Higher-Order Kinetic Term for Controlling Photon Mass in Off-Shell Electrodynamics

In relativistic classical and quantum mechanics with Poincaré-invariant parameter, particle worldlines are traced out by the evolution of spacetime events. The formulation of a covariant canonical framework for the evolving events leads to a dynamical theory in which mass conservation is demoted from a priori constraint to the status of conserved Noether current for a certain class of interactions. In pre-Maxwell electrodynamics—the local gauge theory associated with this framework —events induce five local off-shell fields, which mediate interactions between instantaneous events, not between the worldlines which represent entire particle histories. The fifth field, required to compensate for dependence of gauge transformations on the evolution parameter, enables the exchange of mass between particles and fields. In the equilibrium limit, these pre-Maxwell fields are pushed onto the zero-mass shell, but during interactions there is no mechanism regulating the mass that photons may acquire, even when event trajectories evolve far into the spacelike region. This feature of the off-shell formalism requires the application of some ad hoc mechanism for controlling the photon mass in two opposite physical domains: the low energy motion of a charged event in classical Coulomb scattering, and the renormalization of off-shell quantum electrodynamics. In this paper, we discuss a nonlocal, higher derivative correction to the photon kinetic term, which provides regulation of the photon mass in a manner which preserves the gauge invariance and Poincaré covariance of the original theory. We demonstrate that the inclusion of this term is equivalent to an earlier solution to the classical Coulomb problem, and that the resulting quantum field theory is renormalized.     

Spin Physics and the Theory of Strong Interactions

Taking the Minkowski space as the scene of quantum field theory implies an implicit assumption: the spin plays no dynamical role. This assumption (already challenged by reggeism) should be re-examined in the light of recent advances of experimental spin physics. To make a dynamical role of spin possible, it is proposed to use as a scene for the theory of strong interactions the whole Poincaré group. On the other hand characteristic functions, defined on the Poincare group, provide the only known way to describe the distinctive peculiarity of resonances, namely that a resonance is both one particle and several particles. Regge trajectories are interpreted as evidence for a mass-spin correlation among the virtual hadrons of vacuum. It is conjectured that the form of this correlation is deducible from a central limit theorem on the Poincare group. And that a statistical mechanics of hadronic vacuum is important for strong interaction theory.     

Space-time theories and symmetry groups

This paper addresses the significance of the general class of diffeomorphisms in the theory of general relativity as opposed to the Poincaré group in a special relativistic theory. Using Anderson's concept of an absolute object for a theory, with suitable revisions, it is shown that the general group of local diffeomorphisms is associated with the theory of general relativity as its local dynamical symmetry group, while the Poincaré group is associated with a special relativistic theory as both its global dynamical symmetry group and its geometrical symmetry group. It is argued that the two groups are of equal significance as symmetry groups of their associated theories.     

Particle dynamics on Snyder space

We examine Hamiltonian formalism on Euclidean Snyder space. The latter corresponds to a lattice in the quantum theory. For any given dynamical system, it may not be possible to identify time with a real number parametrizing the evolution in the quantum theory. The alternative requires the introduction of a dynamical time operator. We obtain the dynamical time operator for the relativistic (nonrelativistic) particle, and use it to construct the generators of Poincaré (Galilei) group on Snyder space.     

Space-time and degrees of freedom of the elementary particle

First, a general property of Lie groups is used in the case of the Poincaré group in order to define the one particle phase space. It is eight-dimensional in the general case and six-dimensional for a spinless or massless particle.Embedding the Poincaré group into the similitude group of space-time permits us to interpret the dilatation operator as a dynamical variable. The connection between the similitude group and field equations is discussed. Lurçat's ideas on a possible dynamical role of spin and mass-spin spectra of particles (Regge trajectories) are discussed under the point of view of the degrees of freedom.This work constitutes a completed version of a preprint entitled Classical Hamiltonian Formalism for Spin, Argonne, September, 1966.On leave from Université de Marseille, France. Work supported in part by the National Science Foundation.     

Supergravity in (2 + 1) dimensionsfrom (3 + 1)-dimensional supergravity

In the context of the formalism proposed by Stelle-West and Grignani-Nardelli, it is shown that Chern-Simons supergravity can be consistently obtained as a dimensional reduction of (3 + 1)-dimensional supergravity, when written as a gauge theory of the Poincaré group. The dimensional reductions are consistent with the gauge symmetries, mapping (3 + 1)-dimensional Poincaré supergroup gauge transformations onto (2 + 1)-dimensional Poincaré supergroup ones.     

Cosmological constant and the dynamics of the Bianchi type I models with spin and torsion: A qualitative investigation

By including the cosmological term in the minimum quadratic (Poincaré) gauge theory of gravity, the basic equation set for the homogeneous anisotropic Bianchi type I spinning-fluid models are obtained. For the linear equation of state, using methods of qualitative theory of dynamical systems, we make the complete qualitative analysis of properties of every possible solution of these equations, In particular, some solutions with regular behaviour of the metric and torsion are found.     

Fields on the Poincaré Group: Arbitrary Spin Description and Relativistic Wave Equations

In this paper, starting from a pure group-theoretic point of view, we develop an approach to describing particles with different spins in the framework of a theory of scalar fields on the Poincaré group. Such fields can be considered as generating functions for conventional spin-tensor fields. The case of two, three, and four dimensions are elaborated in detail. Discrete transformations C, P, T are defined for the scalar fields as automorphisms of the Poincaré group. We classify the scalar functions, and obtain relativistic wave equations for particles with definite spin and mass. There exist two different types of scalar functions (which describe the same mass and spin), one related to a finite-dimensional nonunitary representation and the other to an infinite-dimensional unitary representation of the Lorentz subgroup. This allows us to derive both usual finite-component wave equations for spin-tensor fields and positive-energy, infinite-component wave equations.     

Quantum mechanics and the direction of time

In recent papers the authors have discussed the dynamical properties of large Poincaré systems (LPS), that is, nonintegrable systems with a continuous spectrum (both classical and quantum). An interesting example of LPS is given by the Friedrichs model of field theory. As is well known, perturbation methods analytic in the coupling constant diverge because of resonant denominators. We show that this Poincaré catastrophe can be eliminated by a natural time ordering of the dynamical states. We obtain then a dynamical theory which incorporates a privileged direction of time (and therefore the second law of thermodynamics). However, it is only in very simple situations that this time ordering can be performed in an extended Hilbert space. In general, we need to go to the Liouville space (superspace) and introduce a time ordering of dynamical states according to the number of particles involved in correlations. This leads then to a generalization of quantum mechanics in which the usual Heisenberg's eigenvalue problem is replaced by a complex eigenvalue problem in the Liouville space.     

Perturbed Euler top and bifurcation of limit cycles on invariant Casimir surfaces

Analytical perturbations of the Euler top are considered. The perturbations are based on the Poisson structure for such a dynamical system, in such a way that the Casimir invariants of the system remain invariant for the perturbed flow. By means of the Poincaré-Pontryagin theory, the existence of limit cycles on the invariant Casimir surfaces for the perturbed system is investigated up to first order of perturbation, providing sharp bounds for their number. Examples are given.     

Normal forms of reversible dynamical systems

We consider reversible dynamical systems with a fixed point which is also fixed under the reversing involution; we show that applying to such a system the canonical Poincaré-Dulac procedure reducing a dynamical system to its normal form, we obtain a normal form which is still reversible (under the same involution as the original system); conversely, we also show how to obtain all the reversible systems which are reduced to a given reversible form. This allows one to (locally) classify reversible dynamical systems, and reduce their (local) study to that of reversible normal forms.     

The Hamiltonian in relativistic systems of interacting particles

The dynamics of a system of relativistically interacting particles is determined by a set of constraints, some combination of which has been frequently identified with the Hamiltonian. These constraints differ from the generators of the Poincaré transformations, among whichp ₀ generates translations along the time axis and hence is to be considered as the energy of the system. There are thus grounds for consideringP ₀ as the appropriate Hamiltonian. In this paper we establish a close relationship between transformations generated by the constraints and those generated by the Poincaré generators. In particular we find that the true Hamiltonian is a rather complicated but well-defined function ofp ₀ and all the constraints. We show that the generators of the entire algebra of the Poincaré group can be realized in such a fashion that the Hamiltonian is correctly included among them, and such that particle world lines in Minkowski space-time generated by this Hamiltonian transform correctly under the Poincaré group.This work was partially supported by the National Science Foundation Grant No. PHY 79-0887 to Syracuse University and by Grant No. PHY 79-09405 to Yeshiva University.     

Relativistic Nonlocal Quantum Field Theory

A theory is defined to be relativistic if its Hamiltonian, total momenta, and boost's generators satisfy commutation relations of the Poincaré group. Field theories with usual local interactions are known to be relativistic. A simple example of a relativistic nonlocal theory is found. However, it has divergences. Some conditions are obtained which are necessary in order that a nonlocal theory be relativistic and divergenceless.     

Some aspects of the geometry of first-quantized theories

The ECSK and Yang-Mills theories are constructed with emphasis on their fiber bundle structure. In particular, the momentum tensor is derived as the Noether current of translational symmetry. The structure of the ECSK theory as a gauge theory of the Poincaré group is discussed. A theory of a Dirac field exhibiting internal affine symmetry, i.e., full internal Poincaré symmetry, is described. Aspects of the topological-geometric foundations of these theories are discussed, and some intuitive interpretations are presented.     

Non-gravitating scalars and spacetime compactification

We discuss role of partially gravitating scalar fields, scalar fields whose energy–momentum tensors vanish for a subset of dimensions, in dynamical compactification of a given set of dimensions. We show that the resulting spacetime exhibits a factorizable geometry consisting of usual four-dimensional spacetime with full Poincaré invariance times a manifold of extra dimensions whose size and shape are determined by the scalar field dynamics. Depending on the strength of its coupling to the curvature scalar, the vacuum expectation value (VEV) of the scalar field may or may not vanish. When its VEV is zero the higher-dimensional spacetime is completely flat and there is no compactification effect at all. On the other hand, when its VEV is nonzero the extra dimensions get spontaneously compactified. The compactification process is such that a bulk cosmological constant is utilized for curving the extra dimensions.     

Conformal invariance of relativistic equations for arbitrary spin particles

We show that any Poincaré-invariant equation for particles of zero mass and of discrete spin provide a unitary representation of the conformal group, and find an explicit expression of the conformal group generators in terms of Poincaré group generators.     

A self-consistent approach to quantum field theory for extended particles

A notion of quantum space-time is introduced, physically defined as the totality of all flows of quantum test particles in free fall. In quantum space-time the classical notion of deterministic inertial frames is replaced by that of stochastic frames marked by extended particles. The same particles are used both as markers of quantum space-time points as well as natural clocks, each species of quantum test particle thus providing a standard for space-time measurements. In the considered flat-space case, the fluctuations in coordinate values with respect to stochastic frames are described by coordinate probability amplitudes related to irreducible stochastic phase space representations of the Poincaré group. Lagrangian field theory on quantum space-time is formulated. The ensuing equations of motion for interacting fields contain no singularities in their nonlinear terms, and therefore can be handled by methods borrowed from classical nonlinear analysis.Supported in part by an NSERC grant.     

On dynamics analysis of a new chaotic attractor

In this Letter, a new chaotic system is discussed. Some basic dynamical properties, such as Lyapunov exponents, Poincaré mapping, fractal dimension, bifurcation diagram, continuous spectrum and chaotic dynamical behaviors of the new chaotic system are studied, either numerically or analytically. The obtained results show clearly that the system discussed in this Letter is a new chaotic system and deserves a further detailed investigation.     

Constraint algebra from local Poincaré symmetry

A rigorous derivation of the constraint algebra between lapse, shift and Lorentz Hamiltonians is presented assuming that only local Poincaré symmetry constraints are present in the theory. It is also shown that the Dirac-Arnowitt-Deser-Misner form of the Hamiltonian is merely a consequence of the local Poincaré symmetry identities.     

Chaos in classical cosmology

We study the dynamics of a Friedmann-Robertson-Walker universe conformally coupled to a real, self-interacting, massive scalar field. We apply a full set of tools corresponding to dynamical system theory: fixed points, linear stability analysis, resonances study and numerical evaluation of Poincaré sections of the dynamical flux. We can conclude that the chaotic behaviour is possible in the very early universe. In the case of a spatially closed universe we show that the route to chaos is reached by successive breaking of the resonant tori due to the action of 11 resonances.