can quantise the Newtonian potential, which solves theclassical Poisson equation, in terms of (virtual) scalar gravitons, in a similar fashion. However, forthe purpose of studying the strong field regime of gravity, it is more interesting to try and includesome nonlinearities [3] in the classical equation for the gravitational potential (the bootstrappedNewtonian gravity introduced in Refs. [4, 5]) and then express the resulting solutions in terms ofmodified coherent states [6, 7].In this work, we will instead quantise the bootstrapped potential as a scalar field in order todetermine one-loop quantum corrections to its effective field equation. For this purpose, we will firstperform a field redefinition to dispose of the derivative self-interaction and then apply heat kerneltechniques in order to compute the quantum effective action [8–13]. Hopefully, this simplifiedapproach to nonlinearities will help to gain some insight about the quantum dynamics of gravity inthe strong field regime generated by the presence of matter sources.This paper is organised as follows: in Section 2, we show how we can deal with derivativeinteractions non-perturbatively by means of a field redefinition that is able to transform the actioninto a canonical form; in Section 3, we review the unique effective action first introduced in Ref. [8]and which allows us to extend the findings of Section 2 to the quantum level by requiring thequantum action to be covariant off-shell; Section 4 is devoted to the calculation of the quantumaction for bootstrapped gravity. We show that the quantum equation of motion is described by anon-local equation in the infrared regime, which is then solved for an idealized point-like source and2or a pair of point-like sources; we finally draw some conclusions in Section. 5
We start by recalling that the action for the bootstrapped Newtonian potential V = V ( r ) forspherically symmetric systems is given by [4] S [ V ] = − π Z r d r (cid:20) ( V ′ ) π G N (1 − q φ V ) + q B ρ V (1 − q φ V ) (cid:21) , (2.1)where q φ is a (positive) coupling that controls the potential self-interaction and q B is introduced tokeep track of the coupling with the matter source of density ρ . We here want to show that thederivative interaction can be transformed into a canonical kinetic term under a field redefinition,which is only possible because the configuration space turns out to have vanishing curvature. Thiswill allow us to quantize the theory non-perturbatively in q φ , although we will still need to invokeperturbation theory in the coupling constant q B to deal with matter interactions.We first rescale the (dimensionless) gravitational potential [6] φ = V √ G N = r m p ℓ p V , (2.2)the matter density J B = 4 π p G N ρ = 4 π s ℓ p m p ρ , (2.3)and promote the new scalar field φ = φ ( x µ ) as well as J B = J B ( x µ ) to depend on all spacetimecoordinates x µ = ( t, ~x ) . The generalised bootstrapped action then reads S [ φ, J B ] = S [ φ ] + S int [ φ, J B ] , (2.4)where S [ φ ] = Z d x (cid:20) − ∂ µ φ ∂ µ φ + α ( ∂ µ φ ) φ (cid:21) (2.5)is the kinetic part containing a derivative self-interaction, and S int [ φ, J B ] = Z d x ξ ( φ ) J B . (2.6)In order to avoid heavy notation, we also introduced α = 2 q φ s ℓ p m p (2.7) We shall use units with c = 1 , the Newton constant G N = ℓ p /m p and ~ = ℓ p m p . In the present work we neglect the pressure term analysed in Ref. [5], and just note that it could simply beabsorbed into the definition of the matter density. ξ ( φ ) = − q B φ (1 − α φ ) , (2.8)which represents a non-linear coupling to the source J B .The above action contains derivative interactions, which means that the action for the free fieldis not recovered by simply setting q B = q φ = 0 . But one can perform a field redefinition and try toput it in canonical form by diagonalising the whole kinetic Lagrangian density L = (cid:18) −
12 + α φ (cid:19) ∂ µ φ ∂ µ φ . (2.9)As can be seen from Eq. (2.9), for φ > / α the kinetic term changes sign and φ becomes a ghost.For this reason, we will focus on the branch φ < / α . The dependence on φ inside the bracketsthat multiply ( ∂ µ φ ) also indicates that the metric in field space is not trivially flat, which caneither mean that the field space is curved or that the field space is flat but the chosen coordinateis curvilinear. If the field space is flat, then there exists a field transformation which diagonalisesthe kinetic term. Reciprocally, should there be a frame where the kinetic term is diagonal, thenthe field space must be flat. We will show below that this is indeed the case for the Lagrangiandensity (2.9).To put the kinetic term in the canonical form, we need a field redefinition ϕ = ψ ( φ ) such that ∂ µ ϕ = p − α φ ∂ µ φ , (2.10)which is real and non-singular for φ < α , (2.11)and is solved by ϕ = C − α (1 − α φ ) / , (2.12)where C is an arbitrary integration constant. Upon requiring that the transformation reduces tothe identity for α → , we obtain ϕ = ψ ( φ ) = 13 α h − (1 − α φ ) / i . (2.13)Upon inverting the above relation, we get φ = ψ − ( ϕ ) = 12 α h − (1 − α ϕ ) / i , (2.14)which is precisely the relation between the exact “vacuum” solution V c = 14 q φ h − (1 − q φ V N ) / i (2.15) Typical classical solutions are expected to have φ < , so that this condition is trivially satisfied. V N = − G N Mr , (2.16)where M is the source mass. This is consistent with the fact that the Newtonian potential hascanonical kinetic term. In terms of ϕ , the complete Lagrangian density then reads L = − ∂ µ ϕ ∂ µ ϕ + ˜ ξ ( ϕ ) J B , (2.17)where the non-linear coupling ˜ ξ ( ϕ ) = ξ ( ψ − ( ϕ )) is given by ˜ ξ ( ϕ ) = − q B α h − (1 − α ϕ ) / i . (2.18)Since the interaction terms do not contain any derivatives, the Lagrangian density (2.17) for ϕ canbe quantised in the standard way by defining the asymptotic states for the free field ϕ . The quantisation of the theory will be performed using the one particle irreducible (1PI, or quantum)action. The calculation of the 1PI action in our case is a little subtler than the standard calculationsin quantum field theory because of the curvilinear coordinates in the original frame (2.4). Inparticular, the naive definition of the 1PI action is frame dependent off-shell and can only beused if one is interested in the components of the S-matrix, which are calculated on-shell. On theother hand, we are interested in the dynamics of the off-shell mean field in a given state, whosequantum action would produce non-covariant results. Fortunately, a covariant formulation of the1PI action exists as it was introduced in Ref. [8] (see also [9]). In the following, we briefly review suchformulation in one-dimensional field space, which is enough for our purposes since the action (2.4)contains only one degree of freedom.The covariant partition function is given by Z [ J ] = Z D φ e i { S [ φ ]+ [ v [ ¯ φ ] − σ ( ¯ φ,φ ) ] J } , (3.1)where v [ ¯ φ ] is an arbitrary covector field satisfying ∇ ¯ φ v [ ¯ φ ] = 1 with ∇ ¯ φ being the covariant derivativeassociated with the Levi-Civita connection γ ( ¯ φ ) of the field space, the displacement vector is definedby σ ( ¯ φ, φ ) = ∂ ¯ φ ˜ σ ( ¯ φ, φ ) and ˜ σ ( ¯ φ, φ ) is Synge’s world function, which is numerically equal to one-halfof the square of the geodesic distance between φ and ¯ φ . Because v [ ¯ φ ] transforms as a covector and σ ( ¯ φ, φ ) transforms as a covector with respect to its first argument and as a scalar with respect toits second argument, the partition function Z [ J ] is completely covariant under redefinitions of thebackground and of the quantum fields. The covariant relation between the background field andthe mean field is now given by (cid:10) σ ( ¯ φ, φ ) (cid:11) = 0 , (3.2) We shall see later on that the mass M however differs from the Newtonian expectation [4]. ¯ φ = h φ i . The displacement vector can be expanded as − σ ( ¯ φ, φ ) = φ − ¯ φ + 12 γ ( ¯ φ ) (cid:0) φ − ¯ φ (cid:1) + O ( φ − ¯ φ ) , (3.3)where γ ( ¯ φ ) is the one-dimensional Levi-Civita connection in field space, thus one obviously retrievethe naive definition ¯ φ = h φ i for a vanishing connection. The covariant 1PI action is defined as Γ[ ¯ φ ] = W [ J ] − v [ ¯ φ ] J , with δW [ J ] δJ = v [ ¯ φ ] , (3.4)where W [ J ] = − i log Z [ J ] is the generating functional of connected diagrams. From Eqs. (3.1) and(3.4), one obtains e i Γ[ ¯ φ ] = Z D φ e i { S [ φ ]+ σ ( ¯ φ,φ ) ∇ ¯ φ Γ[ ¯ φ ] } . (3.5)To perform the integral in Eq. (3.5), it is best to make the change of variables φ → σ ( ¯ φ, φ ) . (3.6)By expanding the classical action as a covariant Taylor series S [ φ ] = ∞ X n =0 ( − n n ! (cid:8) ∇ σ S [ ¯ φ ] σ ( ¯ φ, φ ) (cid:9) n (3.7)and the quantum action as a loop expansion Γ[ ¯ φ ] = S [ ¯ φ ] + ~ Γ (1) [ ¯ φ ] + ~ Γ (2) [ ¯ φ ] + . . . , (3.8)one can calculate the quantum action order by order in ~ . At one-loop order, we find Γ[ ¯ φ ] = S [ ¯ φ ] + i ~ (cid:0) ∇ S [ ¯ φ ] (cid:1) , (3.9)where Tr denotes the functional trace. The only difference with respect to the standard quantumaction is the appearance of the functional covariant derivative instead of the standard functionalderivative, thus making Γ[ ¯ φ ] a covariant object under field redefinitions. The result in the previous section could be applied to the Lagrangian density (2.17) by interpreting J = J B as the auxiliary source. An alternative procedure would rather be to introduce a differentauxiliary source J , in which case the partition function would be given by Z [ J, J B ] = Z D φ e i { S [ φ,J B ]+ [ v [ ¯ φ ] − σ ( ¯ φ,φ ) ] J } . (4.1) The source J B is meant to generate a non-trivial background field and cannot be made to vanish arbitrarily.
6n this way, J can be simply interpreted like a Lagrange multiplier, which will be set to zero in theend as usual, and needs not necessarily be a physical source. The 1PI action results in e i Γ[ ¯ φ,J B ] = Z D φ e i { S [ φ,J B ]+ σ ( ¯ φ,φ ) ∇ ¯ φ Γ[ ¯ φ,J B ] } (4.2) = Z D ϕ e i { ˜ S [ ϕ,J B ]+ σ ( ψ − ( ¯ ϕ ) ,ψ − ( ϕ )) ∇ ¯ ϕ ˜Γ[ ¯ ϕ,J B ] } , (4.3)where in the second line we used the redefinition (2.14) and denoted ˜ S [ ϕ, J B ] = S [ ψ − ( ϕ ) , J B ] = Z d x (cid:20) − ∂ µ ϕ ∂ µ ϕ + ˜ ξ ( ϕ ) J B (cid:21) , (4.4)and ˜Γ[ ϕ, J B ] = Γ[ ψ − ( ϕ ) , J B ] . As it was already pointed out in Section 2, in the new frame thefield coordinates are Cartesian. The covariant functional derivative thus reduces to the flat form ∇ ϕ = δδϕ , and the displacement vector reduces to the coordinate difference in field space σ ( ψ − ( ¯ ϕ ) , ψ − ( ϕ )) = ψ − ( ¯ ϕ ) − ψ − ( ϕ ) . (4.5)The quantum action to one-loop order then reads ˜Γ[ ¯ ϕ, J B ] = ˜ S [ ¯ ϕ, J B ] + i ~ , (4.6)with ∆ = (cid:3) + ˜ ξ ′′ ( ¯ ϕ ) J B = (cid:3) + q B α (1 − α ¯ ϕ ) − / J B , (4.7)where (cid:3) = η µν ∂ µ ∂ ν denotes the D’Alembert operator in flat spacetime. To calculate the secondterm in Eq. (4.6), we use the Schwinger proper time method to represent the one-loop functionaldeterminant in terms of the heat kernel K of the operator ∆ as Tr log ∆ = − Z ∞ d s Tr K ( s ) s . (4.8)In the presence of a potential term in ∆ , such as P ≡ ˜ ξ ′′ ( ¯ ϕ ) J B in Eq. (4.7), the computation of theexact heat kernel becomes highly non-trivial and it is necessary to rely on approximate methods.Different approximation techniques with different scopes of applicability have been developed [10–13], the most popular one perhaps being the Schwinger-DeWitt expansion in inverse powers of thefield mass. However, the Schwinger-DeWitt expansion obviously breaks down for massless theories,like our action (2.4), only being able to produce the divergent part of the quantum action.A useful tool to study heat kernels in the absence of a mass term is the covariant perturbationtheory [12, 13], which is based on an asymptotic expansion in terms of spacetime curvatures, fibrebundle curvatures (gauge field strengths) and potential terms and can be seen as a resummation ofthe Schwinger-DeWitt expansion. In our case, there are no curvatures present, either in spacetime7r in field space, and the trace of the heat kernel can only be expanded as a series in the potentialterm P as Tr K ( s ) = 1(4 π s ) ω Z d x (cid:2) s P + s P f ( − s (cid:3) ) P + O ( P ) (cid:3) , (4.9)where f ( u ) = 12 Z d t e − t (1 − t ) u . (4.10)Substituting (4.9) in Eq. (4.8) and changing the order of integration, we find the one-loop contri-bution to the 1PI action to second order in the potential term P is given by ˜Γ (1) [ ¯ ϕ ] = q α π ) ω Z d x J B (1 − α ¯ ϕ ) / (cid:20) − ω + 2 − log (cid:18) − (cid:3) µ (cid:19)(cid:21) J B (1 − α ¯ ϕ ) / , (4.11)for ω → . Here µ is a normalisation mass necessary to make the logarithm dimensionless and whosearbitrariness reflects the renormalisation arbitrariness of the one-loop effective action. Note that thedivergence is proportional to (1 − α ¯ ϕ ) − / , whereas the bare action only contains (1 − α ¯ ϕ ) / ,which indicates that the theory is non-renormalisable and only makes sense as an effective fieldtheory. Since the only mass parameter present in the theory is the Planck mass m p , one indeedexpects that the theory breaks down at energies of the order of m p .Albeit being non-renormalisable, the quantum action can and must be renormalised order byorder in the loop expansion in order to produce sensible results. This is done by regarding ˜ S [ ¯ ϕ, J B ] as the bare action and adding counter-terms in an expansion in powers of ~ , to wit ˜ S [ ¯ ϕ, J B ] = ˜ S (0) [ ¯ ϕ, J B ] + ~ ˜ S (1) [ ¯ ϕ, J B ] + O ( ~ ) . (4.12)The ultraviolet divergences in Eq. (4.11) can then be eliminated with the choice ˜ S (1) [ ¯ ϕ, J B ] = ˜ S (1)R [ ¯ ϕ, J B ] − q α π ) ω Z d x J B (1 − α ¯ ϕ ) / (cid:18) − ω + 2 (cid:19) J B (1 − α ¯ ϕ ) / , (4.13)where ˜ S (1)R denotes the action with renormalised coupling constants. Note that we included in ˜ S (1) a non-divergent term, which is local and does not contribute to the infrared physics. Thus wecan eliminate it via a finite renormalisation for convenience. After renormalisation, the infraredequation of motion is finally given by (cid:3) ¯ ϕ = q B J B (1 − α ¯ ϕ ) / + q α ~ π J B (1 − α ¯ ϕ ) / log (cid:18) − (cid:3) µ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) R J B (1 − α ¯ ϕ ) / . (4.14)The subscript R in the log operator is a reminder that we must impose retarded boundary conditionsby replacing the Feynman Green’s function with the retarded one. This procedure, albeit seeminglyad-hoc, results from the in-in path integral formalism and it is required in order to obtain a causalevolution for the mean field [12, 14, 15]. Since the field coordinates in the new frame are Cartesian,8e have ψ − ( ¯ ϕ ) = (cid:10) ψ − ( ϕ ) (cid:11) . Therefore, solutions to Eq. (4.14) will correspond to corrections tothe Newtonian potential in the vacuum state.We should note that we followed a bottom-up approach to quantum gravity, starting froma modified Newtonian theory of gravity, by promoting the Newtonian potential to a scalar fieldand then quantising it. Nothing in this construction suggests that the resulting theory exhibits anygauge invariance, thus we need not worry about the usual complications that arise in gauge theories.However, from a top-down viewpoint, gauge symmetry is required to account for the backgroundindependence of gravity. The reason why gauge symmetry is not realised in bootstrapped gravity isthat we phrased the entire approach in terms of the bootstrapped potential, which is observable and therefore gauge-invariant. Should one try to formulate the theory in terms of non-observablequantities, gauge fixing conditions are expected to become necessary at the quantum level. Acomplete analysis of gauge-invariance requires a reconstruction of the full spacetime metric, whichis however left for future investigations.Since it does not look possible to solve exactly Eq. (4.14), we will expand the solution in powersof the coupling q B as ¯ ϕ = q B ¯ ϕ (1) + q ¯ ϕ (2) + . . . , (4.15)and solve Eq. (4.14) perturbatively up to second order in q B . For the static potential ¯ ϕ = ¯ ϕ ( ~x ) generated by a static source J B = J B ( ~x ) , we find ∇ ¯ ϕ (1) = J B , (4.16) ∇ ¯ ϕ (2) = − α J B ϕ (1) + α ~ π J B log (cid:18) − ∇ µ (cid:19) J B , (4.17)whose solution is ¯ ϕ (1) ( ~x ) = Z d x ′ G ( ~x, ~x ′ ) J B ( ~x ′ ) , (4.18) ¯ ϕ (2) ( ~x ) = − α Z d x ′ G ( ~x, ~x ′ ) J B ( ~x ′ ) ¯ ϕ (1) ( ~x ′ )+ α ~ π Z d x ′ Z d x ′′ G ( ~x, ~x ′ ) J B ( ~x ′ ) L ( ~x ′ , ~x ′′ ) J B ( ~x ′′ ) , (4.19)where G ( ~x, ~x ′ ) = − π | ~x − ~x ′ | (4.20)is the Green function for the Laplace operator ∇ . Moreover, the kernel L of the log operator, log (cid:18) − ∇ µ (cid:19) f ( x ) = Z d x ′ L ( ~x, ~x ′ ) f ( ~x ′ ) , ∀ f ( x ) , (4.21) For instance, the potential determines the radial acceleration of a static particle, which is directly observable.
9s defined as a pseudo-differential operator acting via the Fourier transform, L ( ~x, ~x ′ ) = Z d q (2 π ) e − i ~q · ( ~x − ~x ′ ) log (cid:18) q µ (cid:19) = − π | ~x − ~x ′ | . (4.22)Let us now look at some examples. For the case of a point-like source of mass M and current J B ( x ) = 4 π p G N M δ ( ~x ) , (4.23)we obtain the standard Newtonian potential (2.16) with M = M from Eq. (4.18) by defining r ≡ | ~x | and setting q B = 1 , ¯ V N = p G N ¯ ϕ (1) = − G N M r . (4.24)The correction at order q is likewise obtained from Eq. (4.19) and reads ¯ ϕ (2) = − lim ǫ → α G N M r ǫ (cid:18) − α ~ π ǫ (cid:19) , (4.25)which diverges due to the ultra-localized source (4.23). One instead expects finite results when theDirac delta is replaced by an extended source (whose radius is greater than the Planck length).This is physically expected because strong quantum gravitational effects become important whenone probes Planckian distances and they cannot be accounted for within the realm of effective fieldtheory. Nonetheless, these divergences can be removed by any regularisation method or, formally,by choosing an integration contour which does not enclose the spatial origin. With that in mind,we will drop them out and focus on finite terms, which therefore leads to vanishing corrections forthe Newtonian potential (4.24). By mapping ϕ back to the original field φ according to Eq. (2.14),one then precisely recovers the classical solution V c in Eq. (2.15) with M = M .We will show in our next example that the above divergences indeed reflect short-distance effectstaking place at Planckian scales by considering two point-like sources. This way, an extended bodycan be simulated, with the distance between the sources determining the body’s size. We will alsorecover the result that the mass M in the bootstrapped potential is not equal to the proper mass M of the source. We will next solve Eqs. (4.18) and (4.19) for two massive point-like sources of equal mass M / located at points of coordinate ~x and ~x and contributing to the total source as J B ( x ) = 2 π p G N M [ δ ( ~x − ~x ) + δ ( ~x − ~x )] . (4.26) Notice that we are taking into account the rescalings (2.2) and (2.3). p G N ¯ ϕ (1) = − G N M (cid:18) | ~x − ~x | + 1 | ~x − ~x | (cid:19) . (4.27)If we assume that | ~x − ~x | ≡ R ≪ | ~x − ~x | ≃ | ~x − ~x | ≡ r , we can in fact write p G N ¯ ϕ (1) ≃ − G N M r , (4.28)which reproduces the result in the previous example to first order in q B .For the calculation of ¯ ϕ (2) , we must point out that the product of quantities evaluated at thesame point ~x i , such as δ ( ~x − ~x i ) ¯ ϕ (1) ( ~x i ) , is ill-defined. This is analogous to the situation of theprevious example and only reflects the existence of short-distance effects beyond the grasp of theeffective field theory. We will therefore drop single-point quantities and focus on the cross terms ofquantities evaluated at ~x and ~x . Eq. (4.19) then gives ¯ ϕ (2) = − α G N M | ~x − ~x | (cid:18) − α ~ π | ~x − ~x | (cid:19) (cid:18) | ~x − ~x | + 1 | ~x − ~x | (cid:19) . (4.29)Finally, the total solution to order q reads p G N ¯ ϕ ≃ − q B G N M " q B q φ G N M | ~x − ~x | − q φ ℓ p ~ π m p | ~x − ~x | ! | ~x − ~x | + 1 | ~x − ~x | (cid:19) ≃ − G N M r " q φ G N M | ~x − ~x | − q φ ℓ π | ~x − ~x | ! , (4.30)in which we used Eq. (2.7) to display the original coupling q φ and again set q B = 1 at the end.Note that one recovers the classical Newtonian potential either for q φ = α = 0 or, more formally,when the sources are far away from each other, | ~x − ~x | → ∞ . At first order in q φ ∼ α wehave the classical contribution to the non-linearity introduced in the action (2.5), whereas at thirdorder we find the correction due to (one-loop) quantum gravity. We are also able to interpret thedivergences in Eq. (4.25) for a single point-like particle as indeed originated from the limit ~x → ~x in Eq. (4.30). Although we still had to deal with the single-point divergences mentioned above, theywill presumably yield finite results once a smooth matter source comprising of all macroscopicalsources, is used as opposed to a system of point-like sources.Note that the above expression (4.30) more accurately reproduces the classical solution V c inEq. (2.15) outside an extended source of small compactness G N M ≪ R . In fact, we can considerthe distance between the two point sources as the size R of an extended source, that is R ≃ | ~x − ~x | ,and introduce the modified mass ˜ M ≡ M " q φ G N M R − q φ ℓ π R ! ≃ M − q φ ℓ π R ! , (4.31)11here the relation between M and M is the same as the one found in Ref. [4] for a uniform star ofsmall compactness, modulo a numerical coefficient of order one. We then obtain p G N ¯ ϕ ≃ − G N ˜ Mr . (4.32)By transforming back to the original frame √ G N φ = V , we then recover the classical bootstrappedpotential (2.15) in the vacuum with the mass M now further corrected by a one-loop contribution. We remark that the expression for the mass (4.31) only holds for small compactness of the source(that is for G N M /R ≪ ) and that M > M is precisely a consequence of the nonlinearity includedin the bootstrapped dynamics, as discussed in Refs. [4, 5].Finally, we must emphasize that the above solution is non-perturbative in q φ and should repro-duce all effects due to the non-linear self-coupling of gravity. On the other hand, the coupling tomatter was handled perturbatively. Treating both q φ and q B non-perturbatively is utterly difficult,but one can study non-perturbative effects due to the gravitational self-coupling at the expense ofdealing perturbatively with respect to the coupling to matter. In this paper, we have considered non-linear derivative self-interactions of the Newtonian potentialby allowing the first few post-Newtonian terms to take arbitrary values. Such a theory has beencalled bootstrapped Newtonian gravity. We calculated one-loop quantum corrections to the boot-strapped potential by first promoting the non-relativistic potential to a Lorentz covariant form thatallows the application of quantum field theory techniques in intermediate steps. These intermediatecalculations are obviously supposed to serve only as a guideline for the quantisation of the completetheory of gravity ( e.g. light bending requires a non-scalar gravitational field), thus we must takethe non-relativistic limit in the end, which is all we need for our purposes.We showed that the bootstrapped Newtonian potential is described by a non-local equation ofmotion in the infrared, which is typical of massless theories. We solved it for a point-like sourceand a system of two point-like sources. The latter can be thought of as a rough approximation ofan extended source. Our results recover Newtonian physics in the limit where the sources are farapart and for vanishing derivative interactions α → , as one would expect. The analysis of morerealistic situations, such as a smooth extended source, proves much more challenging already at theclassical level [5] and will be left for future investigations. In any case, the effective equations ofmotion together with the resulting quantum bootstrapped potential permits a better understandingof quantum processes taking place at non-perturbative settings which are important for strong fieldapplications. We note in passing that this contribution is consistently of the same order ℓ as the corrections found in Ref. [16]for the metric generated by a star. cknowledgments R.C. and I.K. are partially supported by the INFN grant FLAG. The work of R.C. has also beencarried out in the framework of activities of the National Group of Mathematical Physics (GNFM,INdAM) and COST action Cantata.
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