Generalizing the ADM Computation to Quantum Field Theory
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Generalizing the ADM Computation to Quantum Field Theory
P. J. Mora ∗ , N. C. Tsamis † and R. P. Woodard , ‡ Department of Physics, University of FloridaGainesville, FL 32611, UNITED STATES Institute of Theoretical Physics & Computational PhysicsDepartment of Physics University of CreteGR-710 03 Heraklion, HELLAS
ABSTRACTThe absence of recognizable, low energy quantum gravitational effects re-quires that some asymptotic series expansion be wonderfully accurate, butthe correct expansion might involve logarithms or fractional powers of New-ton’s constant. That would explain why conventional perturbation theoryshows uncontrollable ultraviolet divergences. We explore this possibility inthe context of the mass of a charged, gravitating scalar. The classical limit ofthis system was solved exactly in 1960 by Arnowitt, Deser and Misner, andtheir solution does exhibit nonanalytic dependence on Newton’s constant.We derive an exact functional integral representation for the mass of thequantum field theoretic system, and then develop an alternate expansion forit based on a correct implementation of the method of stationary phase. Thenew expansion entails adding an infinite class of new diagrams to each orderand subtracting them from higher orders. The zeroth order term of the newexpansion has the physical interpretation of a first quantized Klein-Gordonscalar which forms a bound state in the gravitational and electromagneticpotentials sourced by its own probability current. We show that such boundstates exist and we obtain numerical results for their masses.PACS numbers: 04.60-m ∗ e-mail: [email protected]fl.edu † e-mail: [email protected] ‡ e-mail: [email protected]fl.edu Introduction
The problem of quantum gravity is that perturbative loop corrections toquantum general relativity contain ultraviolet divergences that can only beabsorbed by adding higher derivative counterterms which would make theuniverse decay instantly [1]. The divergence problems are well known andubiquitous: • Einstein + scalar requires a bad counterterm at one loop order [2]; • The same unacceptable one loop counterterm is also needed for Einstein+ Maxwell [3], Einstein + Dirac [4] and Einstein + Yang-Mills [5]; • The Einstein theory by itself requires an unacceptable counterterm attwo loop order [6]; and • Although supergravity is on-shell finite at two loop order [7], and ex-plicit computation shows that N = 8 supergravity is on-shell finite atthree [8] and four loop order [9], all supergravity models are expectedto require unacceptable counterterms by seven loop order [10].Quantum gravity can of course be used as an effective field theory bytreating the bad counterterms as perturbations and then restricting to lowenergy predictions [11] which are insensitive to them. If the AsymptoticSafety scenario [12] is realized, it might even be that the escalating series ofperturbative counterterms does not spoil predictivity at energies below thePlanck scale. However, neither approach provides a fundamental resolution.The problem arises from the tension between four facts [1]: • Continuum Field Theories possess an infinite number of modes; • Quantum Mechanics requires each mode to have a minimum amountof energy; • General Relativity stipulates that stress-energy is the source of gravi-tation; and • Perturbation Theory simply adds up the contribution from each modeat lowest order. 1ne or more of these principles must be sacrificed, and a little thought sug-gests focussing on the last two. There does not seem any way of disputingthe experimental confirmation of quantum mechanics in the matter sectorwhich is responsible for the lowest order divergences of quantum general rel-ativity. And inflationary cosmology makes nonsense of any attempt to invokea nonzero physical cutoff length. Inflation predicts that the universe has ex-panded by the staggering factor of at least 10 [13], so if the physical cutoffis at the Planck length today then it must have been about 10 − m dur-ing primordial inflation. But fossilized quantum gravitational effects fromprimordial inflation have been measured with a fractional strength of about10 − [14], which is inconsistent with so small a physical cutoff length.Superstring theory can be viewed as an attempt to preserve the validity ofperturbation theory by sacrificing general relativity. We wish here to investi-gate the alternative: that the problems of quantum general relativity derivefrom using conventional perturbation theory. We disavow any intention ofseeking the exact solution. There is so far no example of an interacting quan-tum field theory in D = 3 + 1 dimensions which can be solved exactly, andall experience with classical field theory suggests that general relativity is anunlikely candidate to be the first one. What interests us instead is the possi-bility that quantum general relativity has a perfectly finite asymptotic seriesexpansion which is simply not given by conventional perturbation theory.The conventional result for the expectation value of a quantum gravityobservable O with characteristic length R is assumed to take the form, D Ω (cid:12)(cid:12)(cid:12) O (cid:12)(cid:12)(cid:12) Ω E = (cid:16) Tree Order (cid:17)( ∞ X ℓ =1 a ℓ (cid:16) ¯ hGc R (cid:17) ℓ ) , (1)where G is Newton’s constant, ¯ h is Planck’s constant and c is the speed oflight. Support for this form is adduced from the fact that quantum grav-ity has no observable effects at low energies. Even for the smallest dis-tances ever probed, R ∼ − m, the loop counting parameter is minuscule,¯ hG/ ( c R ) ∼ − . But the same thing would be true of a series thatincorporates fractional powers or logarithms such as, D Ω (cid:12)(cid:12)(cid:12) O (cid:12)(cid:12)(cid:12) Ω E = (cid:16) Tree Order (cid:17)( ∞ X ℓ =1 ℓ X m =0 a ℓm (cid:16) ¯ hGc R (cid:17) ℓ h ln (cid:16) ¯ hGc R (cid:17)i m ) . (2)If the actual asymptotic expansion of quantum gravity were to take the form(2) then loop effects at R ∼ − m would still be suppressed by unobserv-2bly small powers of the parameter, (cid:16) ¯ hGc R (cid:17) ln (cid:16) ¯ hGc R (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R ∼ − m ∼ − − . (3)However, trying to force the putative series (2) into the assumed form (1)would result in logarithmically divergent coefficients a ℓ , which is exactly whatexplicit computations reveal.The incorporation of such nonanalytic terms into an asymptotic expan-sion occurs even for very simple physical systems. Consider the canonicalpartition function for a non-interacting, highly relativistic particle of mass m in a three dimensional volume V at temperature k B T = 1 /β , Z = V π ¯ h Z ∞ dp p exp h − β q p c + m c + βmc i , (4)= V π ¯ h c Z ∞ dK ( K + mc ) √ K +2 Kmc exp( − βK ) . (5)When the rest mass energy is small compared to the thermal energy it oughtto make sense to expand in the small parameter x ≡ βmc . But straightfor-ward perturbation theory fails, Z = V π (cid:16) k B T ¯ hc (cid:17) Z ∞ dt t e − t (cid:16) xt (cid:17)r xt , (6)= V π (cid:16) k B T ¯ hc (cid:17) Z ∞ dt t e − t ( xt + 12 (cid:16) xt (cid:17) − ∞ X n =3 ( n − n − n ! (cid:16) − xt (cid:17) n ) . (7)From expression (7) it seems as though the term of order x vanishes, andthat the higher terms have increasingly divergent coefficients with oscillatingsigns. In fact the x term is non-zero, and the apparent divergences merelysignal contamination with logarithms, Z = V π (cid:16) k B T ¯ hc (cid:17) ( x + 12 x − x − x ln( x )+ O (cid:16) x (cid:17)) . (8)Just as we suspect is the case for quantum gravity, the partition functionhas an excellent expansion for small x = βmc ; the terms after order x areindeed smaller than x , they just are not as small as one naively thinks.Rather than attempting to develop a new expansion for an arbitraryquantum gravity observable, we restrict attention here to the self-energy of a3harged, gravitating particle. An exact result for the ¯ h = 0 limit of this sys-tem was obtained in 1960 by Arnowitt, Deser and Misner (ADM) [15]. Theirwork provides strong support both for the possibility that negative gravita-tional interaction energy cancels divergences, and for the possibility that thecorrect asymptotic expansion involves nonanalytic dependence on Newton’sconstant. We review this evidence in section 2. In section 3 we propose analternate expansion scheme for the self-energy of a quantum field-theoreticparticle. How the new expansion reshuffles the diagrams of conventional per-turbation theory is worked out in section 4. We discuss the 0th order termof the new expansion in section 5. Our conclusions comprise section 6. Arnowitt, Deser and Misner showed that perturbation theory breaks down incomputing the self-energy of a classical, charged, gravitating point particle[15]. It is simplest to model the particle as a stationary spherical shell ofradius R , charge e and bare mass M . In Newtonian gravity its rest massenergy would be, M R c = M c + e πǫ R − GM R . (9)It turns out that all the effects of general relativity are accounted for byassuming it is the full mass M R which gravitates, rather than M [15], M R = M + e πǫ Rc − GM R Rc = Rc G s GRc (cid:16) M + e πǫ Rc (cid:17) − Rc G . (10)The perturbative result is obtained by expanding the square root, M pert = M + e πǫ Rc + ∞ X n =2 (2 n − n ! − GRc ! n − M + e πǫ Rc ! n , (11)and shows the oscillating series of increasingly singular terms characteristicof the previous examples. The alternating sign derives from the fact thatgravity is attractive. The positive divergence of order e /R evokes a negativedivergence or order Ge /R , which results in a positive divergence of order G e /R , and so on. The reason these terms are increasingly singular is thatthe gravitational response to an effect at one order is delayed to a higherorder in perturbation theory. 4he correct result is obtained by taking R to zero before expanding inthe coupling constants e and G ,lim R → M R = e πǫ G ! . (12)Like the example of Section 1 it is finite but not analytic in the couplingconstants e and G . Unlike this example, it diverges for small G . This isbecause gravity has regulated the linear self-energy divergence which resultsfor a non-gravitating charged particle.One can understand the process from the fact that gravity has a built-intendency to oppose divergences. A charge shell does not want to contract inpure electromagnetism; the act of compressing it calls forth a huge energydensity concentrated in the nearby electric field. Gravity, on the other hand,tends to make things collapse, especially large concentrations of energy den-sity. The dynamical signature of this tendency is the large negative energydensity concentrated in the Newtonian gravitational potential. In the limitthe two effects balance and a finite total mass results.Said this way, there seems no reason why gravitational interactions shouldnot act to cancel divergences in quantum field theory [16]. It is especiallysignificant, in this context, that the divergences of some quantum field the-ories — such as QED — are weaker than the linear ones which ADM haveshown that classical gravity regulates. The frustrating thing is that one can-not hope to see the cancellation perturbatively. In perturbation theory thegravitational response to an effect at any order must be delayed to a higherorder. This is why the perturbative result (11) consists of an oscillatingseries of ever higher divergences. What is needed is an approximation tech-nique in which gravity knows what is happening in the gauge sector so thegravitational response can keep pace at the same order.A final point of interest is that any finite bare mass drops out of the exactresult (12) in the limit R →
0. This makes for an interesting contrast withthe usual program of renormalization. Without gravity one would pick thedesired physical mass, M phys , and then adjust the bare mass to be whateverdivergent quantity was necessary to give it, M = M phys − e πǫ Rc . (13)5f course the same procedure would work with gravity as well, M = M phys − e πǫ Rc + GM ǫ Rc . (14)The difference with gravity is that we have an alternative: keep M finiteand let the dynamical cancellation of divergences produce a unique result forthe physical mass. The ADM mechanism is in fact the classical realization ofthe old dream of computing a particle’s mass from its self-interactions [17]. The purpose of this section is to explain the new expansion we propose forparticle masses. For simplicity we work in the context of a charged andgravitating scalar field, although the same technique applies to fermions andto Yang-Mills force fields. The Lagrangian is the sum of those for generalrelativity, electrodynamics and a charged scalar, L GR [ g ] = R √− g πG , (15) L EM [ g, A ] = − F ρσ F µν g ρµ g σν √− g , (16) L SC [ g, A, φ, φ ∗ ] = − ( D µ φ ) ∗ D ν φg µν √− g − M φ ∗ φ √− g . (17)Here and henceforth g µν ( t, ~x ) stands for the metric field, with inverse g µν and determinant g ; A µ ( t, ~x ) denotes the electromagnetic vector potentialwith field strength F µν ≡ ∂ µ A ν − ∂ ν A µ ; and φ ( t, ~x ) is the complex scalarfield. The covariant derivative operator is D µ ≡ ∂ µ + ieA µ . The alert readerwill note that the scalar Lagrangian lacks the quartic self-interaction thatwould be required for perturbative renormalizability in flat space. Becausethe charged scalar is anyway not perturbatively renormalizable once gravityhas been included, there does not seem to be any point to including this termfor a first investigation of nonperturbative renormalizability.We employ the usual units of particle physics in which ¯ h = 1 = c , sothat time and space have the dimensions of inverse mass, the charge e is apure number, the Newton constant G is an inverse mass-squared, and thebare mass M is a mass. We shall also sometimes distinguish time and spacearguments — as in φ ( t, ~x ) — and sometimes lump them together into a singlespacetime coordinate x µ = ( t, ~x ) — as in φ ( x ).6ur attitude is that the physical mass M of single scalar states is somecomplicated function of the bare parameters, e , G and M . Our first goal isto derive a formal expression that would give M , assuming we had infinitecomputational ability. We then develop an alternative to the conventionalperturbative expansion for evaluating this formal expression. If all interactions were turned off it would be simple to express the free scalarfield in terms of the operators b † ( ~k ) and a ( ~k ) which create and annihilate oneparticle states with wave number ~k , φ free ( t, ~x ) = Z d k (2 π ) ( √ ω e − iω t + i~k · ~x a ( ~k ) + 1 √ ω e iω t − ~k · ~x b † ( ~k ) ) . (18)Here ω ≡ q k + M is the free energy. We can invert relation (18) to solvefor the annihilation operator using the Wronskian ↔ W µ ≡ ←− ∂ µ − −→ ∂ µ , a ( ~k ) = i √ ω e iω t ↔ W e φ free ( t, ~k ) . (19)Here and henceforth, a tilde over some function denotes its spatial Fouriertransform, e f ( t, ~k ) ≡ Z d x e − i~k · ~x f ( t, ~x ) . (20)In the presence of interactions it is no longer possible to give explicitrelations such as (19) for the operators which create and destroy exact 1-particle states. However, if we temporarily regulate infrared divergences andagree to understand operator relations in the weak sense then it is possible towrite the operators which annihilate outgoing particles and create incomingones as simple limits [18], a out ( ~k ) = lim t + →∞ ie iωt + √ ωZ ↔ W t + e φ ( t + , ~k ) , (21) (cid:16) a in ( ~k ) (cid:17) † = lim t − →−∞ ie − iωt − √ ωZ ↔ W t − e φ ∗ ( t − , ~k ) . (22)Here Z is the field strength renormalization (defined as the amplitude for thefield to create a 1-particle state) and ω is the full energy, ω ≡ √ k + M . (23)7t this stage we do not know the physical mass M ; it is some function of thebare parameters of the theory.Now consider single particle states whose wave functions in the infinitepast and future are ψ ∓ ( ~k ), respectively. We can employ relations (21-22) toderive an expression for the inner product between these states, D ψ out+ (cid:12)(cid:12)(cid:12) ψ in − E = Z d k (2 π ) Zω ψ ∗ + ( ~k ) ψ − ( ~k ) h lim t + →∞ e iωt + ↔ W t + • ih lim t − →∞ e − iωt − ↔ W t − • i × Z d x e − i~k · ~x D Ω out (cid:12)(cid:12)(cid:12) φ ( t + , ~x ) φ ∗ ( t − , ~ (cid:12)(cid:12)(cid:12) Ω in E . (24)One way of computing the physical mass M would be to adjust it to theprecise value for which expression (24) reduces to, D ψ out+ (cid:12)(cid:12)(cid:12) ψ in − E = Z d k (2 π ) Zω ψ ∗ + ( ~k ) ψ − ( ~k ) . (25)This agrees with the usual definition of the mass as the pole of the propagator,but it is problematic owing to infrared divergences.A more direct way of computing the mass is to focus on the second lineof (24) which we can express as the exponent of − i times some complexfunction ξ ( t + , t − , k ), e − iξ ( t + ,t − ,k ) ≡ Z d x e − i~k · ~x D Ω out (cid:12)(cid:12)(cid:12) φ ( t + , ~x ) φ ∗ ( t − , ~ (cid:12)(cid:12)(cid:12) Ω in E . (26)This function ξ ( t + , t − , k ) includes many things — the field strength renor-malization, finite time correlation effects from multiparticle states, and so on— but only the single particle energy grows linearly with the time interval.Dividing by the time interval and then taking it to infinity gives this energy,lim t ± →±∞ " ξ ( t + , t − , k ) t + − t − = √ k + M . (27)Setting k = 0 gives the physical mass we are seeking. This way of computing M avoids the problems of infrared divergences which affect the field strengthrenormalization but not the mass.It is straightforward to write (26) as a functional integral, e − iξ ( t + ,t − ,k ) = Z d x e − i~k · ~x × %& [ dg ][ dA ][ dφ ][ dφ ∗ ] φ ( t + , ~x ) φ ∗ ( t − , ~ e iS GR [ g ]+ iS EM [ g,A ]+ iS SC [ g,A,φ,φ ∗ ] . (28)8e have subsumed the details of gravitational and electromagnetic gaugefixing into the measure factors [ dg ] and [ dA ]. Note that the various actionintegrals in expression (28) go from time t − to t + , S GR [ g ] ≡ Z t + t − dt Z d x L GR [ g ]( t, ~x ) , (29) S EM [ g, A ] ≡ Z t + t − dt Z d x L EM [ g, A ]( t, ~x ) , (30) S SC [ g, A, φ, φ ∗ ] ≡ Z t + t − dt Z d x L SC [ g, A, φ, φ ∗ ]( t, ~x ) . (31) Expression (28) is only formal because there is not yet any way of definingit or evaluating it. One would usually resort to perturbation theory at thispoint. We shall instead integrate out the scalar field, e − iξ ( t + ,t − ,k ) = Z d x e − i~k · ~x × %& [ dg ][ dA ] i ∆[ g, A ] (cid:16) t + , ~x ; t − , ~ (cid:17) e iS GR [ g ]+ iS EM [ g,A ]+ i Γ SC [ g,A ] . (32)The two new quantities this produces are the scalar propagator i ∆[ g, A ]( x ; x ′ )and the scalar effective action Γ SC [ g, A ]. Both can be defined using the scalarkinetic operator in the presence of an arbitrary metric and vector potential, D [ g, A ] ≡ D µ √− gg µν D ν − M √− g . (33)In rough terms, the scalar propagator is i times the functional inverse of D [ g, A ], subject to Feynman boundary conditions, while the effective actionis i times the logarithm of its determinant, D [ g, A ] × i ∆[ g, A ]( x ; x ′ ) = iδ ( x − x ′ ) , (34)Γ SC [ g, A ] ≡ i ln (cid:16) det[ D [ g, A ] − iǫ ] (cid:17) . (35)It will facilitate subsequent work to be more precise about the definition ofof the scalar propagator i ∆[ g, A ]( x ; x ′ ). Consider a general “mode function” u [ g, A ]( t, ~x ; λ ) which obeys the homogeneous equation, D [ g, A ] × u [ g, A ]( x ; λ ) = 0 . (36)9ere “ λ ” is a (possibly continuous) index which labels the solution; it wouldbe the wave vector ~k for flat space and zero potential. We additionally requirethat the set of all solutions obey the canonical normalization condition, − i Z t =const d x q − g ( x ) g ν ( x ) u [ g, A ]( x ; λ ) (cid:16) ←− D ν − −→ D ∗ ν (cid:17) u ∗ [ g, A ]( x ; κ ) = δ λκ . (37)In terms of these mode functions the propagator is, i ∆[ g, A ]( x ; x ′ ) = X λ " θ ( t − t ′ ) u [ g, A ]( x ; λ ) u ∗ [ g, A ]( x ′ ; λ )+ θ ( t ′ − t ) u ∗ [ g, A ]( x ; λ ) u [ g, A ]( x ′ ; λ ) . (38)Note that only the first theta function contributes in the limit we require,lim t + ≫ t − i ∆[ g, A ] (cid:16) t + , ~x ; t − , ~ (cid:17) = X λ u [ g, A ]( t + , ~x ; λ ) u ∗ [ g, A ]( t − , ~ λ ) . (39)Our expression for the physical mass can therefore be written as, M = lim t ± →±∞ it + − t − ! ln "Z d x X λ %& [ dg ][ dA ] × u [ g, A ]( t + , ~x ; λ ) u ∗ [ g, A ]( t − , ~ λ ) e iS GR [ g ]+ iS EM [ g,A ]+ i Γ SC [ g,A ] . (40) The expression (40) we have derived for the physical mass is exact, butformal because no one knows how to evaluate the functional integral. Thatwould ordinarily be done by resorting to conventional perturbation theory.This could be viewed as an application of a simplified variant of the methodof stationary phase to the original functional integral (28). Because ourmodified expansion involves undoing some of the simplifications we digressto review the technique.Recall that the method of stationary phase gives an asymptotic expansionfor integrals of the form, I ≡ Z dz e if ( z ) . (41)10he technique is to first find the stationary point z (which we assume to beunique) such that f ′ ( z ) = 0. One then expands f ( z ) around z , f ( z ) = f ( z ) + 12 f ′′ ( z )( z − z ) + ∞ X n =3 f ( n ) ( z ) n ! ( z − z ) n , (42) ≡ f + 12 f ′′ ( z − z ) + ∆ f ( z − z ) . (43)The next step is shifting to the variable ζ ≡ z − z and expanding e i ∆ f , I = e if Z dζ e i f ′′ ζ ∞ X m =0 m ! h i ∆ f ( ζ ) i m . (44)At this stage the result is still exact, but generally no simpler to evaluatethan the original form. What gives a computable series is the final stepof interchanging integration and summation. It is at this point that theexpansion ceases to be exact and becomes only asymptotic, I −→ e if ∞ X m =0 i m m ! Z dζ e i f ′′ ζ h ∆ f ( ζ ) i m , (45)= e if × e i π q πf ′′ " × if ′′′′ (cid:16) if ′′ (cid:17) + 524 × ( if ′′′ ) (cid:16) if ′′ (cid:17) + · · · . (46)Conventional perturbation theory is a simplified form of this technique,applied to the functional integral (28). The two simplifications are: • The multiplicative factor of φ ( t + , ~x ) φ ∗ ( t − , ~
0) is not included in the ex-ponential, along with the action; and • The stationary field configuration is assumed to be flat space with nocharge fields, g µν ( x ) = η µν , A µ ( x ) = 0 , φ ( x ) = 0 . (47)These two simplification make conventional perturbation theory much sim-pler and more generally applicable than a strict application of the methodof stationary phase because they remove any dependence of the propagatorsand vertices on the operator whose expectation value is being computed — inthis case φ ( t + , ~x ) φ ∗ ( t − , ~ M can contain only integer powersof G and e . The same two simplifications also imply that the gravitationalresponse to an ℓ loop effect in the electromagnetic sector must be delayeduntil ℓ + 1 loop order.Our modified expansion is defined by making three changes. The firstis to integrate out the matter fields and start from the functional integral(40). That is not so important. The important change is the second one:we include the factor u [ g, A ]( t + , ~x ; λ ) × u ∗ [ g, A ]( t − , ~ λ ) in the exponential,along with the gravitational and electromagnetic actions [19]. This makesthe eventual series vastly more complicated but it does three desirable things: • It allows the apparatus of perturbation theory — propagators and ver-tices — to depend on the operator whose expectation value is beingcomputed; • It permits the eventual expansion to involve complicated, nonanalyticfunctions of e and G ; and • It allows the gravitational response to an electromagnetic effect at someorder to occur at the same perturbative order.The final change is that we drop the scalar effective action Γ SC [ g, A ] fromhow we define the expansion. That is, it plays no role in determining thestationary point, the “classical” action, the propagators or the vertices; weregard e i Γ SC as a multiplicative factor, like φφ ∗ was in the conventional ex-pansion. Although this can be done consistently because Γ SC [ g, A ] is gaugeinvariant, there is no physical justification for it. The scalar effective ac-tion incorporates one loop vacuum polarization and its gravitational analog,which may be important effects. Our justification is just the pragmatic onethat including Γ SC [ g, A ] leads to a more complicated set of equations for thestationary point than we presently understand how to solve. The 0th order term in the new expansion can be interpreted as the energy ofa first-quantized Klein-Gordon particle which moves in the electromagneticand gravitational fields that are sourced by its probability current [20]. To see12his, note that the quantum mechanical propagator for such a Klein-Gordonparticle in fixed background fields A µ ( x ) and g µν ( x ) is, P [ g, A ]( x ; x ′ ) = X λ u [ g, A ]( x ; λ ) u ∗ [ g, A ]( x ′ ; λ ) . (48)That means we can evolve the first-quantized wave function from t ′ to t bytaking the inner product with P [ g, A ]( x ; x ′ ) according to relation (37), Z t ′ =const d x ′ q − g ( x ′ ) g ν ( x ′ ) P [ g, A ]( x ; x ′ ) ↔ W ′ ν u [ g, A ]( x ′ ; λ ) = u [ g, A ]( x ; λ ) . (49)In expression (40) we are going from a delta function ψ − ( ~x ′ ) = δ ( ~x ′ ) toa zero momentum plane wave ψ + ( ~x ) = 1. Of course the stationary fieldconfigurations for the metric and the vector potential are just those sourcedby the quantum mechanical particle itself, hence the stated interpretation ofthe 0th order term.Our ability to evaluate the 0th order term depends upon whether or notthe first-quantized particle can form a bound state in its own potentials. Ifnot then we are left with a complicated scattering problem which seems tobe intractable. However, many simplifications are possible if a bound stateforms. First, one can forget about the continuum solutions; the result for M in expression (40) will derive entirely from the bound state with the largestoverlap between the two asymptotic wavefunctions ψ ± . Second, one canspecialize the wave function to take the form, u [ g, A ]( t, ~x ) = e − iEt F ( r ) (50)Third, in solving for the stationary potentials one need only consider a classof metrics and vector potentials which is broad enough to include the eventualsolution. For scalar QED this reduces the potentials from nine functions ofspacetime (after gauge fixing) to only three functions of a single variable, g µν ( x ) dx µ dx ν = − B ( r ) dt + A ( r ) dr + r d Ω , (51) A µ ( x ) dx µ = − Φ( r ) dt . (52)This simplification might also be relevant to the problem of going beyond 0thorder. Although it is impossible to work out propagators for general back-ground fields, it is sometimes possible to derive the propagator for potentialswhich depend upon only a single variable [21].13 final simplification is the possibility of deriving a variational formulafor the bound state energy. Even if the functions F ( r ), A ( r ), B ( r ) and Φ( r )could not be determined exactly, this technique would allow us to explorereasonable approximations for them. We might even optimize the free pa-rameters in the largest class of solutions for which the field propagators canbe worked out. A variational technique would also give an upper bound onthe bound state energy.The reason deriving a variational technique is only a possibility is thatthis system involves gravitation. It is well known that not all of the gravi-tational potentials contribute positive energy [22]. Of course it is preciselythe negative energy potentials that provide the possibility for canceling self-energy divergences! These negative energy potentials do not lead to anyinstability because they are completely constrained variables; that is, theyare determined in terms of the other variables. However, it is the constraintequations which enforces these relations. The Hamiltonian is not boundedbelow before imposing these constraint equations. So being able to derive avariational formalism depends upon identifying the negative energy poten-tials and solving the constraint equations for them. The purpose of this section is to show where the old diagrams end up in thenew expansion. The point of doing this is not to perform the computation;there are more efficient techniques which will be developed in the next section.The point is rather to see that the new expansion offers gravity the chanceof “keeping up” with what is happening in other sectors. We will showthat all the usual diagrams are present, and at the same “loop” order asin conventional perturbation theory. However, they are joined with a vastclass of new diagrams which are added to one order and subtracted fromanother. To see all this we resort to a very simple, zero dimensional modelof the functional integral in expression (28). We first work out what theconventional expansion gives, then derive the zeroth and first order results forthe new expansion. The section closes with a discussion of the implicationsfor the new expansion. 14 .1 Zero Dimensional Model
We can model the functional integral (28) as an ordinary integration over two N -vectors: x i , representing the metric and gauge fields; and y a representingthe scalar. A simple model for the action is, S ( ~x, ~y ) ≡ A ij x i x j + 12 B ab y a y b + 12 C abi x i y a y b . (53)Of course scalar QED includes interactions between two photons and twoscalars, and the action of general relativity contains self-interactions of themetric, as well as infinite order interactions of the metric with the scalar andthe vector potential. However, our model action (53) will suffice for the pur-pose of understanding how conventional perturbation theory is reorganized.The normalization integral and the 2-point function of our model are, J ≡ Z d N x Z d N y e iS ( ~x,~y ) , (54) J ab ≡ J − Z d N x Z d N y y a y b e iS ( ~x,~y ) . (55)It will simplify the notation if we use the script letters A ij and B ab to denotethe matrix inverses of A ij and B ab , A ij A jk = δ ik , B ab B bc = δ ac . (56)When multiplied by i these inverses are the “propagators” of this one dimen-sional quantum field theory. The expansions of the normalization integraland the 2-point function are, J = (2 π ) N e Nπ i q det( A ) det( B ) ( iC abi iC cdj i A ij h i B ab i B cd +2 i B ac i B bd i + O ( C ) ) . (57) J ab = i B ab + 12 i B ac iC cdi i A ij iC efj i B ef i B db + i B ac iC cdi i A ij i B de iC efj i B fb + O ( C ) . (58)Fig. 1 gives the diagrams associated with the expansion (58).Following subsection 3.2, we integrate out the “matter fields” y a , Z d N y e iS ( ~x,~y ) y a y b = e i A ij x i x j " iB + C k x k ab (2 π ) N e Nπ i q det( B + C ℓ x ℓ ) . (59)15 + + Figure 1: Usual expansion for the scalar propagator. Solid lines are scalars,wavy lines represent photons and gravitons.At this stage it is useful to extract the lowest order contribution from thedeterminant, det( B + C ℓ x ℓ ) = det( B ) × det( I − i B · iC ℓ x ℓ ) . (60)It is also useful to contract into “asymptotic state wavefunctions” — theanalogues of ψ ± . These are the N -vectors u a , analogous to ψ + , and v b ,analogous to ψ − . We shall also include a phase so that u a i B ab v b = 1. Thenthe contraction of these two vectors into the field dependent propagator, theanalogue of i ∆[ g, A ]( x ; x ′ ), can be written as, u a × " B + C k x k ab × v b = u a × " iI − i B · iC k x k ac × i B cb × v b , (61)= 1 + u a × ∞ X n =1 "(cid:16) i B · iC k x k (cid:17) n ac × i B cb × v b , (62) ≡ i ∆( ~x ) . (63)Except for the phase, the quantity whose logarithm we wish to take is, J = u a × J ab × v b = q det( A )(2 π ) N e Nπ i Z d N x e i E ( ~x )+ i Γ( ~x ) . (64)Here the exponent E ( ~x ) and the “matter effective action” Γ( ~x ) are, E ( ~x ) = 12 A ij x i x j − i ln h i ∆( ~x ) i , Γ( ~x ) = i h det( I − i B· iC k x k ) i . (65)16 v − u v u vu v + u vu v + u vu v + Figure 2: Expansion for the solution X i to equation (67). Solid lines are y a propagators (scalars), wavy lines represent x i propagators (photons andgravitons). We want to do the stationary phase expansion on expression (64), but treat-ing Γ( ~x ) as higher order. Hence the zeroth order term in the expansion is, J ≡ e i E ( ~X ) = h i ∆( ~X ) i e i A ij X i X j , (66)where the “stationary field configuration” X i is found by solving, ∂ E ( ~x ) ∂x i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ~x = ~X = A ij X j − i ∂i ∆( ~X ) ∂x i i ∆( ~X ) = 0 . (67)Despite the simplicity of our model, equation (67) is too difficult to solveexactly for general A ij , B ab and C abi . However, we are only interested in aperturbative solution — in powers of the “interaction” C abi — and that issimple enough to generate by iteration, X i = i A ij ∂i ∆( ~X ) ∂x j i ∆( ~X ) , (68)= i A ij u h i B iC j i B i v − i A ij u h i B iC j i B i v × u h i B iC k i B i vi A kℓ u h i B iC ℓ i B i v + i A ij u h i B (cid:16) iC j i B iC k + iC k i B iC j (cid:17) i B i vi A kℓ u h i B iC ℓ i B i v + O ( C ) . (69)Fig. 2 gives a diagrammatic representation of the expansion.In evaluating J it is useful to first formally express things in terms of i ∆( ~X ) and i ∆( ~X ) ∂x i X i , i ∆( ~X ) = u h i B · iC k X k · i B i v + u h i B · iC k X k · i B · iC ℓ X ℓ · i B i v + · · · , (70) ∂i ∆( ~X ) ∂x i X i = u h i B · iC k X k · i B i v + 2 u h i B · iC k X k · i B · iC ℓ X ℓ · i B i v + · · · . (71)17 + u vu v − u vu v u vu vu v + Figure 3: Expansion for the 0th order term J . Solid lines are y a propagators(scalars), wavy lines represent x i propagators (photons and gravitons).Using expressions (66) and (67) we can write J as, J = 1 + i ∆( ~X ) − ∂i ∆( ~x ) ∂x i X i + [ ∂i ∆( ~X ) ∂x i X i ] i ∆( ~X )] + · · · , (72)= 1 + 12 u h i B · iC k X k · i B i v + 18 u h i B · iC k X k · i B i v ! + . . . (73)Now just substitute (69) in (73) to obtain, J = 1 + 12 u h i B · iC i · i B i v i A ij u h i B · iC j · i B i v − u h i B · iC i · i B i v i A ij u h i B · iC j · i B i v ! + u h i B · iC i · i B i v × i A ij u h i B · iC j · i B · iC k · i B i v i A kℓ u h i B · iC ℓ · i B i v + O ( C ) . (74)Fig. 3 gives a diagrammatic expansion of J . One loop effects come from the determinantal correction to the Method ofStationary Phase, J ≡ e i E ( ~X ) × det " A ij ∂ E ( ~X ) ∂x j ∂x k − . (75)The matrix whose determinant we must compute is, A ij ∂ E ( ~X ) ∂x j ∂x k = δ ik + " X i ∂i ∆( ~X ) ∂x k − i A ij ∂ i ∆( ~X ) ∂x j ∂x k i ∆( ~X ) ≡ δ ik + M ik . (76)18e can evaluate the determinant in (75) by first expanding in powers of thematrix M ij ,1 q det( I + M ) = e − Tr[ln( I + M )] , (77)= 1 −
12 Tr[ M ] + 18 (cid:16) Tr[ M ] (cid:17) + 14 Tr[ M ] + O ( M ) . (78)The determinantal contribution to (75) is sufficiently complex that it isworthwhile to present results for each term of (78). As before, it is best toexpress these contributions in terms of the full solution X i , −
12 Tr[ M ] = − u h i B · iC k X k · i B i v + i A ij u h i B · iC i · i B · iC j · i B i v + 12 u h i B · iC k X k · i B i v ! − u h i B · iC k X k · i B i v i A ij u h i B · iC i · i B · iC j · i B i v − u h i B · iC k X k · i B · iC ℓ X ℓ · i B i v + i A ij u " i B iC i · i B · iC j · i B · iC k X k + iC i · i B · iC k X k · i B · iC j + iC k X k · i B · iC i · i B · iC j ! i B v + O ( C ) , (79)18 (cid:16) Tr[ M ] (cid:17) = 18 u h i B · iC k X k · i B i v ! − u h i B · iC k X k · i B i v i A ij × u h i B · iC i · i B · iC j · i B i v + 12 i A ij u h i B · iC i · i B · iC j · i B i v ! + O ( C ) , (80)14 Tr[ M ] = 14 u h i B · iC k X k · i B i v ! − u h i B · iC k X k · i B · iC ℓ X ℓ · i B i v + 12 i A ij i A kℓ × u h i B · iC j · i B · iC k · i B i v u h i B (cid:16) iC ℓ · i B · iC i + iC i · i B · iC ℓ (cid:17) i B i v + O ( C ) . (81)The first order contribution to J in our new expansion is, J = e i E ( ~X ) ( det " A ij ∂ E ( ~X ) ∂x j ∂x k − − ) , (82)= − u h i B · iC i · i B i v i A ij u h i B · iC j · i B i v + i A ij u h i B · iC i i B iC j · i B i v + 78 u h i B · iC i · i B i v i A ij u h i B · iC j · i B i v ! − u h i B · iC i · i B i v i A ij u vu v + u v + u vu v − u vu vu v − u v u vu v + u vu v + u vu v + u vu v + u vu v + u vu v + u v Figure 4: Expansion for the 1st order term J . Solid lines are y a propagators(scalars), wavy lines represent x i propagators (photons and gravitons). × u h i B · iC j · i B · iC k · i B i v i A kℓ u h i B · iC ℓ · i B i v − u h i B · iC i · i B i v × i A ij u h i B · iC j · i B i v i A kℓ u h i B · iC k · i B · iC ℓ · i B i v + u h i B · C i · i B i v i A ij i A kℓ × u " i B iC j · i B · iC k · i B · iC ℓ + iC k · i B · iC j · i B · iC ℓ + iC k · i B · iC ℓ · i B · iC j ! i B v + 12 i A ij u h i B · iC j · i B · iC k · i B i v i A kℓ u h i B (cid:16) iC ℓ · i B · iC i + iC i · i B · iC ℓ (cid:17) i B i v + 12 i A ij u h i B · iC i · i B · iC j · i B i v ! + O ( C ) . (83)Fig. 4 gives a diagrammatic representation of the expansion. Let us compare the new expansion J = J + J + · · · with the old one, J = J + J + · · · . The first two terms of the new expansion are expressions(66) and (83). In contrast, the first two terms of the old expansion are, J = 1 , J = i A ij u h i B · iC i · i B · iC j i B i v . (84)It is useful to define the difference between new and old at order ℓ ,∆ J ℓ ≡ J ℓ − J ℓ . (85)20e can read off ∆ J from expression (66),∆ J = 12 u h i B · iC i · i B i v i A ij u h i B · iC j · i B i v − u h i B · iC i · i B i v i A ij u h i B · iC j · i B i v ! u h i B · iC i · i B i v i A ij × u h i B · iC j · i B · iC k · i B i v i A kℓ u h i B · iC ℓ · i B i v + O ( C ) . (86)Note that each contribution to ∆ J is a “tree diagram”; the expansion inpowers of the interaction C could also be viewed as an expansion in numbersof external lines. From expression (83) we see that ∆ J possesses both “tree”and “one loop” contributions,∆ J = − u h i B · iC i · i B i v i A ij u h i B · iC j · i B i v + 78 u h i B · iC i · i B i v i A ij u h i B · iC j · i B i v ! − u h i B · iC i · i B i v i A ij × u h i B · iC j · i B · iC k · i B i v i A kℓ u h i B · iC ℓ · i B i v + O ( C ) , (87)∆ J = − u h i B · iC i · i B i vi A ij u h i B · iC j · i B i v i A kℓ u h i B · iC k · i B · iC ℓ · i B i v + u h i B · C i · i B i v i A ij i A kℓ u " i B iC j · i B · iC k · i B · iC ℓ + iC k · i B · iC j · i B · iC ℓ + iC k · i B · iC ℓ · i B · iC j ! i B v + 12 i A ij u h i B · iC j · i B · iC k · i B i v i A kℓ u h i B (cid:16) iC ℓ × i B · iC i + iC i · i B · iC ℓ (cid:17) i B i v + 12 i A ij u h i B · iC i · i B · iC j · i B i v ! + O ( C ) . (88)Note that the order C contributions to ∆ J and ∆ J cancel. They donot cancel at order C , but the residual C terms are canceled by ∆ J . Sothe new expansion does not move the old diagrams from one order to another,rather it adds a new class of diagrams — with more external lines — to oneorder and subtracts them from higher orders. What we have learned can besummarized by the following observations: • Each term J ℓ in the new expansion can be written as an infinite serieswhich begins at order C ℓ ; • The lowest order term in this series expansion of each J ℓ is the old ℓ -loop term J ℓ ; 21 The new terms in J ℓ involve diagrams with up to and including ℓ loops; • The new terms at each order have more powers of the external statefactors u a and v b ; and • The sum of all the new terms is zero.Although there is no guarantee that the new expansion is free of ultra-violet divergences, it does possess a number of desirable features. Note firstthat ultraviolet divergences are no worse in the new expansion than the oldone because none of the new diagrams in J ℓ have more than ℓ loops. Oursimple model makes no distinction between photons and gravitons but thequantum field theoretic expansion will of course involve both particles. Thenew diagrams — with more powers of the coupling constants e and G , butno more loops — are one way gravity at order ℓ in the new expansion canknow about an order ℓ divergence from the gauge sector. So the new ex-pansion breaks the conundrum of conventional perturbation theory that thegravitational response to a problem at one order cannot come until the nextorder. Note finally, that the infinite sums of new diagrams at any fixed orderof the new expansion allow the possibility of getting nonanalytic dependenceupon G and e . The purpose of this section is the evaluate the 0th order term in the newexpansion of expression (40) for the scalar mass. Recall that this meanssolving the problem of a first quantized Klein-Gordon scalar which forms abound state in the gravitational and electrostatic potentials sourced by itsown probability current. We begin by specializing the Lagrangians, the fieldequations and the normalization condition to the bound state ansatz (50-52). We then note that one of the equations can be solved exactly for thenegative energy gravitational degree of freedom. Substituting this solutioninto the action gives a functional of the remaining fields whose minimizationyields the remaining field equations. This functional serves as the basis for avariational formulation of the problem. The section closes with a numericalsolution. 22 .1 Field Equations
If a bound state forms we can simplify the fields to take the form (50-52), φ = e − iEt F ( r ) , A µ = − Φ( r ) δ µ , (89) g = − B ( r ) , g i = 0 , g ij = π ij + b r i b r j A ( r ) . (90)Here b r i ≡ x i /r is the radial unit vector and π ij ≡ δ ij − b r i b r j is the transverseprojection operator. The nonvanishing components of the affine connectionare,Γ i = b r i (cid:16) B ′ B (cid:17) , Γ i = b r i (cid:16) B ′ A (cid:17) , Γ i jk = b r i b r j b r k A ′ A + b r i π jk (cid:16) A − rA (cid:17) . (91)The nonzero components of the Riemann tensor are, R i j = b r i b r j h − B ′′ B + B ′ B + A ′ B ′ AB i − π ij (cid:16) B ′ rAB (cid:17) , (92) R ijkℓ = h π ik b r j b r ℓ − π iℓ b r j b r k i(cid:16) A ′ rA (cid:17) + h π iℓ b r i b r k − π jk b r i b r ℓ i(cid:16) A ′ rA (cid:17) + h π ik π jℓ − π iℓ π jk i(cid:16) A − r A (cid:17) . (93)Contraction gives the Ricci tensor R µν = R ρµρν , R = B ′′ A − B ′ AB − A ′ B ′ A + A ′ rA , (94) R ij = b r i b r j h − B ′′ B + B ′ B + A ′ B ′ AB + A ′ rA i + π ij h − B ′ rAB + A ′ rA + (cid:16) A − r A (cid:17)i . (95)And contracting that gives the Ricci scalar R = g µν R µν , R = − B ′′ AB + B ′ AB + A ′ B ′ A B + 2 A ′ rA − B ′ rAB +2 (cid:16) A − r A (cid:17) . (96)In specializing the various Lagrangians (15-17) to our ansatz (89-90) it isbest to use spherical coordinates and integrate the angles so that the effectivemeasure factor is, √− g = 4 πr q A ( r ) B ( r ) . (97)23DM long ago worked out the surface term one must add to the HilbertLagrangian (15) for asymptotically flat field configurations [22]. For ourstatic, spherically symmetric geometry the result is, L ADM −→ πG " R √− g − ∂∂r (cid:16) B ′ √− gAB (cid:17) = 18 πG " A ′ rA + (cid:16) A − r A (cid:17) √− g . (98)The other two Lagrangians (16-17) require no surface terms, L EM −→ ǫ Φ ′ AB √− g , (99) L SC −→ B ( E + e Φ) F √− g − A F ′ √− g − M F √− g . (100)Hence the action is, S [ F, Φ , B, A ] = ( t + − t − ) Z ∞ dr √− g ( πG " A ′ rA + (cid:16) A − r A (cid:17) + ǫ Φ ′ AB + ( E + e Φ) F B − F ′ A − M F ) . (101)It is well known that the operation of making an ansatz for the solutiondoes not commute with varying the action to obtain the field equations.The correct procedure is to vary first and then simplify. However, all of theequations one gets by simplifying first and then varying are correct, and itturns out that the only ones we miss are the trivial relations implied by theBianchi identity [23]. Except for the overall factor of ∆ t ≡ ( t + − t − ), thevariations with respect to various fields are,1∆ t δSδF = ∂∂r " F ′ √− gA + 2( E + e Φ) F √− gB − M F √− g , (102)1∆ t δSδ Φ = ∂∂r " ǫ Φ ′ √− gAB + 2 e ( E + e Φ) F √− gB , (103)1∆ t δSδB = ( πG " A ′ rA + (cid:16) A − r A (cid:17) − ǫ Φ ′ AB − ( E + e Φ) F B − F ′ A − M F ) √− g B , (104)1∆ t δSδA = ( πG " − B ′ rAB + (cid:16) A − r A (cid:17) − ǫ Φ ′ AB
24 ( E + e Φ) F B + F ′ A − M F ) √− g A . (105)Our goal is to find F ( r ), Φ( r ), B ( r ) and A ( r ) so as to make each of thevariations (102-105) vanish. However, those equations have solutions for anyconstant E . As always, it is normalizability which puts the “quantum” inquantum mechanics. For our problem the normalization condition is,2 Z ∞ dr √− g ( E + e Φ) F B = 1 . (106)Any field configurations F ( r ), Φ ( r ), B ( r ) and A ( r ) which make all thevariations (102-105) vanish and also obey (106) will only do so for very specialvalues of E . Note that the field equations and normalizability requires A ( r )to approach one at infinity, and F ( r ) to approach zero. The asymptoticvalues of Φ( r ) and B ( r ) are both gauge choices. By choosing the U (1) gaugeparameter to be − t × Φ ∞ we can make Φ( r ) vanish at infinity. By changingtime to t/k we induce the rescalings, B ( r ) −→ k × B ( r ) , (107)Φ( r ) −→ k × Φ( r ) , (108) E −→ k × E . (109)We shall always use this freedom to make B ( r ) approach one at infinity.If we can find a normalized solution F ( r ), Φ ( r ), B ( r ) and A ( r ) thenour zeroth order result for the mass is, M = E − t S [ F , Φ , B , A ] . (110)The first term on the right hand side of (110) is from the the two scalar fields, φ ( t + , ~x ) φ ∗ ( t − , ~ −→ F ( r ) F (0) e − iE ∆ t . (111)The final term in (110) represents the gravitational and electromagnetic con-tribution to the mass. Note that the scalar action vanishes for solutionsbecause the scalar Lagrangian is a surface term which goes to zero, L SC −→ − ∂∂r " F F ′ √− g A . (112)25 .2 A Variational Formalism Solving differential equations is tough, and we are not able to find exactsolutions for all four of the fields. For many bound state problems in quantummechanics the absence of exact solutions is not crippling because variationaltechniques allow one to derive strong bounds on the ground state energy.Such a technique would be simple to formulate for our Klein-Gordon scalarif only the electromagnetic and gravitational potentials were fixed. However,the fact that these potentials are sourced by the Klein-Gordon wave functionitself endows this problem with a slippery, nonlinear character. The presenceof gravitational interactions is especially problematic because some of theconstrained degrees of freedom in gravity possess negative energy. Instabilityis only avoided by constraining these degrees of freedom to obey their fieldequations; attempting to minimize the action with respect to these degreesof freedom would carry one away from the actual solution.There are good reasons for suspecting that the field B ( r ) is the onlynegative energy degree of freedom. In the normal ADM formalism B = N would be the square of the lapse field, and it could be specified arbitrarilyas a choice of gauge. However, B is a dynamical degree of freedom in thisproblem. The structure of our Lagrangian is similar to the usual formalismfor describing cosmological perturbations during primordial inflation [24]. Inthat setting, as for us, the Lagrangian can be written as the sum of a “kinetic”part K and a “potential” part P , L = " KB + P √− g . (113)For us the kinetic and potential parts are, K = ǫ Φ ′ A + ( E + e Φ) F , (114) P = 18 πG " A ′ rA + (cid:16) A − r A (cid:17) − F ′ A − M F . (115)The field equation for B is algebraic and has a trivial solution,1∆ t δSδB = " − KB + P √− g B = 0 = ⇒ B = KP . (116)26ubstituting (116) into (113) allows us to express the action in terms ofjust F ( r ), Φ( r ) and A ( r ), S [ F, Φ , A ] = 8 π ∆ t Z ∞ dr r √ AKP . (117)It is simple to show that varying (117) gives equations (102-103) and (105).Because (117) is positive semi-definite, the problem of extremizing it is likelyto be the same as that of minimizing it. The corresponding normalizationcondition is, 8 π Z ∞ dr r s AKP ( E + e Φ) F = 1 . (118)And our zeroth order result for the scalar mass becomes, M = E − π Z ∞ dr r q A K P . (119)We illustrate the method with simple trial functions for F ( r ), Φ( r ) and A ( r ). We cannot make F ( r ) a spherical shell like ADM, or even a hard sphere,because the factors of F ′ become ill-defined if F ( r ) has a discontinuity. Thenext best thing is to assume the scalar drops linearly to zero within somedistance R , F ( r ) = a ( R − r ) θ ( R − r ) . (120)Comparably simple forms for the potentials are,Φ( r ) = − eθ ( R − r )4 πǫ R − eθ ( r − R )4 πǫ r , (121)1 A ( r ) = h − br i θ ( R − r ) + h − cr + dr i θ ( r − R ) . (122)We can make A ( r ) continuous by choosing, b = cR − dR . (123)The corresponding forms for the kinetic and potential terms are, K ( r ) = (cid:16) E − αR (cid:17) a ( R − r ) θ ( R − r ) + α πr (cid:16) − cr + dr (cid:17) θ ( r − R ) , (124) P ( r ) = " b πG − a h − br + M ( R − r ) i θ ( R − r ) + dθ ( r − R )8 πGr . (125)27ence the potential B ( r ) is, B ( r ) = ( E − αR ) a ( R − r ) θ ( R − r ) b πG − a [1 − br + M ( R − r ) ] + α πr (1 − cr + dr ) θ ( r − R ) d πGr . (126)Enforcing that B ( r ) goes to one at infinity determines the coefficient d , d = αG , (127)where α ≡ e / πǫ ≈ /
137 is the fine structure constant. Enforcing conti-nuity at r = R — which means B ( R ) = 0 — requires the choice, c = R + αGR = ⇒ b = 1 R . (128)At this stage the free parameters in our trial solution are R , a and theenergy E . The next step is to enforce normalizability, which requires,18 π = Z R dr r √ − br ( E − αR ) a ( R − r ) q b πG − a [1 − br + M ( R − r ) ] , (129)= ( RE − α ) R a Z dx x √ − x (1 − x ) q πGR a − [1 − x + R M (1 − x ) ] , (130) ≡ ( RE − α ) R a × I (cid:16) GR a , R M ) . (131)The function I ( x, y ) in equation (131) can be expressed in terms of ellipticintegrals but we may as well treat it as an elementary function and use it toexpress the energy, E = " q πR a I ( GR a , R M ) + α R . (132)We can now regard the two free parameters as, A ≡ RaM , µ ≡ RM . (133)The field action is, S ∆ t = " √ π AJ ( GM A , µ ) q I ( GM A , µ ) + αµ M , (134)28here the new integral is, J ( GM A , µ ) ≡ Z dx x (1 − x ) √ − x s πGM A − [1 − x + µ (1 − x ) ] . (135)We determine the free parameters A and µ by minimizing S/ ∆ t .Expressions (131), (134) and (135) seem very complicated. However,note that because I ( GM A , µ ) is an increasing function of A and µ , and J ( GM A , µ ) is a decreasing function of A and µ , S/ ∆ t is a decreasingfunction of µ at fixed A . Hence S/ ∆ t is minimized, at fixed A , by choosing µ to be the maximum value for which the two integrals remain real, µ max = s πGM A − ⇒ < A < A max = s πGM . (136)At µ = µ max the two integrals become, i ( µ max ) ≡ I ( GM A , µ ) = Z dx x √ − x (1 − x ) q µ [1 − (1 − x ) ]+ x , (137) j ( µ max ) ≡ J ( GM A , µ ) = Z dx x (1 − x ) √ − x q µ [1 − (1 − x ) ]+ x . (138)And the field action (134) takes the form, S ∆ t = "s GM j ( µ max ) q (1+ µ ) i ( µ max ) + αµ max M . (139)At this stage the problem is numerical. Fig. 5 shows S/M ∆ t as a functionof µ max for GM = 0 .
36. The minimum seems to be at about µ max = 0 . R = µ max q GM × √ G ≈ . × √ G , (140) a = s GM π µ max q µ × G ≈ . × G , (141) E = " GM µ vuut µ i ( µ max ) + α q GM µ max √ G ≈ × √ G , (142) M = E − S ∆ t ≈ × √ G . (143)We will see that these results are not very accurate.29 .2 0.4 0.6 0.8 1.0 Μ max S D tM Figure 5: Plot of SM ∆ t versus µ max from equation (139) for GM = 0 . µ max = 0 . It is desirable to check any variational ansatz against a direct, numericalsolution to the problem. Of course computers can only solve for dimensionlessquantities, so it is first necessary to express everything in geometrodynamicalunits, using G to absorb each quantity’s natural units, r = √ G e r , M = f M √ G , E = e E √ G , M = f M √ G , (144) F ( r ) = e F ( e r ) √ G , Φ( r ) = [ e Φ( e r ) − e E ] e √ G , B ( r ) = e B ( e r ) , A ( r ) = e A ( e r ) , (145) q − g ( r ) = G q − e g ( e r ) , K ( r ) = f K ( e r ) G , P ( r ) = e P ( e r ) G . (146)Note that we have absorbed the energy into the electrostatic potential. In allcases we employ a tilde to denote the dimensionless quantity. Geometrody-namic fields such as e F are considered to be functions of the geometrodynamic30
00 400 600 800 r F
200 400 600 800 r - - - - F Figure 6: Plots of the scalar amplitude F ( r ) (in units of M Pl = 1 / √ G ) andthe electrostatic potential Φ( r ) (in units of M Pl /e ) as functions of r (in unitsof 1 /M Pl ). These figures were generated for bare mass M = 0 . M Pl .radius e r . A prime on such a field indicates differentiation with respect to e r ,so we have, F ′ = e F ′ G , Φ ′ = e Φ ′ eG . (147)In these units the four field equations (102-105) take the form, ∂∂ e r " e F ′ √− e g e A + e Φ e F √− e g e B − f M e F q − e g = 0 , (148) ∂∂ e r " e Φ ′ √− e g e A e B + 8 πα e Φ e F √− e g e B = 0 , (149)18 π " e A ′ e r e A + (cid:16) e A − e r e A (cid:17) − e Φ ′ πα e A e B − e Φ e F e B − e F ′ e A − f M e F = 0 , (150)18 π " − e B ′ e r e A e B + (cid:16) e A − e r e A (cid:17) − e Φ ′ πα e A e B + e Φ e F e B + e F ′ e A − f M e F = 0 . (151)(Recall that α ≡ e / πǫ ≈ /
137 is the fine structure constant.) The kineticand potential terms are, f K = e Φ ′ πα e A + e Φ e F , (152)31
00 400 600 800 r B
200 400 600 800 r B Figure 7: Plots of minus the tt component of the metric B ( r ) (dimensionless)as a function of r (in units of 1 /M Pl = √ G ). The right hand figure has anexpanded vertical axis to show the small variation of the field. These figureswere generated for bare mass M = 0 . M Pl . e P = 18 π " e A ′ e r e A ′ + (cid:16) e A − e r e A (cid:17) − e F ′ e A − f M e F . (153)The normalization condition is,2 Z ∞ d e r q − e g e Φ e F e B = 1 . (154)And the final result is, f M = e E − π Z ∞ d e r e r q e A f K e P . (155)The nonlinear nature of this problem requires a special solution strategy.The development of our technique was facilitated by the vast amount of workthat has been done of “boson stars” [25, 26]. There has also been a recentstudy by Carlip of gravitationally bound atoms [27].Our strategy is to begin by evolving equations (148-151) outward from e r = 0, with arbitrary choices for e F (0) > e Φ(0) < e B (0) >
0, and withthe other boundary values at, e F ′ (0) = 0 , e Φ(0) = 0 , e A (0) = 1 . (156)32
00 400 600 800 r A
200 400 600 800 r A Figure 8: Plots of the rr component of the metric A ( r ) (dimensionless) asa function of r (in units of 1 /M Pl = √ G ). The right hand figure has anexpanded vertical axis to show the small variation of the field. These figureswere generated for bare mass M = 0 . M Pl .The choice of e B (0) > e B ( e r ) to approach one at infinity. However,the choice of e Φ(0) essentially gives the energy, and this matters of course.There is zero probability of guessing a true eigenvalue. With the other con-ditions fixed, varying e Φ(0) gives solutions for which e F ( e r ) either becomesnegative (which a magnitude cannot do) or grows at infinity (which a nor-malizable solution cannot do). One knows that a true energy eigenvalue hasbeen bracketed between two different choices of e Φ(0) when the behavior of e F ( e r ) changes from one extreme to the other. Then one closes in on the eigen-value to whatever accuracy is desired. Note that this means cutting off thebehavior of the solution past a certain value of e r , beyond which e F ( e r ) beginsto degenerate.The procedure we have just outlined gives a solution which is normal-izable , but not yet normalized. For that we compute (154) and then eitherincrease or decrease e F (0) as needed. Of course the nonlinear nature of thisproblem means that one does not get a solution by simply multiplying e F ( e r )by a constant! We must instead start from the new e F (0) and again gothrough the process of trapping the energy eigenvalue. However, our evolu-tion programs are efficient enough that this can be done to high accuracy,33nd fairly quickly.Figures 6-8 show the behavior of the fields for f M = 0 .
6. For this baremass the energy is e E ≈ . f M ≈ − . e S SC / ∆ t ≈ × − . Another measure of accuracy comesfrom the finite cutoff at e r = e R cut , occasioned by the finite accuracy of e E .For that bare mass we cut the various integrations off at e R cut = 500, whichcorresponds to a contribution of α/ e R cut ≈ − from the electromagnetictail.The variational results (140-143) obtained in the previous subsection arequite different from the numerical solution. From Fig. 6 one can see thanthe scalar amplitude has roughly the same shape as that of our trial function(120), but with initial height Ga ≈ . Ga ≈ .
7. And the radial extent is about √ G R ≈ √ G R ≈ .
20. There is no chance thatthe discrepancy derives from the numerical solution, the error of which weestimate to be no larger than 10 − . The problem must lie instead with thevariational formalism. Our trial solution seems roughly correct, but it may bethat, like B ( r ), the gravitational potential A ( r ) represents a negative energydirection in field space. In that case minimizing the constrained action wouldtake us away from the actual solution, which seems to be what has happened.Table 1 gives our results for e E and f M for a variety of different bare masses.The most obvious feature is the almost total cancellation between the energyof the scalar wave function and the field action, to give a very small, negativetotal mass. This is physical nonsense because it fails to agree with the massone can read off from asymptotic values of the metric. We believe that theproblem arises from the asymptotic conditions (21-22) — which are certainlyvalid for scattering with other particles — not being right for the study ofself-interactions. We believe that this can be fixed without much change.The other features of our numerical work are: • The energy e E agrees with the mass inferred from the asymptotic valuesof the metric. • There is no bound state unless the bare mass f M exceeds the ADMresult of √ α ≈ .
85 [26]. • The bound state energy e E is in all cases less than the bare mass.34 M e E f M e S SC / ∆ t e R cut . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − e E and the total mass f M fordifferent values of the bare mass f M . Also given are the scalar action e S SC / ∆ t ,which should vanish, and the cutoff radius beyond which the finite accuracyof the energy eigenvalue makes the solution unreliable. All quantities areexpressed in Planck units. 35 The ratio e E/ f M increases with f M and eventually becomes zero [26].Because there would not even be any bound states without gravity, it seemsfair to conclude that the system depends nonanalytically upon G . We have explored the possibility that the apparent problems of quantumgeneral relativity may be artifacts of conventional perturbation theory. Onemight think this unlikely because the absence of recognizable, low energyquantum gravitational phenomena implies that some asymptotic series ex-pansion is wonderfully accurate. However, it may be that the correct seriesinvolves logarithms or fractional powers of Newton’s constant. If that werethe case, trying to re-expand in integer powers of G would result in an esca-lating series of divergences, which is exactly what conventional perturbationtheory shows.We studied this possibility in the context of computing the mass of acharged, gravitating scalar. An exact result for the classical limit of thissystem was derived by ADM in 1960 [15], and it does exhibit both nonanalyticdependence upon G and the breakdown of conventional perturbation theory.If the classical point particle is regulated to be a spherical shell of radius R ,the ADM result is, M R = Rc G s
1+ 2 GM Rc + e G πǫ R c − Rc G . (157)The correct zero radius limit is M = q α/G . Its finiteness results from neg-ative gravitational interaction energy canceling the positive electromagneticenergy. In contrast, the perturbative result is obtained by first expandingthe square root in powers of G and e , which produces a series of ever-higherdivergences with alternating signs. The alternating signs are a signal thatgravity is trying to cancel the electromagnetic self-energy divergence, but thiscancellation can never happen in conventional perturbation theory becausethe gravitational response to a divergence at one order is delayed until oneorder higher. What we need for quantum gravity is an alternate expansionin which the negative gravitational interaction energy has a chance to “keepup” with what is going on in the positive energy sectors.36n section 3 we derived an exact functional integral expression (40) for thescalar mass. We then developed an alternate asymptotic expansion based onthe Method of Stationary Phase, with the full functional integrand — notjust the action — used to determine the stationary point. This is more diffi-cult to implement than conventional perturbation theory, but it is also morecorrect. A simple integral representation for the Bessel function illustratesthe distinction between our approach and that of conventional perturbationtheory, J N ( z ) = 12 π Z π − π dθ e iz sin( θ ) × (cid:16) e − iθ (cid:17) N . (158)In our approach both factors are included in the exponent and the two sta-tionary points are found by minimizing the function f ( θ ) = z sin( θ ) − N θ , f ′ ( θ ± ) = 0 = ⇒ θ ± = ± acos (cid:16) Nz (cid:17) . (159)The values of the function and its second derivative at these points are, f ( θ ± ) = ± h √ z − N − N acos (cid:16) Nz (cid:17)i , f ′′ ( θ ± ) = ∓√ z − N . (160)And the result for the 0th and 1st order contributions is, J N ( z ) −→ s π √ z − N cos h √ z − N − N acos (cid:16) Nz (cid:17) − π i . (161)In contrast, conventional perturbation theory would be based on the function f ( θ ) = z sin( θ ), with the stationary points at θ ± = ± π . The result for the0th and 1st order contributions from conventional perturbation theory is, J N ( z ) −→ s πz cos h z − N π − π i . (162)Section 4 presents an analysis of the new expansion in the context ofa simplified model. We conclude that all the old ℓ loop diagrams appearat ℓ -th order in the new expansion. However, the old ℓ loop diagrams arecombined with an infinite class of new diagrams which possess more externallines and no more than ℓ loops. The new diagrams which are added at ℓ -thorder are all subtracted at higher orders, so we are really adding zero to theusual expansion. Because the new ℓ -th order diagrams have no more than37 loops, the divergences of the new expansion can be no worse than thoseof conventional perturbation theory. Because infinitely many new diagramsare added at each order, the new expansion can depend nonanalytically onNewton’s constant. It also offers a way in which the negative gravitationalinteraction energy can respond, at the same order, to problems in the positiveenergy sectors. These are all desirable features, although it must be admittedthat these is no guarantee at this stage that the new expansion is any betterthan the old one.The analysis of section 4 was done only to understand how the new ex-pansion compares with the old one. There are much better ways of actuallyimplementing the new expansion. We exploit two of these methods in section5 to evaluate the zeroth order result. Our analysis is based on interpreting thezeroth order term as the phase developed by a first-quantized Klein-Gordonscalar moving in the gravitational and electrodynamic potentials which aresourced by its own probability current. The fact that this system reducesto the ADM problem for ¯ h → M is greater than the ADM result of q α/G .We developed solutions for many choices of M above this limit. All of themshow an almost total cancellation between the energy of the scalar wave func-tion and the field energy of the gravitational and electromagnetic potentials.This gives nearly zero for the total mass, which seems to be nonsense. It alsofails to agree with a determination of the scalar mass from the asymptotic38alues of the gravitational potentials.The problem seems to derive from our use of the asymptotic conditions(21-22). Expressed in simple words, these conditions mean that “the fieldsbecome free at asymptotically early and late times”. That is perfectly true(in the weak operator sense and assuming the existence of a mass gap) forinteractions between different particles, which is the usual application [18].However, we are here trying to use the conditions to study interactions of aparticle with itself . These self-interactions would usually be subsumed intoforcing the field strength and mass to come out right by renormalization, butthat is exactly what we are not doing. We believe that when a more accurateprocedure is used to interpolate the single particle states — which might beas simple as including a U (1) gauge string between the two fields to makethem invariant — then the nonsense result for M will go away, and most ofour analysis of the new expansion will be unchanged. Acknowledgements
We are grateful to Stanley Deser for years of guidance and inspiration.We have profited from conversations on this subject with G. T. Horowitz andT. N. Tomaras. This work was partially supported by European Union GrantFP-7-REGPOT-2008-1-CreteHEPCosmo-228644, by FQXi Grant RFP2-08-31, by NSF grant PHY-0855021, and by the Institute for Fundamental The-ory at the University of Florida.
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