Poisson Bracket Relations Research Articles

Classicalization of Quantum Mechanics: Classical Radiation Damping Without the Runaway Solution

In this paper, we review a new treatment of classical radiation damping, which resolves a known contradiction in the Abraham–Lorentz equation that has long been a concern. This radiation damping problem has already been solved in quantum mechanics by the method introduced by Friedrichs. Based on Friedrichs’ treatment, we solved this long-standing problem by classicalizing quantum mechanics by replacing the canonical commutation relation from quantum mechanics with the Poisson bracket relation in classical mechanics.

Integrable coupled bosonic massive Thirring model and its nonlocal reductions

A coupled bosonic massive Thirring model (BMTM), involving an interaction between the two independent spinors, is introduced and shown to be integrable. By incorporating suitable reductions between the field components of the coupled BMTM, five novel integrable models with various type of nonlocal interactions are constructed. Lax pairs satisfying the zero curvature condition are obtained for the coupled BMTM and for each of the related nonlocal models. An infinite number of conserved quantities are derived for each of these models which confirms the integrability of the systems. It is shown that the coupled BMTM respects important symmetries of the original BMTM such as parity, time reversal, global U(1)-gauge and the proper Lorentz transformations. Similarly, all the nonlocal models obtained from the coupled BMTM remain invariant under combined operation of parity and time reversal transformations. However, it is found that only one of the nonlocal models is invariant under proper Lorentz transformation and two other models are invariant under global U(1)-gauge transformation. By using ultralocal Poisson bracket relations among the elements of the Lax operator, it is shown that the coupled BMTM and one of the nonlocal models are completely integrable in the Liouville sense.

Higher derivative Hamiltonians with benign ghosts from affine Toda lattices

We provide further evidence for Smilga’s conjecture that higher charges of integrable systems are suitable candidates for higher derivative theories that possess benign ghost sectors in their parameter space. As concrete examples we study the properties of the classical phase spaces for a number of affine Toda lattices theories related to different types of Kac–Moody algebras. We identify several types of scenarios for theories with higher charge Hamiltonians: some that possess oscillatory, divergent, benign oscillatory and benign divergent behaviour when ghost sectors are present in the quantum theory. No divergent behaviour was observed for which the trajectories reach a singularity in finite time. For theories based on particular representations for the Lie algebraic roots we found an extreme sensitivity towards the initial conditions governed by the Poisson bracket relations between the centre-of-mass coordinate and the charges.

Loop quantum gravity of a spherically symmetric scalar field coupled to gravity with a clock

The inclusion of matter fields in spherically symmetric loop quantum gravity has proved problematic at the level of implementing the constraint algebra including the Hamiltonian constraint. Here we consider the system with the introduction of a clock. Using the abelianizaton technique we introduced in previous papers in the case of gravity coupled to matter, the system can be gauge fixed and rewritten in terms of a restricted set of dynamical variables that satisfy simple Poisson bracket relations. This creates a true Hamiltonian and therefore one bypasses the issue of the constraint algebra. We show how loop quantum gravity techniques may be applied for the vacuum case and show that the Hamiltonian system reproduces previous results for the physical space of states and the observables of a Schwarzchild black hole.

Poisson brackets for some Coulomb branches

We construct Poisson bracket relations between the operators which generate the chiral ring of the Coulomb branch of certain 3d mathcal{N} = 4 quiver gauge theories. In the case where the Coulomb branch is a free space, ADE Klein singularity, or the minimal A2 nilpotent orbit, we explicitly compute the Poisson brackets between the generators using either inherited properties of the abstract Coulomb branch variety, or the expected charges of these operators under the global symmetry (known through use of the monopole formula). We also conjecture Poisson brackets for Higgs branches that originate from 6d theories with tensionless strings or 5d theories with massless instantons for which the HWG is known, based on representation theoretic and operator content constraints known from the study of their magnetic quiver.

Taylor Series and Getting the General Solutions for the Equations of Motion Using Poisson Bracket Relations

In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, called canonical transformations, which map canonical coordinate systems into canonical coordinate systems. In this work we study some examples from the classical mechanics of particles and apply mathematical method for building the equation of motion. In the present paper Poisson Brackets and their properties are presented, by using Poisson brackets and their properties we calculate some brackets. We use the Poisson bracket with Hamiltonians to express the time dependence of a function u (t), the main idea Taylor series is taken as the required solution for equation of motion using the properties of the Poisson Brackets, We have examined examples from the classical mechanics to illustrate the idea such as motion with a constant acceleration, simple harmonic oscillator, freely falling particle. The solutions are compatible with what is known in classical mechanics. The work is fundamental and sheds new light onto classical mechanics. Poisson brackets are a powerful and sophisticated tool in the Hamiltonian formalism of Classical Mechanics. They also happen to provide a direct link between classical and quantum mechanics.

Quasi-bi-Hamiltonian structures and superintegrability: Study of a Kepler-related family of systems endowed with generalized Runge-Lenz integrals of motion

The existence of quasi-bi-Hamiltonian structures for a two-dimen-sional superintegrable \begin{document}$ (k_1,k_2,k_3) $\end{document} -dependent Kepler-related problem is studied. We make use of an approach that is related with the existence of some complex functions which satisfy interesting Poisson bracket relations and that was previously applied to the standard Kepler problem as well as to some particular superintegrable systems as the Smorodinsky-Winternitz (SW) system, the Tremblay-Turbiner-Winternitz (TTW) and Post-Winternitz (PW) systems. We prove that these complex functions are important for two reasons: first, they determine the integrals of motion, and second they determine the existence of some geometric structures (in this particular case, quasi-bi-Hamiltonian structures). All the results depend on three parameters ( \begin{document}$ k_1, k_2, k_3 $\end{document} ) in such a way that in the particular case \begin{document}$ k_1\ne 0 $\end{document} , \begin{document}$ k_2 = k_3 = 0 $\end{document} , the properties characterizing the Kepler problem are obtained. This paper can be considered as divided in two parts and every part presents a different approach (different complex functions and different quasi-bi-Hamil-tonian structures).

Faithful realizations of semiclassical truncations

Realizations of algebras in terms of canonical or bosonic variables can often be used to simplify calculations and to exhibit underlying properties. There is a long history of using such methods in order to study symmetry groups related to collective motion, for instance in nuclear shell models. Here, related questions are addressed for algebras obtained by turning the quantum commutator into a Poisson bracket on moments of a quantum state, truncated to a given order. In this application, canonical realizations allow one to express the quantum back-reaction of moments on basic expectation values by means of effective potentials. In order to match degrees of freedom, faithfulness of the realization is important, which requires that, at least locally, the space of moments as a Poisson manifold is realized by a complete set of Casimir–Darboux coordinates in local charts. A systematic method to derive such variables is presented and applied to certain sets of moments which are important for physical applications. If only second-order moments are considered, their Poisson-bracket relations are isomorphic to the Lie bracket of sp(2N,R), providing an interesting link with realizations of nuclear shell models.

Ehlers symmetry in four dimensions

Starting with the light-cone Hamiltonian for gravity, we perform a field redefinition that reveals a hidden symmetry in four dimensions, namely the Ehlers $SL(2,R)$ symmetry. The field redefinition, which is non-local in space but local in time, acts as a canonical transformation in the Hamiltonian formulation keeping the Poisson bracket relations unaltered. We discuss the electro-magnetic duality symmetry of gravity in the light-cone formalism, which forms the $SO(2)$ subgroup of the Ehlers symmetry. The helicity states in the original Hamiltonian are not in a representation of the enhanced symmetry group. In order to make the symmetry manifest, we make a change of variables in the path integral from the helicity states to new fields that transform linearly under the $SO(2)$ duality symmetry.

Bracket relations for relativity groups

Poisson bracket relations for generators of canonical transformations are derived directly from the Galilei and Poincaré groups of space-time coordinate changes. The method is simple but rigorous. The meaning of each step is clear, because it corresponds to an operation in the group of changes of space-time coordinates. Only products and inverses are used; differences are not used. It is made explicitly clear why constants occur in some bracket relations but not in others and how some constants can be removed, so that in the end there is a constant in the bracket relations for the Galilei group but not for the Poincaré group. Each change of coordinates needs to be only to first order, so matrices are not needed for rotations or Lorentz transformations; simple three-vector descriptions are enough. Conversion to quantum mechanics is immediate. One result is a simpler derivation of the commutation relations for angular momentum directly from rotations. Problems are included.

Quantum volume and length fluctuations in a midi-superspace model of Minkowski space

In a (1+1)-dimensional midi-superspace model for gravitational plane waves, a flat space–time condition is imposed with constraints derived from null Killing vectors. Solutions to a straightforward regularization of these constraints have diverging length and volume expectation values. Physically acceptable solutions in the kinematic Hilbert space are obtained from the original constraint by multiplying with a power of the volume operator and by a similar modification of the Hamiltonian constraint, which is used in a regularization of the constraints. The solutions of the modified Killing constraint have finite expectation values of geometric quantities. Further, the expectation value of the original Killing constraint vanishes, but its moment is non-vanishing. As the power of the volume grows, the moment of the original constraint grows, while the moments of volume and length both decrease. Thus, these states provide possible kinematic states for flat space, with fluctuations. As a consequence of the regularization of operators, the quantum uncertainty relations between geometric quantities such as length and its conjugate momentum do not reflect naive expectations from the classical Poisson bracket relations.

On the derivation of Darboux form for the action- angle dual of trigonometric BCn Sutherland system

Recently Fehér and the author have constructed the action-angle dual of the trigonometric BCn Sutherland system via Hamiltonian reduction. In this paper a reduction- based calculation is carried out to verify the canonical Poisson bracket relations on the phase space of this dual model. Hence the material serves complementary purposes whilst it can also be regarded as a suitable modification of the hyperbolic case previously sorted out by Pusztai.

Symplectic Property of Discrete-Time Dynamical Systems

Several discrete-time dynamical systems, which are described by special difference equations, are shown to have conserved Poisson bracket relations, which we call the symplectic property. Such dyna...

The spin-statistics connection in classical field theory

The spin-statistics connection is obtained for a simple formulation of a classical field theory containing even and odd Grassmann variables. To that end, the construction of irreducible canonical realizations of the rotation group corresponding to general causal fields is reviewed. The connection is obtained by imposing local commutativity on the fields and exploiting the parity operation to exchange spatial coordinates in the scalar product of classical field evaluated at one spatial location with the same field evaluated at a distinct location. The spin-statistics connection for irreducible canonical realizations of the Poincar\'{e} group of spin $j$ is obtained in the form: Classical fields and their conjugate momenta satisfy fundamental field-theoretic Poisson bracket relations for 2$j$ even, and fundamental Poisson antibracket relations for 2$j$ odd

Function group approach to unconstrained Hamiltonian Yang–Mills theory

Starting from the temporal gauge Hamiltonian for classical pure Yang–Mills theory with the gauge group SU(2) a canonical transformation is initiated by parametrizing the Gauss law generators with three new canonical variables. The construction of the remaining variables of the new set proceeds through a number of intermediate variables in several steps, which are suggested by the Poisson bracket relations and the gauge transformation properties of these variables. The unconstrained Hamiltonian is obtained from the original one by expressing it in the new variables and then setting the Gauss law generators to zero. This Hamiltonian turns out to be local and it decomposes into a finite Laurent series in powers of the coupling constant.

Spin and statistics in classical mechanics

The spin–statistics connection is obtained for classical particles. The connection holds within pseudomechanics, a theory of particle motion that extends classical physics to include anticommuting Grassmann variables, and that exhibits classical analogs of both spin and statistics. Classical realizations of Lie groups are constructed in a canonical formalism generalized to include Grassmann variables. The theory of irreducible canonical realizations of the Poincaré group is developed in this framework, with particular emphasis on the rotation subgroup. The behavior of irreducible realizations under time inversion and charge conjugation is obtained. The requirement that the Lagrangian retain its form under the combined operation of charge conjugation and time reversal leads directly to the spin–statistics connection by an adaptation of Schwinger’s 1951 proof to irreducible canonical realizations of the Poincaré group of spin j: Generalized spin coordinates and momenta satisfy fundamental Poisson bracket relations for 2j even, and fundamental Poisson antibracket relations for 2j odd.

How relativity determines the Hamiltonian description of an object in classical mechanics

For a single object, the Poisson bracket relations for the Galilei group or the Poincaré group and the bracket relations that describe how the position should be changed by space translations, rotations, and either Galilei transformations or Lorentz transformations, imply that if the canonical coordinates are the position coordinates, the generators of the relativity group can be put in their familiar forms by a canonical transformation of the canonical momentum, which is a gauge transformation. In particular, the form of the Hamiltonian as a function of the canonical momentum, and the implied relation of the canonical momentum to the velocity and interpretation of the canonical momentum and Hamiltonian as physical quantities, are made more specific and unique by relativity; if Galilei or Lorentz transformations were not used, the Hamiltonian could be any function of the canonical momentum magnitude. Conversely, the same two sets of bracket relations imply that if the generators have their familiar forms, the canonical coordinates must be the position coordinates.

A Class of Integrable Spin Calogero-Moser Systems

We introduce spin Calogero-Moser systems associated with root systems of simple Lie algebras and give the associated Lax representations (with spectral parameter) and fundamental Poisson bracket relations. The associated integrable models (called integrable spin Calogero-Moser systems in the paper) and their Lax pairs are then obtained via Poisson reduction and gauge transformations. For Lie algebras of $A_{n}$-type, this new class of integrable systems includes the usual Calogero-Moser systems as subsystems. Our method is guided by a general framework which we develop here using dynamical Lie algebroids.

Geometry ofW-algebras from the affine Lie algebra point of view

To classify the classical field theories with W-symmetry one has to classify the symplectic leaves of the corresponding W-algebra, which are the intersection of the defining constraint and the coadjoint orbit of the affine Lie algebra if the W-algebra in question is obtained by reducing a Wess-Zumino-Novikov-Witten (WZNW) model. The fields that survive the reduction will obey nonlinear Poisson bracket (or commutator) relations in general. For example, the Toda models are well known theories which possess such a nonlinear W-symmetry and many features of these models can only be understood if one investigates the reduction procedure. In this paper we analyse the SL(n,) case from which the so-called Wn-algebras can be obtained. One advantage of the reduction viewpoint is that it gives a constructive way to classify the symplectic leaves of the W-algebra - for the n = 2 case corresponding to the coadjoint orbits of the Virasoro algebra and for the n = 3 case which gives rise to the Zamolodchikov algebra. Our method, in principle, is capable of constructing explicit representatives on each leaf. Another attractive feature of this approach is the fact that the global nature of the W-transformations can be explicitly described. The reduction method also enables one to determine the classical highest-weight (HW) states which are the stable minima of the energy on a W-leaf. These are important as only to those leaves can a HW representation space of the W-algebra be associated which contains a classical HW state.

Spin Calogero–Moser systems associated with simple Lie algebras

We introduce spin Calogero-Moser systems associated with root systems of simple Lie algebras and give the associated Lax representations (with spectral parameter) and fundamental Poisson bracket relations. Our analysis is based on a group-theoretic framework similar in spirit to the standard classical $r$-matrix theory for constant $r$-matrices.