Correspondence Principle in Quantum Gravity

View metadata, citation and similar papers at core.ac.uk brought to you by CORE

provided by CERN Document Server

ab Kirill A. Kazakov ∗ aMoscow State University, Physics Faculty, Department of Theoretical Physics, 117234, Moscow, Russian Federation bUppsala University, Physics Department, SE-751 21, Uppsala, Sweden

The problem of consistent formulation of the correspondence principle in quantum gravity is considered. The usual approach based on the use of the two-particle scattering amplitudes is shown to be in disagreement with the classical result of General Relativity given by the Schwarzschild solution. It is shown also that this approach fails to describe whatever non-Newtonian interactions of macroscopic bodies. An alternative interpretation of the correspondence principle is given directly in terms of the eﬀective action. Gauge independence of theh ¯0 part of the one-loop radiative corrections to the gravitational form factors of the scalar particle is proved, justifying the interpretation proposed. Application to the black holes is discussed.

Establishing the correspondence between clas- constanth. ¯ Newton’s theory is therefore correctly sical and quantum modes of description in the reproduced in the formal limith ¯ 0. case of the theory of gravity displays features There is, however, a still more important→ aspect quite different from those encountered in other of the correspondence between classical and quan- theories of fundamental interactions. Namely, the tum pictures of gravitation. The Einstein the- usual limiting procedure of the transition from ory, being essentially nonlinear, demands quan- quantum to classical theory turns out to be inap- tum theory to reproduce not only the Newtonian plicable in General Relativity. In electrodynam- form of the particle interaction, but also all the ics, e.g., the quantum potential, defined as the nonlinear corrections predicted by General Rela- Fourier transform (with respect to the momen- tivity. In this respect, the above-mentioned pecu- tum transfer from one particle to another) of a liarity of the gravitational interaction, namely, its suitably normalized two-particle scattering am- proportionality to the masses of particles, is man- plitude, takes the form of the Coulomb law when ifested in the fact (also pointed out in Ref. [1]) the momentum transfer becomes small as com- that, along with true quantum corrections (i.e., pared with the particles’ masses, so the above proportional to the Planck constanth ¯), the loop mentioned procedure is accomplished by tend- contributions also contain classical pieces (i.e., ing the masses to infinity. Contrary to this, in proportional toh ¯0). Thus, an important question General Relativity, the radiative corrections do arises as to relationship between these classical not disappear in the limit: particle masses , loop contributions and the classical predictions because strength of the gravitational interaction→∞ of General Relativity. of particles is proportional to their masses. In In analogous situation in the Yang-Mills the- fact, the relative value of the radiative correc- ories, the correct correspondence between classi- tions to the classical Newton law, corresponding cal and quantum theories is guaranteed by the to the logarithmical contribution to the gravita- fact that all radiative corrections to the parti- tional form factors of the scalar particle, is in- cle form factors disappear in the limit: masses dependent of the scalar particle mass [1]. Yet, , thus providing the complete reduction of a this contribution is proportional to the Planck given→∞ quantum picture to the corresponding nonlinear classical solution. It is claimed in Ref. [1] ∗E-mail: [email protected] 2 that when collected in the course of the con- The post-Newtonian correction here is twice as struction of the gravitational potential from the small as that obtained in [1]. one-particle-reducible Feynman graphs, the afore- Thus, we arrive at the puzzling conclusion that said classical contributions exactly reproduce the in the classical limit, the quantum theory of grav- post-Newtonian terms given by the expansion of ity, being based on the Bohr correspondence prin- the Schwarzschild metric in powers of rg/r, where ciple, does not reproduce the Einstein theory it rg is the gravitational radius. originates from. However, as will be shown presently, the value However, before making such a conclusion, of the potential found in Ref. [1] disagrees with one should question relevance of the definition the classical result given by the Schwarzschild so- of gravitational potential through the scattering lution. The latter has the form amplitudes. It may well turn out that the above 2 discrepancy arises because of the incorrect choice 2 rg 2 2 dr ds = 1 c dt rg of the quantum-field quantity to be traced back − r − 1 r − to the classical potential. r2(dθ2 +sin2 θdϕ2), (1) − In what follows, I give an interpretation of the where θ, ϕ are the standard spherical angles, r is correspondence principle as applied to gravity, 2 the radial coordinate, and rg =2GM/c is the which naturally resolves the above paradox [2]. gravitational radius of a spherically-symmetric Let us note first of all that, from the formal distribution of mass M. The form of ds2 given point of view, the correspondence between any by Eq. (1) is fixed by the requirements gti =0, quantum theory and its classical original is most 2 i = r, θ, ϕ, gθθ = r . To compare the two re- naturally established in terms of the effective ac- sults, however, one− has to transform Eq. (1) to tion rather than the S-matrix. This is because the DeWitt gauge the effective action (generating functional of the one-particle-irreducible Green functions) just co- µν 1 µν η ∂µgνα η ∂αgµν =0, (2) incides with the initial classical action in the tree − 2 approximation. In particular, the nonlinearity of used in Ref. [1]. classical theory (resulting, e.g., in rg/r power cor- The t, θ, ϕ-components of Eq. (2) are already rections to the Newton law in the General Rela- satisfied by solution (1). To meet the remain- tivity) is correctly reproduced by the trees. In ing condition, one substitutes r = f(˜r). A simple the case of quantum gravity, there are still addi- calculation shows that the function f(˜r) should tional contributions of the orderh ¯0, coming from expand as loop corrections. These are given by the gravita- 1 r 1 r 2 tional form factors of the particles, which serve f(˜r)=˜r 1+ g + g + 2 r˜ 2 r˜ ··· as building blocks for the gravitational potential. Instead of constructing the potential, however, let at larger ˜ for Eq. (2) to be satisfied to the order 2 2 us consider them in the framework of the effective rg /r . action method. From the point of view of this In the new coordinates, g00 component of the method, theh ¯0-parts of the particle form factors, Schwarzschild solution takes the following form together with the proper quantum parts of order (the tilde is omitted), ¯h, describe the radiative corrections to the classi- r r2 r3 cal equations of motion of the gravitational field. g =1 g + g + O g 00 − r 2r2 r3 It is non-vanishing of these terms that violates ! the usual Bohr correspondence. There is, how- 0 Taking the square root of g00, we see that the ever, one essential difference between the ¯h loop classical gravitational potential turns out to be terms and the nonlinear tree corrections. Con- equal to sider, for instance, the first post-Newtonian cor- GM G2M 2 Φc(r)= + . (3) − r 2c2r2 3

tors attached). Therefore, when evaluated between the N-particle states, the former is proportional to (m N) (m N)=M 2, while the latter, to m2 N =· M 2·/N. ·If, for instance, the so- lar gravitational· field is considered, the quantum correction is suppressed by a factor of the order 57 mproton/M 10− . This fact suggests≈ the following interpretation of the correspondence principle when applied to the case of gravity: the effective gravitational field (a) produced by a macroscopic body of mass M consisting of N particles turns into the corresponding classical solution of the Einstein equations in the limit M ,N . It is clear→∞ from→∞ the above discussion that the use of the effective action is essential for this in- = + ... terpretation: the quantum potential defined with the help of the scattering amplitudes, being of the form 2 2 q G(M/N) G (M/N) (b) Φ (r)= N + c1N − r c2r2 G3(M/N)3 Figure 1. Diagrams contributing to the first post- +c N + , 2 4 3 Newtonian correction. (a) The tree diagram oc- c r ··· curring because of the nonlinearity of the Einstein for a macroscopic (N-particle) body of mass M, equations. (b) The one-loop form factor. Wavy would fail to reproduce whatever classical poten- lines represent gravitons, solid lines scalar parti- tial other than Newtonian, since cles. GM Φq(r) when N . →− r →∞ rection of the form Thus, the loop corrections of the orderh ¯0 are now considered on an equal footing with the tree G2M 2 const . corrections, and thereby are endowed with direct 2 2 c r physical meaning as describing deviations of the The ”const” receives contributions from the tree spacetime metric from classical solutions of the diagram pictured in Fig. 1(a), as well as from the Einstein equations in the case of finite N. one-loop form factor, Fig. 1(b). If the gravita- Like any other argument trying to assign phys- tional field is produced by only one particle of ical meaning to the effective action, the above mass M, then the two contributions are of the interpretation immediately runs up against the same order of magnitude. problem of its gauge dependence. In spite of They are not, however, if the field is produced being independent of the Planck constant, the by a macroscopic body consisting of a large num- ¯h0 terms originating from the loop diagrams are ber N of particles with mass m = M/N. Be- not gauge-independent a priori. However, there ing responsible for the nonlinearity of Einstein is a strong evidence for that they are gauge- equations, the tree diagram 1(a) is bilinear in independent nevertheless. Namely, as is demon- the energy-momentum tensor T µν of the parti- strated in full detail in [2], theh ¯0 terms of the cles, while the loop diagram 1(b) is only linear one-loop gravitational form factors of the scalar (to be precise, it has only two particle opera- particle, contributing to the first post-Newtonian 4

56 58 correction to the metric, turn out to be indepen- the stars is of the order 10− 10− . This dif- dent of the Feynman gauge parameter ξ weight- fers, however, in the case of− objects consisting ing the gauge condition. A very specific feature of strongly interacting particle. In particular, in of this gauge dependence cancellation must be the limit of an infinitely strong interaction, the emphasized: it holds only for the ¯h0 part of the object is to be considered as a particle, i.e., one form factors. For instance, the logarithmical part has to set N = 1 in Eq. (5). The sign of the of the form factors (which is of the orderh ¯) is first post-Newtonian correction is then opposite gauge-dependent [3]. to that given by the classical General Relativity. Actual value of the one-looph ¯0 contribution to An example of objects of this type is probably the effective metric can be found with the help of supplied by the black holes. Applicability of the the explicit expressions for the gravitational form above results to these objects depends on whether factors of the scalar particle, obtained in Ref. [1]. they can be considered as elementary particles. In the momentum space, It seems that at least certain types of the black holes do behave like normal elementary particles π2G2m2 δgµν (p)= [4]. It should be noted, however, that the very ex- −c2 p2 istence of the horizon is now under question. The − eff qµqν 7pµpν potential Φ (r) may well turn out to be a regu- 3ηµν + p+ , (4) 0 × m2 p2 lar function of r when all theh ¯ loop corrections are taken into account. where qµ is the 4-momentum of the scalar particle If the notion of elementary particle is indeed with mass m. applicable to the black holes, then the above re- With the help of Eq. (4), one can find the effec- sults imply, in particular, that the emission of the tive gravitational potential of a spherically sym- gravitational waves by the black hole binaries is metric body with mass M, consisting of N iden- strongly affected by the quantum contributions tical (scalar) particles with mass m = M/N, in already in the first post-Newtonian approxima- the first post-Newtonian approximation. Namely, tion. The LIGO and VIRGO gravitational wave if the gravitational interaction of the constituent detectors [5], which are now under construction, particles is not too strong (viz., if it can be repre- will hopefully bring light into this issue. sented as a series in powers of G), then one has, in the coordinate space REFERENCES GM G2M 2 G2M 2 Φeff(r)= + . (5) 1. J. F. Donoghue, Phys. Rev. Lett. 72 (1994) r 2c2r2 Nc2r2 − − 2996; Phys. Rev. D50 (1994) 3874; Perturba- Indeed, the tree contribution to the effective field, tive Dynamics of Quantum General Relativ- represented in this equation by the first two ity, invited plenary talk at the ”Eighth Mar- terms, always coincides with the corresponding cel Grossmann Conference on General Rela- classical (Schwarzschild) solution. It is given, tivity”, Jerusalem (1997). therefore, by Eq. (3), all the effects of the particle 2. K. A. Kazakov, Class. Quantum Grav. 18 interactions are taken into account by identifying (2001) 1039. M as the gravitational mass, while account of the 3. K. A. Kazakov, Phys. Rev. D63 (2001) interaction in the loop correction (the third term 044004. in Eq. (5)), which is itself of the order G2, would 4. C. F. E. Holzhey, F. Wilczek, Nucl. Phys. give rise to terms of the order G3, irrelevant to the B380 (1992) 447. present concern. Appearance of the factor 1/N in 5. Description of the LIGO and VIRGO projects the third term was explained earlier. as well as information about their current de- It was mentioned above that the loop contribu- velopment is available at tions to the post-Newtonian corrections are nor- http://www.ligo.caltech.edu mally highly suppressed: their relative value for http://www.virgo.infn.it.