The angular pattern in the hyperfine structure of Xe I and Kr I atoms

Recent reviews of the hyperfine structure of xenon and krypton have highlighted the variety of the values taken by the hyperfine coefficients A and B of these atoms. These variations, as functions of the atomic angular momenta, were however not explained quantitatively. This article shows the simple picture and angular momentum algebra that make it possible to account for the observed trend. The only necessary approximations are to consider the interaction of the outer electron negligible with respect to the coupling of the p5 core with the nucleus, and to assume that the Racah ||(p5)j l[K]J F〉basis, conventionally used for the atomic states of noble gases, makes a fitting description of the hierarchy of their angular momentum couplings. The way the calculation corroborates the apparently erratic values of hyperfine coefficients A and B in Xe I and Kr I shows up as a confirmation of the validity of these approximations.

Fundamentals of Diatomic Molecular Spectroscopy

The interpretation of optical spectra requires thorough comprehension of quantum mechanics, especially understanding the concept of angular momentum operators. Suppose now that a transformation from laboratory-fixed to molecule-attached coordinates, by invoking the correspondence principle, induces reversed angular momentum operator identities. However, the foundations of quantum mechanics and the mathematical implementation of specific symmetries assert that reversal of motion or time reversal includes complex conjugation as part of anti-unitary operation. Quantum theory contraindicates sign changes of the fundamental angular momentum algebra. Reversed angular momentum sign changes are of heuristic nature and are actually not needed in analysis of diatomic spectra. This work addresses sustenance of usual angular momentum theory, including presentation of straightforward proofs leading to falsification of the occurrence of reversed angular momentum identities. This review also summarises aspects of a consistent implementation of quantum mechanics for spectroscopy with selected diatomic molecules of interest in astrophysics and in engineering applications.

Quantum gravity and Riemannian geometry on the fuzzy sphere

We study the quantum geometry of the fuzzy sphere defined as the angular momentum algebra $$[x_i,x_j]=2\imath \lambda _p \epsilon _{ijk}x_k$$ [ x i , x j ] = 2 ı λ p ϵ ijk x k modulo setting $$\sum _i x_i^2$$ ∑ i x i 2 to a constant, using a recently introduced 3D rotationally invariant differential structure. Metrics are given by symmetric $$3 \times 3$$ 3 × 3 matrices g and we show that for each metric there is a unique quantum Levi-Civita connection with constant coefficients, with scalar curvature $$ \frac{1}{2}(\mathrm{Tr}(g^2)-\frac{1}{2}\mathrm{Tr}(g)^2)/\det (g)$$ 1 2 ( Tr ( g 2 ) - 1 2 Tr ( g ) 2 ) / det ( g ) . As an application, we construct Euclidean quantum gravity on the fuzzy unit sphere. We also calculate the charge 1 monopole for the 3D differential structure.

First-order Theory of Fields with Arbitrary Spin

By exploiting the previously exposed results in 2-spinor geometry, a general description of fields of arbitrary spin is exposed and shown to admit a first-order Lagrangian which extends the theory of Dirac spinors. The needed bundle is the fibered direct product of a symmetric ‘main sector’—carrying an irreducible representation of the angular-momentum algebra—and an induced sequence of ‘ghost sectors’. Several special cases are considered; in particular, one recovers the Bargmann-Wigner and Joos-Weinberg equations.

Many-body Dyson equation approach to the seniority model of pairing

As is well known, the single level seniority model of pairing has been solved exactly, since long using angular momentum algebra. It is shown that it can also be solved using the Dyson equation of standard many-body theory. The formalism shows some interesting many-body aspects.

A first-order Lagrangian theory of fields with arbitrary spin

The bundles suitable for a description of higher-spin fields can be built in terms of a 2-spinor bundle as the basic “building block”. This allows a clear, direct view of geometric constructions aimed at a theory of such fields on a curved spacetime. In particular, one recovers the Bargmann–Wigner equations and the [Formula: see text]-dimensional representation of the angular-momentum algebra needed for the Joos–Weinberg equations. Looking for a first-order Lagrangian field theory we argue, through considerations related to the 2-spinor description of the Dirac map, that the needed bundle must be a fibered direct sum of a symmetric “main sector” — carrying an irreducible representation of the angular-momentum algebra — and an induced sequence of “ghost sectors”. Then one indeed gets a Lagrangian field theory that, at least formally, can be expressed in a way similar to the Dirac theory. In flat spacetime, one gets plane-wave solutions that are characterized by their values in the main sector. Besides symmetric spinors, the above procedures can be adapted to anti-symmetric spinors and to Hermitian spinors (the latter describing integer-spin fields). Through natural decompositions, the case of a spin-2 field describing a possible deformation of the spacetime metric can be treated in terms of the previous results.

SINGLE BOSON AND INVERSION BOSON REALIZATIONS OF MULTI-PARAMETER NONLINEARLY DEFORMED ANGULAR MOMENTUM ALGEBRA

Explicit, analytic and closed expressions for boson realizations of the (m+3)-parameter nonlinearly deformed angular momentum algebra [Formula: see text] with its highest power m of polynomial function being arbitrary, which combines and generalizes Witten's two deformation schemes, are investigated in terms of the single boson and the single inversion boson, respectively. For each kind, the unitary Holstein–Primakoff-like realization, the non-unitary Dyson–Maléev-like realization and their connections are respectively discussed. Using these realizations, the corresponding representations of [Formula: see text] as well as their respective acting spaces in the Fock space are obtained.