Canonical Generators Research Articles

Lessons from discrete light-cone quantization for physics at null infinity: bosons in two dimensions

Motivated by issues in the context of asymptotically flat spacetimes at null infinity, we discuss in the simplest example of a free massless scalar field in two dimensions several subtleties that arise when setting up the canonical formulation on a single or on two intersecting null hyperplanes with a special emphasis on the infinite-dimensional global and conformal symmetries and their canonical generators, the free data, a consistent treatment of zero modes, matching conditions, and implications for quantization of massless versus massive fields.

Logarithmic soft graviton theorems from superrotation Ward identities

Soft graviton theorems receive one-loop contributions that are logarithmic in the energy of the soft graviton, and which are closely related to tails of gravitational waveforms. We demonstrate that these logarithmic corrections are encoded in the Ward identity of superrotation symmetries, i.e. they follow from conservation of superrotation charge across spatial infinity i0. Our proof relies on a careful analysis of the radiative phase space admitting such gravitational tails, and the determination of the fluxes through null infinity I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{I} $$\\end{document} that act as canonical generators of superrotations on both gravitational and matter fields. All logarithmic terms are derived from the fluxes through correlations of the supertranslation Goldstone mode, provided care is taken in manipulating gravitationally interacting (i.e. dressed) rather than free fields. In cases where massive particles take part in the scattering process, logarithmic corrections also partly arise from the superrotation charge generator at timelike infinity i±.

Gyroscopic gravitational memory

We study the motion of a gyroscope located far away from an isolated gravitational source in an asymptotically flat spacetime. As seen from a local frame tied to distant stars, the gyroscope precesses when gravitational waves cross its path, resulting in a net ‘orientation memory’ that carries information on the wave profile. At leading order in the inverse distance to the source, the memory consists of two terms: the first is linear in the metric perturbation and coincides with the spin memory effect, while the second is quadratic and measures the net helicity of the wave burst. Both are closely related to symmetries of the gravitational radiative phase space at null infinity: spin memory probes superrotation charges, while helicity is the canonical generator of local electric-magnetic duality on the celestial sphere.

Noncommutative Hamiltonian formalism for noncommutative gravity *

We present a covariant canonical formalism for noncommutative (NC) gravity, and in general for NC geometric theories defined via a twisted -wedge product between forms. Noether theorems are generalized to the NC setting, and gauge generators are constructed in a twisted phase space with -deformed Poisson bracket. This formalism is applied to NC d = 4 vierbein gravity, and allows to find the canonical generators of the tangent space -gauge group.

Kac-Moody symmetry in the light front of gauge theories

We discuss the emergence of a new symmetry generator in a Hamiltonian realisation of four-dimensional gauge theories in the flat space foliated by retarded (advanced) time. It generates an asymptotic symmetry that acts on the asymptotic fields in a way different from the usual large gauge transformations. The improved canonical generators, corresponding to gauge and asymptotic symmetries, form a classical Kac-Moody charge algebra with a non-trivial central extension. In particular, we describe the case of electromagnetism, where the charge algebra is the U(1) current algebra with a level proportional to the coupling constant of the theory, κ = 4π2/e2. We construct bilinear generators yielding Virasoro algebras on the null boundary. We also provide a non-Abelian generalization of the previous symmetries by analysing the evolution of Yang-Mills theory in Bondi coordinates.

Asymptotic mathcal{O} (r) gauge symmetries and gauge-invariant Poincaré generators in higher spacetime dimensions

The asymptotic symmetries of electromagnetism in all higher spacetime dimensions d > 4 are extended, by incorporating consistently angle-dependent u(1) gauge transformations with a linear growth in the radial coordinate at spatial infinity. Finiteness of the symplectic structure and preservation of the asymptotic conditions require to impose a set of strict parity conditions, under the antipodal map of the (d − 2)-sphere, on the leading order fields at infinity. Canonical generators of the asymptotic symmetries are obtained through standard Hamiltonian methods. Remarkably, the theory endowed with this set of asymptotic conditions turns out to be invariant under a six-fold set of angle-dependent u(1) transformations, whose generators form a centrally extended abelian algebra. The new charges generated by the mathcal{O} (r) gauge parameter are found to be conjugate to those associated to the now improper subleading O(r−d+3) transformations, while the standard mathcal{O} (1) gauge transformations are canonically conjugate to the subleading mathcal{O} (r−d+4) transformations. This algebraic structure, characterized by the presence of central charges, allows us to perform a nonlinear redefinition of the Poincaré generators, that results in the decoupling of all of the u(1) charges from the Poincaré algebra. Thus, the mechanism previously used in d = 4 to find gauge-invariant Poincaré generators is shown to be a robust property of electromagnetism in all spacetime dimensions d ≥ 4.

Extremal Kerr black hole entropy in Poincaré gauge theory

We analyze the near horizon symmetry of the extremal Kerr black hole within the framework of Poincar\'e gauge theory (PG) for two important limiting cases: Riemannian and teleparallel solution. We show that the algebra of canonical generators is realized by Virasoro algebra, with central charge which depends on the black hole horizon radius . The conformal entropy of the black hole is obtained via Cardy formula.

Residual gauge symmetry in light-cone electromagnetism

We analyze the residual gauge freedom in light-cone electromagnetism in four dimensions. The standard boundary conditions involved in the so-called lc2formalism, which contains only the two physical degrees of freedom, allow for a subset of residual gauge transformations. We relax the boundary conditions imposed on the fields in order to obtain all the residual gauge transformations. We compute the canonical generators for Poincaré and gauge transformations with these relaxed boundary conditions. This enables us to distinguish between the trivial (proper) and large (improper) gauge transformations in light-cone electromagnetism. We then employ the Newman-Penrose formalism to identify the incoming and outgoing radiation fields. We comment on the quadratic form structure of light-cone Hamiltonians, often encountered in lc2 gauge theories.

Loop-corrected subleading soft theorem and the celestial stress tensor

We demonstrate that the one-loop exact subleading soft graviton theorem automatically follows from conservation of the BMS charges, provided that the hard and soft fluxes separately represent the extended BMS algebra at null infinity. This confirms that superrotations are genuine symmetries of the gravitational S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{S} $$\\end{document}-matrix beyond the semiclassical regime. In contrast with a previous proposal, the celestial stress tensor accounting for the one-loop corrections follows from the gravitational phase space analysis and does not require the addition of divergent counterterms. Moreover, we show that the symplectic form on the radiative phase space factorises into hard and soft sectors, and that the resulting canonical generators precisely coincide with the correct BMS fluxes.

Sub-subleading soft graviton theorem from asymptotic Einstein’s equations

We identify in Einstein gravity an asymptotic spin-2 charge aspect whose conservation equation gives rise, after quantization, to the sub-subleading soft theorem. Our treatment reveals that this spin-2 charge generates a non-local spacetime symmetry represented at null infinity by pseudo-vector fields. Moreover, we demonstrate that the non-linear nature of Einstein’s equations is reflected in the Ward identity through collinear corrections to the sub-subleading soft theorem. Our analysis also provides a unified treatment of the universal soft theorems as conservation equations for the spin-0,-1,-2 canonical generators, while highlighting the important role played by the dual mass.

Consistent Deformations for a Non-Standard D = 6 Topological BF Model from a BRST-Symmetry-Based Cohomological Approach

This paper falls under the heading of constructing consistent self-couplings in topological BF models. Our endeavor is of interest in the context of pure gravity, General Relativity, and super-gravity in Ashtekar formalism, which allow for certain economic formulations in terms of self-coupled BF theories in the presence of certain extra-constraints. More precisely, herein we address the construction of a special class of D=6 self-interactions for a collection of topological BF models with a non-standard field spectrum. Our methodology relies on a deformation method based on the relationship between antifield–BRST symmetry and the non-trivial gauge symmetries of a given field theory and implemented via the computation of certain precise spaces of the local BRST cohomology corresponding to the free limit. This cohomological BRST approach is applied to the starting free model under standard “selection rules” from Quantum Field Theory. Our findings are completely new and reveal a self-interacting topological BF model in D=6 with a complex gauge structure that is entirely read from the expression of the fully deformed solution to the classical master equation (the canonical generator of the antifield–BRST symmetry), and includes a generalization of the famous D=2 gravity in BF formulation.

Superconnected left quasigroups and involutory quandles

In this paper we study the classes of superconnected and superfaithful left quasigroups, that are relevant in the study of Mal’cev varieties of left quasigroups. Then we focus on quandles and in particular to the involutory ones. We extend the main result of [a paper by Erofeeva] to the infinite case and we offer a characterization of several classes of involutory quandles in terms of the properties of the canonical generators of the displacement group, improving the main results of [a paper by Nobusawa].

Contact geometry in superconductors and New Massive Gravity

The defining property of every three-dimensional ε-contact manifold is shown to be equivalent to requiring the fulfillment of London's equation in 2+1 electromagnetism. To illustrate this point, we show that every such manifold that is also K-contact and η-Einstein is a vacuum solution to the most general quadratic-curvature gravity action, in particular of New Massive Gravity. As an example we analyze S3 equipped with a contact structure together with an associated metric tensor such that the canonical generators of the contact distribution are null. The resulting Lorentzian metric is shown to be a vacuum solution of three-dimensional massive gravity. Moreover, by coupling the New Massive Gravity action to Maxwell-Chern-Simons we obtain a class of charged solutions stemming directly from the para-contact metric structure. Finally, we repeat the exercise for the Abelian Higgs theory.

Asymptotic structure of the Rarita-Schwinger theory in four spacetime dimensions at spatial infinity

We investigate the asymptotic structure of the free Rarita-Schwinger theory in four spacetime dimensions at spatial infinity in the Hamiltonian formalism. We impose boundary conditions for the spin-3/2 field that are invariant under an infinite-dimensional (abelian) algebra of non-trivial asymptotic fermionic symmetries. The compatibility of this set of boundary conditions with the invariance of the theory under Lorentz boosts requires the introduction of boundary degrees of freedom in the Hamiltonian action, along the lines of electromagnetism. These boundary degrees of freedom modify the symplectic structure by a surface contribution appearing in addition to the standard bulk piece. The Poincaré transformations have then well-defined (integrable, finite) canonical generators. Moreover, improper fermionic gauge symmetries, which are also well-defined canonical transformations, are further enlarged and turn out to be parametrized by two independent angle-dependent spinor functions at infinity, which lead to an infinite-dimensional fermionic algebra endowed with a central charge. We extend next the analysis to the supersymmetric spin-(1, 3/2) and spin-(2, 3/2) multiplets. First, we present the canonical realization of the super-Poincaré algebra on the spin-(1, 3/2) multiplet, which is shown to be consistently enhanced by the infinite-dimensional abelian algebra of angle-dependent bosonic and fermionic improper gauge symmetries associated with the electromagnetic and the Rarita-Schwinger fields, respectively. A similar analysis of the spin-(2, 3/2) multiplet is then carried out to obtain the canonical realization of the super-Poincaré algebra, consistently enhanced by the abelian improper bosonic gauge transformations of the spin-2 field (BMS supertranslations) and the abelian improper fermionic gauge transformations of the spin-3/2 field.

Braid groups of type ADE, Garside monoids, and the categorified root lattice

We study Artin-Tits braid groups $\mathbb{B}_W$ of type ADE via the action of $\mathbb{B}_W$ on the homotopy category $\mathcal{K}$ of graded projective zigzag modules (which categorifies the action of the Weyl group $W$ on the root lattice). Following Brav-Thomas, we define a metric on $\mathbb{B}_W$ induced by the canonical $t$-structure on $\mathcal{K}$, and prove that this metric on $\mathbb{B}_W$ agrees with the word-length metric in the canonical generators of the standard positive monoid $\mathbb{B}_W^+$ of the braid group. We also define, for each choice of a Coxeter element $c$ in $W$, a baric structure on $\mathcal{K}$. We use these baric structures to define metrics on the braid group, and we identify these metrics with the word-length metrics in the Birman-Ko-Lee/Bessis dual generators of the associated dual positive monoid $\mathbb{B}_{W.c}^\vee$. As consequences, we give new proofs that the standard and dual positive monoids inject into the group, give linear-algebraic solutions to the membership problem in the standard and dual positive monoids, and provide new proofs of the faithfulness of the action of $\mathbb{B}_W$ on $\mathcal{K}$. Finally, we use the compatibility of the baric and $t$-structures on $\mathcal{K}$ to prove a conjecture of Digne and Gobet regarding the canonical word-length of the dual simple generators of ADE braid groups.

Design of Efficient Resonator-Enhanced Electro-Optic Frequency Comb Generators

Resonator-enhanced electro-optic (RE-EO) frequency comb generators produce broad combs by coupling an optical field to a resonator containing a phase modulator driven at a harmonic of the resonator free spectral range (FSR). Recent advances in integration technologies have opened up the possibility of fabricating low-loss, efficient, and tunable ring-based RE-EO comb generators. In this work, we analyze the properties of a canonical ring-based RE-EO comb generator and propose a new dual-ring comb generator to increase comb conversion efficiency, an especially important characteristic for comb-based optical communications systems. After a brief review of RE-EO comb generator properties in the case of resonant operation, i.e., when the optical frequency and the modulation frequency are harmonics of the resonator FSR, we analyze the effect of input optical phase noise and modulation phase noise on the resulting comb. Additionally, we show analytically that in non-resonant operation the optical frequency offset and the modulation frequency offset can be much larger than the linewidth of the resonator, increasing the tolerance to fabrication errors. Then, we develop and validate numerical models to predict the output spectrum in the presence of dispersive waveguides, which cannot be modeled analytically. Using these accurate models, we analyze a dual-ring RE-EO comb generator that uses a small coupling ring to increase the conversion efficiency to 32%, compared to the 1.3% efficiency of a single-ring RE-EO comb generator. We then analyze a point-to-point inter-data center optical link and determine that a dual-ring RE-EO comb generator can support high-capacity coherent links at 20 Tb/s per fiber.

Covariant Hamiltonian for gravity coupled top-forms

We review the covariant canonical formalism initiated by D'Adda, Nelson and Regge in 1985, and extend it to include a definition of form-Poisson brackets (FPB) for geometric theories coupled to $p$-forms, gauging free differential algebras. The form-Legendre transformation and the form-Hamilton equations are derived from a $d$-form Lagrangian with $p$-form dynamical fields $\phi$. Momenta are defined as derivatives of the Lagrangian with respect to the "velocities" $d\phi$ and no preferred time direction is used. Action invariance under infinitesimal form-canonical transformations can be studied in this framework, and a generalized Noether theorem is derived, both for global and local symmetries. We apply the formalism to vielbein gravity in $d=3$ and $d=4$. In the $d=3$ theory we can define form-Dirac brackets, and use an algorithmic procedure to construct the canonical generators for local Lorentz rotations and diffeomorphisms. In $d=4$ the canonical analysis is carried out using FPB, since the definition of form-Dirac brackets is problematic. Lorentz generators are constructed, while diffeomorphisms are generated by the Lie derivative. A "doubly covariant" hamiltonian formalism is presented, allowing to maintain manifest Lorentz covariance at every stage of the Legendre transformation. The idea is to take curvatures as "velocities" in the definition of momenta.

Subshifts, λ-graph bisystems and C⁎-algebras

We introduce a notion of λ-graph bisystem that consists of a pair (L−,L+) of two labeled Bratteli diagrams L−,L+ satisfying certain compatibility condition for labeling their edges. It is a two-sided extension of λ-graph system, that has been previously introduced by the author. Its matrix presentation is called a symbolic matrix bisystem. We first show that any λ-graph bisystem presents subshifts and conversely any subshift is presented by a λ-graph bisystem, called the canonical λ-graph bisystem for the subshift. We introduce an algebraically defined relation on symbolic matrix bisystems called properly strong shift equivalence and show that two subshifts are topologically conjugate if and only if their canonical symbolic matrix bisystems are properly strong shift equivalent. A λ-graph bisystem (L−,L+) yields a pair of C⁎-algebras written OL−+,OL+− that are first defined as the C⁎-algebras of certain étale groupoids constructed from (L−,L+). We study structure of the C⁎-algebras, and show that they are universal unital unique C⁎-algebras subject to certain operator relations among canonical generators of partial isometries and projections encoded by the structure of the λ-graph bisystem (L−,L+). If a λ-graph bisystem comes from a λ-graph system of a finite directed graph, then the associated subshift is the two-sided topological Markov shift (ΛA,σA) by its transition matrix A of the graph, and the associated C⁎-algebra OL−+ is isomorphic to the Cuntz–Krieger algebra OA, whereas the other C⁎-algebra OL+− is isomorphic to the crossed product C⁎-algebra C(ΛA)⋊σA⁎Z of the commutative C⁎-algebra C(ΛA) of continuous functions on the shift space ΛA of the two-sided topological Markov shift by the automorphism σA⁎ induced by the homeomorphism of the shift σA. This phenomenon shows a duality between Cuntz–Krieger algebra OA and the crossed product C⁎-algebra C(ΛA)⋊σA⁎Z.

The center of the reflection equation algebra via quantum minors

We give simple formulas for the elements ck appearing in a quantum Cayley-Hamilton formula for the reflection equation algebra (RE algebra) associated to the quantum group Uq(glN), answering a question of Kolb and Stokman. The ck's are certain canonical generators of the center of the RE algebra, and hence of Uq(glN) itself; they have been described by Reshetikhin using graphical calculus, by Nazarov-Tarasov using quantum Yangians, and by Gurevich, Pyatov and Saponov using quantum Schur functions; however no explicit formulas for these elements were previously known.As byproducts, we prove a quantum Girard-Newton identity relating the ck's to the so-called quantum power traces, and we give a new presentation for the quantum group Uq(glN), as a localization of the RE algebra along certain principal minors.

The K-dual Bessel generators for unitary groups of Hilbert spaces

Structured frames have interested many mathematicians and engineers due to their potential applications. This paper addresses the theory of K-frames associated with unitary groups. Let U be a unitary group for a separable Hilbert space H. Given a K-frame generator η for U, we present an explicit expression of all K-dual Bessel generators of η for U under two cases, and simultaneously, prove that its canonical K-dual Bessel generator corresponds to the most economic presentation of the element from the range of K.