Jacobi Identity Research Articles

Four-field Hamiltonian fluid closures of the one-dimensional Vlasov–Poisson equation

We consider a reduced dynamics for the first four fluid moments of the one-dimensional Vlasov–Poisson equation, namely, fluid density, fluid velocity, pressure, and heat flux. This dynamics depends on an equation of state to close the system. This equation of state (closure) connects the fifth-order moment—related to the kurtosis in velocity of the Vlasov distribution—with the first four moments. By solving the Jacobi identity, we derive an equation of state, which ensures that the resulting reduced fluid model is Hamiltonian. We show that this Hamiltonian closure allows symmetric homogeneous equilibria of the reduced fluid model to be stable.

Classification of 8-dimensional Nilsolitons by Symbolic Computation

In this paper, we develop an algorithm to classify 8 dimensional nilsolitons with simple nilsoliton derivation. We restrict our classifications to the nilsolitons corresponding to singular Gram matrix with nullity 1-3. This work can be considered as a continuation paper to our previous study where we introduced a procedure to classify algebras in dimension 8 that admit simple derivations and singular Gram matrices U. Having the singular Gram matrices, there exists infinitely many solutions to Uv =[1]_m , where the solutions are exactly the squares of the structure constants. Also, the structure constants have to satisfy the Jacobi identity for the algebra to be a Lie algebra. In our previous work, we did not introduce a procedure to create and solve the Jacobi identity(s). In this study, we take care of this issue by using computer algorithms for each index set. Thus, we complete classification of all 8 dimensional in-decomposable nilsolitons with the nullity of corresponding Gram matrix is in the set {0,1,2,3}. We provide several examples to illustrate the algorithm. For the implementation process of the newly introduced algorithm, we use MATLAB R2020b.

On effective field theories with celestial duals

We show that associativity of the tree-level OPE in a celestial CFT imposes constraints on the coupling constants of the corresponding bulk theory. These constraints are the same as those derived in [9] from the Jacobi identity of the algebra of soft modes. The constrained theories are interesting as apparently well-defined celestial CFTs with a deformed w1+∞ symmetry algebra. We explicitly work out the ramifications of these constraints on scattering amplitudes involving gluons, gravitons and scalars in these theories. We find that all four-point amplitudes constructible solely from holomorphic or anti-holomorphic three-point amplitudes vanish on the support of these constraints, which implies that all purely holomorphic or purely anti-holomorphic higher-point amplitudes vanish.

Revisiting the algebraic structure of the generalized uncertainty principle

We compare different formulations of the generalized uncertainty principle that have an underlying algebraic structure. We show that the formulation by Kempf, Mangano, and Mann [Phys. Rev. D 52 (1995)], quite popular for phenomenological studies, satisfies the Jacobi identities only for spin zero particles. In contrast, the formulation proposed earlier by one of us (Maggiore) [Phys. Lett. B 319 (1993)] has an underlying algebraic structure valid for particles of all spins and in this sense seems more fundamental. The latter is also much more constrained, resulting in only two possible solutions, one expressing the existence of a minimum length and the other expressing a form of quantum-to-classical transition. We also discuss how this more stringent algebraic formulation has an intriguing physical interpretation in terms of a discretized time at the Planck scale.

Poisson brackets and truncations in nonlinear reduced fluid models for plasmas

The Hamiltonian structure for an infinite class of nonlinear reduced fluid models, derived from a Hamiltonian drift-kinetic system, is explicitly provided in terms of the N+1 fluid moments evolving in each model of the class, with N an arbitrary positive integer. This improves previous results, in which the existence of the Hamiltonian structure was shown, but the complete explicit expression for the Poisson bracket of each model of the class was not provided. We also show that, whereas the Hamiltonian functional of the fluid models can be derived from that of the drift-kinetic system, by projecting the perturbation of the distribution function onto its truncated series in terms of Hermite polynomials, this is not the case for the Poisson bracket. Indeed, the antisymmetric bilinear form obtained by means of the aforementioned projection, although, interestingly, ”very similar” to the Poisson bracket of the fluid models, turns out to differ from it. The difference is found to reside in the coefficients W(N)lmn of the bilinear form, when the indices are such that l+m+n is even and l≥N+1,m≥N+1,n≥N+1. We show with a counterexample, related to the case N=2, that such bilinear form, in general, does not satisfy the Jacobi identity. We provide a physical interpretation of the set of variables G0,G1,…,GN, in terms of which the Poisson bracket of the fluid models exhibits a direct-sum structure, and point out an analogy between the present fluid reduction problem and the problem of the truncated quantum harmonic oscillator.

The seeds of EFT double copy

We explore the double copy of effective field theories (EFTs), in the recently proposed generalized color-kinematics and Kawai-Lewellen-Tye (KLT) approaches. In the former, we systematically construct scalar numerators satisfying the Jacobi identities from simpler numerator seeds with trace-like permutation properties. This construction has the advantage of being easily applicable to any multiplicity, which we exemplify up to 6-point. It employs the linear map between color factors formed by single traces of generators and by products of the structure constants, which also relates the generalized KLT and color-kinematics formalisms, allowing to produce KLT kernels at arbitrary order in the EFT expansion. At 4-point, we show that all EFT kernels are generated and that they only yield double-copy amplitudes which can also be obtained from the traditional KLT kernel. We perform initial checks suggesting that the same conclusions also hold at 5-point. We focus on single-trace massless scalar EFTs which however also control the higher-derivative corrections to gauge and gravity theories.

Hamiltonian Structures for Integrable Nonabelian Difference Equations

In this paper we extensively study the notion of Hamiltonian structure for nonabelian differential-difference systems, exploring the link between the different algebraic (in terms of double Poisson algebras and vertex algebras) and geometric (in terms of nonabelian Poisson bivectors) definitions. We introduce multiplicative double Poisson vertex algebras (PVAs) as the suitable noncommutative counterpart to multiplicative PVAs, used to describe Hamiltonian differential-difference equations in the commutative setting, and prove that these algebras are in one-to-one correspondence with the Poisson structures defined by difference operators, providing a sufficient condition for the fulfilment of the Jacobi identity. Moreover, we define nonabelian polyvector fields and their Schouten brackets, for both finitely generated noncommutative algebras and infinitely generated difference ones: this allows us to provide a unified characterisation of Poisson bivectors and double quasi-Poisson algebra structures. Finally, as an application we obtain some results towards the classification of local scalar Hamiltonian difference structures and construct the Hamiltonian structures for the nonabelian Kaup, Ablowitz-Ladik and Chen-Lee-Liu integrable lattices.

Jacobi identity in polyhedral products

We show that a relation among minimal non-faces of a fillable complex K yields an identity of iterated (higher) Whitehead products in a polyhedral product over K. In particular, for the (n−1)-skeleton of a simplicial n-sphere, we always have such an identity, and for the (n−1)-skeleton of a (n+1)-simplex, the identity is the Jacobi identity of Whitehead products (n=1) and Hardie's identity for higher Whitehead products (n≥2).

The gauge-natural bilinear brackets on couples of linear vector fields and linear p-forms

We give complete description of all gauge-natural bilinear operators A transforming pairs of couples of linear vector fields and linear p-forms on a vector bundle E into couples of linear vector fields and linear p-forms on E and satisfying the Jacobi identity in Leibniz form.

Fermionic construction in the supersymmetric coset model

It is known previously that the operator product expansion (OPE) between the first [Formula: see text] multiplet and itself contains the second [Formula: see text] multiplet in the supersymmetric coset model. In this paper, by using their realizations in terms of various fermions, we compute the four kinds of OPEs between the first and the second [Formula: see text] multiplets for fixed [Formula: see text] and [Formula: see text], where the group of the coset contains [Formula: see text]. By supersymmetrizing the above OPEs in [Formula: see text] superspace and using the various Jacobi identities between the currents, we determine the [Formula: see text] supersymmetric OPE between the first and the second [Formula: see text] multiplets completely. The right-hand side of this OPE contains the various [Formula: see text] multiplets: the [Formula: see text] singlet [Formula: see text] multiplets of superspin-[Formula: see text] and the [Formula: see text] triplet [Formula: see text] multiplets of superspin-[Formula: see text]. The [Formula: see text] superspace description and the decoupling of the spin-[Formula: see text] current of the [Formula: see text] superconformal algebra are also described.

Weakly nonlocal Poisson brackets: Tools, examples, computations

We implement an algorithm for the computation of Schouten bracket of weakly nonlocal Hamiltonian operators in three different computer algebra systems: Maple, Reduce and Mathematica. This class of Hamiltonian operators encompass almost all the examples coming from the theory of (1+1)-integrable evolutionary PDEs. Program summaryProgram Title:Jacobi (Maple), CDE module cde_weaklynl.red (Reduce, official distribution), nlPVA (Mathematica)CPC Library link to program files:https://doi.org/10.17632/synmrvr74g.1Developer's repository link:https://gdeq.org/Weakly_nonlocal_Poisson_bracketsLicensing provisions: BSD 2-clauseProgramming language: Maple, Reduce (Rlisp), MathematicaSupplementary material: Example program files in the three languages (Maple, Reduce, Mathematica)Nature of problem: Calculating the Jacobi identity for weakly nonlocal Poisson bracketsSolution method: Bringing the Jacobi identity to a canonical formAdditional comments including restrictions and unusual features: Use a 2020 Maple or Mathematica version, or a 2021 Reduce snapshot

Jacobi's Triple-Product Identity and an Associated Family of Theta-Function Identities

The main object of this paper is to present a family of $q$-series identities which involve some of the theta functions of Jacobi and Ramanujan. Each of these (presumably new) $q$-series identities reveals interesting relationships among three of the theta-type functions which stem from the celebrated Jacobi's triple-product identity in a remarkably simple way. The results presented in this paper are motivated essentially by a number of recent works dealing with the subject matter which is investigated herein.

Off-shell extended graphic rule and the expansion of Berends-Giele currents in Yang-Mills theory

Tree-level color-ordered Yang-Mills (YM) amplitudes can be decomposed in terms of (n − 2)! bi-scalar (BS) amplitudes, whose expansion coefficients form a basis of Bern-Carrasco-Johansson (BCJ) numerators. By the help of the recursive expansion of Einstein-Yang-Mills (EYM) amplitudes, the BCJ numerators are given by polynomial functions of Lorentz contractions which are conveniently described by graphic rule. In this work, we extend the expansion of YM amplitudes to off-shell level. We define different types of off-shell extended numerators that can be generated by graphs. By the use of these extended numerators, we propose a general decomposition formula of off-shell Berends-Giele currents in YM. This formula consists of three terms: (i). an effective current which is expanded as a combination of the Berends-Giele currents in BS theory (The expansion coefficients are one type of off-shell extended numerators) (ii). a term proportional to the total momentum of on-shell lines and (iii). a term expressed by the sum of lower point Berends-Giele currents in which some polarizations and momenta are replaced by vectors proportional to off-shell momenta appropriately. In the on-shell limit, the last two terms vanish while the decomposition of effective current precisely reproduces the decomposition of on-shell YM amplitudes with the expected coefficients (BCJ numerators in DDM basis). We further symmetrize these coefficients such that the Lie symmetries are satisfied. These symmetric BCJ numerators simultaneously satisfy the relabeling property of external lines and the algebraic properties (antisymmetry and Jacobi identity).

10D super-Yang-Mills scattering amplitudes from its pure spinor action

Using the pure spinor master action for 10D super-Yang-Mills in the gauge b0V = QΞ, tree-level scattering amplitudes are calculated through the perturbiner method, and shown to match those obtained from pure spinor CFT techniques. We find kinematic numerators made of nested b-ghost operators, and show that the Siegel gauge condition b0V = 0 gives rise to color-kinematics duality satisfying numerators whose Jacobi identity follows from the Jacobi identity of a kinematic algebra.

The replicator dynamics of zero-sum games arise from a novel poisson algebra

We show that the replicator dynamics for zero-sum games arises as a result of a non-canonical bracket that is a hybrid between a Poisson Bracket and a Nambu Bracket. The resulting non-canonical bracket is parameterized both the by the skew-symmetric payoff matrix and a mediating function. The mediating function is only sometimes a conserved quantity, but plays a critical role in the determination of the dynamics. As a by-product, we show that for the replicator dynamics this function arises in the definition of a natural metric on which phase flow-volume is preserved. Additionally, we show that the non-canonical bracket satisfies all the same identities as the Poisson bracket except for the Jacobi identity (JI), which is satisfied for special cases of the mediating function. In particular, the mediating function that gives rise to the replicator dynamics yields a bracket that satisfies JI. This neatly explains why the mediating function allows us to derive a metric on which phase flow is conserved and suggests a natural geometry for zero-sum games that extends the Symplectic geometry of the Poisson bracket and potentially an alternate approach to quantizing evolutionary games.

Hamiltonian structure of the guiding-center Vlasov–Maxwell equations

The Hamiltonian structure of the guiding-center Vlasov-Maxwell equations is presented in terms of a Hamiltonian functional and a guiding-center Vlasov-Maxwell bracket. The bracket, which is shown to satisfy the Jacobi identity exactly, is used to show that the guiding-center momentum and angular-momentum conservation laws can also be expressed in Hamiltonian form.

Next-to-MHV Yang-Mills kinematic algebra

Kinematic numerators of Yang-Mills scattering amplitudes possess a rich Lie algebraic structure that suggest the existence of a hidden infinite-dimensional kinematic algebra. Explicitly realizing such a kinematic algebra is a longstanding open problem that only has had partial success for simple helicity sectors. In past work, we introduced a framework using tensor currents and fusion rules to generate BCJ numerators of a special subsector of NMHV amplitudes in Yang-Mills theory. Here we enlarge the scope and explicitly realize a kinematic algebra for all NMHV amplitudes. Master numerators are obtained directly from the algebraic rules and through commutators and kinematic Jacobi identities other numerators can be generated. Inspecting the output of the algebra, we conjecture a closed-form expression for the master BCJ numerator up to any multiplicity. We also introduce a new method, based on group algebra of the permutation group, to solve for the generalized gauge freedom of BCJ numerators. It uses the recently introduced binary BCJ relations to provide a complete set of NMHV kinematic numerators that consist of pure gauge.

Quantum-classical dynamical brackets

We study the problem of constructing a general hybrid quantum-classical bracket from a partial classical limit of a full quantum bracket. Introducing a hybrid composition product, we show that such a bracket is the commutator of that product. From this we see that the hybrid bracket will obey the Jacobi identity and the Leibniz rule provided the composition product is associative. This suggests that the set of hybrid variables belonging to an associative subalgebra with the composition product will have consistent quantum-classical dynamics. This restricts the class of allowed quantum-classical interaction Hamiltonians. Furthermore, we show that pure quantum or classical variables can interact in a consistent framework, unaffected by no-go theorems in the literature or the restrictions for hybrid variables. In the proposed scheme, quantum backreaction appears as quantum-dependent terms in the classical equations of motion.

Hamiltonian fluid reduction of the 1.5D Vlasov–Maxwell equations

We consider the Vlasov–Maxwell equations with one spatial direction and two momenta. By solving the Jacobi identity, we derive reduced Hamiltonian fluid models for the density, the fluid momenta, and the second order moments, related to the pressure tensor. We also provide the Casimir invariants of the reduced Poisson bracket. We show that the linearization of the equations of motion around homogeneous equilibria reproduces some essential features of the kinetic model namely, the Weibel instability.

Kinematic numerators from the worldsheet: cubic trees from labelled trees

In this note we revisit the problem of explicitly computing tree-level scattering amplitudes in various theories in any dimension from worldsheet formulas. The latter are known to produce cubic-tree expansion of tree amplitudes with kinematic numerators automatically satisfying Jacobi-identities, once any half-integrand on the worldsheet is reduced to logarithmic functions. We review a natural class of worldsheet functions called “Cayley functions”, which are in one-to-one correspondence with labelled trees, and natural expansions of known half-integrands onto them with coefficients that are particularly compact building blocks of kinematic numerators. We present a general formula expressing kinematic numerators of all cubic trees as linear combinations of coefficients of labelled trees, which satisfy Jacobi identities by construction and include the usual combinations in terms of master numerators as a special case. Our results provide an efficient algorithm, which is implemented in a Mathematica package, for computing all tree amplitudes in theories including non-linear sigma model, special Galileon, Yang-Mills-scalar, Einstein-Yang-Mills and Dirac-Born-Infeld.