Solution of the Hartree-Fock-Bogoliubov equations at high angular frequency

We consider solutions to some of the difficulties which arise when Hartree-Fock-Bogoliubov equations are solved at high angular frequency. These are mainly associated with the fact that then instead of particles always decoupling in pairs, may do so singly and hence lead to quasi-particle wavefunctions containing an odd number of particle components in the quasiparticle wave function. These solutions then also lead to a satisfactory, reliable and fast method for connecting the upper and lower branches of the moment of inertia versus square of the angular frequency curve, and usually reproduce the typical S-shape of the curve. We also show the importance of higher (> 0) particle-particle channel coupled angular momentum of the nucleon-nucleon potential for the rotational spectrum of deformed nuclei at high angular momentum. Some unsolved problems and peculiar cases which arise during the solution of Hartree-Fock-Bogoliubov equations are mentioned in the end, which may be a reflection on the unsatisfactory nature of the present method of obtaining higher angular momentum rotational bands.     

Alternative representation of the linear canonical integral transform

Starting with the Iwasawa-type decomposition of a first-order optical system (or ABCD system) as a cascade of a lens, a magnifier, and an orthosymplectic system (a system that is both symplectic and orthogonal), a further decomposition of the orthosymplectic system in the form of a separable fractional Fourier transformer embedded between two spatial-coordinate rotators is proposed. The resulting decomposition of the entire first-order optical system then shows a physically attractive representation of the linear canonical integral transformation, which, in contrast to Collins integral, is valid for any ray transformation matrix.     

Complex mean fields and unnatural parity pairing in the Hartree-Fock-Bogoliubov problem with symmetry-projection before the variation

The Hartree-Fock-Bogoliuboy problem with restoration of the broken symmetries before the variation as it has been formulated recently within the VAMPIR and EXCITED VAMPIR approaches is studied in more detail. Special emphasis is put on the implications of various symmetry restrictions imposed on the quasiparticle transformations. It is shown that the use of essentially complex transformations provides a possibility to account for time-odd unnatural parity pairing correlations in the wavefunctions without being forced to give up axial and time-reversal symmetry. It is illustrated that together with parity and neutron-proton mixing the use of such complex mean fields extends the applicability of the VAMPIR and EXCITED VAMPIR approaches to states with arbitrary spin and parity in both doubly even and doubly odd nuclear systems. The corresponding variational equations are explicitly evaluated and the numerical feasibility of their solution is investigated.     

Towards a loop representation for quantum canonical supergravity

We study several aspects of the canonical quantization of supergravity in terms of the Ashtekar variables. We cast the theory in terms of a GSU(2) connection and we introduce a loop representation. The solution space is similar to the loop representation of ordinary gravity, the main difference being the form of the Mandelstam identities. Physical states are in general given by knot invariants that are compatible with the GSU(2) Mandelstam identities. There is an explicit solution to all the quantum constraint equations connected with the Chern-Simons form, which coincides exactly with the Dubrovnik version of the Kauffman polynomial. This provides for the first time the possibility of finding explicit analytic expressions for the coefficients of that knot polynomial.     

Symmetry-conserving Hartree-Fock-Bogoliubov theory: (II). Number-projected cranked Hartree-Fock-Bogoliubov calculations in the rare earth region

The yrast line of the nuclei ¹⁶⁴Er and ¹⁶⁸Er is investigated within the cranking approximation by a full variation after exact number projection. The results are compared with those of the cranked shell model and of a self-consistent solution without number projection. In particular we discuss the behavior of pairing correlations in the region of very high spins, where a phase transition is expected.     

Helicity representation of the N-N bound-state problem

The two-nucleon bound-state problem described in the momentum space is presented in the helicity state formalism.     

Neutron radii and skins in the Hartree-Fock-Bogoliubov calculations

We examine the systematic predictions for proton and neutron radii in even-even nuclei made by the self-consistent Skyrme-Hartree-Fock-Bogoliubov theory. Such an approach allows us to describe nuclei far from stability, where the spatial extensions of a nuclear system crucially depend on the continuum effects. We concentrate on the influence of spherical shell structure on global behavior of radii. The (N, Z)-localization of neutron and proton skins is discussed.     

Symmetry restoration in Hartree-Fock-Bogoliubov based theories

We present a Pfaffian formula for projection and symmetry restoration for wave functions of the general Bogoliubov form, including quasiparticle excited states and linear combinations of them. This solves a long-standing problem in calculating states of good symmetry, arising from the sign ambiguity of the commonly used determinant formula. A simple example is given of projecting a good particle number and angular momentum from a Bogoliubov wave function in the Fock space of a single j-shell.     

基于相对论Hartree-Fock-Bogoliubov近似的新幻数研究（英文）

基于相对论Hartree-Fock-Bogoliubov （RHFB） 近似分别探索了质量-电荷极限下的超重元素与极端中质比下的奇特原子核中的新幻数问题。研究结果表明,赝自旋对称性的守恒和破缺与超重核区球形幻数结构的形成密切相关,并分别决定了中子与质子的新幻数结构。同时,理论模型之间的差异也与之密切相关。在中重奇特核区,RHFB近似很好地再现了Ca 同位素中的新幻数N = 32,34,其中同位旋矢量道中洛伦兹张量耦合扮演了较为关键的角色。以此为例,研究证明了显式考虑交换（Fock） 项的RHFB 近似的可靠性。Recent applications of the relativistic Hartree-Fock-Bogoliubov (RHFB) approach in exploring the new magicities under extreme conditions are presented for the superheavy elements with the limits of mass and charge and for the exotic nuclei with extreme neutron-to-proton ratios. It is found that the emergence of new magic shells in superheavy region is tightly related to the restoration and violation of pseudo-spin symmetry, respectively for the neutron and proton ones, in which the model deviations are indicated and discussed. In medium-heavy exotic nuclei, the occurrence of new magicity N = 32, 34 in Ca isotopes is well reproduced by the RHFB approach, in which the isovector Lorentz tensor couplings are found to play an essential role. The results exemplify that the RHFB approach, which considers the exchange (Fock) terms explicitly, furnishes a new theoretical instrument for advancing relativistic nuclear mean-field approaches.     

Complex extension of the representation of the symplectic group associated with the canonical commutation relations

The aim of the investigation is to extend the representation of the real symplectic group associated with the canonical commutation relations into the complex symplectic group. It is shown that an extension exists to a semigroup S such that Sp(2n, R) ? S ? Sp(2n, C). The construction of the extension is achieved by extensive use of Bargmann's reproducing kernel space. We are able to give a simple geometric model for the semigroup S: it is exactly the semigroup of all complex symplectic transformations which increase the U(n, n) “length”.     

Faddeev approach to the three-body problem in total-angular-momentum representation

For a system of three charged particles the Faddeev equations are derived in the total-angular-momentum representation. They have the form of coupled sets of partial differential equations in three-dimensional space and can be used to develop new efficient numerical procedures to tackle the three-body Coulomb problem. The asymptotic conditions at large distances corresponding both to binary scattering and bound-state problems are presented. The behaviour of the Faddeev components near the triple and double collision points is studied.     

Analogue of the Weyl representation of algebra of canonical commutation relations in the case of nonphysical particles

An analogue of the Weyl representation of the algebra of canonical commutation relations is proved to exist in the anti-Fock case achieved in Krein space.     

Temperature-dependent Hartree-Fock-Bogoliubov calculations in hot rotating nuclei

Cranked temperature-dependent Hartree-Fock-Bogoliubov (CTHFB) equations in the rotating frame are solved numerically for a realistic case in the rare-earth region, the nucleus ¹⁶⁴Er. The behavior of pairing correlations, shape degrees of freedom and other physical quantities are discussed as a function of angular momentum and temperature.     

On the character of the Hartree-Fock-Bogoliubov solutions in a rotating frame

The nature of the solutions of the Hartree-Fock-Bogoliubov equations in a rotating frame is described. The significance of solutions which involve negative quasi-particle energy is discussed in detail.     

Solution of the relativistic string problem

The quantum dynamics of the free Nambu-Nielson-Susskind string is known to be consistent only in s + 1 dimensional Minkowski space when s = 25 and the intercept α₀ = 1. Ad hoc modifications are required to allow s < 25. The source of this difficulty is pointed out and the system is solved covariantly and ghost free for any value of s > 1 without ad hoc modifications. The intercept value α₀ = 1 is excluded, but α₀ can be fitted to the experimental value. There are only transverse modes, and linearly rising Regge trajectories result as before.     

Solution of the gauge hierarchy problem

We propose a novel solution to the gauge hierarchy problem in theories with softly broken supersymmetry. Quantum effects can resuscitate classically sick theories, producing the large scale from the small supersymmetry breaking scale. We use this mechanism to construct realistic SU(6) and SU(5) GUTs which do not suffer from gauge hierarchy or fine tuning problems.