scholarly journals Application of the Generalized Hamiltonian Dynamics to Spherical Harmonic Oscillators

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1130
Author(s):  
Eugene Oks
Keyword(s):  
Infinite Number ◽  
Spherical Harmonic ◽  
Degrees Of Freedom ◽  
Hamiltonian Dynamics ◽  
Harmonic Oscillators ◽  
General Description ◽  
Stationary States ◽  
Vector Integral ◽  
Classical Formalism ◽  
Atomic And Molecular Physics

Dirac’s Generalized Hamiltonian Dynamics (GHD) is a purely classical formalism for systems having constraints: it incorporates the constraints into the Hamiltonian. Dirac designed the GHD specifically for applications to quantum field theory. In one of our previous papers, we redesigned Dirac’s GHD for its applications to atomic and molecular physics by choosing integrals of the motion as the constraints. In that paper, after a general description of our formalism, we considered hydrogenic atoms as an example. We showed that this formalism leads to the existence of classical non-radiating (stationary) states and that there is an infinite number of such states—just as in the corresponding quantum solution. In the present paper, we extend the applications of the GHD to a charged Spherical Harmonic Oscillator (SHO). We demonstrate that, by using the higher-than-geometrical symmetry (i.e., the algebraic symmetry) of the SHO and the corresponding additional conserved quantities, it is possible to obtain the classical non-radiating (stationary) states of the SHO and that, generally speaking, there is an infinite number of such states of the SHO. Both the existence of the classical stationary states of the SHO and the infinite number of such states are consistent with the corresponding quantum results. We obtain these new results from first principles. Physically, the existence of the classical stationary states is the manifestation of a non-Einsteinian time dilation. Time dilates more and more as the energy of the system becomes closer and closer to the energy of the classical non-radiating state. We emphasize that the SHO and hydrogenic atoms are not the only microscopic systems that can be successfully treated by the GHD. All classical systems of N degrees of freedom have the algebraic symmetries ON+1 and SUN, and this does not depend on the functional form of the Hamiltonian. In particular, all classical spherically symmetric potentials have algebraic symmetries, namely O4 and SU3; they possess an additional vector integral of the motion, while the quantal counterpart-operator does not exist. This offers possibilities that are absent in quantum mechanics.

scholarly journals Higher spins from exotic dualisations

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Nicolas Boulanger ◽  
Victor Lekeu
Keyword(s):  
Infinite Number ◽  
Arbitrary Number ◽  
Degrees Of Freedom ◽  
Young Tableaux ◽  
Spacetime Dimension ◽  
Massless Field ◽  
Free Level ◽  
Higher Spins

Abstract At the free level, a given massless field can be described by an infinite number of different potentials related to each other by dualities. In terms of Young tableaux, dualities replace any number of columns of height hi by columns of height D − 2 − hi, where D is the spacetime dimension: in particular, applying this operation to empty columns gives rise to potentials containing an arbitrary number of groups of D − 2 extra antisymmetric indices. Using the method of parent actions, action principles including these potentials, but also extra fields, can be derived from the usual ones. In this paper, we revisit this off-shell duality and clarify the counting of degrees of freedom and the role of the extra fields. Among others, we consider the examples of the double dual graviton in D = 5 and two cases, one topological and one dynamical, of exotic dualities leading to spin three fields in D = 3.

scholarly journals Canonical transformations for fermions in superanalysis

2014 ◽  
Vol 26 (06) ◽  
pp. 1450009
Author(s):  
Joachim Kupsch
Keyword(s):  
Infinite Number ◽  
Orthogonal Group ◽  
Degrees Of Freedom ◽  
Fock Space ◽  
Continuous Representation ◽  
Canonical Transformations ◽  
Module Extension ◽  
Bogoliubov Transformations

Canonical transformations (Bogoliubov transformations) for fermions with an infinite number of degrees of freedom are studied within a calculus of superanalysis. A continuous representation of the orthogonal group is constructed on a Grassmann module extension of the Fock space. The pull-back of these operators to the Fock space yields a unitary ray representation of the group that implements the Bogoliubov transformations.

scholarly journals Quasi-Lie Brackets and the Breaking of Time-Translation Symmetry for Quantum Systems Embedded in Classical Baths

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 518 ◽  
Author(s):  
Alessandro Sergi ◽  
Gabriel Hanna ◽  
Roberto Grimaudo ◽  
Antonino Messina
Keyword(s):  
Constant Temperature ◽  
Degrees Of Freedom ◽  
Open Quantum Systems ◽  
Hamiltonian Dynamics ◽  
Quantum Systems ◽  
Classical Dynamics ◽  
Open Quantum ◽  
Time Translation ◽  
Translation Symmetry ◽  
Lie Brackets

Many open quantum systems encountered in both natural and synthetic situations are embedded in classical-like baths. Often, the bath degrees of freedom may be represented in terms of canonically conjugate coordinates, but in some cases they may require a non-canonical or non-Hamiltonian representation. Herein, we review an approach to the dynamics and statistical mechanics of quantum subsystems embedded in either non-canonical or non-Hamiltonian classical-like baths which is based on operator-valued quasi-probability functions. These functions typically evolve through the action of quasi-Lie brackets and their associated Quantum-Classical Liouville Equations, or through quasi-Lie brackets augmented by dissipative terms. Quasi-Lie brackets possess the unique feature that, while conserving the energy (which the Noether theorem links to time-translation symmetry), they violate the time-translation symmetry of their algebra. This fact can be heuristically understood in terms of the dynamics of the open quantum subsystem. We then describe an example in which a quantum subsystem is embedded in a bath of classical spins, which are described by non-canonical coordinates. In this case, it has been shown that an off-diagonal open-bath geometric phase enters into the propagation of the quantum-classical dynamics. Next, we discuss how non-Hamiltonian dynamics may be employed to generate the constant-temperature evolution of phase space degrees of freedom coupled to the quantum subsystem. Constant-temperature dynamics may be generated by either a classical Langevin stochastic process or a Nosé–Hoover deterministic thermostat. These two approaches are not equivalent but have different advantages and drawbacks. In all cases, the calculation of the operator-valued quasi-probability function allows one to compute time-dependent statistical averages of observables. This may be accomplished in practice using a hybrid Molecular Dynamics/Monte Carlo algorithms, which we outline herein.

scholarly journals Hidden Nambu mechanics II: Quantum/semiclassical dynamics

2019 ◽  
Vol 2019 (12) ◽  
Author(s):  
Atsushi Horikoshi
Keyword(s):  
Time Evolution ◽  
Quantum Dynamics ◽  
Degrees Of Freedom ◽  
Hamiltonian Dynamics ◽  
Extended Phase ◽  
Nambu Mechanics ◽  
Mechanical Structure ◽  
Wave Packet Dynamics ◽  
Semiclassical Dynamics ◽  
Metastable Potential

Abstract Nambu mechanics is a generalized Hamiltonian dynamics characterized by an extended phase space and multiple Hamiltonians. In a previous paper [Prog. Theor. Exp. Phys. 2013, 073A01 (2013)] we revealed that the Nambu mechanical structure is hidden in Hamiltonian dynamics, that is, the classical time evolution of variables including redundant degrees of freedom can be formulated as Nambu mechanics. In the present paper we show that the Nambu mechanical structure is also hidden in some quantum or semiclassical dynamics, that is, in some cases the quantum or semiclassical time evolution of expectation values of quantum mechanical operators, including composite operators, can be formulated as Nambu mechanics. We present a procedure to find hidden Nambu structures in quantum/semiclassical systems of one degree of freedom, and give two examples: the exact quantum dynamics of a harmonic oscillator, and semiclassical wave packet dynamics. Our formalism can be extended to many-degrees-of-freedom systems; however, there is a serious difficulty in this case due to interactions between degrees of freedom. To illustrate our formalism we present two sets of numerical results on semiclassical dynamics: from a one-dimensional metastable potential model and a simplified Henon–Heiles model of two interacting oscillators.

Systems with Infinite Number of Degrees of Freedom

2014 ◽  
pp. 331-358
Author(s):  
Ilya Feranchuk ◽  
Alexey Ivanov ◽  
Van-Hoang Le ◽  
Alexander Ulyanenkov
Keyword(s):  
Infinite Number ◽  
Degrees Of Freedom

scholarly journals Geometry of Hamiltonian Dynamics with Conformal Eisenhart Metric

2011 ◽  
Vol 2011 ◽  
pp. 1-26 ◽  
Author(s):  
Linyu Peng ◽  
Huafei Sun ◽  
Xiao Sun
Keyword(s):  
Hamiltonian System ◽  
Geometric Structure ◽  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Hamiltonian Dynamics ◽  
Jacobi Field ◽  
Linear Hamiltonian System ◽  
Conformal Metric ◽  
Jacobi Equations ◽  
Special Case

We characterize the geometry of the Hamiltonian dynamics with a conformal metric. After investigating the Eisenhart metric, we study the corresponding conformal metric and obtain the geometric structure of the classical Hamiltonian dynamics. Furthermore, the equations for the conformal geodesics, for the Jacobi field along the geodesics, and the equations for a certain flow constrained in a family of conformal equivalent nondegenerate metrics are obtained. At last the conformal curvatures, the geodesic equations, the Jacobi equations, and the equations for the flow of the famous models, anNdegrees of freedom linear Hamiltonian system and the Hénon-Heiles model are given, and in a special case, numerical solutions of the conformal geodesics, the generalized momenta, and the Jacobi field along the geodesics of the Hénon-Heiles model are obtained. And the numerical results for the Hénon-Heiles model show us the instability of the associated geodesic spreads.

Sign in / Sign up

Export Citation Format

Share Document