Volume 20, Issue 7, July 1979
Index of content:

Path‐integral evaluation of the space–time propagator for quadratic Hamiltonian systems
View Description Hide DescriptionPath‐integral methods are used to derive an exact expression for the space–time propagator for systems with quadratic Hamiltonians. For a certain subclass of such systems, the result is reduced to a simplified closed form. The propagators for several illustrative elementary cases are exhibited in detail.

Internal symmetries of the axisymmetric gravitational fields
View Description Hide DescriptionThe group H of the internal symmetries of the axisymmetric field equations in general relativity is known to be isomorphic to SO(2,1), which is the double covering of the conformal group of the hyperbolic complex plane H. The Ernst potential ξ can then be geometrically understood as a map ξ:R ^{3}/SO(2) → H. The fact that the hyperbolic plane is split into two connected components is used to introduce an algebraic invariant n∈Z ^{+} for every axisymmetric solution. It is shown that under reasonable hypotheses this invariant is related to the number of S ^{1} curves where the manifold is intrinsically singular.

Resolution of Fredholm equations with kernels K (z−z _{0}) by operational calculus
View Description Hide DescriptionWe show that the solutions of a Fredholm equation with kernels K (z−z _{0}) is the transform of its second member in a transformation defined by a differential operator. The calculations of these solutions are then a matter of the powerful operational calculus.

Classical predictive electrodynamics of two charges with radiation: General framework. I
View Description Hide DescriptionOutgoing radiation is introduced in the framework of the classical predictive electrodynamics using Lorentz–Dirac’s equation as a subsidiary condition. In a perturbative scheme in the charges the first radiative ’’self‐terms’’ of the accelerations, momentum and angular momentum of a two charge system without external field are calculated.

Classical predictive electrodynamics of two charges with radiation: Energy and 3‐momentum balance and scattering cross sections. II
View Description Hide DescriptionWe deal with a classical predictive mechanical system of two spinless charges where radiation is considered and there are no external fields. The terms ^{(2,2)} P _{ a } ^{α} of the expansion in the charges of the Hamilton–Jacobi momenta are calculated. Using these, together with known previous results, we can obtain the p ^{α} _{ a } up to the fourth order. Then we have calculated the ’’radiated’’ energy and the 3‐momentum in a scattering process as functions of the impact parameter and the incident energy for the former and 3‐momentum for the latter. Scattering cross‐sections are also calculated. Good agreement with well known results, including those of quantum electrodynamics, has been found.

Current responses of all orders in a collisionless plasma. I. General theory
View Description Hide DescriptionAn arbitrary electromagnetic perturbation of a general solution of the relativistic Vlasov–Maxwell equations is considered. The nonlinear current responses are expressed in a form which in particular is an all order manifestation of the Manley–Rowe relations. A coordinate free formalism is used, starting with a representation of Minkowski space in terms of abstract linear algebra, and all formulas are intrinsically covariant. In the method used to derive the current responses the perturbation of particle orbits rather than of distribution functions is calculated.

Current responses of all orders in a collisionless plasma. II. Homogeneous plasma
View Description Hide DescriptionExpressions for the admittance tensors of all orders are obtained for a relativistic magnetized Vlasov–Maxwell plasma. Symmetries, from which the Manley–Rowe relations follow, are explicit in the expressions obtained. The treatment is covariant.

Weak quantization in a nonperturbative model
View Description Hide DescriptionThe concepts of extended operator convergence and of spectral concentration are used to study rigorously a class of simple models for the tunnel effect and the laser. We compute exactly the asymptotic decay times of the eigenmodes, and we prove their link with the line width of the corresponding resonances.

Representations of the Poincaré group for relativistic extended hadrons
View Description Hide DescriptionRepresentations of the Poincaré group are constructed from the relativistic harmonic oscillatorwave functions which have been effective in describing the physics of internal quark motions in the relativistic quark model. These wave functions are solutions of the Lorentz‐invariant harmonic oscillatordifferential equation in the ’’cylindrical’’ coordinate system moving with the hadronic velocity in which the time‐separation variable is treated separately. This result enables us to assert that the hadronic mass spectrum is generated by the internal quark level excitation, and that the hadronic spin is due to the internal orbital angular momentum. An addendum relegated to PAPS contains discussions of detailed calculational aspects of the Lorentz transformation, and of solutions of the oscillator equation which are diagonal in the Casimir operators of the homogeneous Lorentz group. It is shown there that the representation of the homogeneous Lorentz group consists of solutions of the oscillator partial differential equation in a ’’spherical’’ coordinate system in which the Lorentz‐invariant Minkowskian distance between the constituent quarks is the radial variable.

Bifurcate Killing horizons
View Description Hide DescriptionIt is shown that an analytic spacetime with a bifurcate Killing horizon is locally symmetric with respect to the axis of rotation. It is also shown that if the surface gravity of a Killing horizon is a nonzero constant, then there exists a local prolongation (extension) of the spacetime that contains a bifurcate Killing horizon.

Decay of local correlations and absence of phase transitions
View Description Hide DescriptionIf the local truncated correlation functions of a system of statistical mechanics decay in a prescribed manner the limiting pressure becomes differentiable with respect to the activity. Criterions of clustering of the local truncated m‐point correlation functions are shown to lead to a pressure being element of C ^{ n } for arbitrary n and m=1,...,n+1.

On stationary, axially symmetric solutions of Jordan’s five‐dimensional, unified theory
View Description Hide DescriptionIt is shown that the association of a linear eigenvalue problem for solutions of Einstein’s equations admitting a two‐parameter Abelian group of isometries can be extended to Jordan’s five‐dimensional, unified theory admitting three commuting Killing vectors. The reduction to a two‐dimensional problem, the derivation of infinitely many conservation laws and the generation of one‐parameter families of solutions can thereby be transcribed almost literally.

Repulsive and attractive timelike singularities in vacuum cosmologies
View Description Hide DescriptionSpherically symmetric cosmologies whose big bang is partially spacelike and partially timelike are constrained to occur only in the presence of certain types of matter, and in such cosmologies the timelike part of the big bang is a negative‐mass singularity. In this paper examples are given of cylindrically symmetric cosmologies whose big bang is partially spacelike and partially timelike. These cosmologies are vacuum. In some of them, the timelike part of the big bang is clearly a (generalized) negative‐mass singularity, while in others it is a (generalized) positive‐mass singularity.

On angular momentum of stationary gravitating systems
View Description Hide DescriptionA theorem is proved to the effect that, for isolated gravitating systems in equilibrium, the definition of total angular momentum involving fields at null infinity agrees with that involving fields at spatial infinity.

The application of group theory to generate new representability conditions for rotationally invariant density matrices
View Description Hide DescriptionNew linear conditions are derived which must be satisfied by a two‐body density matrix. In the derivation, the ideas of Davidson and McRae are extended so that full use is taken of the symmetries of the system. The coefficients of the linear form are determined by means of reduction of a chosen group in a physically meaningful chain of its subgroups.

Segal quantization of dynamical systems
View Description Hide DescriptionSegal quantization, usually thought of and used as a tool for quantizing kinematical frameworks, is extended to (finite‐dimensional) dynamical systems, i.e., to kinematical frameworks plus dynamical motions applied to them. Such a procedure allows us to classify quantum dynamical systems, and to understand how physical inequivalence appears in spite of von Neumann unitary equivalence, for the former is also grounded on the evolution operators of the systems being different functions of the labeled observables of the systems. Second quantization is also examined and is shown to be just one possible procedure of quantization, which can only be used in a particular class of cases.

On some representations of the Poincaré group on phase space
View Description Hide DescriptionSome representations of the Poincaré group, by functions on phase space are studied both for classical as well as quantum relativistic systems. The classical representations are identified with certain canonically induced representations, and the quantum representations are then obtained on the same Hilbert space.Equations of motion on phase space are also developed.

Representations of the duals of gauge field tensors
View Description Hide DescriptionClassical electromagnetic fields admit a perfectly symmetrical description in terms of the electromagnetic field tensors and the duals thereof, this phenomenon being a simple consequence of Maxwell’sequations for free fields. Since the Yang–Mills equations for free gauge fields possess a somewhat analogous structure, the possibility of a similar symmetrical description of gauge fields is investigated. It is shown that source‐free Yang–Mills equations do, in fact, imply the existence of a new gauge in terms of which the duals of the field tensors admit a representation; however, the latter does not in general possess a structure which is identical with that of the field tensors in terms of the original gauge.

Summation of divergent series by order dependent mappings: Application to the anharmonic oscillator and critical exponents in field theory
View Description Hide DescriptionWe study numerically a method of summation of divergent series based on an order dependent mapping. We consider the example of a simple integral, of the ground‐state energy of the anharmonic oscillator and of the critical exponents of φ_{3} ^{4}field theory. In the case of the simple integral convergence can be rigorously proven, while in the other examples we can only give heuristic arguments to explain the properties of the transformed series. For the anharmonic oscillator we have compared our results to an accurate numerical solution (10^{−23}) of the Schrödinger equation. For the critical exponents we have verified the consistency of our results with those obtained before from methods using a Borel transformation.

On the sound field due to a moving source in a superfluid
View Description Hide DescriptionExpressions are obtained for the sound pressure and temperature fields due to the motion of a monopole point source in a superfluid. The acoustic power spectrum is also given for all the Mach number ranges associated with the problem.