The Place of the Eigenvalues

What a spectrum is being asked to do

Chapter 3 ended with an effective mass read off the curvature of a dispersion relation — that is, off a spectrum. Chapter 4 will open with probability currents and mode structure, which is also a spectrum. The temptation at this junction is to treat eigenvalues as a single explanatory substance: a hidden list of numbers that the world is secretly reading from. That is the same error the volume tracks elsewhere, in its most seductive form, because a spectrum genuinely does organise the behaviour of every linear system in the book.

The discipline is to ask, each time, what job the eigenvalues are being given. Four jobs appear in this literature. They are not interchangeable, and the difference between them is the difference between proving a theorem, reporting a measurement, calibrating an apparatus, and drawing a picture.

Four functions, four claim strengths

The spectral conjecture

The oldest and most ambitious proposal is that the nontrivial zeros are literally the eigenvalues of some self-adjoint Hamiltonian. The argument is short and unusually clean: a self-adjoint operator on a Hilbert space has a real spectrum; if the imaginary parts of the zeros are that spectrum, every zero sits on Re(s) = 1/2, and the Hypothesis follows. The classical dynamics underlying such an operator is expected to be chaotic, with periodic orbits whose periods are logarithms of primes — the Berry–Keating picture and its successors.

Status: open. No operator has been constructed whose self-adjointness is rigorous and whose spectrum is proven to be exactly the zeros. Every physical system in this volume that produces zeta-like numbers produces them because it was engineered to; none of them is a candidate for this operator. The distinction matters because the two claims are separated by the whole width of a proof.

Critical loci, not energy levels

The most precise statement available for the current experimental situation is the second row of the table. In the dynamical-quantum-phase-transition work, the zeros are not the spectrum of anything. An engineered many-body Hamiltonian is built so that its time evolution encodes properties of the zeta function; the zeros then mark the critical times at which the measured observables — the accumulated phase factor, the Loschmidt amplitude — vanish or become non-analytic. Tune the inverse temperature to β = 1/2 and the transitions land on the zeros; move off that line and they do not.

The eigenvalues of the driving Hamiltonian shape the dynamics, but they are not the zeros. This is a correspondence, demonstrated in a system chosen to exhibit it, and it is genuinely more than a picture: the critical line acquires a physical signature that can be measured rather than only computed. It is also not a proof, and nothing about the setup constrains zeros that have not been fed into it.

The essay Humans: Depressed, Demoralized, Unprepared carries the full citation for this result and the reason it resonates with the direction of the wider work.

Quasienergies as a readout

In periodically driven systems the relevant quantities are quasienergies rather than energies. Earlier trapped-ion work shaped a drive waveform so that the evolution froze — a quasienergy degeneracy — precisely when a drive amplitude matched a Riemann zero. The zeros are thereby displayed: the apparatus is a very expensive way of confirming numbers that were already tabulated to hundreds of digits. That is not a criticism of the experiments, which are beautiful control demonstrations. It is a statement of what they establish, which is that a quantum system can be tuned to a number, not that the number came from a quantum system.

The acoustic analogue

Acoustic metamaterials are the cheapest place to see all of this working, and the chapter before this one already listed them as a proxy. In a phononic lattice the eigenvalues are the band frequencies of the unit cell; the stop bands, the negative effective mass density near a resonance, and the negative bulk modulus of a Helmholtz-resonator array are all read directly off that spectrum. Sound is genuinely steered, cloaked, and lensed by structures designed this way, and the engineering is mature.

Photonic crystals do the same in electromagnetism. A periodic dielectric lattice opens a band gap in which propagation through the bulk is forbidden; a line of removed scatterers becomes a defect waveguide, and the field follows that allowed channel even through a sharp bend. The demonstration is visual, the coupling is purely electromagnetic, and the lattice constants are optical. It is a textbook case of a spectrum being converted into geometric function inside its proper validity band.

Two things about acoustics are worth carrying into the fluid chapter. First, the acoustic case is where the analogue-gravity correspondence is exact rather than suggestive: sound waves in a barotropic, irrotational, inviscid flow obey a wave equation governed by an effective metric of Lorentzian signature, and a transonic flow supplies a horizon. The eigenvalue structure of that problem is a real geometry — of the acoustic metric, not of spacetime.

Second, the acoustic effective mass density that goes negative is a lattice bookkeeping quantity, exactly as in Chapter 3. Nothing in an acoustic array constrains gravity; the coupling deficit named there is untouched by any amount of phononic success. What acoustics provides is a working laboratory for how a borrowed spectrum behaves when the borrowing is legitimate — which is why it is the right control case to hold beside the gravitational one.

Where effective field theory stops

Every construction in this part of the volume is an effective theory: a description valid below some cutoff, with the physics above the cutoff absorbed into coefficients. That framing is what licenses the borrowing, and it also states the borrowing's limit. Three limits recur.

The first is wavelength. A metamaterial description holds only while the wavelength is long compared with the unit cell; push toward the cell size and the effective parameters stop being parameters and the spectrum becomes the whole answer. The second is amplitude. The linearised field admits superposition, which is what makes an impedance or a band structure definable at all; the full Einstein equations do not, so the medium picture has no strong-field continuation. The third is the derivative expansion. Treating general relativity as an effective field theory works to finite order below the Planck scale, with higher-curvature corrections suppressed by powers of energy over that scale — and it supplies no access to the regime where the corrections stop being corrections.

Read together, the three say something useful about eigenvalues in particular. A spectrum computed inside an effective theory is trustworthy for the questions the effective theory was built to answer, and silent beyond them. When a mode structure is offered as evidence about the deep theory, the cutoff is being quietly deleted. That deletion is the mechanism by which a proxy becomes a goal, and it is the subject of Chapter 18.