Volume 16, Issue 11, November 1975
Index of content:

The λφ^{4} _{3} Euclidean quantum field theory in a periodic box
View Description Hide DescriptionThe ultraviolet cutoff (the lattice cutoff) normalized Schwinger functions converge as the ultraviolet cutoff (the lattice cutoff) is removed. The limit Schwinger functions are the moments of the normalized physical measure. As a consequence of the lattice approximation, the Lee–Yang theorem and various correlation inequalities hold for the λφ^{4} _{3}field theory in a periodic box.

New Jacobian ϑ functions and the evaluation of lattice sums
View Description Hide DescriptionThe properties of some infinite series are discussed. They are used to evaluate some two‐ and higher‐dimensional lattice sums.

Self‐adjoint operators, derivations and automorphisms on C*‐algebras
View Description Hide DescriptionWe discuss how the concept of C*‐algebras with a strongly continuous one parameter group of automorphisms can be realized, if the automorphism is implemented by an unbounded operator.

Calculation of special functional integrals
View Description Hide DescriptionFor the integration with respect to a Gaussian weak distribution of a special class of functionals φ defined on L ^{2} _{[0,1]} explicit formulas are derived. The φ are defined as nonnegative integer powers of a continuous linear functional B and the integration is performed under the condition that another continuous linear functional A assumes a given value.

The Weyl correspondence and path integrals
View Description Hide DescriptionThe method of Weyl transforms is used to rigorously derive path integral forms for position and momentum transition amplitudes from the time‐dependent Schrödinger equation for arbitrary Hermitian Hamiltonians. It is found that all paths in phase space contribute equally in magnitude, but that each path has a different phase, equal to 1/h/ times an ’’effective action’’ taken along it. The latter is the time integral of p⋅−h (p,q), h (p,q) being the Weyl transform of the Hamiltonian operator H, which differs from the classical Hamiltonian function by terms of order h/^{2}, vanishing in the classical limit. These terms, which can be explicitly computed, are zero for relatively simple Hamiltonians, such as (1/2M)[P−e A (Q)]^{2}+V (Q), but appear when the coupling of the position and momentum operators is stronger, such as for a relativistic spinless particle in an electromagnetic field, or when configuration space is curved. They are always zero if one opts for Weyl’s rule for forming the quantum operator corresponding to a given classical Hamiltonian. The transition amplitude between two position states is found to be expressible as a path integral in configuration space alone only in very special cases, such as when the Hamiltonian is quadratic in the momenta.

Invariant imbedding and the Fredholm integral equation
View Description Hide DescriptionThe standard method of solving the Fredholm integral equation is either to use Neumann series or to convert to an equivalent algebraic equation via the eigenvalues. This paper uses an idea similar to that given by Ueno and others. The integral equation is transformed into a Cauchy problem which can be approximated effectively by modern high speed digital computer.

Ergodic properties of a particle moving elastically inside a polygon
View Description Hide DescriptionThe flow of a classical particle bouncing elastically inside an arbitrary polygon is investigated. If every interior angle is a rational multiple of π, there exists precisely one isolating integral in addition to the energy; this integral is described in detail; any possible third integral is nonisolating. If one or more interior angles is an irrational multiple of π, the second integral becomes everywhere nonisolating and non‐Lebesgue‐measurable, i.e., the second integral disappears. The flow of two hard points bouncing elastically in a finite one‐dimensional box is equivalent to the flow of a point particle moving elastically inside a right triangle having interior angle tan^{−1} (m _{2}/m _{1})^{1/2}, so the preceding remarks apply to this model. Nonrigorous arguments are given in support of the notion that the polygon model is ergodic and mixing, but is not a C‐system.

The 2:1 anisotropic oscillator, separation of variables and symmetry group in Bargmann space
View Description Hide DescriptionWe present a detailed analysis of the separation of variables for the time‐dependent Schrödinger equation for the anisotropicoscillator with a 2:1 frequency ratio. This reduces essentially to the time‐independent one, where the known separability in Cartesian and parabolic coordinates applies. The eigenvalue problem in parabolic coordinates is a multiparameter one which is solved in a simple manner by transforming the system to Bargmann’s Hilbert space. There, the degeneracy space appears as a subspace of homogeneous polynomials which admit unique representations of a solvable symmetry algebras _{3} in terms of first order operators. These representations, as well as their conjugate representations, are then integrated to indecomposable finite‐dimensional nonunitary representations of the corresponding group S _{3}. It is then shown that the two separable coordinate systems correspond to precisely the two orbits of the factor algebras _{3}/u (1) [u (1) generated by the Hamiltonian] under the adjoint action of the group. We derive some special function identitites for the new polynomials which occur in parabolic coordinates. The action of S _{3} induces a nonlinear canocical transformation in phase space which leaves the Hamiltonian invariant. We discuss the differences with previous works which present s u (2) as the algebra responsible for the degeneracy of the two‐dimensional anisotropicoscillator.

Space averaging techniques of determinantal measures
View Description Hide DescriptionTechniques for evaluating approximate space‐averages of determinantal measures are discussed. The concept of combinatorial structure diagrams is introduced and investigated.

G‐Hilbert bundles
View Description Hide DescriptionA notion of Hilbert bundle is proposed which leads to the construction of a ’’big’’ Hilbert spaceH starting from a family of Hilbert spaces. For this, such a family is equipped with a suitable structure, called Borel field structure. A meaningful relationship is established between the Borel structures which can be defined on the union of the Hilbert spaces of the family and the Borel field structures with which the family can be equipped. For a topological groupG, the structure of G‐Hilbert bundle is defined linking in a suitable way a Hilbert bundle with actions of G. In the framework of a G‐Hilbert bundle, a continuous unitary representation of G in H can be constructed. The transitive G‐Hilbert bundles which are often used in the theory of induced representations of groups are shown to be a subclass of the class of the G‐Hilbert bundles which are proposed in this paper.

Unequal mass spinor‐spinor Bethe–Salpeter equation
View Description Hide DescriptionCoupled radial equations are derived for the ladder approximation Bethe–Salpeter equation describing a system of two spin‐ (1/2) particles of unequal masses interacting to form a bound state of total mass zero. The numerical behavior of the coupling parameter λ as a function of the mass ratio is examined for known analytical equal‐mass solutions. In addition a perturbation method is employed to investigate the behavior of λ for small values of the exchange mass.

Kinks and extensions
View Description Hide DescriptionKinks, homotopically nontrivial lightcone fields on R ^{4}, can be black holes without curvature singularities, satisfying the weak energy condition. Kinks and all other spherically symmetric stationary spacetimes on R ^{4} with roots of g _{00} have incomplete geodesics which need extension. For simple roots the Kruskal extension method works and the topology around each root is that of the Kruskal manifold. For multiple roots another extension method is given, based on symmetry, and another topology.

Neutron transport problems in a spherical shell
View Description Hide DescriptionThe density transform method has been extended to cover spherically symmetric transport problems in a spherical shell. The density transform is expanded in plane geometry normal modes and explicit singular integral equations are derived for the expansion coefficients. It is shown that the Green’s function method, introduced by Case e t a l., gives the same representation of total flux. The singular integral equations for the expansion coefficients are rederived using the analytic properties of some sectionally holomorphic functions introduced by the above authors.

The Gel’fand states of certain representations of U (n) and the decomposition of products of representations of U (2)
View Description Hide DescriptionThe representations of U (n), as realized by Bargmann and Moshinsky on spaces of polynomials (’’boson calculus’’), are the main subject of this paper. We consider them from a global point of view, pointing out the connection with induced representations. To compute the detailed structure of the representations, we find the reproducing kernels of the function spaces and the operators that connect them according to Weyl’s branching law. Using these results, we compute the boson polynomials of representations of U (3), and arrange them in a generating function. We extend this generating function to the boson polynomials of representations of U (n) of the form 〈 (m _{1} m _{2}0⋅⋅⋅0) 〉. By considering these polynomials from a different viewpoint, we are able to obtain an explicit decomposition of the Kronecker product of n−1 representations of S U (2).

On finite mass renormalizations in the two‐dimensional Yukawa model
View Description Hide DescriptionIn the Mathews–Salam formulas for the (space–time cutoff) Schwinger functions of Y _{2}, no restriction on finite mass renormalizations for the boson is necessary.

Average multiplicity and the zeros of multiparticle generating function
View Description Hide DescriptionWe have obtained several theorems exhibiting relations between the asymptotic behavior of the average multiplicity 〈n〉 and the distribution of zeros of a generating function for multiparticle production cross sections.

Resonantly coupled nonlinear evolution equations
View Description Hide DescriptionA differential matrix eigenvalue problem is used to generate systems of nonlinear evolution equations. They model triad, multitriad, self‐modal, and quartet wave interactions. A nonlinear string equation is also recovered as a special case. A continuum limit of the eigenvalue problem and associated evolution equations are discussed. The initial value solution requires an investigation of the corresponding inverse‐scattering problem.

A new solution of the Einstein–Maxwell equations
View Description Hide DescriptionA space–time is determined which is a solution of the Einstein–Maxwell equations for a nonsingular electromagnetic field and for which the electromagnetic field tensor is weakly parallelly propagated along its principal null directions. A coordinate system is given in which the metric depends upon one essential arbitrary constant. The space–time admits a four‐parameter simply transitive group of motions and its Weyl tensor is of Petrov type I.

Macroscopic and microscopic, coherent and incoherent variables
View Description Hide DescriptionQuantitative figures of merit for microscopicity (resolving power) and for coherence (the necessity of simultaneous observations of large volumes) are defined operationally. Such indices are not assignable to observables (which are equivalence classes of observation procedures) but to observation procedures. As an application, the problem of reversibility is reconsidered. Known qualitative arguments explaining the difficulty of creating certain processes are made quantitative. A conjectured theorem states that the difficulty of preparing a state at t=0 so that a particular target situation is observed at t=τ increases monotonically with τ.

Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics
View Description Hide DescriptionWe show, by making use of the functional integral technique, that, for a large class of useful quantum statistical systems, the partition function is, with respect to the coupling constant, the Laplace transform of a positive measure. As a consequence, we derive an infinite set of monotonicly converging upper and lower bounds to it. In particular, the lowest approximation appears to be identical to the Gibbs–Bogolioubov variational bound, while the next approximations, for which we give explicit formulas for the first few ones, lead to improve the previous bound. The monotonic character of the variational successive approximations allows a new approach towards the thermodynamical limit.