Historical research guide · 1125 · 2006
From Quantum Hydrodynamics to Quantum Gravity
Grigory E. Volovik
In one minute
Volovik's rapporteur article asks what a laboratory liquid can teach a theory of gravity, and answers it with a calculation rather than an analogy. Both hydrodynamics and general relativity are perfect classical theories, he notes, and general relativity can be viewed as a theory of hydrodynamic type in which the collective variables are the metric fields; quantised, both share deep features including quadratic divergences, and quantising hydrodynamics is a problem as old as quantising gravity. He then works Landau's quantum hydrodynamics through to the vacuum-energy question. In a liquid, the analogue of the vacuum energy density obeys the equation of state that pressure equals minus energy density — the same equation of state the cosmological constant obeys. And for an isolated liquid in equilibrium with nothing pressing on it from outside, that quantity is exactly zero, with no fine tuning anywhere in the derivation. Add matter and it is no longer zero: thermal excitations push, the vacuum pushes back, and the vacuum energy settles at one third of the matter energy density.
Connection
Chapter 5's central move is that the enormous mismatch between the calculated and observed vacuum energy is not a failure of arithmetic but a sign the wrong sum was being done. This is the paper that shows what the right sum looks like in a system you can actually cool and measure, and it states the conclusion the chapter needs: the vacuum energy is naturally determined by macroscopic quantities, rather than by a huge microscopic Planck energy scale.
From the source
General relativity can be viewed as a theory of hydrodynamic type in which the collective variables are the metric fields, and at the quantum level quantum hydrodynamics and quantum gravity share many common features — both, for example, have quadratic divergences.Section 1, Introduction
In a quantum liquid the analogue of the vacuum energy density obeys the equation of state that the vacuum pressure equals minus the vacuum energy density — the same relation the cosmological constant obeys in cosmology.Section 2, Eq. 3; restated in Section 4.3, Eq. 51
For an equilibrium liquid in the absence of any external environment, that vacuum energy density is exactly zero, and Volovik states the generality plainly: the nullification of vacuum energy occurs for any non-disturbed equilibrium vacuum. Nothing is tuned to make it happen — it follows from the liquid being self-sustained and in equilibrium at zero external pressure.Section 2, Eq. 9 and the sentence following it; Section 2 close
Add matter and the vacuum responds. Thermal phonons produce a radiation pressure that the vacuum must compensate, and the vacuum energy density settles at one third of the matter energy density — the back reaction of the vacuum to relativistic matter, obtained entirely within quantum hydrodynamics.Section 4.3, Eqs. 52 and 53
The consequence Volovik draws is that the vacuum energy is naturally determined by macroscopic quantities rather than by the huge microscopic Planck energy scale — which is the point of contact with the cosmological-constant problem, since an effective theory that sums zero-point energies gives an answer many orders of magnitude too big in the liquid exactly as it does in gravity.Section 4.3, close; Section 5.2, the passage citing Weinberg's review
A second hint is a constraint on any microscopic theory of quantum gravity: it must contain an additional parameter beyond h-bar, c and Newton's constant, giving a dimensionless number that distinguishes microscopic theories sharing the same macroscopic behaviour. Volovik quotes Bjorken on the requirement — the emergence can only work if there is an extremely small expansion parameter in the game.Section 5.2
The way in
https://arxiv.org/abs/gr-qc/0612134How to cite it
Grigory E. Volovik (2006) From Quantum Hydrodynamics to Quantum Gravity doi:10.48550/arXiv.gr-qc/0612134