Historical research guide · 1124 · 2002
Sakharov's induced gravity: a modern perspective
Matt Visser
In one minute
Sakharov's 1967 note was three pages and four formulas, and Visser's article is the piece that made it usable again: it translates the idea into modern language and sorts the versions of it now in circulation. The framework is set out in five steps. Take a Lorentzian manifold; assume nothing about how its geometry moves, leaving it free to flap in the breeze; do not quantise gravity, but quantise everything else; work out the one-loop effective action for the matter fields on that background; and read off what it contains. It contains, uninvited, a cosmological-constant term, an Einstein-Hilbert term and curvature-squared terms. Visser then shows that this single calculation supports four quite different research programmes depending on what you demand of it — Sakharov's own demand of one-loop dominance, Pauli's of one-loop finiteness, Frolov and Fursaev's of one-loop calculability, and plain renormalizability's of calculable one-loop changes. Each imposes its own constraints on what particles may exist, and he gives the coefficients in a table.
Connection
Chapter 3 rests on the claim that gravity can be induced rather than fundamental, and this is where that claim is stated in language a reader can follow and check. It also supplies the chapter's sharpest line: whatever view you take of the four programmes, even if Einstein gravity is not there at zero loops, it is automatically generated at one loop.
From the source
The basic idea is that gravity is not fundamental in the sense of particle physics: general relativity emerges from quantum field theory in roughly the same sense that hydrodynamics or continuum elasticity theory emerges from molecular physics.Abstract; Section 1, Introduction
The central observation is a calculation, not a conjecture. Quantise matter on a classical curved background and the one-loop effective action automatically contains a cosmological-constant term, a term proportional to the curvature scalar — the Einstein-Hilbert action — and curvature-squared terms.Section 1, Eqs. 1 and 2
The same one-loop formulae support four different programmes, and the difference between them is what each demands: Sakharov demands one-loop dominance, Pauli one-loop finiteness, Frolov and Fursaev one-loop calculability, and ordinary renormalizability demands only that certain differences be calculable.Section 2, close; Sections 4 to 7; Section 9, Summary and Discussion
On Sakharov's own reading — set every tree-level constant to zero, let one-loop physics dominate, take dimensionless numbers of order one and cut off at the Planck scale — the reciprocal of Newton's constant is given by the square of the cutoff multiplied by a supertrace over the particle spectrum. That conjectured link between Newton's constant and particle physics is what drew the attention, and Visser notes over 260 citations as of February 2002.Section 4, Eq. 29
The finiteness route puts hard constraints on what matter may exist. Pauli's conditions are a forerunner of supersymmetry and are not satisfied by the standard model, and Visser carries Pauli's own verdict on them: these requirements are so extensive that it is rather improbable that they are satisfied in reality.Section 5, Steps 1 to 3, with Table 1; the quoted Pauli remark at the close of Section 5
The conclusion all four routes share is the one that matters for an emergent-gravity programme: even if Einstein gravity is not there at zero loops, it will automatically be generated at one loop — so deriving the inverse square law is not the hard step for a candidate theory of quantum gravity, and gravity was never quantised anywhere in the argument.Section 9, Summary and Discussion
The way in
https://arxiv.org/abs/gr-qc/0204062How to cite it
Matt Visser (2002) Sakharov's induced gravity: a modern perspective doi:10.1142/S0217732302006886