Historical research guide · 0811 · 2002
The quantum vacuum and the cosmological constant problem
S.E. Rugh · H. Zinkernagel
In one minute
Svend Erik Rugh and Henrik Zinkernagel take the single largest number in physics and ask where it comes from. Quantum field theory says empty space is a vacuum state whose fields never stop fluctuating, and adding those fluctuations up gives an energy density enormously larger than the one astronomers actually measure in the expansion of the universe. Rugh and Zinkernagel trace that mismatch from Nernst in 1916 to the supernova results of the late 1990s, and then do something more useful than restating it: they take the problem apart to see which pieces are measurement and which are assumption. Their sharpest observation is that every laboratory demonstration of vacuum energy — Casimir plates, the Lamb shift, spontaneous emission — is read out through matter, so the experiments cannot yet tell us whether the energy belongs to space itself. And the bridge that would connect the two sides, quantum energy sourcing curvature, has never been tested at all.
Connection
Chapter 2 argues that the vacuum is a real, structured medium, and this paper is the most careful account of exactly which experiment is still missing before that becomes a measurement rather than an inference. Chapter 13’s honest ledger needs the number it names — the gap of roughly one hundred and twenty orders of magnitude — stated precisely, together with the two assumptions it rests on.
From the source
Adding the zero-point energies of the electromagnetic field modes up to an ultraviolet cut-off at the electroweak scale of about 100 GeV gives a vacuum energy density of roughly ten to the forty-sixth erg per cubic centimetre, which already exceeds the observational bound on the total vacuum energy density by about fifty-five orders of magnitude.Section ’Estimates of the QED zero-point energy’, preprint page 13
Carrying the same sum up to the Planck energy of about ten to the nineteenth GeV gives roughly ten to the one hundred and fourteenth erg per cubic centimetre — more than one hundred and twenty orders of magnitude above what large-scale cosmology allows. That figure is the cosmological constant problem in its usual form.Section ’Estimates of the QED zero-point energy’, preprint page 14, and the bound at equations 2 and 3
A vacuum state that looks the same to every observer must have an energy-momentum tensor proportional to the metric, which makes the vacuum mathematically a perfect fluid whose pressure equals minus its energy density. That equation of state is what allows vacuum energy and Einstein’s cosmological constant to be treated as the same quantity.Section ’The vacuum energy-momentum tensor in general relativity’, equations 13 and 14
Every experiment usually cited as showing vacuum energy to be real — the Casimir force, the Lamb shift, spontaneous emission, the anomalous magnetic moment of the electron — is a measurement made on a material system, so from those results alone it is difficult to decide whether the fluctuations belong to empty space or are produced by the apparatus. Bohr and Rosenfeld reached the same conclusion in 1933, and the authors note that Schwinger’s source theory reproduces the Casimir result without invoking zero-point energy at all.Section ’Further considerations on the measurability of the vacuum energy’, preprint pages 20 to 21
The bridge between the two theories has never been tested. The semi-classical equation in which the expectation value of the quantum energy-momentum tensor sources spacetime curvature has no experimental support, and in a curved background the vacuum is not generally well defined, because no two free-falling observers need agree on which state contains no particles.Section ’Vacuum in curved spacetime’, equation 15
The proposed resolutions sort into a small map: change general relativity, quantize it, reinterpret the quantum field theory vacuum so that its energy is not a real energy, extend the Standard Model, or build one framework containing both. Supersymmetry is the cleanest of these, because boson and fermion contributions cancel exactly — until the symmetry is broken, when the large energy returns. In 2002 no consensus existed that any route worked.Sections on solution types 1 to 3 and the anthropic discussion, preprint pages 31 to 35
The way in
https://doi.org/10.1016/S1355-2198(02)00033-3How to cite it
S.E. Rugh, H. Zinkernagel (2002) The quantum vacuum and the cosmological constant problem doi:10.1016/S1355-2198(02)00033-3