Zero-mass Particles Research Articles

The Use of Zero-Mass Particles in Analytical and Multi-Body Dynamics: Sphere Rolling on An Arbitrary Surface

Abstract Zero-mass particles are, as a rule, never used in analytical dynamics, because they lead to singular mass matrices. However, recent advances in the development of the explicit equations of motion of constrained mechanical systems with singular mass matrices permit their use under certain circumstances. This paper shows that the use of such particles can be very efficacious in some problems in analytical dynamics that have resisted easy, general formulations, and in obtaining the equations of motion for complex multi-body systems. We explore the ease and simplicity that suitably used zero-mass particles can provide in formulating and simulating the equations of motion of a rigid, non-homogeneous sphere rolling under gravity, without slipping, on an arbitrarily prescribed surface. Computational results comparing the significant difference in the motion of a homogeneous sphere and a non-homogeneous sphere rolling down an asymmetric arbitrarily prescribed surface are obtained, along with measures of the accuracy of the computations. While the paper shows the usefulness of zero-mass particles applied to the classical problem of a rolling sphere, the development given is described in a general enough manner to be applicable to numerous other problems in analytical and multi-body dynamics that may have much greater complexity.

Towards Science Unification through Number Theory

The Number Theory comes back as the heart of unified Science, in a Computing Cosmos using the bases 2;3;5;7 whose two symmetric combinations explain the main lepton mass ratios. The corresponding Holic Principle induces a symmetry between the Newton and Planck constants which confirm the Permanent Sweeping Holography Bang Cosmology, with invariant baryon density 3/10, the dark baryons being dephased matter-antimatter oscillation. This implies the DNA bi-codon mean isotopic mass, confirming to 0.1 ppm the electron-based Topological Axis, whose terminal boson is the base 2 c-observable Universe in the base 3 Cosmos. The physical parameters involve the Euler idoneal numbers and the special Fermat primes of Wieferich (bases 2) and Mirimanoff (base 3). The prime numbers and crystallographic symmetries are related to the 4-fold structure of the DNA bi-codon. The forgotten Eddington’s proton-tau symmetry is rehabilitated, renewing the supersymmetry quest. This excludes the concepts of Multiverse, Continuum, Infinity, Locality and Zero-mass Particle, leading to stringent predictions in Cosmology, Particle Physics and Biology.

Unified Theory of Everything and QQD

The fully Unified Theory of {Strong, EM, Weak, Gravity} interactions all arising from the unique source of electronic charges has been established. The theory proves that the Nature is astutely simple in essence, and the most of the recently promoted circuitous physics seem to be incorrect in the new light. In the proposed theory, the 100% of matter and energy content of the Universe, including the dark matter and energy, is predicted in terms of a single integer parameter L in one-to-one agreement with the observation. To boot, the {Z, W} bosons are proven to be compound states with their innate heavy masses; they never have been zero-mass particles, and the Higgs boson is uncalled for. The unified theory also has explicated the ever mysterious strong interaction number “137”, and leads to QQD (Quantum Quark Dynamics) parallel to the QED. The QQD gives, for example, very simple explanation of “Asymptotic Freedom” without requiring the unwarranted and arduously complex QCD.

Gerald Stanford Guralnik

Gerald Stanford Guralnik

Einstein is Newton with Space Curved

The two defining features of the Einsteinian gravity are that it is self-interactive and it links universally to all particles, including zero-mass particles. In the process of obtaining the Schwarzschild solution for gravitational field of a mass point, we demonstrate how these features are incorporated? We also show, unlike the Newtonian gravity, why gravitational potential in the Schwarzschild solution can have its zero only at infinity and nowhere else, i.e. it is determined absolutely. Further we consider particle orbits to expose certain insightful and subtle points of concept and principle.

GAUGE INVARIANCE AND THE GOLDSTONE THEOREM

This paper was originally created for and printed in the "Proceedings of seminar on unified theories of elementary particles" held in Feldafing, Germany from July 5 to 16, 1965 under the auspices of the Max-Planck-Institute for Physics and Astrophysics in Munich. It details and expands upon the 1964 Guralnik, Hagen, and Kibble paper demonstrating that the Goldstone theorem does not require physical zero mass particles in gauge theories.

Asymptotic behavior of solutions of the fragmentation equation with shattering: An approach via self-similar Markov processes

The subject of this paper is a fragmentation equation with nonconservative solutions, some mass being lost to a dust of zero-mass particles as a consequence of an intensive splitting. Under some assumptions of regular variation on the fragmentation rate, we describe the large time behavior of solutions. Our approach is based on probabilistic tools: the solutions to the fragmentation equation are constructed via nonincreasing self-similar Markov processes that continuously reach 0 in finite time. Our main probabilistic result describes the asymptotic behavior of these processes conditioned on nonextinction and is then used for the solutions to the fragmentation equation. We note that two parameters significantly influence these large time behaviors: the rate of formation of "nearly-1 relative masses" (this rate is related to the behavior near 0 of the L\'evy measure associated with the corresponding self-similar Markov process) and the distribution of large initial particles. Correctly rescaled, the solutions then converge to a nontrivial limit which is related to the quasi-stationary solutions of the equation. Besides, these quasi-stationary solutions, or, equivalently, the quasi-stationary distributions of the self-similar Markov processes, are fully described.

Simple five-dimensional wave equation for a Dirac particle

A first-order relativistic wave equation is constructed in five dimensions. Its solutions are eight-component spinors, interpreted as single-particle fermion wave functions in four-dimensional space-time. Use of a “cylinder condition” (the removal of explicit dependence on the fifth coordinate) reduces each eight-component solution to a pair of degenerate four-component spinors. It is shown that, when the cylinder condition is applied, the results obtained from the new equation are the same as those obtained from the Dirac equation. Without the cylinder condition, on the other hand, the equation implies the existence of a scalar potential, and for zero-mass particles it leads to a four-dimensional fermionic equation analogous to Maxwell’s equation with sources.

Quantum walks and orbital states of a Weyl particle

The time-evolution equation of a one-dimensional quantum walker is exactly mapped to the three-dimensional Weyl equation for a zero-mass particle with spin $1∕2$, in which each wave number $k$ of the walker's wave function is mapped to a point $\mathbf{q}(k)$ in the three-dimensional momentum space and $\mathbf{q}(k)$ makes a planar orbit as $k$ changes its value in $[\ensuremath{-}\ensuremath{\pi},\ensuremath{\pi})$. The integration over $k$ providing the real-space wave function for a quantum walker corresponds to considering an orbital state of a Weyl particle, which is defined as a superposition (curvilinear integration) of the energy-momentum eigenstates of a free Weyl equation along the orbit. Konno's novel distribution function of a quantum walker's pseudovelocities in the long-time limit is fully controlled by the shape of the orbit and how the orbit is embedded in the three-dimensional momentum space. The family of orbital states can be regarded as a geometrical representation of the unitary group U(2) and the present study will propose a new group-theoretical point of view for quantum-walk problems.

Restricted N+1-Body Problem: Existence and Stability Of Relative Equilibria

We consider n bodies (with equal mass m) disposed at the vertices of a regular n-gon and rotating rigidly around an additional mass m 0(at its center) with a constant angular velocity (relative equilibrium). In the present paper, we prove results on the existence and on the linear stability of equilibrium positions for a zero-mass particle submitted to the gravitational field generated by the previous system.

Universality of low-energy scattering in 2+1 dimensions

For any relativistic quantum field theory in 2+1 dimensions, with no zero mass particles, and satisfying the standard axioms, we establish a remarkable low-energy theorem. The $S$-wave phase shift, ${\ensuremath{\delta}}_{0}(k),$ $k$ being the c.m. momentum, vanishes as either ${\ensuremath{\delta}}_{0}\ensuremath{\rightarrow}c/\mathrm{ln}(k/m)$or ${\ensuremath{\delta}}_{0}\ensuremath{\rightarrow}{O(k}^{2})$ as $\stackrel{\ensuremath{\rightarrow}}{k}0.$ The constant $c$ is universal and $c=\ensuremath{\pi}/2.$ This result follows only from the rigorously established analyticity and unitarity properties for 2-particle scattering. This kind of universality was first noted in non-relativistic potential scattering, albeit with an incomplete proof which missed, among other things, an exceptional class of potentials where ${\ensuremath{\delta}}_{0}(k)$ is ${O(k}^{2})$ near $k=0.$ We treat the potential scattering case with full generality and rigor, and explicitly define the exceptional class. Finally, we look at perturbation theory in ${\ensuremath{\varphi}}_{3}^{4}$ and study its relation to our non-perturbative result. The remarkable fact here is that in $n$-th order the perturbative amplitude diverges like $(\mathrm{ln}{k)}^{n}$ as $\stackrel{\ensuremath{\rightarrow}}{k}0,$ while the full amplitude vanishes as $(\mathrm{ln}{k)}^{\ensuremath{-}1}.$ We show how these two facts can be reconciled.

Symbolic calculus on the nilpotent orbits of SO 0(1,2)

The coadjoint conical orbits in so(1,2) ∗ ⋍ su(1,1) ∗ are the phase spaces of the zero mass particles on the two-dimensional (anti-)de Sitter space-time. It contains also, as an open dense subset, the phase space for massless particles on one-dimensional Minkowski space-time when one identifies the Poincaré group to a subgroup of the conformal group SO 0(2,2)⋍SO 0(1,2)×SO 0(1,2). On the other hand, the quantum representation associated to these systems is an indecomposable extension of the first term of the discrete series of representations of SO 0(1,2) ⋍ SU(1,1)/ Z 2. We present in this paper a symbol map linking this representation and this orbit. This calculus is invariant and behaves correctly in the classical limit. As a result we have obtained a conformally invariant symbolic calculus for massless particles on (anti-)de Sitter or Minkowski space-time in 1+1 dimension.

Quantization of the nilpotent orbits in so(1,2)* and massless particles on (anti-)de Sitter space–time

The nontrivial nilpotent orbits in so(1,2)*≂su(1,1)* are the phase spaces of the zero mass particles on the two-dimensional (anti-)de Sitter space–time. As is well known, the lack of global hyperbolicity (respectively, stationarity) for the anti-de Sitter (respectively, de Sitter) space–time implies that the canonical field quantization of its free massless field is not uniquely defined. One might nevertheless hope to get the one-particle quantum theory directly from an appropriate ‘‘first’’ quantization of the classical phase space. Unfortunately, geometric quantization (the orbit method) does not apply to the above orbits and a naive canonical quantization does not yield the correct result. To resolve these difficulties, we present a simple geometric construction that associates to them an indecomposable representation of SO0(1,2) on a positive semidefinite inner product space. It is shown that quotienting out its one-dimensional invariant subspace yields the first term of the holomorphic discrete series of representations of SO0(1,2)≂SU(1,1)/Z2. We interpret these results physically by showing that the above positive semidefinite inner product space is naturally isomorphic to a space of solutions of the conformally invariant zero-mass Klein–Gordon equation on the (anti)-de Sitter space–time, equipped with the usual Klein–Gordon inner product, obtained by integrating over a suitable spacelike hypersurface. As such, our version of geometric quantization selects in a natural way a quantization of the free massless particle. We show it is conformally [i.e., SO0(2,2)] invariant and behaves correctly in the classical limit.

Dark matter around binary galaxies: formation of collimated flows

We study here the dynamics of an extended shell of relatively low-mass (almost zero-mass) particles around massive binary systems by computer simulations in the framework of approximately restricted three-body problem with a set of several initial conditions concerning the massesM 1 andM 2 of the binary components surrounded byN test particles in uniform random distribution on a spherical envelope of radiusR expanding with a velocityV. We apply this model to binary galaxy systems with a halo of baryonic dark matter, e.g., massive black holes, globular clusters, and giant molecular clouds. It is shown that, initially, the shell expands homologously with decreasing velocity and then, falls back into the system forming zones of compressed matter. At some moment there could be a collapse or these particles onto the heavier component of the binary. Further in time, a number of particles escape from the system. We consider a number of different models with different initial parameters. For models with smallerR andV, about one-half of the particles escape from the system; while for larger values the shell disrupts as a whole. The escaping particles form a collimated flow in the plane of the orbit of the binary. The position of the flow and the directions of motions depend on the position of the heavier component of the binary at the moment of the closest approach of the particles and on the ratioM 1/M 2.

The model-independent approach to quantum gravity: implications of asymptotic flatness

The action for the gravitational field in the in-vacuum state is expanded on the basis of nonlocal curvature invariants with arbitrary form factors. It is shown that, for the solution of the expectation-value equations to be asymptotically flat, the form factors should have precisely one-loop behaviour at small ◊. The coefficients of these behaviours measure the rate of the energy radiation through the future null infinity and, at the same time, determine the spectral weights of zero-mass particles in the vacuum. The flux of the vacuum radiation through the future null infinity is calculated to lowest order in the curvature without any assumption about the quantum dynamical model.

Monte Carlo phase space evaluation

A new, efficient method of generating Monte Carlo phase space integrals for multi-particle systems is derived. By using the analytically exact integrals for zero-mass particles as a comparison function, we generate the integrals for massive particles with an extremely high efficiency. We thus can generate unit weight events, simulating actual experimental data. The events are generated in two distinct steps. First, effective masses, conserving energy and momentum are generated. Next, the effective masses are allowed to sequentially “decay” via two-body models, and the decay products are then given an angular distribution, in their own center-of-mass system. The resulting two momenta are then Lorentz transformed into the appropriate laboratory system. Our method is contrasted to traditional methods of generating similar events, where the “dice” are thrown “flat” in effective mass, and comparisons of efficiency are made. For large numbers of particles and very high energies, the improvement is enormous. In addition, new methods of generating distribution functions are introduced.

General solutions and scaling violation for fragmentation with mass loss

Explicit solutions of a linear rate equation lead to an improved understanding of fragmentation with discrete and continuous mass loss. Discrete mass loss provides a general expression for the overall mass loss rate which accounts for mass loss in the shattering regime, where runaway fragmentation for small particles produces a phase of zero-mass particles. An explicit solution for the recession regime, where small particles lose large fractions of their mass to surface recession, shows that the total number of particles increases with time until the typical particles become small enough to lose all of their mass (and disappear) rather than to break. A general series solution is presented for fragmentation with continuous and discrete mass loss. A proof that continuous mass loss precludes dynamic scaling is presented.

Exact and asymptotic scaling solutions for fragmentation with mass loss.

Exact and asymptotic solutions to a linear rate equation for fragmentation with mass loss are presented. Solutions for spatially discrete random bond annihilation illustrate the mutual exclusiveness of the fragmentation and recession terms in the rate equation. Exact solutions for deterministic equal fragment recession show that continuous mass loss between fragmentation events can be approximated by discrete mass loss during fragmentation events when this mass loss is small. Evidence that continuous and discrete mass loss preclude shattering mass loss, the loss of mass to zero-mass particles due to runaway fragmentation, is presented. General asymptotic scaling forms, general solutions reflecting arbitrary initial conditions, and explicit scaling solutions are derived.

Kinetics of fragmentation

A general discussion of the kinetics of continuous, irreversible fragmentation processes is presented. For a linear process, where particle breakup is driven by an external force, the authors develop a scaling theory to describe the evolution of the cluster size distribution. They treat the general case where the breakup rate of a cluster of mass x varies as xlambda . When lambda >0, corresponding to larger clusters more likely to break up, the scaled cluster size distribution, phi (x), decays with the scaled mass, x, as x-2 exp(-axlambda ), as x. For small mass, phi (x) has the log-normal form, exp(-a ln2 x), if the breakup kernel has a small-size cutoff, while phi (x) has a power-law tail in the absence of a cutoff. They also show that a conventional scaling picture applies only for the case lambda >0. For lambda >0, they develop an alternative formulation for the cluster size distribution, in which the typical mass scale is determined by the initial condition. In this regime, they also investigate the nature of a 'shattering' transition, where mass is lost to a 'dust' phase of zero-mass particles. They also study the kinetics of a nonlinear, collision-induced fragmentation process. They analyse the asymptotic behaviour of a simple-minded class of models in which a two-particle collision results in either: (1) both particles splitting into two equal pieces, (2) only the larger particle splitting in two, or (3) only the smaller particle splitting. They map out the kinetics of these models by scaling arguments and by analytic and numerical solutions of the rate equations. Scaling is found to hold for different ranges of homogeneity index for the three models.

Wigner’s little group and decomposition of Lorentz transformations

It is shown how an arbitrary Lorentz transformation can be expressed in terms of elements of Wigner’s little group and its cosets. This yields a natural parametrization for the little group, while its coset members turn out to be helicity-preserving transformations. The associated Wigner angle and its relation to the actual change in helicity are discussed. Finally, the extension to zero-mass particles shows how the little group becomes a gauge transformation in that limit.