Relativistic Phase Space Research Articles

Frames, the -duality in Minkowski space and spin coherent states

In the spirit of some earlier work on building coherent states for the PoincarA© group in one space and one time dimension, we construct here analogous families of states for the full PoincarA© group, for representations corresponding to mass m > 0 and arbitrary integral or half-integral spin. Each family of coherent states is defined by an affine section in the group and constitutes a frame. The sections, in their turn, are determined by particular velocity vector fields, the latter always appearing in dual pairs. Geometrically, each family of coherent states is related to the choice of a Riemannian structure on the forward mass hyperboloid or, equivalently, to the choice of a certain parallel bundle in the relativistic phase space. The large variety of coherent states obtained tempts us to believe that there is rich scope here for application to spin-dependent problems in atomic and nuclear physics, as well as to image reconstruction problems, using the discretized versions of these frames. © 1996 IOP Publishing Ltd.

Reply to "Comment on 'Nonlinear resonance and chaos in the relativistic phase space for driven nonlinear systems' "

As a Reply to the Comment by Luchinsky et al. [preceding paper, Phys. Rev. E 53, 4240 (1996)] on the paper by Kim and Lee [Phys. Rev. E 52, 473 (1995)], we point out that the formation of zero-dispersion nonlinear resonance can always be observed in a relativistic hard oscillator driven by a periodic force. We also report results of a computation of resonance energies for a driven relativistic Duffing oscillator using the formula suggested by Luchinsky et al. \textcopyright{} 1996 The American Physical Society.

Comment on "Nonlinear resonance and chaos in the relativistic phase space for driven nonlinear systems"

Kim and Lee (Phys. Rev. E 52, 473; 1995) report relativity-induced resonances in periodically driven oscillators. We comment that zero-dispersion nonlinear resonance (ZDNR) will occur in some of the systems considered, outline the physical origins of the ZDNR, and propose an explanation of a discrepancy noted by Kim and Lee between their theoretical and numerical values of the energy at the stationary stable points of Poincare sections.

Nonlinear resonance and chaos in the relativistic phase space for driven nonlinear systems.

We investigate the relativistic dynamics of a time-driven nonlinear system by analyzing the resonance structure in the relativistic phase space. It is shown that, when relativistic effects become appreciable, resonances that exist in the nonrelativistic description may be shifted or suppressed and resonances that are absent in the nonrelativistic description may be induced to appear. When these relativity-induced resonances overlap, the system exhibits chaotic relativistic motion. Numerical data that demonstrate such relativistic effects are presented with model systems of the driven Duffing oscillator and the driven Morse oscillator.

Octopus: An efficient phase space mapping for light particles

I present a generator for relativistic phase space that incorporates much of the effect of typical experimental cuts, and which is suitable for use in Monte Carlo calculations of cross sections for high-energy hadron-hadron or electron-positron scattering experiments.

CONFORMAL THEORIES, CURVED PHASE SPACES, RELATIVISTIC WAVELETS AND THE GEOMETRY OF COMPLEX DOMAINS

We investigate some aspects of complex geometry in relation with possible applications to quantization, relativistic phase spaces, conformal field theories, general relativity and the music of two and three-dimensional spheres.

A GAUGE MODEL DESCRIBING N RELATIVISTIC PARTICLES BOUND BY LINEAR FORCES

A relativistic model of N particles bound by linear forces is obtained by applying the gauging procedure to the linear canonical symmetries of a simple (rudimentary) nonrelativistic N-particle Lagrangian extended to relativistic phase space. The new (gauged) Lagrangian is formally Poincaré invariant, the Hamiltonian is a linear combination of firstclass constraints which are closed with respect to Poisson brackets and generate the localized canonical symmetries. The "gauge potentials" appear as the Lagrange multipliers of the constraints. Gauge fixing and quantization of the model are also briefly discussed.

Closing and abelizing operatorial gauge algebra generated by first class constraints

A canonical transformation of operators of relativistic phase space is constructed, which abelizes the basis of the operator-valued gauge algebra of first class constraints. The new operators of contraints commute among themselves. The new Hamiltonian commutes with the constraints. Dynamics of the new operators is physically equivalent to the initial one.

Quantization as geometry in phase space

The geometrical interpretation of quantum mechanics in relativistic phase space proposed by this writer leads, under the sole assumption of a connection from which commutation rules between covariant derivatives ensue that reproduce Heisenberg’s, to an 8-dimensional metric space which contains and generalizesall of first quantization. Some of the new results (to be tested) are: Existence of a maximal proper acceleration; The reality condition on geodesic lines is identical with Bohr-Sommerfeld quantization; The Klein-Gordon and Dirac equations thus generalized give positive mass spectra lying on Regge trajectories; There exists (only in 8 dimensions) a natural supersymmetry (Cartan’s triality); Pure spinors and octonions appear as natural tools for describing particles, and for deeper analyses.

Consistent formulation of relativistic dynamics for massive spin-zero particles in external fields

It is shown that in the context of stochastic-phase-space formulations of quantum mechanics a nonlocal type of nonrelativistic as well as relativistic dynamics for a massive spin-zero particle moving in an external electromagnetic field can be formulated in ${L}^{2}(\ensuremath{\Gamma})$ space. In a sharp-point limit $s\ensuremath{\rightarrow}+0$ the nonrelativistic dynamics merges into the conventional local model. The relativistic dynamics is gauge invariant, it can be formulated covariantly in the relativistic phase space ${M}_{m}$, it possesses a covariant propagator, it leaves the space of positive-energy solutions of the Klein-Gordon equation invariant, and it gives rise to a covariant and conserved probability current on stochastic phase space. Furthermore, for states describing a particle moving at nonrelativistic speeds in relation to a given inertial frame, the relativistic propagation contracts into its nonrelativistic counterpart. The sharp-point limit $s\ensuremath{\rightarrow}+0$ of the relativistic $S$ matrix also exists in each such inertial frame. The necessity of interactions acting at nonsharp configuration points for a consistent formulation of covariant relativistic one-particle dynamics is traced to the question of the nonexistence of covariant position operators in relativistic theories. The mandatory character of this feature is shown to be firmly rooted in the theory of quantum measurement.

Algorithm for the numerical evaluation of the n-particle relativistic phase space integral in invariant variables

A previously derived analytical formulation of the n-particle relativistic momentum phase space integral in invariant variables is simplified. The final equations are eminently suited to machine evaluation and have been incorporated into a Monte Carlo computer program (described elsewhere). The main new feature of this work is the derivation of a set of recurrence relations that allows for the rapid evaluation of dependent invariants in terms of independent ones even in the exceptional regions of the physical region.

Recursion relation for the relativistic quantum phase space

Recursion relation for the relativistic quantum phase space

The role of the symmetric group in the cluster decomposition of the relativistic phase space

The role of the symmetric group in the cluster decomposition of the relativistic phase space

Ideal relativistic quantum gas and invariant phase space

The density of states of an ideal relativistic gas with Bose-Einstein or Fermi-Dirac statistics is written in closed form as a cluster decomposition over the usual phase-space integrals with Boltzmann statistics. We stress the difference between invariant phase space and invariant momentum space; only the former gives the correct description of ideal gases. Our results are formulated for the microcanonical, grand microcanonical, canonical and grand canonical ensembles in invariant phase space and invariant momentum space. A whole section is devoted to a numerical study, which reveals that while under normal conditions thermodynamical quantities (derived from the logarithm of the density of states) acquire only very small corrections, the densities (phase-space integrals) themselves are significantly changed. This holds in particular for elementary particle applications (statistical models, phase-space considerations) where the effects on phase space of Bose-Einstein and/or Fermi-Dirac statistics can almost never be neglected; however, in these applications, the number of equal particles is so low that the cluster decompositions can be explicitly evaluated.

Phase-space symmetries of a relativistic plasma

Relativistic plasmas phase space symmetries in Minkowski space, noting variance of plasma interaction with electromagnetic field

Lorentz-invariant Markov processes in relativistic phase space

This paper deals with certain “random motions” permitted by the special theory of relativity; that is, with probability measures on sets of trajectories on which speeds are less than or equal to the speed c of light (which we take equal to 1 throughout). We deal with classes of such measures indexed by possible initial states, related to each other by the Lorentz group (implying certain invariance conditions for individual measures).

Connexion between Local and Global Physics

IF relativistic quantum mechanics can be founded on the ‘principle of triality’ as I have claimed elsewhere1,2 then this ought to be reflected in the global structure of physics. Locally, triality means the in variance of a form trilinear in phase functions and two kinds of wave functions. Globally the latter are replaced by the expectation values of phase functions and globally defined fields. So the question becomes, what kind of triality is possible in the relativistic phase space E8, considered as a quasi-classical object ?

Choice of invariant variables and analytical properties of many-point functions

A previously derived analytical formulation of the n-particle relativistic momentum phase space integral in invariant variables is simplified. The final equations are eminently suited to machine evaluation and have been incorporated into a Monte Carlo computer program (described elsewhere). The main new feature of this work is the derivation of a set of recurrence relations that allows for the rapid evaluation of dependent invariants in terms of independent ones even in the exceptional regions of the physical region.

The pion-pion interaction in τ decay

The momentum dependence of the τ-decay rate deviates considerably from that predicted by the relativistic phase space factor and Coulomb corrections. The difference is attributed here to the final state pion pion interaction. Three different phenomenological analyses are made to determine the T = 0 and T = 2 s-state pion-pion force required for consistency with τ and τ′ data: a scattering length approximation, an independent pair approximation for an exponential potential, and a Born approximation for a Yukawa potential. The results of all three approximations agree where they are applicable and indicate a weak or repulsive T = 0 force and an attractive T = 2 force.