An Explicit Construction of Casimir Operators and Eigenvalues : I

We give a general method to construct a complete set of linearly independent Casimir operators of a Lie algebra with rank N. For a Casimir operator of degree p, this will be provided by an explicit calculation of its symmetric coefficients $g^{A_1,A_2,.. A_p}$. It is seen that these coefficients can be descibed by some rational polinomials of rank N. These polinomials are also multilinear in Cartan sub-algebra indices taking values from the set $I_0 = {1,2,.. N}$. The crucial point here is that for each degree one needs, in general, more than one polinomials. This in fact is related with an observation that the whole set of symmetric coefficients $g^{A_1,A_2,.. A_p}$ is decomposed into sum subsets which are in one to one correspondence with these polinomials. We call these subsets clusters and introduce some indicators with which we specify different clusters. These indicators determine all the clusters whatever the numerical values of coefficients $g^{A_1,A_2,.. A_p}$ are. For any degree p, the number of clusters is independent of rank N. This hence allows us to generalize our results to any value of rank N. To specify the general framework explicit constructions of 4th and 5th order Casimir operators of $A_N$ Lie algebras are studied and all the polinomials which specify the numerical value of their coefficients are given explicitly.Comment: 14 pages, no figures, revised version, to appear in Jour.Math.Phy

On an Alternative Parametrization for the Theory of Complex Spectra

The purpose of this letter is threefold : (i) to derive, in the framework of a new parametrization, some compact formulas of energy averages for the electrostatic interaction within an (nl)N configuration, (ii) to describe a new generating function for obtaining the number of states with a given spin angular momentum in an (nl)N configuration, and (iii) to report some apparently new sum rules, actually a by-product of (i), for SU(2) > U(1) coupling coefficients.Comment: Published in Physics Letters A 147, 417-422 (1990

Solvability of eigenvalues in jn configurations

Eigenvalues of eigenstates in jn configurations (n identical nucle- ons in the j -orbit) are functions of two-body energies. In some cases they are linear combinations of two-body energies whose coe+/-cients are independent of the interaction and are rational non-negative num- bers. It is shown here that a state which is an eigenstate of any two-body interaction has this solvability property. This includes, in particular, any state with spin J if there are no other states with this J in the jn configuration. It is also shown that eigenstates with solvable eigenvalues have definite seniority v and thus, exhibit partial dynamical symmetry

Casimir operators of the exceptional group G2

We calculate the degree 2 and 6 Casimirs operators in explicit form, with the generators of G2 written in terms of the subalgebra A2Comment: 10 p., MAD/TH/93-05, (LaTex

Evidence for higher order QED in e+ e- pair production at RHIC

A new lowest order QED calculation for RHIC e+ e- pair production has been carried out with a phenomenological treatment of the Coulomb dissociation of the heavy ion nuclei observed in the STAR ZDC triggers. The lowest order QED result for the experimental acceptance is nearly two standard deviations larger than the STAR data. A corresponding higher order QED calculation is consistent with the data.Comment: 4 pages, 4 figures, latex, revte

On characteristic equations, trace identities and Casimir operators of simple Lie algebras

Two approaches are developed to exploit, for simple complex or compact real Lie algebras g, the information that stems from the characteristic equations of representation matrices and Casimir operators. These approaches are selected so as to be viable not only for `small' Lie algebras and suitable for treatment by computer algebra. A very large body of new results emerges in the forms, a) of identities of a tensorial nature, involving structure constants etc. of g, b) of trace identities for powers of matrices of the adjoint and defining representations of g, c) of expressions of non-primitive Casimir operators of g in terms of primitive ones. The methods are sufficiently tractable to allow not only explicit proof by hand of the non-primitive nature of the quartic Casimir of g2, f4, e6, but also e.g. of that of the tenth order Casimir of f4.Comment: 39 pages, 8 tables, late

Symmetry and Supersymmetry in Nuclear Pairing: Exact Solutions

Pairing plays a crucial role in nuclear spectra and attempts to describe it has a long history in nuclear physics. The limiting case in which all single particle states are degenerate, but with different s-wave pairing strengths was only recently solved. In this strong coupling limit the nuclear pairing Hamiltonian also exhibits a supersymmetry. Another solution away from those limits, namely two non-degenerate single particle states with different pairing strengths, was also given. In this contribution these developments are summarized and difficulties with possible generalizations to more single particle states and d-wave pairing are discussed.Comment: 6 pages of LATEX, to be published in the Proceedings of the "10th Int. Spring Seminar on Nuclear Physics: New Quests in Nuclear Structure", Vietri Sul Mare, May 21-25, 201

New Relations for Coefficients of Fractional Parentage--the Redmond Recursion Formula with Seniority

We find a relationship between coefficients of fractional parentage (cfp) obtained on the one hand from the principal parent method and on the other hand from a seniority classification. We apply this to the Redmond recursion formula which relates $n \to n+1$ cfp's to $n-1 \to n$ cfp's where the principal parent classification is used. We transform this to the seniority scheme. Our formula differs from the Redmond formula inasmuch as we have a sum over the possible seniorities for the $n \to n+1$ cfp's, whereas Redmond has only one term.Comment: RevTex4, 17 pages; added Appendix A, with proof for the new relation; corrected Eqs.(26),(38), and (39

Multipair approach to pairing in nuclei

The ground state of a general pairing Hamiltonian for a finite nuclear system is constructed as a product of collective, real, distinct pairs. These are determined sequentially via an iterative variational procedure that resorts to diagonalizations of the Hamiltonian in restricted model spaces. Different applications of the method are provided that include comparisons with exact and projected BCS results. The quantities that are examined are correlation energies, occupation numbers and pair transfer matrix elements. In a first application within the picket-fence model, the method is seen to generate the exact ground state for pairing strengths confined in a given range. Further applications of the method concern pairing in spherically symmetric mean fields and include simple exactly solvable models as well as some realistic calculations for middle-shell Sn isotopes. In the latter applications, two different ways of defining the pairs are examined: either with J=0 or with no well-defined angular momentum. The second choice reveals to be more effective leading, under some circumstances, to solutions that are basically exact.Comment: To appear in Physical Review

Revealing Fundamental Physics from the Daya Bay Neutrino Experiment using Deep Neural Networks

Experiments in particle physics produce enormous quantities of data that must be analyzed and interpreted by teams of physicists. This analysis is often exploratory, where scientists are unable to enumerate the possible types of signal prior to performing the experiment. Thus, tools for summarizing, clustering, visualizing and classifying high-dimensional data are essential. In this work, we show that meaningful physical content can be revealed by transforming the raw data into a learned high-level representation using deep neural networks, with measurements taken at the Daya Bay Neutrino Experiment as a case study. We further show how convolutional deep neural networks can provide an effective classification filter with greater than 97% accuracy across different classes of physics events, significantly better than other machine learning approaches