Resonances, scattering theory, and rigged Hilbert spaces

The problem of decaying states and resonances is examined within the framework of scattering theory in a rigged Hilbert space formalism. The stationary free,''in,'' and ''out'' eigenvectors of formal scattering theory, which have a rigorous setting in rigged Hilbert space, are considered to be analytic functions of the energy eigenvalue. The value of these analytic functions at any point of regularity, real or complex, is an eigenvector with eigenvalue equal to the position of the point. The poles of the eigenvector families give origin to other eigenvectors of the Hamiltonian: the singularities of the ''out'' eigenvector family are the same as those of the continued S matrix, so that resonances are seen as eigenvectors of the Hamiltonian with eigenvalue equal to their location in the complex energy plane. Cauchy theorem then provides for expansions in terms of ''complete'' sets of eigenvectors with complex eigenvalues of the Hamiltonian. Applying such expansions to the survival amplitude of a decaying state, one finds that resonances give discrete contributions with purely exponential time behavior; the background is of course present, but explicitly separated. The resolvent of the Hamiltonian, restricted to the nuclear space appearing in the rigged Hilbert space, can be continued across the absolutely continuous spectrum; the singularities of the continuation are the same as those of the ''out'' eigenvectors. The free, ''in'' and ''out'' eigenvectors with complex eigenvalues and those corresponding to resonances can be approximated by physical vectors in the Hilbert space, as plane waves can. The need for having some further physical information in addition to the specification of the total Hamiltonian is apparent in the proposed framework. The formalism is applied to the Lee–Friedrichs model and to the scattering of a spinless particle by a local central potential. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.

Gaussian-Maxwell beams

Gaussian-beam-type solutions to the Maxwell equations are constructed by using results from relativistic front analysis, and the propagation characteristics of these beams are analyzed. The rays of geometrical optics are shown to be the trajectories of energy flow, as given by the Poynting vector. The longitudinal components of the field vectors in the direction of the beam axis, though small, are shown to be essential for a consistent description.

Anisotropic Gaussian Schell-model beams: Passage through optical systems and associated invariants

Anisotropic Gaussian Schell-model (AGSM) fields and their transformation by first-order optical systems (FOS’s) forming Sp(4,R) are studied using the generalized pencils of rays. The fact that Sp(4,R), rather than the larger group SL(4,R), is the relevant group is emphasized. A convenient geometrical picture wherein AGSM fields and FOS’s are represented, respectively, by antisymmetric second-rank tensors and de Sitter transformations in a (3+2)-dimensional space is developed. These fields are shown to separate into two qualitatively different families of orbits and the invariants over each orbit, two in number, are worked out. We also develop another geometrical picture in a (2+1)-dimensional Minkowski space suitable for the description of the action of axially symmetric FOS’s on AGSM fields, and the invariants, now seven in number, are derived. Interesting limiting cases forming coherent and quasihomogeneous fields are analyzed.

Representation and properties of para-Bose oscillator operators. II. Coherent states and the minimum uncertainty states

The energy, position, and momentum eigenstates of a para-Bose oscillator system were considered in paper I. Here we consider the Bargmann or the analytic function description of the para-Bose system. This brings in, in a natural way, the coherent states ||z;alpha> defined as the eigenstates of the annihilation operator ?. The transformation functions relating this description to the energy, position, and momentum eigenstates are explicitly obtained. Possible resolution of the identity operator using coherent states is examined. A particular resolution contains two integrals, one containing the diagonal basis ||z;alpha><−z;alpha||. We briefly consider the normal and antinormal ordering of the operators and their diagonal and discrete diagonal coherent state approximations. The problem of constructing states with a minimum value of the product of the position and momentum uncertainties and the possible alpha dependence of this minimum value is considered. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.

Embedding Of Non-Simple Lie-Groups, Coupling-Constant Relations And Non-Uniqueness Of Models Of Unification

A general derivation of the coupling constant relations which result on embedding a non-simple group like SU L (2) @ U(1) in a larger simple group (or graded Lie group) is given. It is shown that such relations depend only on the requirement (i) that the multiplet of vector fields form an irreducible representation of the unifying algebra and (ii) the transformation properties of the fermions under SU L (2). This point is illustrated in two ways, one by constructing two different unification groups containing the same fermions and therefore have same Weinberg angle; the other by putting different SU L (2) structures on the same fermions and consequently have different Weinberg angles. In particular the value sin~0=3/8 is characteristic of the sequential doublet models or models which invoke a large number of additional leptons like E 6, while addition of extra charged fermion singlets can reduce the value of sin ~ 0 to 1/4. We point out that at the present time the models of grand unification are far from unique.

Differences in the Behavior of Luteinizing Hormones of Various Species at the Rat Gonadal Cell Receptor Site

The ability of different LH-like hormones, such as hCG, PMSG/equine (e) CG, ovine (o) LH, eLH, and rat (r) LH, to bind to and stimulate steroidogenesis in two types of rat gonadal cells was studied under the same experimental conditions. In both Leydig and granulosa cells, the maximal steroidogenic responses elicited by optimal doses of different LHs present during a 2-h incubation were comparable. However, if the cells were exposed to the different LHs for a brief period and then subjected to interference with hormone action by removing the unbound hormone from the medium by washing or adding specific antisera, differences were observed in the amount of steroid produced during subsequent incubation in hormone-free medium. Thus, in the case of hCG, either of these procedures carried out at 15 or 30 min of incubation had little inhibitory effect on the amount of steroid produced at 2 h, the latter being similar to that produced by cells incubated in the continued presence of hCG for 2 h. With eCG and rLH, the effect was dramatic, in that there was a total inhibition of subsequent steroidogenic response. In cells exposed to eLH and oLH, inhibition of subsequent steroidogenesis due to either removal of the free-hormone or addition of specific antisera at 15 or 30 min was only partial. Although all of the antisera used were equally effective in inhibiting the steroidogenic response to respective gonadotropins when added along with hormones at the beginning of incubation, differences were observed in the degree of inhibition of this response when the same antisera were added at later times of incubation. Thus, when antisera were added 60 min after the hormone, the inhibition of steroidogenesis was total (100%) for eCG, partial (10–40%) for eLH and oLH, and totally lacking in cells treated with hCG. From this, it appears that hCG bound to the receptor probably becomes unavailable for binding to its antibody with time, while in the case of eCG and other LHs used, the antibody can still inhibit the biological activity of the hormone. Studies with 125I-labeled hormones further supported the conclusion that hCG differs from all other LHs in being most tightly bound and, hence, least dissociable, while eCG and rLH dissociate most readily; oLH and eLH can be placed in between these hormones in the extent of their dissociability. (Endocrinology 116: 597–603,1985)

Superfluid vacuum carrying real Einstein-de Broglie waves

An earlier superfluid aether model pairing fermionic and antifermionic fields is invoked to explain Rauch's time dependent neutron interference results which now suggest that microobjects are waves and particles simultaneously. The covariant superfluid provides a medium which carries real Einstein-de Broglie “pilot” waves. Further consequences are discussed.

Follicle stimulating hormone secretion and dominant follicle growth during treatment of Bos indicus heifers with intra-vaginal progesterone releasing devices, oestradiol benzoate, equine chorionic gonadotrophin and prostaglandin F-2 alpha

The aim of this study was to investigate the effects on follicle stimulating hormone (FSH) secretion and dominant follicle (OF) growth, of treatment of Bos indicus heifers with different combinations of intra-vaginal progesterone releasing devices (IPRD), oestradiol benzoate (ODB), PGF(2 alpha), and eCG. Two-year-old Brahman (BN; n=30) and Brahman-cross (BNX; n=34) heifers were randomly allocated to three IPRD-treatments: (i) standard-dose IPRD [CM 1.56 g; 1.56 g progesterone (P-4); n = 17]; (ii) half-dose IPRD (CM 0.78 g; 0.78 g p(4); n=15); (iii) half-dose IPRD + 300 IU eCG at IPRD removal (CM 0.78 g+G; n=14); and, (iv) non-IPRD control (2 x PGF(2 alpha); n=18) 500 mu g cloprostenol on Days -16 and -2. IPRD-treated heifers received 250 mu g PGF(2 alpha) at IPRD insertion (Day 10) and IPRD removal (Day -2) and 1 mg ODB on Day -10 and Day -1. Follicular dynamics were monitored daily by trans-rectal ultrasonography from Day -10 to Day 1. Blood samples for determination of P-4 were collected daily and samples for FSH determination were collected at 12 h intervals from Day -9 to Day -2. A significant surge in concentrations of FSH was observed in the 2 x PGF(2 alpha), treatment 12 h prior and 48 h after follicular wave emergence, but not in the IPRD-treated heifers. Estimated mean concentrations of total plasma P-4 during the 8 days of IPRD insertion was greater (P<0.001) in the CM 1.56 g P-4 treated heifers compared to the CM 0.78 g P-4 treated heifers (18.38 ng/ml compared with 11.09 ng/ml, respectively). A treatment by genotype interaction (P=0.036) was observed in the mean plasma P4 concentration in heifers with no CL during IPRD insertion, whereby BN heifers in the CM 1.56 g treatment had greater plasma P-4 than the BNX heifers on Days-9, -7, -6, -5, and -4. However, there was no genotype effect in the CM 0.78 g +/- G or the 2 x PGF(2 alpha) treatment. Treatment had no effect on the DF growth from either day of wave emergence (P=0.378) or day of IPRD removal (P=0.780) to ovulation. This study demonstrates that FSH secretion in B. indicus heifers treated with a combination of IPRD's and ODB to synchronise ovulation was suppressed during the period of IPRD insertion but no significant effect on growth of the DF was observed. (C) 2013 Elsevier B.V. All rights reserved.

Non-Abelian monopoles break color. I. Classical mechanics

Monopoles which are sources of non-Abelian magnetic flux are predicted by many models of grand unification. It has been argued elsewhere that a generic transformation of the "unbroken" symmetry group H cannot be globally implemented on such monopoles for reasons of topology. In this paper, we show that similar topological obstructions are encountered in the mechanics of a test particle in the field of these monopoles and that the transformations of H cannot all be globally implemented as canonical transformations. For the SU(5) model, if H is SU(3)C×U(1)em, a consequence is that color multiplets are not globally defined, while if H is SU(3)C×SU(2)WS×U(1)Y, the same is the case for both color and electroweak multiplets. There are, however, several subgroups KT, KT′,… of H which can be globally implemented, with the transformation laws of the observables differing from group to group in a novel way. For H=SU(3)C×U(1)em, a choice for KT is SU(2)C×U(1)em, while for H=SU(3)C×SU(2)WS×U(1)Y, a choice is SU(2)C×U(1)×U(1)×U(1). The paper also develops the differential geometry of monopoles in a form convenient for computations.

Non-Abelian monopoles break color. II. Field theory and quantum mechanics

Many grand unified theories (GUT's) predict non-Abelian monopoles which are sources of non-Abelian (and Abelian) magnetic flux. In the preceding paper, we discussed in detail the topological obstructions to the global implementation of the action of the "unbroken symmetry group" H on a classical test particle in the field of such a monopole. In this paper, the existence of similar topological obstructions to the definition of H action on the fields in such a monopole sector, as well as on the states of a quantum-mechanical test particle in the presence of such fields, are shown in detail. Some subgroups of H which can be globally realized as groups of automorphisms are identified. We also discuss the application of our analysis to the SU(5) GUT and show in particular that the non-Abelian monopoles of that theory break color and electroweak symmetries.

Karl Josef Scheuermann Collection 1981-1986

Photocopies of correspondence between Scheuermann, Ernst Cramer, Cardinal Ratzinger, Reuben Hecht on Pope John XXIII and his alleged repentance prayer.

Paraxial Maxwell beams: transformation by general linear optical systems

Extending the work of earlier papers on the relativistic-front description of paraxial optics and the formulation of Fourier optics for vector waves consistent with the Maxwell equations, we generalize the Jones calculus of axial plane waves to describe the action of the most general linear optical system on paraxial Maxwell fields. Several examples are worked out, and in each case it is shown that the formalism leads to physically correct results. The importance of retaining the small components of the field vectors along the axis of the system for a consistent description is emphasized.

Leo Baeck Medal awarded to Axel Springer Coins and Medals

"Presented to Axel Springer devoted friend and generous benefactor of the Leo Baeck Institute 1977"

Superpropagator for a Nonpolynomial Field

By using a method originally due to Okubo we calculate the momentum-space superpropagator for a nonpolynomial field U(x)=1 / [1+fφ(x)] both for a massless and a massive neutral scalar φ(x) field. For the massless case we obtain a representation that resembles the weighted superposition of propagators for the exchange of a group of scalar fields φ(x) as is intuitively expected. The exact equivalence of this representation with the propagator function which has been obtained earlier through the use of the Fourier transform of a generalized function is established. For the massive case we determine the asymptotic form of the superpropagator.

Comparison of the pregnancy rates and costs per calf born after fixed-time artificial insemination or artificial insemination after estrus detection in Bos indicus heifers

This study compared pregnancy rates (PRs) and costs per calf born after fixed-time artificial insemination (FTAI) or AI after estrus detection (i.e., estrus detection and AI, EDAI), before and after a single PGF2α treatment in Bos indicus (Brahman-cross) heifers. On Day 0, the body weight, body condition score, and presence of a CL (46% of heifers) were determined. The heifers were then alternately allocated to one of two FTAI groups (FTAI-1, n = 139) and (FTAI-2, n = 141) and an EDAI group (n = 273). Heifers in the FTAI groups received an intravaginal progesterone-releasing device (IPRD; 0.78 g of progesterone) and 1 mg of estradiol benzoate intramuscularly (im) on Day 0. Eight days later, the IPRD was removed and heifers received 500 μg of PGF2α and 300 IU of eCG im; 24 hours later, they received 1 mg estradiol benzoate im and were submitted to FTAI 30 to 34 hours later (54 and 58 hours after IPRD removal). Heifers in the FTAI-2 group started treatment 8 days after those in the FTAI-1 group. Heifers in the EDAI group were inseminated approximately 12 hours after the detection of estrus between Days 4 and 9 at which time the heifers that had not been detected in estrus received 500 μg of PGF2α im and EDAI continued until Day 13. Heifers in the FTAI groups had a higher overall PR (proportion pregnant as per the entire group) than the EDAI group (34.6% vs. 23.2%; P = 0.003), however, conception rate (PR of heifers submitted for AI) tended to favor the estrus detection group (34.6% vs. 44.1%; P = 0.059). The cost per AI calf born was estimated to be $267.67 and $291.37 for the FTAI and EDAI groups, respectively. It was concluded that in Brahman heifers typical of those annually mated in northern Australia FTAI compared with EDAI increases the number of heifers pregnant and reduces the cost per calf born.