Volume 19, Issue 9, September 1978
Index of content:

Quantum dynamical semigroups and multipole relaxation of a spin in isotropic surroundings
View Description Hide DescriptionWe derive and discuss three different parametrizations of the generator of a dynamical semigroup which describes the Markovian relaxation of a spin j under the influence of isotropic surroundings. The relevant parametrizations that we consider are the strengths of the polarities of the interaction, the relaxation rates of the different multipoles and the transition probabilities per unit time among the Zeeman sublevels. The results are model independent and allow us to derive a set of relations and inequalities for the transition probabilities and for the relaxation rates whose validity is not bound to any specific assumption concerning the mechanisms which govern the relaxation.

Generation of stationary axisymmetric Einstein–Maxwell fields
View Description Hide DescriptionA method is presented which allows one to generate new solutions from old ones. Kinnersley’s functions u, v, w associated with the old solutions must be stationary, axially‐symmetric, and satisfy a linear equationa*u+b*v−c*w=0. The method generalizes work of Bonner and Misra e t a l. who obtained a electric/magnetic dipole solution from the Kerr solution.

Theorem on the representations of SO(n) groups
View Description Hide DescriptionIt is shown that SO(n) groups (n≳5) that possess a c‐number Lorentz Casimir operator F do not have infinite‐dimensional representations. Accordingly these groups cannot support new positive‐energy wave equations of Staunton type. In an appendix Staunton’s spin‐1/2 positive‐energy wave equation is extended, in an analogous manner to that used by Rarita and Schwinger (RS) on Dirac’s equation, to yield positive‐energy wave equations for arbitrary spin. It is noted that analogous auxiliary conditions to those of RS do not hold. Stationary solutions for low spin are listed.

Field equations and integrability conditions for special type N twisting gravitational fields
View Description Hide DescriptionWe let SNT designate a special kind of twisting type Ngravitational field solution of the Einstein field equations, viz., one for which there exists a real scalar field χ which satisfies (∇_{[α} k _{β}−k _{[α}∇_{β}χ) (∇_{γ]} k _{δ}) t ^{δ}=0, where k _{α} is a principal null vector, t ^{α} t _{α} =t ^{α} k _{α}=0, and t ^{α} t _{α}*=1. We obtain some theorems which provide necessary and sufficient criteria for an SNT to be the twisting type N metric discovered by I. Hauser, in Phys. Rev. Lett. 33, 1112 (1974). Field equations, a hierarchy of integrability conditions, and analytic techniques which are applicable to the quest for a new SNT are given. Two tractable SNT subcases are considered in detail.

On the application of the generalized quantal Bohr–Sommerfeld quantization condition to single‐well potentials with very steep walls
View Description Hide DescriptionThis investigation concerns some model potentials with very steep walls, for which the quantal Bohr–Sommerfeld h a l f‐i n t e g e r quantization condition, if necessary generalized to correspond to modified phase‐integral approximations of arbitrary order, and used without or with higher‐order corrections included, can be used for obtaining, very accurately, the energy eigenvalues of the bound states, apart possibly from the lowest ones. One of the cases treated is the potential proportional to cot^{2} x, for which a modified Bohr–Sommerfeld half‐integer quantization condition yields the energy eigenvalues exactly. That the Bohr–Sommerfeld h a l f‐i n t e g e r quantization condition is applicable to potentials with very steep walls may at first sight seem surprising in view of the well‐known fact that the energy eigenvalues of a square‐well potential with infinitely high walls are obtained exactly from the Bohr–Sommerfeld i n t e g e r quantization condition, i.e., the quantization condition obtained by replacing (s+1/2) π by (s+1) π in the right‐hand member of the Bohr–Sommerfeld half‐integer quantization condition. From the study of a potential with horizontal bottom and linearly rising walls, which goes over into a square well when the inclination of the walls tends to zero, it can easily be understood, why the h a l f‐i n t e g e r quantization condition is appropriate when the steep walls have a finite slope.

On phase‐integral quantization conditions for bound states in one‐dimensional smooth single‐well potentials
View Description Hide DescriptionPrevious work by the present authors on phase‐integral quantization conditions for single‐well potentials is extended and generalized. Exact as well as approximate quantization conditions are considered and investigated more fully than previously. Not only their necessity but also their sufficiency is treated. An improvement of the estimate of the error of the generalized quantal Bohr–Sommerfeld quantization condition is also given.

The tensor virial theorem in quantum mechanics
View Description Hide DescriptionA quantum mechanical generalization of the scalar virial theorem is derived and specialized to atoms and molecules in the Born–Oppenheimer approximation. The theorem is the quantum mechanical counterpart to Chandrasekhar’s classical tensor virial theorem. The usual scalar virial equation follows by tensor contraction. One possible application is the introduction of more than one scale factor in a trial wavefunction. The scaling method proposed involves different stretchings for the different spatial coordinates. This is in contrast to the standard method of using the scalar virial theorem where the stretching is the same in all directions. An example is given where the introduction of multiple scale factors and the imposition of the tensor virial theorem yields a better result than the usual procedure of subjecting the wavefunction to a single scale transformation and imposing the scalar virial theorem.

Quantum theory and Hilbert space
View Description Hide DescriptionTwo theorems are proven which show that any orthomodular partially ordered set (hence, in particular, the orthologic of questions on a physical system) can be embedded in the lattice of closed subspaces of a Hilbert space in such a way that the standard trace formula of quantum theory can be used to calculate all probabilities. Possible conclusions from these results and from the existence of counterexamples to stronger conjectures are then discussed.

On the Kerr–Tomimatsu–Sato family of solutions with nonintegral distortion parameter
View Description Hide DescriptionThe generalization of the Kerr–Tomimatsu–Sato family of solutions for gravitational fields of spinning masses to the case of the arbitrary positive nonintegral distortion parameter δ is conjectured.

Search for periodic Hamiltonian flows: A generalized Bertrand’s theorem
View Description Hide DescriptionA complete classification is given of the two‐dimensional Hamiltonian systems (whose Hamilton–Jacobi equation separates in Cartesian or polar coordinates) which admit strictly periodic motions for open sets of initial conditions (completely degenerate systems). Any of the systems which are separable in Cartesian coordinates turn out to be canonically equivalent to some anisotropic harmonic oscillator. In the polar case our results provide a generalization of a celebrated theorem of Bertrand. It is proven that all the completely degenerate systems fall into two families. These families are characterized by the semiclassical inverse ’’spectral functions’’n _{1} J _{1}+n _{2} J _{2}=F (H) =α (−H)^{−1/2}−β, n _{1} J _{1}+n _{2} J _{2}=F (H) =αH−β (α, β real positive constants) and contain, as central symmetric cases, the Kepler system and the isotropic harmonic oscillator, respectively. Qualitative and higher symmetry properties of these systems are also discussed.

Light‐cone finite normal products
View Description Hide DescriptionA graphical subtraction procedure for constructing the perturbative Green functions of light‐cone finite, multiply localized products of fields is proposed. The existence of the Green functions as tempered distributions is proved, together with the properties of light‐cone finiteness and localization on a line segment. The derivation of light cone expansions is sketched, but not treated in detail.

A time dependent generalization of the multiple scattering formalism
View Description Hide DescriptionThe multiple scattering formalism is generalized to include the restricted class of time dependent potentials in which each scattering center is moving with an arbitrary, but constant, velocity.

Variational principles on rth order jets of fibre bundles in field theory
View Description Hide DescriptionThe Hamilton and the modified Hamilton variational principles in classical field theory are studied for physical systems described by Lagrangian and Hamiltonian densities depending on arbitrary order derivatives of the field. These principles are established on the fibre bundles J ^{r}(E), J ^{1}(J ^{ r−1}(E)), J ^{1*}(J ^{ r−1}(E)). This is accomplished by defining an appropriate Poincaré–Cartan form. This form is also required in the definition of the associated symmetry problem and in the explicit construction of the Noether currents.

Vector bundles, rth order Noether invariants and canonical symmetries in Lagrangian field theory
View Description Hide DescriptionWe present a formulation of the ’’canonical’’ transformations of the Dirac theory by using the bundle of r‐jets associated with the Dirac vector bundle. This allows us, by means of a variational principle previously introduced, to study in a natural way (through an appropriate Noether theorem) the role which these ’’canonical’’ theories play in the definition of ’’new’’ symmetries. The limit r→∞ corresponds to the equivalence between the canonical and the ordinary symmetries.

Nuclearity of some spaces of C ^{∞} vectors in induced representations
View Description Hide DescriptionWe give conditions on irreducible unitary induced representations of inhomogeneous Lie groups ensuring nuclearity of the space of differentiable vectors. We study the consistency of these conditions. Finally we show that our conditions hold almost everywhere with respect to the Plancherel measure for some generalizations of the Poincaré group.

Slow motion approximation in predictive relativistic mechanics. I. Approximated dynamics up to order c ^{−4}
View Description Hide DescriptionWe obtain the most general two‐body system relativistic invariant having a Newtonian limit up to c ^{−4} order. In particular, it includes the scalar, gravitational, and other well‐known dynamics up to c ^{−2} order. It also includes to c ^{−4} order the dynamics obtained recently by Lapiedra and Mas in predictive relativistic time symmetric electrodynamics.

Stochastic quantization of wave fields and its application to dissipatively interacting fields
View Description Hide DescriptionThe stochastic quantization procedure, proposed by Nelson, is further developed to include the problem of field quantization. To maintain the mathematical rigor in treating infinitely many degrees of freedom, basic notions of the nonstandard analysis are adopted. As an application of the present method, a quantum mechanical description of dissipatively interacting fields, such as the laser electric field in the lossy cavity, is investigated.

Unified theory of direct interaction between particles, strings, and membranes
View Description Hide DescriptionA model of generalized relativistic membranes that contains as special cases, particles, geometric strings and geometric membranes and other new one‐ and two‐dimensional objects is studied. The equations of motion for such objects in direct interaction are studied. The constraints on the interaction due to the freedom of gauge of the free model are solved. A scalar, a vectorial, and a tensorial type of interaction are discussed. Conservation theorems associated with Poincaré invariance of the action are studied, as well as the generalization for action‐at‐a‐distance theories of the action and reaction law.

General algebraic theory of identical particle scattering
View Description Hide DescriptionWe consider the nonrelativistic N‐body scattering problem for a system of particles in which some subsets of the particles are identical. We demonstrate how the particle identity can be included in a general class of linear integral equations for scattering operators or components of scattering operators. The Yakubovskii, Yakubovskii–Narodestkii, Rosenberg, and Bencze–Redish–Sloan equations are included in this class. Algebraic methods are used which rely on the properties of the symmetry group of the system. Operators depending only on physically distinguishable labels are introduced and linear integral equations for them are derived. This procedure maximally reduces the number of coupled equations while retaining the connectivity properties of the original equations.

Energy–momentum tensor symmetries and concomitant conservation laws. I. Einstein‐massless‐scalar (meson) field
View Description Hide DescriptionSymmetries of energy–momentum tensorsT in a Riemannian space–time are defined by infinitesimal mappings x̄ ^{ i } =x ^{ i }+ξ^{ i }(x) δa where the mapping vector ξ^{ i } is determined by the symmetry condition L_{ξ}(g ^{ w/2}T) =0, (g ^{ w/2}T) is a relative tensor of weight w, g≡ absolute value of the metrical determinant, and L_{ξ} is the Lie derivative with respect to the vector ξ^{ i }). The existence of such symmetry vectors ξ^{ i } leads to concomitant conservation laws in the form of conserved vector currentsJ ^{ i } for both special and general relativity. The currentsJ ^{ i } will be explicit functions of the energy momentum tensorT and the symmetry vector ξ^{ i }. The symmetries and conservation laws so obtained will in general differ from the familiar Trautman formulation. The theory is applied to obtain symmetries and conserved currents for a class of conformally flat solutions of the Einstein‐massless‐scalar (meson) field equations.