Volume 19, Issue 8, August 1978
Index of content:

Inönü–Wigner contractions of the real four‐dimensional Lie algebras
View Description Hide DescriptionAll Inönü–Wigner contractions of the real four‐dimensional Lie algebras are found. The results are summarized in tables.

Mesonic test fields and spacetime cohomology
View Description Hide DescriptionWe prove the following theorem: Classical spin‐0 and ‐1 mesonic test fields that can be constructed on a given spacetime manifold determine at least some of its de Rham cohomological structure. We explore this result and give some examples. The extension of the present technique to higher spin is also discussed.

Continuum S state wavefunctions for the Debye potential by Ecker–Weizel approximations
View Description Hide DescriptionAnalytical expressions for the continuum eigenfunctions for the Debye (Yukawa) or shielded Coulomb potential are derived by using Ecker–Weizel approximations. The results are used to obtain the S matrix in closed form.

Constraints of the Lorentz–Dirac equation
View Description Hide DescriptionBaylis and Huschilt pointed out that the Lorentz–Dirac equation with the usual constraint [?^{μ}(τ) →0 as τ→∞] for physical solution, may permit two or more solutions to some problems. We impose another constraint to make the physical solution unique.

The asymptotic behavior of bound eigenfunctions of Hamiltonians for single‐variable systems
View Description Hide DescriptionAn extremely simple technique for examining the asymptotic behavior of bound eigenfunctions of Hamiltonians for single‐variable systems is presented. A few simple examples are studied.

Projected basis set for the irreducible representation {2^{ N/2−S },1^{2S }}ofU(n)
View Description Hide DescriptionThe transformation properties of a projected basis for the irreducible representation (IR), {2^{ N/2−S },1^{2S }}, of U(n) under the elementary generators of the group have been studied. It has been found that these transformations are identical (to within a phase factor) with those of the standard bases spanning the given IR. The correspondence between this basis set and the standard basis set has also been indicated.

The Lorentz group in the oscillator realization. I. The group SO(2,1) and the transformation matrices connecting the SO(2) and SO(1,1) bases
View Description Hide DescriptionThe unitary transformation connecting the SO(2) and SO(1,1) bases for the principal and discrete series of representations of the three‐dimensional Lorentz group is determined by using the oscillator representation technique. The Hilbert space and the SO(1,1) basis, in this realization, have a simple appearance while the compact basis appears as the solution of an ordinary differential equation reducible to the confluent hypergeometric equation by a simple substitution. The Taylor expansion of this solution obtained by the use of certain functional identities yields the continuous spectrum of the SO(1,1) representations and the unitary transformation from the compact to the noncompact basis after the Sommerfeld–Watson transformation.

A simple derivation of the Onsager–Machlup formula for one‐dimensional nonlinear diffusion process
View Description Hide DescriptionTransforming the Fokker–Planck equation into a self‐adjoint form the Onsager–Machlup formula for a one‐dimensional nonlinear diffusion process is derived rigorously. By approximating the Wiener measure by an n‐gate cylinder measure, an equation of motion for the most probable path is also derived.

Recoupling coefficients of the symmetric group involving outer plethysms
View Description Hide DescriptionIn a series of papers we have examined the properties of certain double coset matrix elements (DCME) in the representations of the symmetric group S _{ N } that act as recoupling coefficients for outer products carried out via alternate subgroup sequences. In this paper we examine these same properties using symmetrized outer products in S _{ N }, which are also known as outer plethysms. The notions of double coset representative, symbol, and matrix element are extended to this case using the theory of semidirect products and little groups. The recoupling coefficients between bases symmetry adapted with respect to the usual outer product and the outer plethysm are examined in detail. Because of the Weyl–Schur construction of irreducible tensors, the recoupling theory of S _{ N } is central to a unified recoupling theory of the general linear group and its subgroups.

Recoupling coefficients of the general linear group in bases adapted to shell theories
View Description Hide DescriptionRecoupling coefficients for tensor representations of the general linear group Gl(n) are identified with analogous quantities in representations of the symmetric group S _{ N }. Two basis labeling schemes in Gl(n) are considered: (a) uses weights and outer product labels from S _{ N }, and (b) uses outer plethysms in S _{ N } and labels with respect to some elementary subgroup, usually SU(2). Scheme (a) corresponds to a generalized Gel’fand–Tsetlin basis and is the one usually adopted in elementary particletheories. Scheme (b) corresponds to the basis usually adopted in nuclear and atomic shell theory. The transformation between the two equivalent bases is identified with certain weighted double coset matrix elements (WDCME) of S _{ N }. Racah factors are generalized isoscalar factors in scheme (a) and have previously been identified with certain WDCME in that basis. In scheme (b) Racah factors determine the coefficients of fractional parentage (CFP) and are here identified with certain double coset matrix elements (DCME) of S _{ N }. Identification of these recoupling coefficients with the analogous quantities in S _{ N } exposes new symmetries and orthogonality properties of the coefficients which follow from the representation theory of S _{ N }. Some particular examples are verified by coefficients evaluated using well established techniques for SU(2).

Unitarily equivalent multiparticle Hamiltonian systems yielding equal scattering for corresponding states
View Description Hide DescriptionConsider a pair of nonrelativistic N‐particle (N?2) systems with unitarily equivalent Hamiltonians H and W*H W. Typically, W*H W involves nonlocal multiparticle interactions even when only local pair interactions are present in H. For the respective cases when H involves short‐range interactions alone and also when it involves long‐range ones, sufficient conditions are established in order for a scattering amplitude of the first system pertaining to any states in given entry and exit channels to equal the amplitude of the second system pertaining to corresponding states. This is accomplished by a time‐dependent approach. The correspondence in question assigns to each channel state of the first system a channel state of the second system in a bijective and intuitively natural manner. Nontrivial examples are given of unitary operators W for which the above equality holds for all channels of these systems. This applies to a large class of interactions in H, including interactions with suitable long‐range parts. The present work is the theoretical foundation of a new method of the authors, discussed elsewhere, for investigating many‐body nuclear forces phenomenologically.

Ground state representation of the infinite one‐dimensional Heisenberg ferromagnet. III. Scattering theory
View Description Hide DescriptionThis article gives a complete description of the scattering for the spin 1/2 Heisenberg ferromagnetic chain in its ground state representation.

Conditional probabilities and statistical independence in quantum theory
View Description Hide DescriptionThe problem of defining conditional probabilities and the notion of statistical independence in quantum theory is analyzed. It is shown that (unlike in classical probability theory) the conditional probabilities of a given set of events can be determined only if the sequence of all the experiments performed on the system is also specified. Such a specification is necessary also for the concept of statistical independence to become physically meaningful.

Entropy and phase transitions in partially ordered sets
View Description Hide DescriptionWe define the entropy function S (ρ) =lim_{ n→∞}2n ^{−2}ln N (n,ρ), where N (n,ρ) is the number of different partial order relations definable over a set of n distinct objects, such that of the possible n (n−1)/2 pairs of objects, a fraction ρ are comparable. Using rigorous upper and lower bounds for S (ρ), we show that there exist real numbers ρ_{1} and ρ_{2};.083<ρ_{1}⩽1/4 and 3/8⩽ρ_{2}<48/49; such that S (ρ) has a constant value (ln2)/2 in the interval ρ_{1}⩽ρ⩽ρ_{2}; but is strictly less than (ln2)/2 if ρ⩽.083 or if ρ⩾48/49. We point out that the function S (ρ) may be considered to be the entropy function of an interacting ’’lattice gas’’ with long‐range three‐body interaction, in which case, the lattice gas undergoes a first order phase transition as a function of the ’’chemical activity’’ of the gas molecules, the value of the chemical activity at the phase transition being 1. A variational calculation suggests that the system undergoes an infinite number of first order phase transitions at larger values of the chemical activity. We conjecture that our best lower bound to S (ρ) gives the exact value of S (ρ) for all ρ.

On the higher approximations of the K‐harmonics method
View Description Hide DescriptionExact formulas for the calculation of two‐body central interactions in the K‐harmonics method are derived. These formulas hold for any approximations and account for diagonal and off‐diagonal matrix elements as well.

Bifurcation from rotationally invariant states
View Description Hide DescriptionBifurcation in the presence of the rotation group is investigated. The covariant bifurcationequations are derived using the familiar angular momentum operators of quantum mechanics.Variational methods are also discussed. It is shown that the quadratic terms either vanish for odd l or possess a gradient structure for even l. This result is generalized to the case of an arbitrary simply reducible group. Applications to problems in geophysics and elasticity theory are discussed.

Systems of differential inequalities and stochastic differential equations. IV
View Description Hide DescriptionConsider the system of stochastic functional differential equationsx′ (t,ω) =f (t,x (t,ω), x _{ t }(ω),ω), x _{ t } _{ 0 }(ω) =φ_{0}(ω), where f (t,x (t,ω), x _{ t }(ω),ω) is a product measurable n‐dimensional random vector functional whenever x (t,ω) is a product measurable random function, and it satisfies the desired regularity conditions to assure the existence of solution process. By developing systems of random differential inequalities, very general comparison theorems in the framework of a vector Lyapunov function are obtained, and, furthermore, sufficient conditions are given for the stability of solutions in probability, in the mean and with probability one.

The Feynman maps and the Wiener integral
View Description Hide DescriptionBy introducing the family of Feynman maps F^{ s }, we show that our earlier definition of the Feynman path integral F=F^{1} can be obtained as the analytic continuation of the Wiener integral E =F^{−i }. This leads to some new results for the Wiener and Feynman integrals. We establish a translation and Cameron–Martin formula for the Feynman maps F^{ s }, having applications to nonrelativistic quantum mechanics. We also estalish a (weak) dominated convergence theorem for F^{1}=F.

The theory of superselection rules. I. A class of inequivalent, irreducible *‐representations of the canonical commutation relations of the free electromagnetic field
View Description Hide DescriptionWe begin here the rigorous construction of new superselection sectors for the free quantum electromagnetic field by exhibiting a wide class of inequivalent irreducible *‐representations of the canonical commutation relations of the electromagnetic field. The *‐representations constructed here satisfy all the axioms of Haag and Kastler, except possibly Poincaré covariance. In a forthcoming paper, the construction of the new superselection sectors is completed with the study of the spectrum condition for the *‐representations.

The optical group and its subgroups
View Description Hide DescriptionThe optical group Opt(3,1) is a ten‐dimensional maximal subgroup of the conformal group of space–time, characterized by the fact that it leaves a lightlike vector subspace in Minkowski space invariant. Thus it is the group underlying the symmetry structure of the parton model in particle physics. The present article is devoted to a complete classification of all closed connected subgroups of Opt(3,1). A list of representatives of all Lie subalgebras of the algebra opt(3,1) is given in the form of tables and many of their properties are established (their invariants, normalizers, isomorphism classes, etc.). Most of the subalgebras of opt(3,1) are also contained in the similitude algebra sim(3,1). We discuss a method for extracting the ’’new’’ subalgebras of opt(3,1) from the list; these will go over into a future list of subalgebras of the conformal Lie algrebra itself.