Volume 17, Issue 5, May 1976
Index of content:

Canonical transforms, separation of variables, and similarity solutions for a class of parabolic differential equations
View Description Hide DescriptionUsing the method of canonical transforms, we explicitly find the similarity or kinematical symmetry group, all ’’separating’’ coordinates and invariant boundaries for a class of differential equations of the form [α∂^{2}/∂q ^{2}+βq ∂/∂q+γq ^{2}+δq+ε∂/∂q+ζ] u (q,t) =−i (∂/∂t) u (q,t),or of the form [α′ (∂^{2}/∂q ^{2}+μ/q ^{2})+β′q∂/∂q+γ′q ^{2}] u (q,t) =−i (∂/∂t) u (q,t), for complex α,β,..., γ′. The first case allows a six‐parameter WSL(2,R) invariance group and the second allows a four‐parameter O(2) ⊗ SL(2,R) group. Any such differential equation has an invariant scalar product form which, in the case of the heat equation, appears to be new. The proposed method allows us to work with the group, rather than the algebra, and reduces all computation to the use of 2×2 matrices.

The relationship between the normalization coefficient and dispersion function for the multigroup transport equation
View Description Hide DescriptionAn explicit formula for the discrete Case normalization coefficient is presented in terms of functions related to the dispersion function. These functions are easily determined and provide the normalization coefficient without need of prior evaluation of the eigenvectors.

Models of Zermelo Frankel set theory as carriers for the mathematics of physics. I
View Description Hide DescriptionThis paper is a first attempt to explore the relationship between physics and mathematics ’’in the large.’’ In particular, the use of different Zermelo Frankel model universes of sets (ZFC models) as carriers for the mathematics of quantum mechanics is discussed. It is proved that given a standard transitive ZFC model M, if, inside M, B (H_{ M }) is the algebra of all bounded linear operators over a Hilbert spaceH_{ M }, there exists, outside M, a Hilbert spaceH and an algebraB (H), along with isometric monomorphisms U _{ M } and V _{ M } from H_{ M } into H and from B (H_{ M }) into B (H). U _{ M } and V _{ M } are used to relate quantum mechanics based on M to quantum mechanics based on the usual ZFC model. It is then shown that, contrary to what one would expect, all ZFC models may not be equivalent as carriers for the mathematics of physics. In particular, it is proved that if one requires that an outcome sequence, associated with an infinite repetition of measuring a question observable on a system prepared in some state, be random, and if a strong definition of randomness is used, then the minimal standard ZFC model cannot be a carrier for the mathematics of quantum mechanics.

Models of Zermelo Frankel set theory as carriers for the mathematics of physics. II
View Description Hide DescriptionThis paper continues the study of the use of different models of ZF set theory as carriers for the mathematics of quantum mechanics. The basic tool used here is the construction of Cohen extensions of ZFC models by use of Boolean valued ZFC models [C=axiom of choice]. Let M be a standard transitive ZFC model. Inside M, B (H_{ M }) is the algebra of all bounded linear operators over some Hilbert spaceH_{ M }. It is shown that with each state ρ in B (H_{ M }) and projection operator ο in B (H_{ M }) one can associate a unique Boolean valued ZFC model M^{B} _{ρο}. B_{ρο} is the algebra of all Borel subsets of {0,1}^{ω}, the set of all infinite 0–1 sequences, modulo sets of P _{ρο} =⊗p _{ρο} measure zero with p _{ρο}({1}) =Trρο in M. Let Ψ_{ M } and Φ_{ M } be respective maps from the sets of state preparation and question measuring procedures into B (H_{ M }). Let M=M_{0}, the minimal standard transitive ZFC model. It is then shown that with each state preparation procedure s ‐ Dom(Ψ_{ M } _{ 0 }) and each question measuring procedure q ‐ Dom(Φ_{ M } _{ 0 }) and with each infinite repetition (t s q) of doing s and q at times t (0), t (1),..., if the definition of randomness is sufficiently strong, one can associate the Cohen extension M_{0}[ψ_{ t s q }] of M_{0} by ψ_{ t s q }. ψ_{ t s q } is the random outcome sequence associated with (t s q). A third condition, in addition to the two given in the previous paper, is then given which must be satisfied if a ZFC model M is to serve as a carrier for the mathematics of quantum mechanics. In essence it says that for each pair (t s q) and (w u k) of distinct infinite repetitions of doing s and q and of doing u and k with s, u ‐ Dom(Ψ_{ M }) and q, k ‐ Dom(Φ_{ M }), the two outcome sequences ψ_{ t s q } and ψ_{ w u k } are mutually statistically independent. It is then shown that for a strong definition of independence, corresponding to the definition of randomness used previously, no Cohen extension M_{0}[ψ_{ t s q }] of M_{0} can serve as the carrier for the mathematics of quantum mechanics.

Conformal transformation for the Coulter–Weinberg form of the equations for mass zero spin‐2 field
View Description Hide DescriptionIn this work we show that it is impossible to introduce a third‐rank tensor potential that preserves the conformal covariance of the mass zero spin‐2 field equations in the Coulter–Weinberg scheme.

Charge quantization and canonical quantization
View Description Hide DescriptionDirac’s charge quantization condition is derived by means of a canonical quantization procedure of an enlarged classical phase space in which the interaction constant is a dynamical variable. The charge quantization condition follows by imposing a superselection rule. The method avoids string singularities and does not depend on spherical symmetry. The charge quantization condition is due solely to the topology of the enlarged classical configuration space.

On the dynamics of particles in a bounded region: A measure theoretical approach
View Description Hide DescriptionAn existence theorem is proven for the solution of the differential equations of motion of a finite number of particles moving in a bounded piecewise regular region and mutually interacting via C ^{1} forces. it is shown that the elastic reflection laws uniquely determine a Lebesque measuresolution of the differential equations of motion (with elasticboundary conditions). The Lebesgue measure is invariant so that an extension of the Liouville theorem to non‐Hamiltonian flows is obtained. A natural representation of the time evolution is given as a flow upward from a base under a ’’ceiling’’ function.

Conditioning of states
View Description Hide DescriptionA system of axioms for the state space of a quantum system is proposed which, together with the concept of conditioning a state by the occurrence of an event, leads to the construction of the standard orthomodular events system.

Algebraically special H‐spaces
View Description Hide DescriptionAll diverging algebraically special solutions of the complex vacuum Einstein equations which are left (or right) conformally flat (H‐spaces) are found explicitly. These metrics contain four arbitrary functions of two variables.

A class of states on the boson–fermion algebra
View Description Hide DescriptionOn the tensor product C*‐algebra of bosons and fermions a class of states determined by the two‐point functions is proved to exist. It is indicated how they induce a class of states on the CCR algebra which are not quasifree, but determined by their two‐ and four‐point functions.

U(5) ⊆O(5) ⊆O(3) and the exact solution for the problem of quadrupole vibrations of the nucleus
View Description Hide DescriptionOver twenty years ago A. Bohr discussed the quantum mechanical problem of the quadrupole vibrations in the liquid drop model of the nucleus. States of definite angular momentumL could not be obtained exactly except when L=0,3. In the present paper we indicate how we can determine states for arbitrary angular momentumL and definite number of quanta ν in terms of polynomials of the creation operators characterized by irreducible representation (IR) of the chain of groups U(5) ⊆O(3). We furthermore characterize the states by a definite IR λ of O(5) by replacing the creation operators by traceless ones. These states are fully determined by an extra label μ that gives the number of triplets of traceless creation operators coupled to angular momentum zero. We show then how all the wavefunctions of the problem discussed by Bohr can be obtained in a recursive fashion and briefly discuss some of their applications.

An approximant for democratic representation of all Born terms
View Description Hide DescriptionWe propose an approximant which attempts to reconstruct a solution starting from the Born terms of a formal power series. The approximant follows closely the Fredholm solutions to Born–Neumann series of completely continuous operators and their finite rank approximations. We show that if the Fredholm solution is written as a combination of Born terms, it tends to become independent of the expansion parameter λ asymptotically. The proposed approximants then appear as solutions to differential equations approximately satisfied by the formal power series, a feature they share with kernel of finite rank approximations. All Born terms are put on equal footing and their respective weight is determined independently of λ by initial conditions; thus knowledge of the solution and (N−1) derivatives at one point is necessary. Analytic and crossing‐symmetry properties are preserved by the proposed approximant, but unitarity is not insured and has to be examined specifically. Its error structure and its properties are studied and compared to Padé approximants.

A multipoint interpolation method based on variational principles for functionals of the solution to linear equations
View Description Hide DescriptionA method is derived for using variational expressions to interpolate among known values of a functional of the solution to linear equations. For linear functionals of the solution to an inhomogeneous equation, the interpolation expression is exact at N distinct points when N distinct functions are used, each of which is the solution of the underlying Euler equation. Two point variational interpolation is derived to interpolate on the value of an eigenvalue using the Rayleigh quotient. Illustrative examples are given based on neutron transport studies of fusion reactor blanket systems and applications to sensitivity and optimization studies in reactor theory are discussed.

Scattering of scalar waves from a Schwarzschild black hole
View Description Hide DescriptionThe scattering of scalar waves from a Schwarzschild black hole is investigated for wavelengths much less than the graviational radius (r_{s}). Explicit expressions for scattering parameters are obtained for two cases: high angular momenta and low angular momenta. In the first case we obtain the phase shifts and absorption coefficient with the JWKB method. The elastic differential cross section and the total absorption cross section are also calculated. For low angular momenta we present a method based in the DWBA (distorted wave Born approximation). With this method, the phase shifts and the absorption coefficients are obtained.

A geometric proof of no‐interaction theorems
View Description Hide DescriptionNo‐interaction theorems are proven, using the methods of modern differential geometry, and an example of a Hamiltonian yielding relativistic canonical equations of motion with interaction is presented.

The Kirkwood–Salsburg equation and phase transitions
View Description Hide DescriptionA perturbation expansion solution of the Kirkwood–Salsburg equation is used to investigate the hard sphere phase transition. A necessary condition for the phase transition identical in form to one obtained by Kirkwood is obtained. However, it is shown that this condition is not sufficient and that other conditions must also be satisfied.

Rigorous derivation of the Kirkwood–Monroe equation for small activity
View Description Hide DescriptionWe show, for small values of z, that the solution of the Kirkwood‐Salsburg equation approaches, in the norm topolgy, the solution of the Kirkwood‐Monroe and van Kampen equations if the potential of interaction is the Kac potential φ (x_{12}) =γ^{ s }g(γx_{12}) and the limit γ→0 is taken. We have to assume that the function g is bounded and absolutely integrable and that Σ_{ i≠j } (γx _{ i j }) ⩾−m B (B<∞), the the sum being performed over all pairs of the m particles.

Spontaneously broken symmetry and cosmological constant
View Description Hide DescriptionA solution of the Einstein equation with cosmological term produced by spontaneously broken symmetry is presented. The solution implies that the universe will recontract.

Bethe–Salpeter spinor equation at P _{μ}=0 and SO(5) spinor spherical harmonics
View Description Hide DescriptionWe study the B−S equaiton, in the ladder approximation, for the zero energy bound states of a spinor and a scalar particle interacting via the exchange of a massless scalar particle. Constructing and using a complete set of SO(5) spinor spherical harmonics, we find the SO(5) degenerate spectrum of the coupling constant and the bound state amplitudes up to a normalization constant. It turns out that the SO(5) symmetry is broken by these amplitudes in a peculiar way.

Asymptotic solutions and conservation laws for the nonlinear Schrödinger equation. I
View Description Hide DescriptionIn the absence of solitons, the nonlinear Schrödinger equation has an asymptotic solution which decays in time as t ^{−1/2}, and contains two arbitrary functions (in the amplitude and phase, respectively). For appropriate initial data, the amplitude function is uniquely determined in terms of the initial data by the conservation laws; the other function is undetermined. This method determines the leading two terms in each of the asymptotic expansions for the amplitude and phase, but no more. The method makes no direct use of th Marchenko integral equations.