arXiv subjects
Publications and source records attributed to Thomas L. Curtright.
Scale invariant scattering suggests that all Bernoulli numbers B_{2n} can be naturally partitioned, i.e., written as particular finite sums of same-signed, monotonic, rational numbers. Some properties of these rational numbers are discussed here, especially in the limit of large n.
Non-relativistic quantum mechanical scattering from an inverse square potential in two spatial dimensions leads to a novel representation of the Bernoulli numbers.
In the holographic picture, the Brout-Englert-Higgs (BEH) mechanism in d-dimensional Yang-Mills theories is conjectured to provide a Higgs-like mechanism for gravity in d+1 dimensions, resulting in massive (or massless) gravitons in IR (or UV) completions. Accordingly, one could imagine dual (magnetic-type) fields of massive gravitons in the IR (low-energy) limit that are coupled to the curl of their own energy-momentum, as well as to the rotation of matter fields on large scales. This hypothesis, which might solve cosmological issues currently ascribed to dark matter and dark energy, needs to be examined by the future Laser Interferometer Space Antenna (LISA) mission using observations of gravitational waves emitted from extragalactic sources.
Elementary features of galileon models are discussed at an introductory level. Following a simple example, a general formalism leading to a hierarchy of field equations and Lagrangians is developed for flat spacetimes. Legendre duality is discussed. Implicit and explicit solutions are then constructed and analyzed in some detail. Galileon shock fronts are conjectured to exist. Finally, some interesting general relativistic effects are studied for galileons coupled minimally to gravity. Spherically symmetric galileon and metric solutions with naked curvature singularities are obtained and are shown to be separated from solutions which exhibit event horizons by a critical curve in the space of boundary data.
The state space of a generic string bit model is spanned by $N\times N$ matrix creation operators acting on a vacuum state. Such creation operators transform in the adjoint representation of the color group $U(N)$ (or $SU(N)$ if the matrices are traceless). We consider a system of $b$ species of bosonic bits and $f$ speciesof fermionic bits. The string, emerging in the $N\to\infty$ limit, identifies $P^+=mM\sqrt{2}$ with $M$ the bit number operator and $P^-=H\sqrt{2}$ with $H$ the system Hamiltonian. We study the thermal properties of this string bit system in the case $H=0$, which can be considered the tensionless string limit: the only dynamics is restricting physical states to color singlets. Then the thermal partition function ${\rm Tr} e^{-βmM}$ can be identified, putting $x=e^{-βm}$, with a generating function $χ_0^{bf}(x)$, for which the coefficient of $x^n$ in its expansion about $x=0$ is the number of color singlets with bit number $M=n$. This function is a purely group theoretic object, which is well-studied in the literature. We show that at $N=\infty$ this system displays a Hagedorn divergence at $x=1/(b+f)$ with ultimate temperature $T_H=m/\ln(b+f)$. The corresponding function for finite $N$ is perfectly finite for $0<x<1$, so the $N=\infty$ system exhibits a phase transition at temperature $T_H$ which is absent for any finite $N$. We demonstrate that the low temperature phase is unstable above $T_H$. The lowest-order $1/N$ asymptotic correction, for $x\to1$ in the high temperature phase, is computed for large $N$. Remarkably, this is related to the number of labeled Eulerian digraphs with $N$ nodes. Systematic methods to extend our results to higher orders in $1/N$ are described.
Any nonsingular function of spin j matrices always reduces to a matrix polynomial of order 2j. The challenge is to find a convenient form for the coefficients of the matrix polynomial. The theory of biorthogonal systems is a useful framework to meet this challenge. Central factorial numbers play a key role in the theoretical development. Explicit polynomial coefficients for rotations expressed either as exponentials or as rational Cayley transforms are considered here. Structural features of the results are discussed and compared, and large j limits of the coefficients are examined.
A general group element for the fundamental representation of SU(3) is expressed as a second order polynomial in the hermitian generating matrix H, with coefficients consisting of elementary trigonometric functions dependent on the sole invariant det(H), in addition to the group parameter.
Group elements of SU(2) are expressed in closed form as finite polynomials of the Lie algebra generators, for all definite spin representations of the rotation group. The simple explicit result exhibits connections between group theory, combinatorics, and Fourier analysis, especially in the large spin limit. Salient intuitive features of the formula are illustrated and discussed.
A simple trace-coupled Galileon model is shown to admit spherically symmetric static solutions with naked spacetime curvature singularities.
We suggest that galileon theories should have an additional self-coupling of the fields to the trace of their own energy-momentum tensor. We explore the classical features of one such model, in flat 4D spacetime, with emphasis on solutions that are scalar analogues of gravitational geons. We discuss the stability of these scalar geons, and some of their possible signatures, including shock fronts.
Functional conjugation methods are used to analyze the global structure of various renormalization group trajectories, and to gain insight into the interplay between continuous and discrete rescaling. With minimal assumptions, the methods produce continuous flows from step-scaling σ functions, and lead to exact functional relations for the local flow β functions, whose solutions may have novel, exotic features, including multiple branches. As a result, fixed points of σ are sometimes not true fixed points under continuous changes in scale, and zeroes of β do not necessarily signal fixed points of the flow, but instead may only indicate turning points of the trajectories.
Holographic functional methods are introduced as probes of discrete time-stepped maps that lead to chaotic behavior. The methods provide continuous time interpolation between the time steps, thereby revealing the maps to be quasi-Hamiltonian systems underlain by novel potentials that govern the motion of a perceived point particle. Between turning points, the particle is strictly driven by Hamiltonian dynamics, but at each encounter with a turning point the potential changes abruptly, loosely analogous to the switchbacks on a mountain road. A sequence of successively deepening switchback potentials explains, in physical terms, the frequency cascade and trajectory folding that occur on the particular route to chaos revealed by the logistic map.
A Hilbert space metric is found for the SU(2|1)-invariant `superflag' Landau models, parametrized by integer 2N' and real number M, such that the Hilbert space norm is positive definite. The spectrum of these unitary super-Landau models is determined. The M=0 case yields a unitary Landau model on the supersphere SU(2|1)/U(1|1) with U(1) charge 2N=2N'+1. For the generic unitary superflag model, the manifest SU(2|1) symmetry is dynamically enhanced to SU(2|2); this is the `spherical' analog of the hidden worldline supersymmetry found previously in the planar limit.
The nonlinear flow equations discussed recently by Bender and Feinberg are all reduced to the well-known Euler equation after change of variables.
Phase space is a framework ideally suited for quantizing superintegrable systems through the use of deformation methods, as illustrated here by applications to de Sitter and chiral particles. Within this framework, Nambu brackets elegantly incorporate the additional quantum invariants of such models. New results are presented for the non-Abelian quantization of these brackets.