Analogue gravity is a research programme which investigates analogues of
general relativistic gravitational fields within other physical systems,
typically but not exclusively condensed matter systems, with the aim of gaining
new insights into their corresponding problems Analogue models of (and for)
gravity have a long and distinguished history dating back to the earliest years
of general relativity In this review article we will discuss the history,
aims, results, and future prospects for the various analogue models We start
the discussion by presenting a particularly simple example of an analogue
model, before exploring the rich history and complex tapestry of models
discussed in the literature The last decade in particular has seen a
remarkable and sustained development of analogue gravity ideas, leading to some
hundreds of published articles, a workshop, two books, and this review article
Future prospects for the analogue gravity programme also look promising, both
on the experimental front (where technology is rapidly advancing) and on the
theoretical front (where variants of analogue models can be used as a
springboard for radical attacks on the problem of quantum gravity)

Analogue Gravity

The functional determinant of an eigenvalue sequence, as defined by zeta regularization, can be simply evaluated by quadratures. We apply this procedure to the Selberg trace formula for a compact Riemann surface to find a factorization of the Selberg zeta function into two functional determinants, respectively related to the Laplacian on the compact surface itself, and on the sphere. We also apply our formalism to various explicit eigenvalue sequences, reproducing in a simpler way classical results about the gamma function and the BarnesG-function. Concerning the latter, our method explains its connection to the Selberg zeta function and evaluates the related Glaisher-Kinkelin constantA.

https://projecteuclid.org/download/pdf_1/euclid.cmp/1104159315

Spectral functions, special functions and the Selberg zeta function

Comparison between formulae for the counting functions of the heights tn of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the tn are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian Hcl. Many features of Hcl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the tn have a similar structure to those of the semiclassical En; in particular, they display random-matrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the tn can be computed accurately from formulae with quantum analogues. The Riemann-Siegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian Hcl=XP.

/pdf/the-riemann-zeros-and-eigenvalue-asymptotics-29974mp4rc.pdf

The Riemann Zeros and Eigenvalue Asymptotics

Quantum systems with finite Hilbert space are considered, and phase-space methods like the Heisenberg–Weyl group, symplectic transformations and Wigner and Weyl functions are discussed. A factorization of such systems in terms of smaller subsystems, based on the Chinese remainder theorem, is studied. The general formalism is applied to the case of angular momentum. In this context, SU(2) coherent states are used for analytic representations. Links between the theory of finite quantum systems and other fields of research are discussed.

https://iopscience.iop.org/article/10.1088/0034-4885/67/3/R03/pdf

Quantum systems with finite Hilbert space

The way in which quantum eigenstates are influenced by the closed orbits of a chaotic classical system is analysed in phase space x ═ ( q, p ) through the spectral Wigner function W ( x ; E, ∊ ). This is a sum over Wigner functions of eigenstates within a range ∊ of energy E . In the classical limit, W is concentrated on the energy surface and smoothly distributed over it. Closed orbits provide oscillatory corrections (scars) for which explicit semiclassical formulae are calculated. Each scar is a fringe pattern decorating the orbit. As x moves off the energy surface the fringes form an Airy pattern with spacing of order h ⅔ . As x moves off the closed orbit the fringes form a complex gaussian with spacing h ½ .

N.L. Balazs

Papers

Chaos on the pseudosphere

Effect of a Gravitational Field, Due to a Rotating Body, on the Plane of Polarization of an Electromagnetic Wave