Does Gravity Screen UV Divergences?

This article falls in the category of technical, but potentially important. And the fact that this is framed as a homage to Stanley Deser, one of the leading late 20th century GR theorists, means that it could gain traction, in the face of sociological rooted resistance to paradigm breaking ideas in astrophysics. 

Ultraviolet (i.e. high energy) divergences are a major barrier to quantum gravity theories, rendering gravity non-renormalizable. But this article explores the possibility that non-perturbative approaches to GR could solve that problem, which has been a major roadblock in quantum gravity theories. Deur's approach to explaining dark matter and dark energy phenomena also involve resort to non-perturbative GR effects.

In the peculiar manner by which physicists reckon descent, this article is by a "child" and "grandchild" of the late Stanley Deser. We begin by sharing reminiscences of Stanley from over 40 years. Then we turn to a problem which was dear to his heart: the prospect that gravity might nonperturbatively screen its own ultraviolet divergences and those of other theories. After reviewing the original 1960 work by ADM, we describe a cosmological analogue of the problem and then begin the process of implementing it in gravity plus QED.

R. P. Woodard and B. Yesilyurt, "The Other ADM" arXiv:2410.05213 (October 7, 2024).

The introduction begins as follows (in selected parts):

Stanley Deser was the Grand Old Man of quantum gravity. Everyone in the f ield knew him, and the vast majority of us loved him. His life was a testament to the persistence of scientific inquiry, optimism and simple humanity over the course of a turbulent century. . . . He graduated college at 18, and took a Ph.D from Harvard at age 22. In a career spanning seven decades he is credited with hundreds of publications, including 8 papers and a book written after the age of 90. 

. . . 

Gravity needed him: despite the lovely geometrical formulation of its early days, general relativity was not then a proper field theory. There was no canonical formalism, with its careful enumeration of the degrees of freedom and their contribution to the total energy. Hence there was no way to prove classical stability, no way to develop numerical integration techniques and no way to even begin thinking about quantization. In a memorable sequence of papers with Dick Arnowitt and Charlie Misner, Stanley sorted out the gauge issue, identified canonical variables and defined an energy functional for asymptotically flat geometries. Stanley would return to the problem of gravitational energy later on in his career. 

With Claudio Teitelboim (now Bunster) he established the stability of supergravity. And he collaborated with Larry Abbott to prove the classical stability of gravity with a positive cosmological constant. Stanley was a consummate collaborator: 

• With David Boulware he showed that endowing the graviton with a mass inevitably results in a ghost mode, provided that the theory has a smooth perturbative limit. 

• With Peter van Nieuwenhuizen he extended the work of ’t Hooft and Veltman to show that renormalizability is lost at one loop when general relativity is combined with either electromagnetism, Yang-Mills theory, or Dirac fermions. 

• With Bruno Zumino he showed that consistently coupling a spin 3/2 gravitino to gravity produces a locally supersymmetric theory. The two then applied their formalism to string theory and to the breaking of supersymmetry. 

• With Mike Duff and Chris Isham he identified the first true conformal anomaly, which led to a classification scheme with Adam Schwimmer. 

• With Roman Jackiw and Stephen Templeton he showed that adding a dimensionally-reduced Chern-Simons term to Yang-Mills or gravity results in massive particles of spin 1 and 2, respectively. 

• With Gerard ’t Hooft and Roman Jackiw he explained how to understand general relativity in 2 + 1 dimensions. 

• With Cedric Deffayet and Gilles Esposito-Farese he extended flat space Galileons to curved space. . . .

Then the two authors turn to a problem which long fascinated Stanley: the possibility that quantum gravity might regulate its own ultraviolet divergence problem and even those of other theories. We first review ADM’s prescient and thought-provoking demonstration that classical gravitation cancels the self-energy divergence of a point charge. This is “the other ADM” of our title. We then describe a quantum field theoretic analogue of the same basic effect in the context of inflationary cosmology. The next section discusses the prospects for implementing the ADM mechanism in general relativity plus QED on an asymptotically flat background. 

Part 3 of the paper explains this ADM mechanism:

The idea that gravity might regulate divergences is based on the fact that gravitational interaction energy is negative. 

For example, this makes the mass of the Earth-Moon system slightly smaller than the sum of their masses, even when one includes the kinetic energy of their orbital motion, 

M(EM)=M(E)+M(M)−(GM(E)M(M)/(2c^2R)). (1) 

The decrease works out to about 6 × 10^11 kilograms, which makes for a fractional reduction of 10^−13. Note that the fractional reduction becomes larger as the orbital radius R decreases. In 1960 Arnowitt, Deser and Misner quantified the mechanism in the context of a classical (in the sense of non-quantum) charged and gravitating point particle. Although they solved the full general relativistic constraints and then computed the ADM mass, their result can be understood using a simple model that they devised. 

Suppose the particle has a bare mass M(0) and charge Q, and is regulated as a spherical shell of radius R. Then its rest mass energy might be expressed as, 

M(R)c^2 = M(0)c^2 + Q^2/(8πε(0)R)−(GM^2(R)/(2R)), (2) 

where the single concession to relativity is that the Newtonian gravitational interaction energy has been evaluated using the total mass. Of course the quadratic equation (2) can be solved to give, 

M(R) = c^2R/G(√(1+(2GM(0)/(Rc^2)) + (GQ^2/(4πε(0)R^2c^4))) − 1). (3)

The unregulated limit is finite and independent of the bare mass,  

lim R→0 M(R) = √(Q^2/(4πε(0)G)). (4)

Three crucial points about the result (4) deserve mention: 

• It is finite; (3) (4) 

• It is independent of the bare mass M(0), as long as that is finite; and 

• It is nonperturbative. 

Of course finiteness results from the fact that gravitational interaction energy is negative. This is evident from expression (2). The Q^2/(8πǫ(0)R) term means that compressing a shell of charge costs energy, however, the −GM^2/(2R) term signals that gravity is able to pay the bill, no matter how high. 

The fact that any fixed M(0) drops out is also evident from expression (2). Note that this is not at all how a conventional particle physicist would have approached the problem. Our conventional colleague would have regarded the total mass M as a measured quantity and then required the bare mass to depend upon the regulating parameter R so as to force the result to agree with measurement, 

M(0)(R) = M(meas) − Q^2/(8πε(0)Rc^2) + GM(meas)^2/(2Rc^2) . (5) 

That is how renormalization works. It is unavoidable without gravity, but the presence of gravity opens up the fascinating prospect of computing fundamental particle masses from first principles. Setting Q = e in expression (4) gives an impossibly large result for the electron, 

√(e^2/(4πε(0)G) = √(e^2/(4πε(0)ℏc) × √(ℏc/G) = √α × M(Planck). (6)

However, it is well known that quantum field theoretic effects often the linear self-energy divergence of a classical electron to alogarithmic divergence. 

This opens the possibility of the true relation containing exponentials. For example, one gets within a factor of four with, 

M(electron)=√α x M(Planck) × exp(− 1/(e^1 x α)) ≈ 0.134 MeV, (7)

[Ed. the measured value of the electron mass is 0.51099895000 ± 0.00000000015 MeV]

where e^1≈2.71828 is the base of the natural logarithm. The electron also carries weak charge, which should enter at some level. Perhaps all fundamental particle masses can be computed from first principles? One might even hope that the mysterious generations of the Standard Model appear as “excited states” in such a picture.

The nonperturbative nature of the ADM mechanism is evident from the fact that (4) goes like the square root of the fine structure constant and actually diverges as Newton’s constant goes to zero. The perturbative result comes from expanding the square root of (3) in powers of Q^2 and G, 

M(R)= (M(0)+ Q^2/(8πε(0)Rc^2)) x [1 

− 1/4(2GM(0)/(Rc^2) + GQ^2/(4πε(0)R2c4)) 

+ 1/8 (2GM(0)/(Rc^2) + GQ^2/(4πε(0)R^2c^4)^2 

− 5/64(2GM(0)/(Rc^2) + GQ^2/4πε(0)R^2c^4)^3 

+ ...] . (8) 

This is a series of ever-higher divergences. Of course perturbation theory becomes invalid for large values of the expansion parameter, 

2GM(0)/(Rc^2) + GQ^2/(4πε(0)R^2c^4). (9) 

Perhaps the same problem invalidates the use of perturbation theory in quantum general relativity, which would show cancellations like (4) if only we could devise a better approximation scheme? It is hopeless trying to perform a genuinely nonperturbative computation. However, a glance at the expansion (8) shows what goes wrong with conventional perturbation theory: gravity has no chance to“keep up” with the gauge sector. The lowest electromagnetic divergence is Q^2/(8πε(0)Rc^2), whereas gravity’s first move to cancel comes at order −[Q^2/(4πε(0)Rc^2)]^2 × G/(8Rc^2).

What is needed is are organization of perturbation theory in which the gravitational response comes at the“same order”as the gravitational response. Several studies have searched for such an expansion without success. 

The paper concludes with the material quoted below:

Stanley Deser was a great physicist and a good man who left the world a better place. Section 1 reviews some of his most important contributions to physics while section 2 presents personal reminiscences from one of his students. The remainder of the paper is devoted to one motivation for Deser’s early fascination with quantum gravity: the possibility that it might regulate its own divergences and those of other theories. This possibility arises because the gravitational interaction energy is negative, and sourced by the same sectors which diverge.

Section 3 reviews the example ADM discovered of how classical (that is, non-quantum) general relativity cancels the famous linear divergence of a point charged particle. The final result (4) is not only finite but also independent of the bare mass, as long as that is finite. This raises the fascinating prospect of not only solving the problem of quantum gravity but also computing fundamental particle masses from first principles. It is impossible to overstate the revolution this would work on our perception of quantum gravity. From a sterile issue of logical consistency, without observable consequences at ordinary energies, and only perturbatively small effects even at the fantastic scales of primordial inflation, quantum gravity would be thrust to center stage. Every measurement of a fundamental particle mass would represent a sensitive check. One might even hope that the mysterious 2nd and 3rd generations of the Standard Model emerged as excited states of the 1st generation.

Of course there is a catch: one must make the calculation nonperturbatively in an interacting quantum field theory. This is evident from how its classical limit (4) depends upon α and G. There seems little hope of ever being able to perform an exact computation in an interacting 3 + 1 dimensional quantum field theory. What is needed instead is a way of reorganizing conventional perturbation theory so that the negative energy constrained degrees of freedom have a chance to “keep up” with the positive energy, unconstrained degrees of freedom. The key to this seems to be solving the Hamiltonian Constraint. 

Section 4 describes how one accomplishes just that in the theory of primordial inflation. Fittingly, the solution (24) is given using ADM variables. Although it is not clear if this form regulates the usual ultraviolet divergences of gravity with a scalar, the weak field expansion (25) of the gauge-fixed and constrained action does show an ADM-like erasure of the scalar perturbation except for those parts protected by the nonzero first slow roll parameter ǫ.

Section 5 represents our initial attempt to implement the ADM mechanism for Quantum Electrodynamics + General Relativity. Although it is clear that the Hamiltonian Constraint can be solved exactly, a number of issues remain, the most important of which is how to extract “background” parts of the kinetic and potential energies so as to keep quantum corrections small. We look forward to further study of this system.