Recently, Galajinsky, Lechtenfeld and Polovnikov proposed an elegant group-theoretical transformation of the generic conformal-invariant mechanics to the free one. Considering the classical counterpart of this transformation, we relate this transformation with the Weil model of Lobachewsky space.
Higher-derivative analogues of multidimensional conformal particle and many-body conformal mechanics are constructed. Their Newton-Hooke counterparts are derived by applying appropriate coordinate transformations.
This short note is some obvious mathematical addendum to our papers on Wilson loops on polygon-like contours with circular edges \cite{Dorn:2020meb,Dorn:2020vzj}. Using the technique of osculating spheres and circles we identify the conformal invariants characterising the kinks (cusps) of generic piecewise smooth curves in 3-dimensional space.
Based on the results in [Nucl. Phys. B 866 (2013) 212], we consider a way to construct a higher-derivative mechanical model which possesses the $l$-conformal Galilei symmetry. The dynamical system describes generalized Pais-Uhlenbeck oscillator in the presence of an effective external field.
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所）；Saint Petersburg State University（圣彼得堡国立大学）；National Research University Higher School of Economics（俄罗斯国立高等经济大学）
We investigate how the Lax-Novikov integral in the perfectly invisible $PT$-regularized zero-gap quantum conformal and superconformal mechanics systems affects on their (super)-conformal symmetries. We show that the expansion of the conformal symmetry with this integral results in a nonlinearly extended generalized Shrödinger algebra. The $PT$-regularized superconformal mechanics systems in the phase of the unbroken exotic nonlinear $\mathcal{N}=4$ super-Poincaré symmetry are described by nonlinearly super-extended Schrödinger algebra with the $osp(2|2)$ sub-superalgebra. In the partially brok…
We prove a conjecture on uniqueness and existence of the irregular vertex operators of rank $r$ introduced in our previous paper. We also introduce ramified irregular vertex operators of the Virasoro algebra. As applications, we give conjectural formulas for series expansions of Painlevé III and II tau functions in terms of our ramified irregular conformal blocks.
We present a relationship between the Calogero-Moser particles confined in harmonic oscillator potentials and a representation theory of the infinite dimensional Lie algebra which is a semi-direct sum of Virasoro algebra and its module. More precisely, it is a correspondence of excited states of the model and singular vectors in Verma modules over the algebra. This is found by a free field realization of the time evolution operator of the model. We investigate the Verma modules and some explicit example of singular vectors are given.
It is shown that, if the energy in the Schwarzian mechanics (SM) is equal to the coupling constant in the de Alfaro-Fubini-Furlan (DAFF) model, there exists a link between these two systems. In particular, the equation of motion, SL(2,R)-symmetry transformations and the corresponding conserved charges of SM can be derived from those of the DAFF model by applying a coordinate transformation of a special type, while the general solution of the DAFF system maps to the velocity function of SM. A way to reproduce this link via the method of nonlinear realizations is presented. Schwarzian counterpar…
The dynamical symmetries of $1+1$-dimensional Matrix Partial Differential Equations with a Calogero potential (with/without the presence of an extra oscillatorial De Alfaro-Fubini-Furlan, DFF, damping term) are investigated. The first-order invariant differential operators induce several invariant algebras and superalgebras. Besides the $sl(2)\oplus u(1)$ invariance of the Calogero Conformal Mechanics, an $osp(2|2)$ invariant superalgebra, realized by first-order and second-order differential operators, is obtained. The invariant algebras with an infinite tower of generators are given by the u…
Asymptotic Freedom, Dimensional Transmutation, and an Infra-red Conformal Fixed Point for the $δ$-Function Potential in 1-dimensional Relativistic Quantum Mechanics
We study deformations of the quantum conformal mechanics of De Alfaro-Fubini-Furlan with rational additional potential term generated by applying the generalized Darboux-Crum-Krein-Adler transformations to the quantum harmonic oscillator and by using the method of dual schemes and mirror diagrams. In this way we obtain infinite families of isospectral and non-isospectral deformations of the conformal mechanics model with special values of the coupling constant $g=m(m+1)$, $m\in {\mathbb N}$, in the inverse square potential term, and for each completely isospectral or gapped deformation given b…
Inspired by the results in a recent paper by G. Galloway and C. Vega (see arXiv:1712.00785), we investigate a number of geometric consequences of the existence of a timelike conformal Killing vector field on a globally hyperbolic spacetime with compact Cauchy hypersurfaces, especially in connection with the so-called Bartnik's splitting conjecture. In particular we give a complementary result to the main theorem in Galloway and Vega's paper.
In this paper we aim at presenting a concise but also comprehensive study of time-dependent (tdependent) Hamiltonian dynamics on a locally conformal symplectic (lcs) manifold. We present a generalized geometric theory of canonical transformations and formulate a time-dependent geometric Hamilton-Jacobi theory on lcs manifolds. In contrast to previous papers concerning locally conformal symplectic manifolds, here the introduction of the time dependency brings out interesting geometric properties, as it is the introduction of contact geometry in locally symplectic patches. To conclude, we show e…
Renormalization of the inverse square potential usually breaks its classical conformal invariance. In a strongly attractive potential, the scaling symmetry is broken to a discrete subgroup while, in a strongly repulsive potential, it is preserved at quantum level. In the intermediate, weak-medium range of the coupling, an anomalous length scale appears due to a flow of the renormalization group away from a critical point. We show that potentials with couplings in the strongly-repulsive and in the weak-medium ranges can be related by a dynamical supersymmetry. Imposing SUSY invariance unifies t…
In quantum statistical mechanics, closed many-body systems that do not exhibit thermalization after an arbitrarily long time in spite of the presence of interactions are called as many-body localized systems, and recently have been vigorously investigated. After a brief review of this topic, we consider a many-body interacting quantum system in one dimension, which has conformal symmetry and integrability. We exactly solve the system and discuss its thermal or non-thermal behavior.
In this article we explain that several integrable mechanical billiards in the plane are connected via conformal transformations. We first remark that the free billiard in the plane are conformal equivalent to infinitely many billiard systems defined in central force problems on a particular fixed energy level. We then explain that the classical Hooke-Kepler correspondence can be carried over to a correspondence between integrable Hooke-Kepler billiards. As part of the conclusion we show that any focused conic section gives rise to integrable Kepler billiards, which brings generalizations to a…
Remarks related to the paper of Rafael de la Madrid: "On the inconsistency of the Bohm-Gadella theory with quantum mechanics", JPhysA 39, No. 29, 9255-9268 (2006)
We investigate dynamics of probe particles moving in the near-horizon limit of (2N+1)-dimensional extremal Myers-Perry black hole with arbitrary rotation parameters. We observe that in the most general case with nonequal nonvanishing rotational parameters the system admits separation of variables in N-dimensional ellipsoidal coordinates. This allows us to find solution of the corresponding Hamilton-Jacobi equation and write down the explicit expressions of Liouville constants of motion.
Working within the framework of parity-time-symmetric quantum mechanics we look into the possibility of entanglement generation and demonstrate that the feature of non-violation of no-signaling principle may hold for the simplest non-trivial case of bipartite systems. Basically our arguments are based on the computation of the reduced density matrix of one party to justify that the entropy of the other does not change.
We study classical and quantum hidden symmetries of a particle with electric charge $e$ in the background of a Dirac monopole of magnetic charge $g$ subjected to an additional central potential $V(r)=U(r) +(eg)^2/2mr^{2}$ with $U(r)=\tfrac{1}{2}mω^2r^2$, similar to that in the one-dimensional conformal mechanics model of de Alfaro, Fubini and Furlan (AFF). By means of a non-unitary conformal bridge transformation, we establish a relation of the quantum states and of all symmetries of the system with those of the system without harmonic trap, $U(r)=0$. Introducing spin degrees of freedom via a …
We study the Klein four-group $(K_4)$ symmetry of the time-dependent Schrödinger equation for the conformal mechanics model of de Alfaro-Fubini-Furlan (AFF) with confining harmonic potential and coupling constant $g=ν(ν+1)\geq -1/4$. We show that it undergoes a complete or partial (at half-integer $ν$) breaking on eigenstates of the system, and is the automorphism of the $\mathfrak{osp}(2,2)$ superconformal symmetry in super-extensions of the model by inducing a transformation between the exact and spontaneously broken phases of $\mathcal{N}=2$ Poincaré supersymmetry. We exploit the $K_4$ symm…
Any conformally invariant energy associated with a curve possesses tension-free equilibrium states which are self-similar. When this energy is the three dimensional conformal arc-length, these states are the natural spatial generalizations of planar logarithmic spirals. In this paper, a geometric framework is developed to construct these states explicitly using the conservation laws associated with the symmetry. The tension along a curve, conserved in equilibrium, is first constructed. While the tension itself is not invariant, the statement of its conservation is. By projecting the conservati…
We investigate dynamics of probe particles moving in the near-horizon limit of (2N+1)-dimensional extremal Myers-Perry black hole (in the cases of N=3,4,5) with arbitrary rotation parameters. Very recently it has been shown arXiv:1703.00713v1 [hep-th] that in the most general case with nonequal nonvanishing rotational parameters the system admits separation of variables in N-dimensional ellipsoidal coordinates. We wrote down the explicit expressions of Liouville integrals of motion, given in arXiv:1703.00713v1 [hep-th] in ellipsoidal coordinates, in initial "Cartesian" coordinates in seven, ni…
Self-similar curves arise naturally as the tension-free equilibrium states of conformally invariant bending energies. The simplest example is the Möbius invariant conformal arc-length on planar curves, dependent on the Frenet curvature $κ$ through its first derivative with respect to arc-length. There are four conserved currents associated with this invariance: the tension and torque associated with Euclidean invariance, as well as scalar and vector currents reflecting invariance under scaling and special conformal transformations respectively. If the tension vanishes, all equilibrium states a…
This is an idiosyncratic survey of statistical fluid mechanics centering on the Hopf functional differential equation. Using the Burgers equation for illustration we review several functional integration approaches to theory of turbulence. We notice in particular that some important contributions have been brought about by researchers working on wave propagation in random media, among which Uriel Frisch is not an exception. We also discuss a particular finite-dimensional approximation for the Burgers equation.
Using a result of Longo and Xu, we show that the anomaly arising from a cyclic permutation orbifold of order 3 of a holomorphic conformal net $\mathcal A$ with central charge $c=8k$ depends on the "gravitational anomaly" $k\pmod 3$. In particular, the conjecture that holomorphic permutation orbifolds are non-anomalous and therefore a stronger conjecture of Müger about braided crossed $S_n$-categories arising from permutation orbifolds of completely rational conformal nets are wrong. More general, we show that cyclic permutations of order $n$ are non-anomalous if and only if $3\nmid n$ or $24|c…
We present a characterization of $2$-dimensional Lorentzian manifolds with constant Ricci scalar curvature. It is well known that every $2$-dimensional Lorentzian manifolds is conformally flat, so we rewrite the Ricci scalar curvature in terms of the conformal factor and we study the solutions of the corresponding differential equations. Several remarkable examples are provided.
