General relativity or gravitons?

Question: How is the existence of a graviton consistent with the GR paradigm of gravity as a purely geometrical effect?

Answer: Ontologically, it’s not! Gravitons are predicated on a quantum field-theoretic formulation of gravity, while spacetime curvature is the corresponding classical description. By analogy, the electromagnetic force may be alternatively described in terms of the exchange of virtual bosons (in QFT), or in terms of electromagnetic waves (in classical electromagnetism); these are fundamentally different paradigms, but are epistemically consistent in the sense that the former (quantum electrodynamics) reduces to the latter (classical electrodynamics) in the appropriate limit.

That said, it is possible to show that the classical electromagnetic and gravitational forces must correspond — in the language of Lorentz-invariant quantum particle (as opposed to field) theory — to the transmission of a massless virtual particle with helicity {\pm1} and {\pm2}, respectively. In particular, as shown in a beautiful paper by Boulware and Deser in 1975 [1], “a quantum particle description of local (non-cosmological) gravitational phenomena necessarily leads to a classical limit which is just a metric theory of gravity. [If] only helicity {\pm2} gravitons are included, the theory is precisely Einstein’s general relativity…” This implies that Einstein’s theory enjoys a sort of quantum uniqueness (at least at tree level: it is entirely possible that the high-frequency behaviour of gravitons differs substantially from the (experimentally probed) low-energy regime of effective field theory).

The remarkable aspect of this correspondence is that one sees the emergence of a metric theory from a non-geometrical, flat-space formulation. Perhaps this will shed light on the notion of “emergent spacetime” from other non-geometrical precepts (namely, entanglement)?

To begin, consider the description of the world entirely in terms of S-matrix elements (or rather, the generalizations thereof necessitated by zero-mass particles, i.e., soft theorems). Observation of a force then implies the existence of a mediating particle whose exchange produces it. Since the effective potential for the exchange of a massive particle is {V\sim e^{-mr}/r}, the experimental {1/r} gravitational potential implies that the graviton must be massless, at least to within experimental accuracy.

Establishing the spin is more subtle. It must be an integer, since the Pauli exclusion principle prevents any virtual particle obeying Fermi-Dirac statistics from conspiring in sufficient numbers to produce a classical force. (The keyword here is “virtual”. Spin-{1/2} electrons, for example, are not force carriers in this paradigm; that role belongs to the integer bosons). We can also rule out spin {\pm1}, since a vector exchange would result in repulsion between like charges—masses, in this case.

(As an aside, the term “vector boson” for particles of spin {\pm1} arises from the fact that in quantum field theory, the component of a (massive) particle’s spin along any axis can take one of three values: {0}, {\pm\hbar}. Thus the dimension of the space of spin states is the same as that of a vector in three-dimensional space, and in fact can be shown to form a representation of SU(2), the corresponding group of rotations).

Ruling out a scalar particle — that is, spin 0 — can be done by considering the bending of light by a gravitational field. In particular, as shown in Boulware and Deser’s paper [1], the scattering angle for a photon interacting with a massive object (such as the sun) via a scalar graviton depends on both the momentum of the photon and its polarization. But experiments reveal no such dependence.

Finally, the possibility of spin greater than 2 was quashed by Weinberg’s 1964 paper [2], though we shall not repeat the arguments here. The graviton must therefore be a spin-2 particle. Furthermore, one can show that a finite-range exchange is untenable on both experimental and theoretical grounds [3,4]. We therefore conclude that the gravitational force corresponds, in the special relativistic scattering paradigm, to the exchange of massless spin-2 virtual bosons with infinite range.

The above is essentially the argument that classical (Einstein) gravity implies the existence of a massless spin-2 virtual particle. What about the other direction, namely that this graviton uniquely leads to Einstein gravity in the classical limit?

Unfortunately this direction is substantially more technical, so we will only summarize the argument here. The basic idea is that the virtual exchange of helicity {\pm2} gravitons is governed by second-rank tensor vertices, which correspond to matrix elements of the stress-energy tensor of a local field theory. In particular, Boulware and Deser show [1] that the graviton can only couple (to other particles as well as itself) via a conserved stress-energy tensor; otherwise, the graviton is free (i.e., it couples to nothing).

While Boulware and Deser’s analysis is only at tree level, it suffices to show that in the low-frequency limit, Einstein gravity follows uniquely from special relativistic scattering theory combined with a few observational constraints. Philosophically, this implies that one can view Einstein’s theory — and by extension, the intuitively-pleasing conception of gravity as the curvature of spacetime — as a phenomenological theory for macroscopic interactions. It is by no means necessary that this same geometrical interpretation continue to hold at small distances and times—such as at the Compton wavelength of a given interaction, where quantum effects are expected to be relevant. And while we have yet to understand the UV nature of gravity, the above picture of low-energy, geometrical gravity as a purely phenomenological descriptor, however much it may baffle one’s intuition, is tantalizingly in line with the emergent spacetime paradigm.


[1] D. G. Boulware and S. Deser, “Classical General Relativity Derived from Quantum Gravity,” Annals Phys. 89 (1975) 193.

[2] S. Weinberg, “Photons and Gravitons in s Matrix Theory: Derivation of Charge Conservation and Equality of Gravitational and Inertial Mass,” Phys. Rev. 135 (1964) B1049–B1056.

[3] H. van Dam and M. J. G. Veltman, “Massive and massless Yang-Mills and gravitational fields,” Nucl. Phys. B22 (1970) 397–411.

[4] D. G. Boulware and S. Deser, “Can gravitation have a finite range?,” Phys. Rev. D6 (1972) 3368–3382.

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