Curved Phase Space Research Articles

Operator quantization of dynamical systems with curved phase space

The operator quantization method is extended to dynamical systems with curved phase space by treating them effectively as those with a flat phase space of the double dimension and effective second-class constraints.

Hopf algebras for physics at the Planck scale

Applies ideas of non-commutative geometry to reformulate the classical and quantum mechanics of a particle moving on a homogeneous spacetime. The reformulation maintains an interesting symmetry between observables and states in the form of a Hopf algebra structure. In the simplest example both the dynamics and quantum mechanics are completely determined by the Hopf algebra consideration. The simplest example is a two-parameter algebra-co-algebra KAB which is the unique Hopf algebra extension, such as is possible, of the self-dual Hopf algebra of functions on flat phase space C(R*R). In the limit (A=0,B) the author recovers functions on a curved phase space with curvature proportional to B2 and in another limit (A= infinity , B= infinity ), A/B=h(cross), the author recovers quantum mechanics on R>or=0 with an absorbing wall at the origin. The algebra in this way corresponds to a toy model of quantum mechanics of a particle in one space dimension combined with gravity-like forces. It has an interesting Z2 symmetry interchanging A to or from B, and thereby, in some sense, the quantum element with the geometric element. The compatibility conditions that are solved are a generalisation of the classical Yang-Baxter equations.

Spatially inhomogeneous structures in a one-dimensional array of Brussellators

Considering a one-dimensional array of diffusively coupled Brussellators, we examine the bifurcations of the steady homogeneous solution to steady but spatially inhomogeneous structures. Several important aspects of such bifurcations are related to branching (see below) of the fixed point of a four-dimensional conservative map. In addition to the usual period doubling (‘‘wavelength doubling’’ in our terminology) observed in case of two-dimensional maps, the four-dimensional map is seen to exhibit a new type of branching, namely, giving rise to an invariant curve in phase space. In case of subthreshold branching of either type, the steady homogeneous solution bifurcates to either a ‘‘wavelength-two’’ or a quasiperiodic solution. In case of superthreshold branching, on the other hand, bifurcations involving more complicated spatial and temporal behavior are possible. Other relevant questions are dealt with.

Classical aspects of the laser excitation of a Morse oscillator

A Morse oscillator in an intense laser field is studied classically. A new approach to solving the classical equations of motion, based on construction of a map from initial conditions to final conditions after one period of oscillation of the laser, is given. The evolution of phase-space curves over many periods is then examined by repeatedly applying the map. Phase-space curves, as a function of time, are seen to exhibit some very interesting behaviour, including spirals or “whorls”. The pendulum model of non-linear resonance is shown to describe this behaviour reasonably well and, in fact, can be used to crudely interpret dynamical trends, such as the average energy absorption as a function of initial vibrational state.

Conserved noether currents and path intergrals for dynamical groups

We consider path integrals for the dynamical group SU(1,1) based on its associated coherent states. For relevant systems the classical dynamics is oscillator-like in a curved phase space. The classical lagrangian exhibits an “internal” symmetry whose conserved Noether currents are in the SU(1,1) Lie algebra.

Path integral for coherent states of the dynamical group SU(1,1)

Path integrals over coherent states of the dynamical group (noninvariance group) SU(1,1) are constructed. From the continuous limit the relevant classical dynamics is extracted and is shown to take place in a curved phase space of the form of a Lobachevskii plane. Applications are made to the harmonic oscillator, a model of superfluid helium, the Morse oscillator, and the hydrogen atom. It is shown that when SU(1,1) is the relevant dynamical group the motion will appear oscillator-like on the Lobachevskii plane.

Extended Dirac particles and their spectra in curved phase space

Extended Dirac particles and their spectra in curved phase space

A semiclassical treatment of path integrals for the spin system

Starting with path integrals in the SU(2) coherent state representation, the semiclassical approximation of the propagator for the spin system is investigated. By extending the idea of the semiclassical expansion method, which was developed in the usual phase-space path integrals, to the path integrals in the curved phase space, which is characteristic of the SU(2) coherent states, we obtain a closed form for the semiclassical propagator. As an application, we discuss the semiclassical quantization condition for the spin system.

Semiclassical quantization of a spin system by coherent state path integrals

By using the path integral in the SU(2) coherent state representation, we investigate the semiclassical quantization for an interacting spin system. We arrive at the Bohr-Sommerfeld like quantization rule in curved phase space.

Extended particles and their spectra in curved phase space

Extended particles and their spectra in curved phase space

Path integral in the representation of SU(2) coherent state and classical dynamics in a generalized phase space

Path integral in the representation of coherent state for the simplest semisimple Lie group SU(2) and its classical consequences are investigated. Using the completeness relation of the coherent state, we derive a path integral expression for the transition amplitude which connects a pair of SU(2) coherent states. In the classical limit we arrive at a canonical equation of motion in a ’’curved phase space’’ (two-dimensional sphere) which reproduces the ordinary Euler’s equation of a rigid body when applied to a rotator.

Models of Gross-Neveu type are quantization of a classical mechanics with nonlinear phase space

It is shown that the expansion in powers of 1/N characteristic of Gross-Neveu type models is of quasiclassical nature, 1/N taking part of the Planck constant. The limity classical mechanics has curved phase space that does not admit introduction of naturally canonically conjugated coordinates.

Lagrangian for Diffusion in Curved Phase Space.

A Lagrangian is obtained by deriving the path-integral representation of the diffusion process. It can be applied, e.g., to nonequilibrium thermodynamics and to quantized motion in general relativity. In second quantization it is shown to lead to a particularly well-behaved energy-momentum tensor as a source of gravity.

Lagrangian for Diffusion in Curved Phase Space

A Lagrangian is obtained by deriving the path-integral representation of the diffusion process. It can be applied, e.g., to nonequilibrium thermodynamics and to quantized motion in general relativity. In second quantization it is shown to lead to a particularly well-behaved energy-momentum tensor as a source of gravity.

Age structure effects in predator-prey interactions

A mathematical model of predator-prey interactions is proposed which incorporates both age structure in the predators and density dependence in the prey. The properties of the model are investigated by a linearized analysis, which enables the conditions for stability to be formulated. The analysis indicates that for a substantial domain of parameter space, a stable equilibrium is possible with the prey well below its carrying capacity. The effect of violating the stability conditions on the behaviour of the model was investigated by computer simulation. Two further types of behaviour are noted in which coexistence is possible. The first is a two point limit cycle in which young and old predators occur in alternate time periods. The second involves a limit cycle in which the component population trajectories lie on closed curves in phase space.

The Unitary Group U(m) as the Symmetry Group of Curved Phase Space

By describing U(m) as the symmetry group of a curved phase space, namely, by describing U(m) as the group of holonomy of a Hamiltonian space endowed with a Riemannian metric, it is shown that the Lie algebra of such a group is prior to any explicit Hamiltonian assumption.

Experimental Study of Double-Charged Pion Production in (π−,p) Collisions at 900 MeV

Eight hundred and forty events of the kind ${\ensuremath{\pi}}^{\ensuremath{-}}+p\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}+p+{\ensuremath{\pi}}^{\ensuremath{-}}+{\ensuremath{\pi}}^{+}$ produced in the 20-in. BNL hydrogen bubble chamber by 900-MeV pions have been unambiguously identified using spatial reconstruction and kinematic fitting programs as well as ionization density estimates. The ${\ensuremath{\pi}}^{+}$,${\ensuremath{\pi}}^{\ensuremath{-}}$ and ${\ensuremath{\pi}}^{\ensuremath{-}}$,${\ensuremath{\pi}}^{\ensuremath{-}}$ combined mass distributions can be fitted by smooth curves, with no deviation beyond statistical fluctuations; no indication has been found of any prominent pion-pion resonance in this interaction, which covers a mass range up to 610 MeV. The ${\ensuremath{\pi}}^{+}$,$p$ combined mass distribution differs markedly from the four-body phase-space curve, but can be well fitted by weighting the ${\ensuremath{\pi}}^{+}$,$p$ total cross-section curve at each point according to the amount of phase space available for production of an isobar of corresponding mass. Assuming that the interaction proceeds exclusively via formation of the (${\ensuremath{\pi}}^{+},p$) isobar, one can get a good fit to the ${\ensuremath{\pi}}^{\ensuremath{-}}$,${\ensuremath{\pi}}^{\ensuremath{-}}$ mass spectrum. This isobar model is also consistent with all of the observed angle and momentum distributions for both pions and protons. The momentum distributions show no indication of any pion-pion-proton resonance in the range up to 1550 MeV, or of any three-pion resonance in the range up to 750 MeV. The cross section for the events studied was measured and found to be (0.33\ifmmode\pm\else\textpm\fi{}0.04) mb.