[en] This book constitutes the proceedings of an International School of Supersymmetry held in Mexico City in 1981. Lectures presented include an introduction to supersymmetry (symmetries in relativistic quantum field theory, supersymmetry in quantum field theory, Dirac matrices and Majorana spinors, supersymmetric Yang-Mills theory, scalar multiplet and auxiliary fields, supergravity, N=1 supersymmetric theories, extended supersymmetry algebras, representations of extended supersymmetry, N=4 supersymmetric Yang-Mills theory, extended supergravity), superfields (irreducible representations and chiral superfields, invariants and ''tensor calculus,'' gauge superfield, N=1 supergravity), grand unification with and without supersymmetry (supersymmetric models), Yang-Mills theories with global and local supersymmetry (Higgs and Superhiggs effect in unified field theories), and supergroups and their representations (fermion and Grassmann numbers, supertrace and superdeterminant, harmonic oscillator representation, the Tilde operator, eigenvalues of Casimir operators, branching rules, Kac-Dynkin diagrams and supertableaux)

[en] Crystallized Schroedinger cat states (male and female) are introduced on the base of extension of group construction for the even and odd coherent states of the electromagnetic field oscillator. The Wigner and Q functions are calculated and some are plotted for C₂, C₃, C₄, C₅, C_3v Schroedinger cat states. Quadrature means and dispersions for these states are calculated and squeezing and correlation phenomena are studied. Photon distribution functions for these states are given explicitly and are plotted for several examples. A strong oscillatory behavior of the photon distribution function for some field amplitudes is found in the new type of states

[en] A simple interaction potential for nuclear reactions is considered. It is found that this permits a clear understanding of the physical ideas underlying the current theoretical models used to explain the nature of 'molecular states'. The latter have been proposed as a possible explanation for the intermediate structure observed in the cross sections of some light-ion reactions. (author)

[en] We show that linear time-dependent invariants in spin or creation and annihilation operators can be used to determine the quantum solutions of non-stationary systems. These constants of motion are determined by the corresponding classical equations of motion. Also we establish classes of exactly solvable Hamiltonians parametrized in terms of the SU(2) and Sp(2) time-dependent group elements

[en] We derive compact expressions for the matrix elements of all relevant boson operators in the Interacting Boson Approximation Model (IBA) of Iachello and Arima, on the basis of eigenfunctions characterized by irreducible representations of the chain of groups U(6) contains U(5) contains O(5) contains O(3). Using the group theoretical analysis of a previous publication we evaluate in closed form the matrix elements of the Casimir operators of O(6) and SU(3) and of the operators associated with electromagnetic transitions and quadrupole moments. (author)

[en] A contracted version of the symplectic model, with the raising and lowering generators of Sp(6,R) replaced by boson creation and annihilation operators, is presented. The symmetry of this scheme is shown to be U_b(6) x U_s(3) where U_s(3) is the familiar 0ℎω Elliott shell-model symmetry and U_b(6) is the group of the six-dimensional oscillator that is generated by bilinear products of the boson (l=0 and 2) creation and annihilation operators. While this boson symmetry is realized in the same way as the U(6) group of the IBA model, they are different theories because in the present case the bosons are associated with intershell excitations and not intrashell ones. This scheme is also known as the U(3) boson model. Mathematical justification for the simplifying assumptions is provided through an application of the group deformation mechanism. A simple hamiltonian that takes into account the shell structure, couplings to major shells through a quadrupole-quadrupole interaction, and a residual rotor term is expressed in terms of generators of the U_b(6) x U_s(3) model. Calculated excitation spectra and E2 transition strengths for the ds-shell nuclei ²⁰Ne, ²²Ne and ²⁴Mg are compared with the available experimental numbers. (orig.)

[en] A shell-model description of the structure of low-energy states of heavy deformed nuclei is used to predict the number and strengths of 1⁺ states that couple to the ground state via a strong M1 transition. Results are given for the magnetic moments of 2_g⁺ states of various Gd, Er, U, and Pu isotopes and also the M1 transition strengths 1⁺ → O_g⁺, 1⁺ → 2_g⁺, and 1⁺ → 2_γ⁺ for several rare earth and actinide nuclei. (orig.)

[en] A simple FORTRAN program called ROTXSU3, which determines eigenvalues of the quantum rotor and the corresponding algebraic SU(3) model Hamiltonian, is introduced. General analytic expressions for matrix elements of the two Hamiltonians are given. These results are used to establish the equivalence of the SU(3) and rotor theories in the min(λ, μ) >> L limit, where λ and μ are SU(3) representation labels and L is the angular momentum. The results can also be used to study group expansion and deformation mechanism since T₅ x SO(3), the symmetry group of the quantum rotor, is a contraction of SU(3). A mapping between eigenvalues of the invariant operators of the two theories gives a relationship between the parameters of their Hamiltonians. This mapping also leads to a shell-model interpretation of the β and γ shape variables of the collective model. (orig.)

[en] The irreducible representation labels λ and μ of the SU(3) shell models are related to the shape variables β and γ of the collective model by invoking a linear mapping between eigenvalues of invariant operators of the two theories. All but one parameter of the theory is fixed if the shell-model result is required to reproduce the collective-model geometry. And for one special value of the remaining free parameter there is a simple linear relationship between the eigenvalues, λ_α, of the quadrupole matrix of the collective model and the SU(3) representation labels: λ₁=(-λ+μ)/3, λ₂=-(λ+2μ+3)/3, λ₃=(2λ+μ+3)/3. The correspondence between hamiltonians that describe rotations in each theory is also given. Results are shown for two cases, ²⁴Mg and ¹⁶⁸Er, to demonstrate that the simplest mapping yields excellent results for both energies and transition rates. For λ and/or μ large, the (β,γ) ↔ (λ,μ) correspondence introduced here reduces to the symplectic shell-model result. (orig.)

[en] It is well known how to expand in spherical harmonics the gradient of a radial function in turn multiplied by a spherical harmonic. This expansion involves the use of the Wigner--Eckart theorem for the familiar O(3) is contained inO(2) chain of groups, and leads to Wigner coefficients in the formula together with reduced matrix elements which are simple first order differential operators in the radial variable. In the present paper we extend the above analysis to the application of the momentum operator π/sub m/ to functions of the collective coordinates α/sub m/, m=2,1,0,-1,-2 associated with quadrupole vibrations. The spherical harmonics are now replaced by the complete but nonorthonormal set of functions chi/sup lambda//sub s/LM, characterized by the irreducible representations lambda,L,M of the O(5) is contained inO(3) is contained inO(2) chain of groups as well as by an extra labelling index s, that were derived in a previous publication. The application of the gradient to a product of a function F (β), β²=Σ/sub m/α/sub m/α/sup m/, by chi/sup lambda//sub s/LM requires an extension of the Wigner--Eckart theorem for the nonorthonormal basis. Results similar to the ones mentioned in the previous paragraph are obtained, though, of course, now we will have Wigner coefficients in the O(5) is contained in (3) is contained inO(2) chain which have already been derived and programmed. With the help of the gradient formula we discuss the effect of the operators [π x π]/sup L//sub m/, L=0,2,4, [α x π]/sup L//sub m/, L=1,3 on basis of the O(5) is contained inO(3) chain of groups and indicate some of their applications