Volume 17, Issue 12, December 1976
Index of content:

N‐level systems in contact with a singular reservoir. II
View Description Hide DescriptionWe study an N‐level system coupled linearly to an infinite quasifree Fermi or Bose reservoir in the vacuum state or in a state corresponding to an arbitrary temperature. We show that the singular reservoir limit can be performed in the vacuum state and at infinite temperature, thus leading to a completely positive Markovian reduced time evolution for the system, which, in the infinite temperature case, preserves the central state. On the other hand, no such limit is possible for KMS states (finite temperature) and at zero temperature. Some extension to norm‐continuous semigroups of an infinite‐dimensional B (H) is possible.

Some properties of canonical invariantly relaxing (CIR) systems
View Description Hide DescriptionIn this paper, two general properties of the conserved discrete states CIR systems have been found: (1) the equal spacing of and a general expression for the characteristic roots of their transition‐rates‐matrices; (2) a general formulation for their conditional probabilities. The discussion is also extended to the disappearing discrete‐states CIR systems which include the infinite level harmonic oscillators as a special case.

A note on energy bounds for boson matter
View Description Hide DescriptionTwo new proofs are given of the Dyson and Lenard lower bound for the energy of matter with boson electrons. Another result is a new inequality for the two‐point correlation function.

Shock waves in relativistic magnetohydrodynamics under general assumptions
View Description Hide DescriptionWe present a rigorous theory of magnetohydrodynamical shock waves in the framework of a given curved space–time, under general assumptions corresponding both to a plasma and to a condensed medium. The results can be of use in astrophysics. We prove the timelike character of the wavefronts, the main thermodynamic inequalities, the relative location of the speeds of the shock waves with respect to the magnetosonic and Alfvén speeds, and show some existence and uniqueness theorems. In particular, we show that there can exist initial states giving slow shocks, but no weak shocks.

Differential equations satisfied by Fredholm determinants and application to the inversion formalism for parameter dependent potentials
View Description Hide DescriptionWe consider the two first order differential operators A _{ x }=μ (x) ∂/∂x+λ (x), B _{ x }=−(∂/∂x) μ (x)+λ (x), associate two kernels f, g satisfying both well‐defined boundary conditions and A _{ x } f=B _{ y } g, A _{ y } f=B _{ x } g, and construct the Fredholm determinants corresponding to these kernels. From these determinants we can build up solutions of second order differential equations. These solutions have an interpretation in the Schrödinger inversion formalism. For instance, for the inversion at fixed angular momentuml, these solutions for k=0 correspond to the classical Gel’fand–Levitan and Marchenko equations, whereas, for k≠0, they correspond to k dependent potentials. Similarly, for the inversion at fixed k, these solutions for l=0 correspond to the classical Regge–Newton equations, whereas, for l≠0, they correspond to l‐dependent potentials. More generally we show, in a generalized inversion formalism, how parameter dependent potentials appear very naturally in the theory.

Uniqueness of perturbation of a Reissner–Nordström black hole
View Description Hide DescriptionCoupled gravitational and electromagnetic perturbations of a Reissner–Nordström black hole are analyzed using the Newman–Penrose formalism. It is shown that χ^{ B } _{1}(≡3ψ_{2}φ^{ B } _{0}−2φ_{1}ψ^{ B } _{1}) or χ^{ B } _{−1}(≡3ψ_{2}φ^{ B } _{2}−2φ_{1}ψ^{ B } _{3}) determines the perturbations except for those corresponding to an infinitesimal change in the mass, charge, and angular momentum parameters of the balck hole.

Zeros of the grand canonical partition function: Generalization of a result of Lee and Yang
View Description Hide DescriptionA simple sufficiency condition for the zeros of a polynomial of grand partition function form to lie entirely on the unit circle in the complex fugacity (z) plane is rigorously proven. The condition has two parts: the canonical partition function Q _{ n }(M) is symmetric, Q _{ n }(M) =Q _{ M−n }(M), and is bounded above by the binomial coefficient (^{ M } _{ n }). This represents a generalization of the condition given by Lee and Yang in the context of the Ising model and the proof is independent of theirs. Necessity of the condition is trivially proven.

Intrinsic geometry of Killing trajectories
View Description Hide DescriptionWe use the Frenet–Serret formalism to study the intrinsic geometry of Killing trajectories that are admitted by an arbitrary n‐dimensional Riemannian space. The intrinsic quantities associated with these curves, i.e. their curvatures, are found to be constants of the motion that can be evaluated in terms of Hankel determinants. The results are then applied to curves in real quantum mechanics.

Radial integrals with finite energy loss for Dirac–Coulomb functions
View Description Hide DescriptionAnalytic results for radial integrals over products of Dirac–Coulomb functions and the radial part of the electromagnetic Green’s function are expressed in terms of a matrix generalization of the gamma function. This matrix gamma function has many useful properties, including a recurrence relation similar to that of the gamma function, and provides a compact easily manipulated method of evaluating the Dirac–Coulomb radial integrals. These results can be used to calculate the virtual and real photon spectra associated with electron scattering from the nucleus.

The frequency dependent electrical conductivity for disordered alloys: Application of an abstract Hilbert space generalization of Feenberg’s perturbation theory
View Description Hide DescriptionAn abstract Hilbert space with a particularly convenient scalar product is introduced to permit a generalization of Feenberg’s rearrangement method of perturbation theory to be applied to thermal Green’s function calculations. This method has the advantage of treating averages (either thermal or configuration) rigorously from the start. Explicit calculations are done for the frequency dependent electrical conductivity for alloys with diagonal disorder at zero temperature. Three practical approaches are discussed: (1) the Gram–Schmidt orthogonalization procedure, (2) a trick which depends on the Hermitian character of the polarization operator, and (3) a general procedure for using nonorthogonal basis vectors to expand the Feenberg formulas. To second order in the scattering strength, a new expression for the conductivity is found which is valid for all frequencies. This expression agrees with earlier perturbation theory results when the frequency is very small or very large.

On a solution of the Einstein–Maxwell equations admitting a nonsingular electromagnetic field
View Description Hide DescriptionSolutions of the Einstein–Maxwell equations are investigated for which the electromagnetic field is nonsingular and weakly parallelly propagated along its principal null congruences. It is shown that a subclass of these solutions admits an invertible two‐dimensional Abelian group of motions. A weaker characterization of a recently found solution is thereby obtained.

A note on certain multiple integrals
View Description Hide DescriptionAn analytical expression is given for the multiple integral ∫⋅⋅⋅∫dμ (x _{ n+1}) dμ (x _{ n+2}) ⋅⋅⋅dμ (x _{ N }) Π_{1⩽j<k⩽N }‖x _{ j } −x _{ k }‖^{β}, where 0⩽n⩽N, β=1,2, or 4 and the positive measure dμ (x) is such that all its moments exist, ∫dμ (x) x ^{ j }<∞, j=0,1,2,⋅⋅⋅. The case dμ (x) =d x, for −1⩽x⩽1, and dμ (x) =0, for ‖x‖≳1, is given as an example. In the limit N→∞ the correlation functions of this example, the so‐called Legendre ensembles, coincide with those of the circular or the Gaussian ensembles of random matrices.

Some basic properties of Killing spinors
View Description Hide DescriptionThe concept of Killing spinor is analyzed in a general way by using the spinorial formalism. It is shown, among other things, that higher derivatives of Killing spinors can be expressed in terms of lower order derivatives. Conformal Killing vectors are studied in some detail in the light of spinorial analysis: Classical results are formulated in terms of spinors. A theorem on Lie derivatives of Debever–Penrose vectors is proved, and it is shown that conformal motion in vacuum with zero cosmological constant must be homothetic, unless the conformal tensor vanishes or is of type N. Our results are valid for either real or complex space–time manifolds.

The intrinsic spinorial structure of hyperheavens
View Description Hide DescriptionFollowing Plebański and Robinson, complex V _{4}’s which admit a congruence of totally null surfaces are shown to have coordinates which, in pairs, have a spinor structure which generates the usual spinor structure of the 2‐forms over the space. This structure allows Einstein’s vacuum equations to fracture into three triples and a singlet, which allow for easy reduction of the entire set to one nonlinear partial differential equation needed for consistency. An inhomogeneous GL̃ (2,C) group of coordinate transformations, constrained to leave the tetrad form invariant, is constructed and used to simplify the equations and clarify the geometrical meaning of the parameters introduced during the integration process.

Lie series and invariant functions for analytic symplectic maps
View Description Hide DescriptionSymplectic maps (canonical transformations) are treated from the Lie algebraic point of view using Lie series and Lie algebraic techniques. It is shown that under very general conditions an analytic symplectic map can be written as a product of Lie transformations. Under certain conditions this product of Lie transformations can be combined to form a single Lie transformation by means of the Campbell–Baker–Hausdorff theorem. This result leads to invariant functions and generalizes to several variables a classic result of Birkhoff for the case of two variables. It also provides a new approach since the connection between symplectic maps, Lie algebras, invariant functions, and Birkhoff’s work has not been previously recognized and exploited. It is expected that the results obtained will be applicable to the normal form problem in Hamiltonian mechanics, the use of the Poincaré section map in stability analysis, and the behavior of magnetic field lines in a toroidalplasma device.

Monopoles, vortices and the geometry of the Yang–Mills bundles
View Description Hide DescriptionA topological classification of monopoles and vortices is formulated in terms of fibre bundles. The distinction between Dirac and ’t Hooft monopoles is made in the light of the energy finiteness problem. Finite‐length vortices with Dirac monopoles at the end points are also discussed.

On the stationary axisymmetric Einstein–Maxwell field equations
View Description Hide DescriptionWe show the existence of a formal identity between Einstein’s and Ernst’s stationary axisymmetric gravitational field equations and the Einstein–Maxwell and the Ernst equations for the electrostatic and magnetostatic axisymmetric cases. Our equations are invariant under very simple internal symmetry groups, and one of them appears to be new. We also obtain a method for associating two stationary axisymmetric vacuum solutions with every electrostatic known.