• #### Lectures on quantization of gauge systems

A gauge system is a classical field theory where among the fields there are connections in a principal G-bundle over the space-time manifold and the classical action is either invariant or transforms appropriately with respect to the action of the gauge group. The lectures are focused on the path integral quantization of such systems. Here two main examples of gauge systems are Yang-Mills and Chern-Simons.
ManifoldFeynman diagramsYang-Mills theoryGraphAsymptotic expansionChern-Simons theoryPartition functionPath integralCritical pointField theory...
• #### A Free-Field Lagrangian for a Gauge Theory of the CPT Symmetry

A simplified mathematical approach is presented and used to find a suitable free-field Lagrangian to complete previous work on constructing a gauge theory of CPT transformations.
Gauge theorySpin connectionCPT-symmetryFree fieldGeneral relativityDark matterGauge fieldLorentz transformationQuantum theoryPerturbation theory...
• #### Double-Scaling Limit of the O(N)-Symmetric Anharmonic Oscillator

In an earlier paper it was argued that the conventional double-scaling limit of an O(N)-symmetric quartic quantum field theory is inconsistent because the critical coupling constant is negative and thus the integral representing the partition function of the critical theory does not exist. In this earlier paper it was shown that for an O(N)-symmetric quantum field theory in zero-dimensional spacetime one can avoid this difficulty if one replaces the original quartic theory by its PT-symmetric analog. In the current paper an O(N)-symmetric quartic quantum field theory in one-dimensional spacetime [that is, O(N)-symmetric quantum mechanics] is studied using the Schroedinger equation. It is shown that the global PT-symmetric formulation of this differential equation provides a consistent way to perform the double-scaling limit of the O(N)-symmetric anharmonic oscillator. The physical nature of the critical behavior is explained by studying the PT-symmetric quantum theory and the corresponding and equivalent Hermitian quantum theory.
Quantum field theoryCoupling constantHamiltonianCritical valueCritical pointPartition functionQuantum mechanicsField theoryEigenfunctionFunctional integration...
• #### Non-Hermitian extension of gauge theories and implications for neutrino physicsver. 2

An extension of QED is considered in which the Dirac fermion has both Hermitian and anti-Hermitian mass terms, as well as both vector and axial-vector couplings to the gauge field. Gauge invariance is restored when the Hermitian and anti-Hermitian masses are of equal magnitude, and the theory reduces to that of a single massless Weyl fermion. An analogous non-Hermitian Yukawa theory is considered, and it is shown that this model can explain the smallness of the light-neutrino masses and provide an additional source of leptonic CP violation.
Standard ModelNeutrinoGauge invarianceQuantum electrodynamicsSterile neutrinoYukawa couplingGauge fieldNeutrino physicsSelf-energyFlavour...
• #### Convergent and Divergent Series in Physicsver. 2

The aim of this review, based on a series of four lectures held at the 22nd "Saalburg" Summer School (2016), is to cover selected topics in the theory of perturbation series and their summation. The first part is devoted to strategies for accelerating the rate of convergence of convergent series, namely Richardson extrapolation, and the Shanks transformations, and also covers a few techniques for accelerating the convergence of Fourier series. The second part focuses on divergent series, and on the tools allowing one to retrieve information from them. These techniques include Euler summation, Borel summation, generic summation, and the method of continued functions, including in particular the Pad\'e theory based on continued fractions. Finally, a brief discussion of Stieltjes series and functions is given in order to study the convergence properties of Pad\'e approximants.
HamiltonianWeight functionBorel summationPerturbation theoryComplex planeMain sequence starCauchy problemEuler numberRiemann zeta functionBernoulli number...
• #### Quasi-Exactly Solvable Systems and Orthogonal Polynomials

This paper shows that there is a correspondence between quasi-exactly solvable models in quantum mechanics and sets of orthogonal polynomials $\{ P_n\}$. The quantum-mechanical wave function is the generating function for the $P_n (E)$, which are polynomials in the energy $E$. The condition of quasi-exact solvability is reflected in the vanishing of the norm of all polynomials whose index $n$ exceeds a critical value $J$. The zeros of the critical polynomial $P_J(E)$ are the quasi-exact energy eigenvalues of the system.
Orthogonal polynomialsHamiltonianWeight functionEigenfunctionQuantum mechanicsCritical valueNonnegativeOverdetermined systemEuler numberBernoulli polynomials...
• #### Exactly Solvable Quantum Mechanics

A comprehensive review of exactly solvable quantum mechanics is presented with the emphasis of the recently discovered multi-indexed orthogonal polynomials. The main subjects to be discussed are the factorised Hamiltonians, the general structure of the solution spaces of the Schroedinger equation (Crum's theorem and its modifications), the shape invariance, the exact solvability in the Schroedinger picture as well as in the Heisenberg picture, the creation/annihilation operators and the dynamical symmetry algebras, coherent states, various deformation schemes (multiple Darboux transformations) and the infinite families of multi-indexed orthogonal polynomials, the exceptional orthogonal polynomials, and deformed exactly solvable scattering problems.
EigenfunctionWavefunctionQuantum mechanicsWronskianJacobi polynomialsOrthogonal polynomialsHarmonic oscillatorSolitonDiscrete symmetryWeight function...
• #### Dirac Equation with vector and scalar potentials via Supersymmetry in Quantum Mechanics

In this work, a spin $\frac 12$ relativistic particle described by a generalized potential containing both the Coulomb potential and a Lorentz scalar potential in Dirac equation is investigated in terms of the generalized ladder operators from supersymmetry in quantum mechanics. This formalism is applied for the generalized Dirac-Coulomb problem, which is an exactly solvable potential in relativistic quantum mechanics. We obtain the energy eigenvalues and calculate explicitly the energy eigenfunctions for the ground state and the first excited state.
SupersymmetryQuantum mechanicsEigenfunctionHamiltonianLadder operatorWave equationBound stateExact solutionDirac operatorPrimary star in a binary system...
• #### Classical and Quantum Mechanics via Lie algebrasver. 2

The goal of this book is to present classical mechanics, quantum mechanics, and statistical mechanics in an almost completely algebraic setting, thereby introducing mathematicians, physicists, and engineers to the ideas relating classical and quantum mechanics with Lie algebras and Lie groups. The book emphasizes the closeness of classical and quantum mechanics, and the material is selected in a way to make this closeness as apparent as possible. Much of the material covered here is not part of standard textbook treatments of classical or quantum mechanics (or is only superficially treated there). For physics students who want to get a broader view of the subject, this book may therefore serve as a useful complement to standard treatments of quantum mechanics. Almost without exception, this book is about precise concepts and exact results in classical mechanics, quantum mechanics, and statistical mechanics. The structural properties of mechanics are discussed independent of computational techniques for obtaining quantitatively correct numbers from the assumptions made. The standard approximation machinery for calculating from first principles explicit thermodynamic properties of materials, or explicit cross sections for high energy experiments can be found in many textbooks and is not repeated here.
Quantum mechanicsStatistical mechanicsAlgebraic setStructural propertiesLie algebraMaterialsClassical mechanicsThermodynamicsLie groupEnergy...
• #### A short introduction to the quantum formalism[s]

These notes are an elaboration on: (i) a short course that I gave at the IPhT-Saclay in May-June 2012; (ii) a previous letter on reversibility in quantum mechanics. They present an introductory, but hopefully coherent, view of the main formalizations of quantum mechanics, of their interrelations and of their common physical underpinnings: causality, reversibility and locality/separability. The approaches covered are mainly: (ii) the canonical formalism; (ii) the algebraic formalism; (iii) the quantum logic formulation. Other subjects: quantum information approaches, quantum correlations, contextuality and non-locality issues, quantum measurements, interpretations and alternate theories, quantum gravity, are only very briefly and superficially discussed. Most of the material is not new, but is presented in an original, homogeneous and hopefully not technical or abstract way. I try to define simply all the mathematical concepts used and to justify them physically. These notes should be accessible to young physicists (graduate level) with a good knowledge of the standard formalism of quantum mechanics, and some interest for theoretical physics (and mathematics). These notes do not cover the historical and philosophical aspects of quantum physics.
Quantum mechanicsQuantum logicQuantum gravityCausalityQuantum measurementQuantum correlationQuantum field theoryOperator algebraSpecial relativityStatistics...
• #### Alternative Perspective on Quantum Tunneling and Instantonsver. 2

We present a new way to compute and interpret quantum tunneling in a 1-D double-well potential. For large transition time we show that the quantum action functional gives an analytical expression for tunneling amplitudes. This has been confirmed by numerical simulations giving relative errors in the order of 1e-5. In contrast to the classical potential, the quantum potential has a triple-well if the classical wells are deep enough. Its minima are located at the position of extrema of the ground state wave function. The striking feature is that a single trajectory with a double instanton reproduces the tunneling amplitude. This is in contrast to the standard instanton approach, where infinitely many instantons and anti-instatons have to be taken into account. The quantum action functional is valid in the deep quantum regime in contrast to the semi-classical regime where the standard instanton approach holds. We compare both approaches via numerical simulations. While the standard instanton picture describes only the transition between potential minima of equal depth, the quantum action may give rise to instantons also for asymmetric potential minima. Such case is illustrated by an example.
InstantonTunneling amplitudesQuantum tunnelingNumerical simulationExcited stateHamiltonianBound stateDense nuclear matterDynamical timeCondensed matter physics...
• #### Chaotic instantons and enhancement of tunneling in kicked double-well system with time-reversal symmetry

Chaotic instanton approach is used to describe dynamical tunneling in kicked double well system. Effective Hamiltonian for the kicked system is obtained using matrix expansion formula for operator exponent and exploited to construct an approximation for chaotic instanton solution. This approximation is used for derivation of the ground quasienergy splitting dependence on both the perturbation strength and frequency. Results of numerical calculations for corresponding ground quasienergy splitting dependencies based on Floquet theory are in good agreement with the derived analytical formula in a wide range of perturbation parameters.
InstantonHamiltonianPhase spaceTime-reversal symmetryNumerical simulationWave packetFloquet theoryRegularizationElliptic integralTunnelling rate...
• #### Supersymmetry and Quantum Mechanicsver. 2

In the past ten years, the ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems. In particular, there is now a much deeper understanding of why certain potentials are analytically solvable and an array of powerful new approximation methods for handling potentials which are not exactly solvable. In this report, we review the theoretical formulation of supersymmetric quantum mechanics and discuss many applications. Exactly solvable potentials can be understood in terms of a few basic ideas which include supersymmetric partner potentials, shape invariance and operator transformations. Familiar solvable potentials all have the property of shape invariance. We describe new exactly solvable shape invariant potentials which include the recently discovered self-similar potentials as a special case. The connection between inverse scattering, isospectral potentials and supersymmetric quantum mechanics is discussed and multi-soliton solutions of the KdV equation are constructed. Approximation methods are also discussed within the framework of supersymmetric quantum mechanics and in particular it is shown that a supersymmetry inspired WKB approximation is exact for a class of shape invariant potentials. Supersymmetry ideas give particularly nice results for the tunneling rate in a double well potential and for improving large $N$ expansions. We also discuss the problem of a charged Dirac particle in an external magnetic field and other potentials in terms of supersymmetric quantum mechanics. Finally, we discuss structures more general than supersymmetric quantum mechanics such as parasupersymmetric quantum mechanics in which there is a symmetry between a boson and a para-fermion of order $p$.
SupersymmetryHamiltonianQuantum mechanicsSuperpotentialEigenfunctionBound stateWentzel-Kramers-Brillouin approximationPath integralHarmonic oscillatorQuantization...
• #### Dark ages reionization & galaxy formation simulation XII: Bubbles at dawn

Direct detection of regions of ionized hydrogen (HII) has been suggested as a promising probe of cosmic reionization. Observing the redshifted 21-cm signal of hydrogen from the epoch of reionization (EoR) is a key scientific driver behind new-generation, low-frequency radio interferometers. We investigate the feasibility of combining low-frequency observations with the Square Kilometre Array and near infra-red survey data of the Wide-Field Infrared Survey Telescope to detect cosmic reionization by imaging HII bubbles surrounding massive galaxies during the cosmic dawn. While individual bubbles will be too small to be detected, we find that by stacking redshifted 21-cm spectra centred on known galaxies, it will be possible to directly detect the EoR at $z \sim 9-12$, and to place qualitative constraints on the evolution of the spin temperature of the intergalactic medium (IGM) at $z \geq 9$. In particular, given a detection of ionized bubbles using this technique, it is possible to determine if the IGM surrounding them is typically in absorption or emission. Determining the globally-averaged neutral fraction of the IGM using this method will prove more difficult due to degeneracy with the average size of HII regions.
Intergalactic mediumHydrogen 21 cm lineReionizationEpoch of reionizationSignal to noise ratioLine of sightSquare Kilometre ArrayLuminosityGrismGalaxy Formation...
• #### Radiation Backgrounds at Cosmic Dawn: X-Rays from Compact Binariesver. 2

We compute the expected X-ray diffuse background and radiative feedback on the intergalactic medium (IGM) from X-ray binaries prior and during the epoch of reionization. The cosmic evolution of compact binaries is followed using a population synthesis technique that treats separately neutron stars and black hole binaries in different spectral states and is calibrated to reproduce the observed X-ray properties of galaxies at z<4. Together with an updated empirical determination of the cosmic history of star formation, recent modeling of the stellar mass-metallicity relation, and a scheme for absorption by the IGM that accounts for the presence of ionized HII bubbles during the epoch of reionization, our detailed calculations provide refined predictions of the X-ray volume emissivity and filtered radiation background from "normal" galaxies at z>6. Radiative transfer effects modulate the background spectrum, which shows a characteristic peak between 1 and 2 keV. While the filtering of X-ray radiation through the IGM slightly increases the mean excess energy per photoionization, it also weakens the radiation intensity below 1 keV, lowering the mean photoionization and heating rates. Numerical integration of the rate and energy equations shows that the contribution of X-ray binaries to the ionization of the bulk IGM is negligible, with the electron fraction never exceeding 1%. Direct HeI photoionizations are the main source of IGM heating, and the temperature of the largely neutral medium in between HII cavities increases above the temperature of the cosmic microwave background (CMB) only at z<10, when the volume filling factor of HII bubbles is already >0.1. Therefore, in this scenario, it is only at relatively late epochs that the bulk of neutral intergalactic hydrogen may be observable in 21-cm emission against the CMB.
Intergalactic mediumMetallicityHigh-mass x-ray binaryStar formation rateLow-mass X-ray binaryReionizationCosmic microwave backgroundStellar massLuminosityPhotoionization...
• #### How Zwicky already ruled out modified gravity theories without dark matterver. 2

Various theories, such as MOND, MOG, Emergent Gravity and $f(R)$ theories avoid dark matter by assuming a change in General Relativity and/or in Newton's law. Galactic rotation curves are typically described well. Here the application to galaxy clusters is considered, focussed on the good lensing and X-ray data for A1689. As a start, the no-dark-matter case is confirmed to work badly: the need for dark matter starts near the cluster centre, where Newton's law is still supposed to be valid. This leads to the conundrum discovered by Zwicky, which is likely only solvable in his way, namely by assuming additional (dark) matter. Neutrinos with eV masses serve well without altering the successes in (dwarf) galaxies.
Dark matterNeutrinoModified Newtonian DynamicsAbell 1689Strong gravitational lensingEmergent gravityBrightest cluster galaxyMass profileWeak lensingTheories of gravity...
• #### Subaru High-z Exploration of Low-Luminosity Quasars (SHELLQs). II. Discovery of 32 Quasars and Luminous Galaxies at 5.7 < z < 6.8

We present spectroscopic identification of 32 new quasars and luminous galaxies discovered at 5.7 < z < 6.8. This is the second in a series of papers presenting the results of the Subaru High-z Exploration of Low-Luminosity Quasars (SHELLQs) project, which exploits the deep multi-band imaging data produced by the Hyper Suprime-Cam (HSC) Subaru Strategic Program survey. The photometric candidates were selected by a Bayesian probabilistic algorithm, and then observed with spectrographs on the Gran Telescopio Canarias and the Subaru Telescope. Combined with the sample presented in the previous paper, we have now identified 64 HSC sources over about 430 deg2, which include 33 high-z quasars, 14 high-z luminous galaxies, 2 [O III] emitters at z ~ 0.8, and 15 Galactic brown dwarfs. The new quasars have considerably lower luminosity (M1450 ~ -25 to -22 mag) than most of the previously known high-z quasars. Several of these quasars have luminous (> 10^(43) erg/s) and narrow (< 500 km/s) Ly alpha lines, and also a possible mini broad absorption line system of N V 1240 in the composite spectrum, which clearly separate them from typical quasars. On the other hand, the high-z galaxies have extremely high luminosity (M1450 ~ -24 to -22 mag) compared to other galaxies found at similar redshift. With the discovery of these new classes of objects, we are opening up new parameter spaces in the high-z Universe. Further survey observations and follow-up studies of the identified objects, including the construction of the quasar luminosity function at z ~ 6, are ongoing.
QuasarHyper Suprime-CamLuminositySubaru telescopeBrown dwarfTelescopesBayesianPoint spread functionPhotometryIntergalactic medium...
• #### Great Optically Luminous Dropout Research Using Subaru HSC (GOLDRUSH). I. UV Luminosity Functions at $z \sim 4-7$ Derived with the Half-Million Dropouts on the 100 deg$^2$ Sky

We study the UV luminosity functions (LFs) at $z\sim 4$, $5$, $6,$ and $7$ based on the deep large-area optical images taken by the Hyper Suprime-Cam (HSC) Subaru strategic program (SSP). On the 100 deg$^2$ sky of the HSC SSP data available to date, we make enormous samples consisting of a total of 579,555 dropout candidates at $z\sim 4-7$ by the standard color selection technique, 348 out of which are spectroscopically confirmed by our follow-up spectroscopy and the other studies. We obtain the UV LFs at $z \sim 4-7$ that span a very wide UV luminosity range of $\sim 0.002 - 100 \, L_{\rm UV}^\ast$ ($-26 < M_{\rm UV} < -14$ mag), combining UV LFs of our program and the ultra-deep Hubble Space Telescope legacy surveys. We derive three parameters of the best-fit Schechter function, $\phi^\ast$, $M_{\rm UV}^\ast$, and $\alpha$, of the UV LFs in the magnitude range where the AGN contribution is negligible, and find that $\alpha$ and $\phi^\ast$ decrease from $z\sim 4$ to $7$ with no significant evolution of $M_{\rm UV}^\ast$. Because our HSC SSP data bridge the LFs of galaxies and AGNs with great statistical accuracies, we carefully investigate the bright ends of the galaxy UV LFs that are estimated by the subtraction of the AGN contribution either with the spectroscopy results or the best-fit AGN UV LFs. We find that the bright ends of the galaxy UV LFs cannot be explained by the Schechter function fits at $> 2 \sigma$ significance, and require either double power-law functions or modified Schechter functions considering the gravitational lensing magnification bias.
Luminosity functionHyper Suprime-CamActive Galactic NucleiSchechter functionMilky WaySubaru telescopeGravitational lensingUV luminosity functionQuasarLuminosity...
• #### Johann Faulhaber and sums of powers

Early 17th-century mathematical publications of Johann Faulhaber contain some remarkable theorems, such as the fact that the $r$-fold summation of $1^m,2^m,...,n^m$ is a polynomial in $n(n+r)$ when $m$ is a positive odd number. The present paper explores a computation-based approach by which Faulhaber may well have discovered such results, and solves a 360-year-old riddle that Faulhaber presented to his readers. It also shows that similar results hold when we express the sums in terms of central factorial powers instead of ordinary powers. Faulhaber's coefficients can moreover be generalized to factorial powers of noninteger exponents, obtaining asymptotic series for $1^{\alpha}+2^{\alpha}+...+n^{\alpha}$ in powers of $n^{-1}(n+1)^{-1}$.
Generating functionalBernoulli polynomialsBernoulli numberNonnegativeElementary theoryStirling numberFormal power seriesArithmeticCatalan numberBinomial coefficient...
• #### What is Mathematics and What Should it Be

This article, dedicated, with admiration to Reuben Hersh, for his forthcoming 90th birthday, argues that mathematics today is not yet a science, but that it is high time that it should become one.
RankAndromeda IArchimedesAlgebraic numberFinite differenceScannerCoolingArithmeticDeityMathematical proof...
• #### Entanglement isn't just for spin

Quantum entanglement occurs not just in discrete systems such as spins, but also in the spatial wave functions of systems with more than one degree of freedom. It is easy to introduce students to entangled wave functions at an early stage, in any course that discusses wave functions. Doing so not only prepares students to learn about Bell's theorem and quantum information science, but can also provide a deeper understanding of the principles of quantum mechanics and help fight against some common misconceptions. Here I introduce several pictorial examples of entangled wave functions that depend on just two spatial variables. I also show how such wave functions can arise dynamically, and describe how to quantify their entanglement.
EntanglementQuantum mechanicsCat stateDegree of freedomBell's theoremQuantum entanglementPhysics EducationWave packetUnit of energyExcited state...
• #### Relativity without tears

Special relativity is no more a new revolutionary theory but firmly established cornerstone of modern physics. The teaching of special relativity, however, still follows its presentation as it was unfolded historically, trying to convince subjects of this teaching that Newtonian physics is natural but incorrect and special relativity is its paradoxical but correct amendment. I argue in this article in favor of logical instead of historical trend in teaching of relativity and that special relativity is neither paradoxical nor correct (in the absolute sense of nineteen century) but the most natural expected description of real space-time around us valid for all practical purposes. This last circumstance constitutes a profound mystery of modern physics better known as the cosmological constant problem.
KinematicsNaturalnessLorentz transformationSubgroupSymmetry groupEllipticityGeneral relativityCausalityInfinitesimalHomogenization...
• #### Notes on the quantum mechanical derivation of the Wallis formula for $\pi$

Some comments on the Friedmann and Hagen's quantum mechanical derivation of the Wallis formula for $\pi$ are given. In particular, we demonstrate that Lorentz trial function, instead of the Gaussian one used by Friedmann and Hagen, also leads to the Wallis formula. The anatomy of the integrals, leading to the appearance of the Wallis ratio, are carefully revealed.
Hydrogen atomExpectation ValueAnatomyHamiltonianGamma functionCorrespondence principleQuantum mechanicsOrbital angular momentum of lightMathematical proofCompleteness...
• #### Lectures on Mathematical Cosmology

We present mathematical details of several cosmological models, whereby the topological and the geometrical background will be emphasized.
• #### A new proof of Birkhoff's theorem

Assuming SO(3)-spherical symmetry, the 4-dimensional Einstein equation reduces to an equation conformally related to the field equation for 2-dimensional gravity following from the Lagrangian L = R^(1/3). Solutions for 2-dimensional gravity always possess a local isometry because the traceless part of its Ricci tensor identically vanishes. Combining both facts, we get a new proof of Birkhoff's theorem; contrary to other proofs, no coordinates must be introduced. The SO(m)-spherically symmetric solutions of the (m+1)-dimensional Einstein equation can be found by considering L = R^(1/m) in two dimensions. This yields several generalizations of Birkhoff's theorem in an arbitrary number of dimensions, and to an arbitrary signature of the metric.
Birkhoff theoremRicci tensorEinstein field equationsIsometry groupIsometryScalar fieldScale-invariant gravityManifoldOrientationCovariance...
• #### Minimal Extension of Einstein's Theory: The Quartic Gravityver. 2

We study structure of solutions of the recently constructed minimal extensions of Einstein's gravity in four dimensions at the quartic curvature level. The extended higher derivative theory, just like Einstein's gravity, has only a massless spin-two graviton about its unique maximally symmetric vacuum. The extended theory does not admit the Schwarzschild or Kerr metrics as exact solutions, hence there is no issue of Schwarzschild type singularity but, approximately, outside a source, spherically symmetric metric with the correct Newtonian limit is recovered. We also show that for all Einstein space-times, square of the Riemann tensor (the Kretschmann scalar or the Gauss-Bonnet invariant) obeys a non-linear scalar Klein-Gordon equation.
General relativityGravitonCurvatureQuantum gravityCosmological constantCurvature tensorCosmological parametersExact solutionCosmologyKretschmann scalar...
• #### Kerr-Schild--Kundt Metrics are Universalver. 2

We define (non-Einsteinian) universal metrics as the metrics that solve the source-free covariant field equations of generic gravity theories. Here, extending the rather scarce family of universal metrics known in the literature, we show that the Kerr-Schild--Kundt class of metrics are universal. Besides being interesting on their own, these metrics can provide consistent backgrounds for quantum field theory at extremely high energies.
Covariant derivativeRankCurvature tensorTheories of gravityAnti de Sitter spaceQuantum field theoryRicci tensorScalar curvatureGeneral relativityWeyl tensor...
• #### Shortcuts to high symmetry solutions in gravitational theories

We apply the Weyl method, as sanctioned by Palais' symmetric criticality theorems, to obtain those -highly symmetric -geometries amenable to explicit solution, in generic gravitational models and dimension. The technique consists of judiciously violating the rules of variational principles by inserting highly symmetric, and seemingly gauge fixed, metrics into the action, then varying it directly to arrive at a small number of transparent, indexless, field equations. Illustrations include spherically and axially symmetric solutions in a wide range of models beyond D=4 Einstein theory; already at D=4, novel results emerge such as exclusion of Schwarzschild solutions in cubic curvature models and restrictions on independent'' integration parameters in quadratic ones. Another application of Weyl's method is an easy derivation of Birkhoff's theorem in systems with only tensor modes. Other uses are also suggested.
CurvatureVariational principleTensor mode fluctuationsGauge fixingGeneral relativityBirkhoff theoremSchwarzschild metricVector mode fluctuationEinstein-Hilbert actionEuler-Lagrange equation...
• #### De/re-constructing the Kerr Metric

We derive the Kerr solution in a pedagogically transparent way, using physical symmetry and gauge arguments to reduce the candidate metric to just two unknowns. The resulting field equations are then easy to obtain, and solve. Separately, we transform the Kerr metric to Schwarzschild frame to exhibit its limits in that familiar setting.
• #### Covariant canonical quantizationver. 2

We present a manifestly covariant quantization procedure based on the de Donder--Weyl Hamiltonian formulation of classical field theory. This procedure agrees with conventional canonical quantization only if the parameter space is $d=1$ dimensional time. In $d>1$ dimensions, covariant canonical quantization requires a fundamental length scale, and any bosonic field generates a spinorial wave function, leading to the emergence of spinors as a byproduct of quantization. We provide a probabilistic interpretation of the wave functions for the fields, and apply the formalism to a number of simple examples. These show that covariant canonical quantization produces both the Klein-Gordon and the Dirac equation, while also predicting the existence of discrete towers of identically charged fermions with different masses. Covariant canonical quantization can thus be understood as a `first' or pre-quantization within the framework of conventional QFT.
• #### Gravity a la Born-Infeldver. 3

A simple technique for the construction of gravity theories in Born-Infeld style is presented, and the properties of some of these novel theories are investigated. They regularize the positive energy Schwarzschild singularity, and a large class of models allows for the cancellation of ghosts. The possible correspondence to low energy string theory is discussed. By including curvature corrections to all orders in alpha', the new theories nicely illustrate a mechanism that string theory might use to regularize gravitational singularities.
• #### Birkhoff for Lovelock Redux

We show succinctly that all metric theories with second order field equations obey Birkhoff's theorem: their spherically symmetric solutions are static.
• #### Canonical Analysis and Stability of Lanczos-Lovelock Gravityver. 2

We perform a space-time analysis of the D>4 quadratic curvature Lanczos-Lovelock (LL) model, exhibiting its dependence on intrinsic/extrinsic curvatures, lapse and shifts. As expected from general covariance, the field equations include D constraints, of zeroth and first time derivative order. In the "linearized" - here necessarily cubic - limit, we give an explicit formulation in terms of the usual ADM metric decomposition, incidentally showing that time derivatives act only on its transverse-traceless spatial components. Unsurprisingly, pure LL has no Hamiltonian formulation, nor are even its - quadratic - weak field constraints easily soluble. Separately, we point out that the extended, more physical R+LL, model is stable - its energy is positive - due to its supersymmetric origin and ghost-freedom.
CurvatureHamiltonianCanonical analysisSuperstringGeneral covarianceSupergravityEinstein-Hilbert actionExtrinsic curvatureEmbeddingBirkhoff theorem...
• #### Stressless Schwarzschildver. 4

This self-contained pedagogical derivation of the Schwarzchild solution, in "3 + 1" formulation and conformal spatial gauge, (almost) avoids all affinity, curvature and index gymnastics.
CurvatureCovarianceHomogenizationCotton tensorEinstein field equationsUniversal propertySchouten tensorChristoffel symbolsBirkhoff theoremAbsorbance...
• #### Time and Chaos in General Relativity

The study of dynamics in general relativity has been hampered by a lack of coordinate independent measures of chaos. Here we present a variety of invariant measures for quantifying chaotic dynamics in relativity by exploiting the coordinate independence of fractal dimensions. We discuss how preferred choices of time naturally arise in chaotic systems and how the existence of invariant signals of chaos allow us to reinstate standard coordinate dependent measures. As an application, we study the Mixmaster universes and find it to exhibit transient soft chaos.
ChaosGeneral relativityFractalFractal dimensionPhase spaceStrangenessEntropyAttractorTopological entropy in physicsMultifractality...
• #### Schwarzschild and Birkhoff a la Weylver. 2

We provide a simple derivation of the Schwarzschild solution in General Relativity, generalizing an early approach by Weyl, to include Birkhoff's theorem: constancy of the mass; its deeper, Hamiltonian, basis is also given. Our procedure is illustrated by a parallel derivation of the Coulomb field and constancy of electric charge, in electrodynamics.
General relativityHamiltonianElectrodynamicsBirkhoff theoremSchwarzschild metricMagnetic monopoleGauge invarianceEinstein field equationsLinearized gravityZero mode...
• #### On the discovery of Birkhoff's theoremver. 2

Birkhoff showed in 1923 that the Schwarzschild solution for the metric from a point particle was also valid in the a priori non-static case as long as the spherical symmetry was maintained. This theorem was actually discovered and published two years earlier by an unknown Norwegian physicist, J.T. Jebsen. His life and scientific career is briefly chronicled.
Birkhoff theoremSchwarzschild metricEinstein field equationsGravitational radiationGravitational fieldsElectrodynamicsSpherical collapse modelVariational principleLine elementMass distribution...
• #### How I got to work with Feynman on the covariant quark model

In the period 1968 - 1974 I was a graduate student and then a postdoc at Caltech and was involved with the developments of the quark and parton models. Most of this time I worked in close contact with Richard Feynman and thus was present from the parton model was proposed until QCD was formulated. A personal account is presented how the collaboration took place and how the various stages of this development looked like from the inside until QCD was established as a theory for strong interactions with the partons being quarks and gluons.
PartonField theoryDeep inelastic scatteringFragmentationLight conesStatisticsCERNQuantum field theoryStrong interactionsFermilab...
• #### Status and developments of event generators

Event generators play a crucial role in the exploration of LHC physics. This presentation summarizes news and plans for the three general-purpose pp generators HERWIG, PYTHIA and SHERPA, as well as briefer notes on a few other generators. Common themes, such as the matching and merging between matrix elements and parton showers, are highlighted. Other topics include a historical introduction, from the Lund perspective, and comments on the role of MCnet.
• #### Electromagnetic Cascades as Probes of Cosmic Magnetism

The existence of intergalactic magnetic fields (IGMFs) is an open problem in cosmology and has never been unambiguously confirmed. High-energy gamma rays emitted by blazars are unique probes of cosmic magnetism, as their interactions with pervasive radiation fields generate a short-lived charged component sensitive to intervening magnetic fields. Spatial and temporal properties of the secondary gamma rays generated in the electromagnetic cascade can provide information about the strength, power spectrum, and topology of IGMFs. To probe these fields, detailed simulations of gamma-ray propagation in the intergalactic medium are necessary. In this work the effects of magnetic fields on the spectrum and arrival directions of gamma rays are studied using three-dimensional simulations, emphasising the particular case of helical IGMFs.
Intergalactic magnetic fieldElectromagnetic cascadeBlazarHelicityMagnetismExtragalactic background lightCosmologyEarthCoherence lengthHelical magnetic field...
• #### DeepStack: Expert-Level Artificial Intelligence in No-Limit Pokerver. 3

Artificial intelligence has seen several breakthroughs in recent years, with games often serving as milestones. A common feature of these games is that players have perfect information. Poker is the quintessential game of imperfect information, and a longstanding challenge problem in artificial intelligence. We introduce DeepStack, an algorithm for imperfect information settings. It combines recursive reasoning to handle information asymmetry, decomposition to focus computation on the relevant decision, and a form of intuition that is automatically learned from self-play using deep learning. In a study involving 44,000 hands of poker, DeepStack defeated with statistical significance professional poker players in heads-up no-limit Texas hold'em. The approach is theoretically sound and is shown to produce more difficult to exploit strategies than prior approaches.
Imperfect informationBest response strategySubgameInformation setPerfect informationArtificial intelligenceNash equilibriumTrunkingNeural networkHidden layer...
• #### Amplitudes for Astrophysicists I: Known Knownsver. 2

The use of quantum field theory to understand astrophysical phenomena is not new. However, for the most part, the methods used are those that have been developed decades ago. The intervening years have seen some remarkable developments in computational quantum field theoretic tools. In particle physics, this technology has facilitated calculations that, even ten years ago would have seemed laughably difficult. It is remarkable, then, that most of these new techniques have remained firmly within the domain of high energy physics. We would like to change this. As alluded to in the title, this is the first in a series of papers aimed at showcasing the use of modern on-shell methods in the context of astrophysics and cosmology. In this first article, we use the old problem of the bending of light by a compact object as an anchor to pedagogically develop these new computational tools. Once developed, we then illustrate their power and utility with an application to the scattering of gravitational waves.
HelicityGravitonFeynman diagramsPropagatorScattering amplitudeQuantum field theoryLittle groupSpinor-helicity formalismVirtual particlePolarization vector...
• #### A Computer Algebra Package for Polynomial Sequence Recognition

The software package developed in the MS thesis research implements functions for the intelligent guessing of polynomial sequence formulas based on user-defined expected sequence factors of the input coefficients. We present a specialized hybrid approach to finding exact representations for polynomial sequences that is motivated by the need for an automated procedures to discover the precise forms of these sums based on user guidance, or intuition, as to special sequence factors present in the formulas. In particular, the package combines the user input on the expected special sequence factors in the polynomial coefficient formulas with calls to the existing functions as subroutines that then process formulas for the remaining sequence terms already recognized by these packages. The factorization-based approach to polynomial sequence recognition is unique to this package and allows the search functions to find expressions for polynomial sums involving Stirling numbers and other special triangular sequences that are not readily handled by other software packages. In contrast to many other sequence recognition and summation software, the package not provide an explicit proof, or certificate, for the correctness of these sequence formulas -- only computationally guided educated guesses at a complete identity generating the sequence over all $n$. The thesis contains a number of concrete, working examples of the package that are intended to both demonstrate its usage and to document its current sequence recognition capabilities.
SageBinomial coefficientStirling numberEulerian numberStirling numbers of the first kindBernoulli numberArithmetic progressionStirling numbers of the second kindHarmonic numberLegendre polynomials...
• #### Continued Fractions for Square Series Generating Functionsver. 4

We consider new series expansions for variants of the so-termed ordinary geometric square series generating functions originally defined in the recent article titled "Square Series Generating Function Transformations" (arXiv: 1609.02803). Whereas the original square series transformations article adapts known generating function transformations to construct integral representations for these square series functions enumerating the square powers of $q^{n^2}$ for some fixed non-zero $q$ with $|q| < 1$, we study the expansions of these special series through power series generated by Jacobi-type continued fractions, or J-fractions. We prove new exact expansions of the $h^{th}$ convergents to these continued fraction series and show that the limiting case of these convergent generating functions exists. We also prove new infinite $q$-series representations of special square series expansions involving square-power terms of the series parameter $q$, the $q$-Pochhammer symbol, and double sums over the $q$-binomial coefficients. Applications of the new results we prove within the article include new $q$-series representations for the ordinary generating functions of the special sequences, $r_p(n)$, and $\sigma_1(n)$, as well as parallels to the examples of the new integral representations for theta functions, series expansions of infinite products and partition function generating functions, and related unilateral special function series cited in the first square series transformations article.
Pochhammer symbolJacobi theta functionsBinomial coefficientTheta functionPartition functionGamma functionRamanujan theta functionFinite differenceDerivative expansionComplete sequence...
• #### Twenty combinatorial examples of asymptotics derived from multivariate generating functionsver. 2

Let $\{a_\rr : \rr \in (\Z^+)^d \}$ be a $d$-dimensional array of numbers, for which the generating function $F(\zz) := \sum_\rr a_\rr \zz^\rr$ is meromorphic in a neighborhood of the origin. For example, $F$ may be a rational multivariate generating function. We discuss recent results that allow the effective computation of asymptotic expansions for the coefficients of $F$. Our purpose is to illustrate the use of these techniques on a variety of problems of combinatorial interest. The survey begins by summarizing previous work on the asymptotics of univariate and multivariate generating functions. Next we describe the Morse-theoretic underpinnings of some new asymptotic techniques. We then quote and summarize these results in such a way that only elementary analyses are needed to check hypotheses and carry out computations. The remainder of the survey focuses on combinatorial applications, such as enumeration of words with forbidden substrings, edges and cycles in graphs, polyominoes, and descents in permutations. After the individual examples, we discuss three broad classes of examples, namely functions derived via the transfer matrix method, those derived via the kernel method, and those derived via the method of Lagrange inversion. These methods have the property that generating functions derived from them are amenable to our asymptotic analyses, and we describe further machinery that facilitates computations for these classes of examples.
• #### Graph Summarization: A Surveyver. 2

While advances in computing resources have made processing enormous amounts of data possible, human ability to identify patterns in such data has not scaled accordingly. Thus, efficient computational methods for condensing and simplifying data are becoming vital for extracting actionable insights. In particular, while data summarization techniques have been studied extensively, only recently has summarizing interconnected data, or graphs, become popular. This survey is a structured, comprehensive overview of the state-of-the-art methods for summarizing graph data. We first broach the motivation behind and the challenges of graph summarization. We then categorize summarization approaches by the type of graphs taken as input and further organize each category by core methodology. Finally, we discuss applications of summarization on real-world graphs and conclude by describing some open problems in the field.
GraphSocial networkAdjacency matrixSupernova/Acceleration ProbeCommunity detectionOptimizationDirected graphTaxonomyPrivacyGraph clustering...
• #### Thermodynamic Origin of the Null Energy Conditionver. 2

We derive the classical null energy condition, understood as a constraint on the Ricci tensor, from the second law of thermodynamics applied locally to Bekenstein-Hawking entropy associated with patches of null congruences. The derivation provides evidence that the null energy condition, which has usually been regarded as a condition on matter, is fundamentally a property of gravity.
Null energy conditionEntropyHorizonSecond law of thermodynamicsEinstein field equationsInfinitesimalBekenstein-Hawking entropyCoarse grainingGeneral relativityEmergent gravity...