Historical research guide · 0589 · 2020
Zeta Functions and the Cosmos—A Basic Brief Review
Emilio Elizalde
In one minute
Emilio Elizalde, of the Spanish National Research Council, is one of the people who made zeta-function regularisation into a working tool, and this review is his plain-spoken tour of it. The problem it solves is the one at the centre of vacuum physics: add up the energy every mode of every field carries at its floor and the total runs away to infinity. Euler’s zeta function assigns such runaway sums a finite value — this is where the famous result that one plus two plus three and onward equals minus one twelfth comes from — and Elizalde shows how physicists use that to get real, measurable numbers for the Casimir force and for vacuum energy in curved space. The second half turns to the cosmos: the cosmological constant and the vacuum’s zero-point energy enter Einstein’s equations in the same place, and the crude Planck-scale estimate misses the measured value by some 123 orders of magnitude. He then works a model where topology and a light scalar field give a Casimir contribution of the right size.
Connection
This is the mathematics that turns the zero-point field of chapter 2 from a divergent sum into a number you can compare with a measurement, and it is written by one of its architects. It also states, from inside mainstream cosmology, the two propositions chapters 6 and 13 are built on: vacuum energy is a candidate for the force accelerating the universe, and the size of the gap between the estimate and the measurement is the open question.
From the source
The zero-point energy of a quantum system is one half of the sum of the eigenvalues of its Hamiltonian, and that sum converges only in very special cases. Generically it is a divergent series or integral that must be regularised by mathematical means, and the finite result must then be given a physical interpretation.Section 1, Introduction, Equation 1
Zeta-function regularisation assigns a finite value to such a divergent sum through analytic continuation on the complex plane — the route by which one plus one plus one and onward is evaluated as minus one half, and one plus two plus three and onward as minus one twelfth. These are not tricks: the results obtained this way match the most precise experiments in high-energy physics to more than fifteen digits, and the Casimir-effect measurements agree well too.Section 3.2, The Zeta Function as a Summation Method
The cosmological constant and the fluctuations of the quantum vacuum enter the same term of Einstein’s field equations. Elizalde writes the stress-energy contribution as a possible universal constant, Λc² over eight pi G, plus the summed zero-point energies of the field modes divided by the volume — one geometric, one quantum, added in the same place.Section 4.4, Equations 18 and 19
Estimating the vacuum fluctuation energy with a cut-off at the Planck scale and comparing it with the value inferred from the cosmic acceleration originally gave a discrepancy of some 123 orders of magnitude. Several theories of very different kinds have reduced it somewhat, but the discrepancy continues to be very large. What to watch: this is the gap any account of vacuum energy has to close, and Elizalde names the newest attempts at it by Unruh, Leonhardt, Carlip and Santos.Section 6.1, opening; Section 6.3
A worked toy model gets the right size. Assume the bulk cosmological constant vanishes, add a cosmic scalar field of very small mass, and let space carry a non-trivial topology with one or two compactified extra dimensions of Planck scale. The Casimir energy density that follows from the boundary conditions comes out at about 10⁻¹⁰ erg per cubic centimetre — the order of magnitude the astronomical observations give.Section 6.1, Two Examples of the Use of the Zeta Function Method
Moving boundaries create real particles out of the quantum vacuum — the dynamical Casimir effect. Elizalde’s group showed that a perfectly reflecting mirror is unphysical, since any real mirror becomes transparent at high enough frequency, and that for partially transmitting mirrors the number of created particles is finite and their energy always positive, the whole process expressible by an analytic matrix. The boundaries of the visible universe are expanding, which naturally calls for the effect.Section 6.2, A Brief Update on the Dynamical Casimir Effect
The way in
https://doi.org/10.3390/universe7010005How to cite it
Emilio Elizalde (2020) Zeta Functions and the Cosmos—A Basic Brief Review doi:10.3390/universe7010005