WKB Expansion Research Articles

Scalar QNM spectra of Kerr and Reissner-Nordström revealed by eigenvalue repulsions in Kerr-Newman

Recent studies of the gravito-electromagnetic frequency spectra of Kerr-Newman (KN) black holes have revealed two families of quasinormal modes (QNMs), namely photon sphere modes and near-horizon modes. However, they can only be unambiguously distinguished in the Reissner-Nordström (RN) limit, due to a phenomenon called eigenvalue repulsion (also known as level repulsion, avoided crossing or the Wigner-Teller effect), whereby the two families can interact strongly near extremality. We find that these features are also present in the QNM spectra of a scalar field in KN, where the perturbation modes are described by ODEs and thus easier to explore. Starting from the RN limit, we study how the scalar QNM spectra of KN dramatically changes as we vary the ratio of charge to angular momentum, all the way until the Kerr limit, while staying at a fixed distance from extremality. This scalar field case clarifies the (so far puzzling) relationship between the QNM spectra of RN and Kerr black holes and the nature of the eigenvalue repulsions in KN, that ultimately settle the fate of the QNM spectra in Kerr. We study not just the slowest-decaying QNMs (both for ℓ = m = 0 and ℓ = m = 2), but several sub-dominant overtones as well, as these turn out to play a crucial role understanding the KN QNM spectra. We also give a new high-order WKB expansion of KN QNMs that typically describes the photon sphere modes beyond the eikonal limit, and use a matched asymptotic expansion to get a very good approximation of the near-horizon modes near extremality.

Parity violating gravitational waves at the end of inflation

Inflaton-vector interactions of the type ϕFF̃ have provided interesting phenomenology to tackle some of current problems in cosmology, namely the vectors could constitute the dark matter component.It could also lead to possible signatures imprinted in a gravitational wave spectrum.Through this coupling, a rolling inflaton induces an exponential production of the transverse polarizations of the vector field, having a maximum at the end of inflation when the inflaton field velocity is at its maximum.These gauge particles, already parity asymmetric, will source the tensor components of the metric perturbations, leading to the production of parity violating gravitational waves.In this work we examine the vector particle production in the weak coupling regime, integrating the gauge mode amplitudes spectrum during the entirety of its production and amplification epochs, until the onset of radiation domination. Finally, we calculate the gravitational wave spectrum combining the vector mode analytical solution, the WKB expansion, valid only during the amplification until horizon crossing, and the numerical solution obtained at the beginning of radiation domination when the modes cease to grow.

On Generalized WKB Expansion of Monodromy Generating Function

We study symplectic properties of the monodromy map of the Schrödinger equation on a Riemann surface with a meromorphic potential having second-order poles. At first, we discuss the conditions for the base projective connection, which induces its own set of Darboux homological coordinates, to imply the Goldman Poisson structure on the character variety. Using this result, we extend the paper [Theoret. and Math. Phys. 206 (2021), 258-295, arXiv:1910.07140], by performing generalized WKB expansion of the generating function of monodromy symplectomorphism (the Yang-Yang function) and computing its first three terms.

Study of the Inflationary Spectrum in the Presence of Quantum Gravity Corrections

After a brief review of the different approaches to predicting the possible quantum gravity corrections to quantum field theory, we discuss in some detail the formulation based on a Gaussian reference frame fixing. Then, we utilize this scenario in the determination of the inflationary spectrum of primordial perturbations. We consider the quantization of an inhomogeneous, free, massless scalar field in a quasi-classical isotropic Universe by developing a WKB expansion of the dynamics of the next order in the Planckian parameter, with respect to the one at which standard QFT emerges. The quantum gravity corrections to the scale-invariant spectrum are discussed in a specific primordial cosmological setting and then in a general minisuperspace formalism, showing that there is no mode-dependent effect, and thus the scale invariant inflationary spectrum is preserved. This result is discussed in connection to the absence of a matter backreaction on the gravitational background in the considered paradigm.

The Exact WKB analysis and the Stokes phenomena of the Unruh effect and Hawking radiation

The physical observables of quantum theory can be described by perturbation theory, which is often given by diverging power series. This divergence is connected to the existence of non-perturbative phenomena, where resurgence allows us to study this connection. Applying this idea to the WKB expansion, the exact WKB analysis gives a clear connection to non-perturbative phenomena. In this paper, we apply the exact WKB analysis to the Unruh effect and Hawking radiation. The mechanism we found in this paper is similar to the Schwinger effect of a constant electric field, where the background is static but the Stokes phenomenon appears in the temporal part. Comparing this with a sonic black hole, our calculations show a clear discrepancy between them. Then, we briefly explain how quantum backreactions can be included in the exact WKB formalism.

WKB Approaches to Restore Time in Quantum Cosmology: Predictions and Shortcomings

In this review, we analyse different aspects concerning the possibility to separate a gravity-matter system into a part which lives close to a quasi-classical state and a “small” quantum subset. The considered approaches are all relying on a WKB expansion of the dynamics by an order parameter and the natural arena consists of the Bianchi universe minisuperspace. We first discuss how, limiting the WKB expansion to the first order of approximation, it is possible to recover for the quantum subsystem a Schrödinger equation, as written on the classical gravitational background. Then, after having tested the validity of the approximation scheme for the Bianchi I model, we give some applications for the quantum subsystem in the so-called “corner” configuration of the Bianchi IX model. We individualize the quantum variable in the small one of the two anisotropy degrees of freedom. The most surprising result is the possibility to obtain a non-singular Bianchi IX cosmology when the scenario is extrapolated backwards in time. In this respect, we provide some basic hints on the extension of this result to the generic cosmological solution. In the last part of the review, we consider the same scheme to the next order of approximation identifying the quantum subset as made of matter variables only. This way, we are considering the very fundamental problem of non-unitary morphology of the quantum gravity corrections to quantum field theory discussing some proposed reformulations. Instead of constructing the time dependence via that one of the classical gravitational variables on the label time as in previous works, we analyse a recent proposal to construct time by fixing a reference frame. This scheme can be reached both introducing the so-called “kinematical action”, as well as by the well-known Kuchar–Torre formulation. In both cases, the Schrödinger equation, amended for quantum gravity corrections, has the same morphology and we provide a cosmological implementation of the model, to elucidate its possible predictions.

Revisit of Macroscopic Dynamics for Some Non-equilibrium Chemical Reactions from a Hamiltonian Viewpoint

Most biochemical reactions in living cells are open systems interacting with environment through chemostats to exchange both energy and materials. At a mesoscopic scale, the number of each species in those biochemical reactions can be modeled by a random time-changed Poisson processes. To characterize macroscopic behaviors in the large number limit, the law of large numbers in the path space determines a mean-field limit nonlinear reaction rate equation describing the dynamics of the concentration of species, while the WKB expansion for the chemical master equation yields a Hamilton–Jacobi equation and the Legendre transform of the corresponding Hamiltonian gives the good rate function (action functional) in the large deviation principle. In this paper, we decompose a general macroscopic reaction rate equation into a conservative part and a dissipative part in terms of the stationary solution to the Hamilton–Jacobi equation. This stationary solution is used to determine the energy landscape and thermodynamics for general chemical reactions, which particularly maintains a positive entropy production rate at a non-equilibrium steady state. The associated energy dissipation law at both the mesoscopic and macroscopic levels is proved together with a passage from the mesoscopic to macroscopic one. A non-convex energy landscape emerges from the convex mesoscopic relative entropy functional in the large number limit, which picks up the non-equilibrium features. The existence of this stationary solution is ensured by the optimal control representation at an undetermined time horizon for the weak KAM solution to the stationary Hamilton–Jacobi equation. Furthermore, we use a symmetric Hamiltonian to study a class of non-equilibrium enzyme reactions, which leads to nonconvex energy landscape due to flux grouping degeneracy and reduces the conservative–dissipative decomposition to an Onsager-type strong gradient flow. This symmetric Hamiltonian implies that the transition paths between multiple steady states (rare events in biochemical reactions) is a modified time reversed least action path with associated path affinities and energy barriers. We illustrate this idea through a bistable catalysis reaction and compute the energy barrier for the transition path connecting two steady states via its energy landscape.

Weakly Nonlinear Multiphase Geometric Optics for Hyperbolic Quasilinear Boundary Value Problems: Construction of a Leading Profile

We investigate in this paper the existence of the leading profile of a WKB expansion for quasilinear initial boundary value problems with a highly oscillating forcing boundary term. The framework is weakly nonlinear, as the boundary term is of order O(e) where the frequencies are of order O(1/e). We consider here multiple phases on the boundary, generating a countable infinite number of phases inside the domain, and we therefore use an almost periodic functional framework. The major difficulties of this work are the lack of symmetry in the leading profile equation and the occurrence of infinitely many resonances (opposite to the simple phase case studied earlier) The leading profile is constructed as the solution of a quasilinear problem, which is solved using a priori estimates without loss of derivatives. The assumptions of this work are illustrated with the example of isentropic Euler equations in space dimension two.

Quantization of hyper-elliptic curves from isomonodromic systems and topological recursion

We prove that the topological recursion formalism can be used to compute the WKB expansion of solutions of second order differential operators obtained by quantization of any hyper-elliptic curve. We express this quantum curve in terms of spectral Darboux coordinates on the moduli space of meromorphic sl2-connections on P1 and argue that the topological recursion produces a 2g-parameter family of associated tau functions, where 2g is the dimension of the moduli space considered. We apply this procedure to the 6 Painlevé equations which correspond to g=1 and consider a g=2 example.

Nonunitarity problem in quantum gravity corrections to quantum field theory with Born-Oppenheimer approximation

The problem of time is one of the most relevant open issues in canonical quantum gravity. Although there is a huge literature about this topic, a commonly accepted solution has not been found yet. Here, we focus on the semiclassical approach to the problem of time, that has the main goal of reproducing quantum field theory on a fixed WKB background accounting also for quantum gravity corrections. We analyze the different choices of the expansion parameter and discuss the problems arising in previous proposals, where a non-unitary evolution emerges as an effect of quantum gravity corrections. In this work we develop a new approach to solve this problem by performing the WKB expansion with the introduction of the so-called kinematical action as a clock for quantum matter, that allows to recover a unitary dynamics.

WKB expansion for a Yang–Yang generating function and the Bergman tau function

We study symplectic properties of the monodromy map of second-order equations on a Riemann surface whose potential is meromorphic with double poles. We show that the Poisson bracket defined in terms of periods of the meromorphic quadratic differential implies the Goldman Poisson structure on the monodromy manifold. We apply these results to a WKB analysis of this equation and show that the leading term in the WKB expansion of the generating function of the monodromy symplectomorphism (the Yang–Yang function introduced by Nekrasov, Rosly, and Shatashvili) is determined by the Bergman tau function on the moduli space of meromorphic quadratic differentials.

Multi-boundary correlators in JT gravity

We continue the systematic study of the thermal partition function of Jackiw-Teitelboim (JT) gravity started in [arXiv:1911.01659]. We generalize our analysis to the case of multi-boundary correlators with the help of the boundary creation operator. We clarify how the Korteweg-de Vries constraints arise in the presence of multiple boundaries, deriving differential equations obeyed by the correlators. The differential equations allow us to compute the genus expansion of the correlators up to any order without ambiguity. We also formulate a systematic method of calculating the WKB expansion of the Baker-Akhiezer function and the ’t Hooft expansion of the multi-boundary correlators. This new formalism is much more efficient than our previous method based on the topological recursion. We further investigate the low temperature expansion of the two-boundary correlator. We formulate a method of computing it up to any order and also find a universal form of the two-boundary correlator in terms of the error function. Using this result we are able to write down the analytic form of the spectral form factor in JT gravity and show how the ramp and plateau behavior comes about. We also study the Hartle-Hawking state in the free boson/fermion representation of the tau-function and discuss how it should be related to the multi-boundary correlators.

On the Holton–Lindzen–Plumb model for mean flow reversals in stratified fluids

AbstractThe Holton–Lindzen–Plumb (HLP) model describes the spontaneous emergence of mean flow reversals in stratified fluids. It has played a central role in understanding the Quasi‐Biennial Oscillation of equatorial winds in Earth's stratosphere and has arguably become a linchpin of wave–mean flow interaction theory in geophysical and astrophysical fluid dynamics. The derivation of the model's equation from primary equations follows from several assumptions, including quasi‐linear approximations, WKB expansion of the wavefield, simplifications of boundary‐layer terms, among others. Starting from the two‐dimensional, non‐rotating, Boussinesq equations, we present in this paper a self‐consistent derivation of the HLP model and show the existence of a distinguished limit for which all approximations remains valid. We furthermore discuss the important role of boundary conditions, and the relevance of this model to describe secondary bifurcations associated with a quasi‐periodic route to chaos.

3d mathcal{N} = 4 OPE coefficients from Fermi gas

The partition function of a 3d mathcal{N} = 4 gauge theory with rank N can be computed using supersymmetric localization in terms of a matrix model, which often can be formulated as an ideal Fermi gas with a non-trivial one-particle Hamiltonian. We show how OPE coefficients of protected operators correspond in this formalism to averages of n-body operators in the Fermi gas, which can be computed to all orders in 1/N using the WKB expansion. We use this formalism to compute OPE coefficients in the U(N)k× U(N)−k ABJM theory as well as the U(N ) theory with one adjoint and Nf fundamental hypermultiplets, both of which have weakly coupled M-theory duals in the large N and finite k or Nf regimes. For ABJM we reproduce known results, while for the Nf theory we compute the all orders in 1/N dependence at finite Nf for the coefficient cT of the stress tensor two-point function.

Transfer matrix of a truncated cone with viscothermal losses: application of the WKB method

The propagation in tubes with varying cross section and wall visco-thermal effects is a classical problem in musical acoustics. To treat this aspect, the first method is the division in a large number of short cylinders. The division in short conical frustums with uniform averaged wall effects is better, but remains time consuming for narrow tubes and low frequencies. The use of the WKB method for the transfer matrix of a truncated cone without any division is investigated. In the frequency domain, the equations due to Zwikker and Kosten are used to define a reference result for a simplified bassoon by considering a division in small conical frustums. Then expressions of the transfer matrix at the WKB zeroth and the second orders are derived. The WKB second order is good at higher frequencies. At low frequencies, the errors are not negligible, and the WKB zeroth order seems to be better. This is due to a slow convergence of the WKB expansion for the particular case: the zeroth order can be kept if the length of the missing cone is large compared to the wavelength. Finally, a simplified version seems to be a satisfactory compromise.

Wavefunctions, integrability, and open strings

It has been recently conjectured that the exact eigenfunctions of quantum mirror curves can be obtained by combining their WKB expansion with the open topological string wavefunction. In this paper we give further evidence for this conjecture. We present closed expressions for the wavefunctions in the so-called maximally supersymmetric case, in various geometries. In the higher genus case, our conjecture provides a solution to the quantum Baxter equation of the corresponding cluster integrable system, and we argue that the quantization conditions of the integrable system follow from imposing appropriate asymptotic conditions on the wavefunction. We also present checks of the conjecture for general values of the Planck constant.

WKB EXPANSIONS FOR HYPERBOLIC BOUNDARY VALUE PROBLEMS IN A STRIP: SELFINTERACTION MEETS STRONG WELL-POSEDNESS

In this article we are interested in the rigorous construction of WKB expansions for hyperbolic boundary value problems in the strip $\mathbb{R}^{d-1}\times [0,1]$. In this geometry, a new inversibility condition has to be imposed to construct the WKB expansion. This new condition is due to selfinteraction phenomenon which naturally appear when several boundary conditions are imposed. More precisely, by selfinteraction we mean that some rays can regenerated themselves after some rebounds against the sides of the strip. This phenomenon is not new and has already been studied in Benoit (Geometric optics expansions for hyperbolic corner problems, I: self-interaction phenomenon, Anal. PDE9(6) (2016), 1359–1418), Sarason and Smoller (Geometrical optics and the corner problem, Arch. Rat. Mech. Anal.56 (1974/75), 34–69) in the corner geometry. In this framework the existence of such selfinteracting rays is linked to specific geometries of the characteristic variety and may seem to be somewhat anecdotal. However for the strip geometry such rays become generic. The new inversibility condition, used to construct the WKB expansion, is a microlocalized version of the one characterizing the uniform in time strong well-posedness (Benoit, Lower exponential strong well-posedness of hyperbolic boundary value problems in a strip (preprint)). It is interesting to point here that such a situation already occurs in the half space geometry (Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math.23 (1970), 277–298).

Massive nonminimally coupled scalar field in Reissner-Nordström spacetime: Long-lived quasinormal modes and instability

Here we show that the phenomenon of arbitrarily long-lived quasinormal modes (called quasiresonances) of a massive scalar field in the vicinity of a black hole is not an artifact of the test field approximation, but takes place also when the (derivative) coupling of a scalar field with the Einstein tensor is taken into consideration. We observe that at large coupling and high multipole numbers, the growing modes appear in the spectrum, which are responsible for the eikonal instability of the field. For small coupling, when the configuration is stable, there appear the purely imaginary quasinormal modes which are nonperturbative in the coupling constant. At the sufficiently small coupling the nonminimal scalar field is stable and the asymptotic late-time tails are not affected by the coupling term. The accurate calculations of quasinormal frequencies for a massive scalar field with the derivative coupling in the Reissner-Nordstr\"om black-hole background are performed with the help of Frobenius method, time-domain integration and WKB expansion.

Connecting the Dunham Expansion to the Dissociation Limit for Interatomic Potentials: Application to Lennard-Jones m- n Potentials.

Dunham generated the expansion for energy levels of a rotating, vibrating diatomic molecule from an expansion of the potential about the equilibrium position. For partition functions, however, the energy levels are needed all the way to dissociation. Analytic Morse oscillator energies are not very useful because the exponential decay of the Morse potential is much too short-ranged for any physical system. The longer-range Lennard-Jones 12-6 potential could be used, but quantum energies have not previously been conveniently fit. I show how Dunham coefficients begin a set of asymptotic functions for any interaction potential, one function arising from each successive term in the WKB expansion. I apply this to the family of Lennard-Jones m- n (LJ m- n) potentials with an R-m repulsive term and R-n attractive term ( m > n) and demonstrate how m can be used as a parameter to adjust either the equilibrium distance or harmonic frequency. I present an empirical parametrization of LJ m- n vibrotor energies starting with Dunham coefficients generated from four terms in the WKB expansion. This information is combined with data from numerically solved energies and asymptotic limits to fit the functions all the way to dissociation. One can also treat exp-6 and similar model potentials with different repulsive parts using the same method because the expansion form is controlled by the long-range part of the potential.

Feynman diagrams, ribbon graphs, and topological recursion of Eynard-Orantin

We consider two seemingly unrelated problems, calculation of the WKB expansion of the harmonic oscillator wave functions and counting the number of Feynman diagrams in QED or in many-body physics and show that their solutions are both encoded in a single enumerative problem: calculation of the number of certain types of ribbon graphs. In turn, the numbers of such ribbon graphs as a function of the number of their vertices and edges can be determined recursively through the application of the topological recursion of Eynard-Orantin to the algebraic curve encoded in the Schrödinger equation of the harmonic oscillator. We show how the numbers of these ribbon graphs can be written down in closed form for any given number of vertices and edges. We use these numbers to obtain a formula for the number of N-rooted ribbon graphs with e edges, which is the same as the number of Feynman diagrams for 2N-point function with e + 1 − N loops.