Angular Momentum Algebra Research Articles

The centre of the Dunkl total angular momentum algebra

For a finite dimensional representation V of a finite reflection group W, we consider the rational Cherednik algebra Ht,c(V,W) associated with (V,W) at the parameters t≠0 and c. The Dunkl total angular momentum algebra Ot,c(V,W) arises as the centraliser algebra of the Lie superalgebra osp(1|2) containing a Dunkl deformation of the Dirac operator, inside the tensor product of Ht,c(V,W) and the Clifford algebra generated by V.We show that when dim⁡V≥3 and for every value of the parameter c, the centre of Ot,c(V,W) is isomorphic to a univariate polynomial ring. Notably, the generator of the centre changes depending on whether or not (−1)V is an element of the group W. Using this description of the centre, and using the projection of the pseudo scalar from the Clifford algebra into Ot,c(V,W), we establish results analogous to “Vogan's conjecture” for a family of operators depending on suitable elements of the double cover W˜.

Spin-flip dynamics in core-excited states in the basis of irreducible spherical tensor operators.

Recent experimental advances in ultrafast science have put different processes occurring on the electronic timescale below a few femtoseconds in focus. In the present theoretical work, we demonstrate how the transformation and propagation of the density matrix in the basis of irreducible spherical tensors can be conveniently used to study sub-few fs spin-flip dynamics in core-excited transition metal compounds. With the help of the Wigner-Eckart theorem, such a transformation separates the essential dynamical information from the geometric factors governed by the angular momentum algebra. We show that an additional reduction can be performed by the physically motivated truncation of the spherical tensor basis. In particular, depending on the degree of coherence, the ultrafast dynamics can be considered semi-quantitative in the notably reduced spherical basis when only the total populations of the basis states of the given spin are of interest. Such truncation should be especially beneficial when the number of high-spin basis states is vast, as it reduces computational costs.

A new look at the Dirac quantization condition

The angular momentum of any quantum system should be unambiguously quantized. We show that such a quantization fails for a pure Dirac monopole due to a previously overlooked field angular momentum from the monopole-electric charge system coming from the magnetic field of the Dirac string and the electric field of the charge. Applying the point-splitting method to the monopole-charge system yields a total angular momentum which obeys the standard angular momentum algebra, but which is gauge variant. In contrast it is possible to properly quantize the angular momentum of a topological ’t Hooft–Polyakov monopole plus charge. This implies that pure Dirac monopoles are not viable – only ’t Hooft–Polyakov monopoles are theoretically consistent with angular momentum quantization and gauge invariance.

Pseudogauge freedom and the SO(3) algebra of spin operators

The energy-momentum and spin tensors for a given theory can be replaced by alternative expressions that obey the same conservation laws for the energy, linear momentum, as well as angular momentum but, however, differ by the local redistribution of such quantities (with global energy, linear momentum, and angular momentum remaining unchanged). This arbitrariness is described in recent literature as the pseudogauge freedom or symmetry. In this letter, we analyze several pseudogauges used to formulate the relativistic hydrodynamics of particles with spin 12 and conclude that the canonical version of the spin tensor has an advantage over other forms as only the canonical definition defines the spin operators that fulfill the SO(3) algebra of angular momentum. This result sheds new light on the results encountered in recent papers demonstrating pseudogauge dependence of various physical quantities. It indicates that for spin-polarization observables, the canonical version is fundamentally better suited for building a connection between theory and experiment.

Controlling Rotationally Resolved Two-Dimensional Infrared Spectra with Polarization.

Recent advancements in infrared frequency combs will enable facile recording of coherent two-dimensional infrared spectra of gas-phase molecules with rotational resolution (RR2DIR). Using time-dependent density-matrix perturbation theory and angular momentum algebra techniques, we derive new polarization conditions unique to freely rotating molecules and absent in the condensed phase. These polarization conditions can be used to suppress parts of 2DIR rovibrational response, clarifying complicated RR2DIR spectra. With the polarization control methods described here, RR2DIR spectroscopy can be a powerful tool for studying complex gas mixtures of polyatomic molecules.

Theory of rotationally resolved two-dimensional infrared spectroscopy including polarization dependence and rotational coherence dynamics

Two-dimensional infrared (2DIR) spectroscopy is widely used to study molecular dynamics but it is typically restricted to solid and liquid phase samples and modest spectral resolution. Only recently, its potential to study gas-phase dynamics is beginning to be realized. Moreover, the recently proposed technique of cavity-enhanced 2D spectroscopy using frequency combs and developments in multi-comb spectroscopy are expected to dramatically advance capabilities for acquisition of rotationally-resolved 2DIR spectra. This demonstrates the need for rigorous and quantitative treatment of rotationally-resolved, polarization-dependent third-order response of gas-phase samples. In this article, we provide a rigorous and quantitative description of rotationally-resolved 2DIR spectroscopy using density-matrix, time-dependent perturbation theory and angular momentum algebra techniques. We describe the band and branch structure of 2D spectra, decompose the molecular response into polarization-dependence classes, use this decomposition to derive and explain special polarization conditions and relate the liquid-phase polarization conditions to gas-phase ones. Furthermore, we discuss the rotational coherence dynamics during waiting time.

Richard Newland Dixon. 25 December 1930—25 May 2021

Richard Newland Dixon began his independent scientific career as a molecular spectroscopist in Sheffield, where he demonstrated new ways to measure, analyse and interpret the electronic spectra of small gas phase species. He later moved to the chair of theoretical chemistry at the University of Bristol, where he embraced new experimental methods, particularly techniques enabled by continuous wave and pulsed lasers, and honed his theoretical skills in angular momentum algebra and time-dependent quantum mechanics. This blend of experimental and theoretical expertise allowed him to make pioneering contributions in the field of gas phase molecular photodissociation dynamics. Richard will be remembered as a polite and modest man, and a very sharp scientist. Among his many lasting scientific achievements, he will be remembered for developing strategies for determining state-resolved correlations between the vectorial properties of the products of molecular photofragmentation processes, and for being one of the first to recognize the near ubiquitous importance of non-adiabatic couplings between potential energy surfaces in rationalizing the dynamics of such dissociations.

Dirac Operators for the Dunkl Angular Momentum Algebra

We define a family of Dirac operators for the Dunkl angular momentum algebra depending on certain central elements of the group algebra of the Pin cover of the Weyl group inherent to the rational Cherednik algebra. We prove an analogue of Vogan's conjecture for this family of operators and use this to show that the Dirac cohomology, when non-zero, determines the central character of representations of the angular momentum algebra. Furthermore, interpreting this algebra in the framework of (deformed) Howe dualities, we show that the natural Dirac element we define yields, up to scalars, a square root of the angular part of the Calogero-Moser Hamiltonian.

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 p 5 core with the nucleus, and to assume that the Racah 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.

Generalized Bloch model: A theory for pulsed magnetization transfer.

The paper introduces a classical model to describe the dynamics of large spin-1/2 ensembles associated with nuclei bound in large molecule structures, commonly referred to as the semi-solid spin pool, and their magnetization transfer (MT) to spins of nuclei in water. Like quantum-mechanical descriptions of spin dynamics and like the original Bloch equations, but unlike existing MT models, the proposed model is based on the algebra of angular momentum in the sense that it explicitly models the rotations induced by radiofrequency (RF) pulses. It generalizes the original Bloch model to non-exponential decays, which are, for example, observed for semi-solid spin pools. The combination of rotations with non-exponential decays is facilitated by describing the latter as Green's functions, comprised in an integro-differential equation. Our model describes the data of an inversion-recovery magnetization-transfer experiment with varying durations of the inversion pulse substantially better than established models. We made this observation for all measured data, but in particular for pulse durations smaller than 300μs. Furthermore, we provide a linear approximation of the generalized Bloch model that reduces the simulation time by approximately a factor 15,000, enabling simulation of the spin dynamics caused by a rectangular RF-pulse in roughly 2μs. The proposed theory unifies the original Bloch model, Henkelman's steady-state theory for MT, and the commonly assumed rotation induced by hard pulses (i.e., strong and infinitesimally short applications of RF-fields) and describes experimental data better than previous models.

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 a heuristic nature and are actually not needed in analysis of diatomic spectra. This review 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 summarizes 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]=2imath lambda _p epsilon _{ijk}x_k modulo setting sum _i x_i^2 to a constant, using a recently introduced 3D rotationally invariant differential structure. Metrics are given by symmetric 3 times 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). 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.

Matrix elements of unitary group generators in many-fermion correlation problem. II. Graphical methods of spin algebras

This second instalment continues our survey inter-relating various approaches to the evaluation of matrix elements (MEs) of the unitary group generators, their products, and spin-orbital U(2n) generators, in the basis of the electronic Gel’fand–Tsetlin (G–T) [or Gel’fand–Paldus (G–P) or P-] states spanning the carrier spaces of the two-column irreducible representations (irreps) of the unitary group approach (UGA) to the many-electron correlation problem. Exploiting an intimate relationship between the U(n), $${\mathcal {S}}_N$$ , and SU(2) groups we rely here on the SU(2) formalism within the confines of UGA rather than on an earlier approach relying on the permutation group $${\mathcal {S}}_N$$ . The key is a close relationship between the P-states of UGA and the Yamanouchi–Kotani (Y–K) states of SU(2) that differ only in a phase. This enables one to exploit the powerful graphical techniques of spin (or angular momentum) algebras as first developed by Jucys. We show the usefulness and power of these techniques in the problem of UGA ME evaluation which, moreover, leads automatically to their segmentation, first derived for single generators by Shavitt and for generator products by Boyle and Paldus. We also show how the same technique can be successfully employed for MEs of spin-dependent U(2n) generators in the UGA P-basis.

The Dirac H-atom using the phase as an extra variable

In this paper, an algebraic solution for the bound states of the relativistic hydrogen atom is presented. The method discussed here adds an operator associated with the phase of the energy eigenstates to the set of variables of the problem. In terms of this set, appropriate ladder operators are constructed in order to express the full solution of the Dirac hydrogen equation. These ladder operators are used to form the Lie algebra of a su(1, 1) group, in the same way as that applied to angular momentum algebra and the ladder operators L±. The elements of the vector space associated with the representation of this algebra are related to a generalization of the Laguerre polynomials of the non integer index, also known as Sonine polynomials. In addition, we find that the eigenvalues of the operator constructed with the sum of the square of the three su(1, 1) generators gives precisely the relativistic energy spectrum of the hydrogen atom, including its angular momentum dependency.

GUP and its Application to Angular Momentum Algebra

The generalization of the Heisenberg uncertainty principle has been proposed by different theories of quantum gravity. This deformation affects the quantum phenomena. In this work, we consider the modification of the linear momentum based on the non-locality, and by using it, we calculate the deformed commutation relation between position and momentum. Then, we show how this generalization of the commutation relation affects the angular momentum algebra and its spectrum.

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.

Monogamous nature of Dicke-class of states with two distinct Majorana spinors

Monogamy relations place restrictions on the shareability of quantum corellations in multipartite states. Being an intrinsic quantum feature, monogamy property throws light on {\emph{residual}} entanglement, an entanglement which is not accounted for by the pairwise entanglement in the state. Expressed in terms of suitable pairwise entanglement measures such as concurrence, the monogamy inequality leads to the evaluation of {\emph{tangle}}, a measure of residual entanglement. In this work, we explore monogamy relations in pure symmetric multiqubit states constituted by two distinct spinors, the so-called {\emph{Dicke-class}} of states. Pure symmetric $N$-qubit states constituted by permutation of two orthogonal qubits form the well-known Dicke states. Those $N$-qubit pure symmetric states constructed by permutations of two non-orthogonal qubits are a one-parameter class of generalized Dicke states. With the help of Majorana geometric representation and angular momentum algebra, we analyze the bounds on monogamy inequality, expressed in terms of squared concurrence/squared negativity of partial transpose. We show that the states with equal distribution of the two spinors are more monogamous and hence possess larger residual entanglement when compared to other inequivalent classes with different degeneracy configurations.

τ decay into a pseudoscalar and an axial-vector meson

We study theoretically the decay $\tau^- \to \nu_\tau P^- A$, with $P^-$ a $\pi^-$ or $K^-$ and $A$ an axial-vector resonance $b_1(1235)$, $h_1(1170)$, $h_1(1380)$, $a_1(1260)$, $f_1(1285)$ or any of the two poles of the $K_1(1270)$. The process proceeds through a triangle mechanism where a vector meson pair is first produced from the weak current and then one of the vectors produces two pseudoscalars, one of which reinteracts with the other vector to produce the axial resonance. For the initial weak hadronic production we use a recent formalism to account for the hadronization after the initial quark-antiquark pair produced from the weak current, which explicitly filters G-parity states and obtain easy analytic formulas after working out the angular momentum algebra. The model also takes advantage of the chiral unitary theories to evaluate the vector-pseudoscalar amplitudes, where the axial-vector resonances were obtained as dynamically generated from the VP interaction. We make predictions for invariant mass distribution and branching ratios for the channels considered.

Line shape and D(*)D¯(*) probabilities of ψ(3770) from the e+e−→DD¯ reaction

We have performed a calculation of the $D\bar D$, $D\bar D^\ast$, $D^\ast\bar D$, $D^\ast\bar D^\ast$ components in the wave function of the $\psi(3770)$. For this we make use of the $^3P_0$ model to find the coupling of $\psi(3770)$ to these components, that with an elaborate angular momentum algebra can be obtained with only one parameter. Then we use data for the $e^+e^-\to D\bar D$ reaction, from where we determine a form factor needed in the theoretical frame work, as well as other parameters needed to evaluate the meson-meson selfenergy of the $\psi(3770)$. Once this is done we determine the $Z$ probability to still have a vector core and the probability to have the different meson components. We find $Z$ about $80\sim85\%$, and the individual meson-meson components are rather small, providing new empirical information to support the largely $q\bar q$ component of vector mesons, and the $\psi(3770)$ in particular.

A simple approach to solve the time independent Schröedinger equation for 1D systems

A simple algebraic approach based on the well known angular momentum SU(2) algebra is presented to describe 1D systems for arbitrary potentials. The approach is based on the dimension increase of the 1D harmonic oscillator space through the addition of a scalar boson, keeping constant the total number of bosons. In this new space the realization of the coordinate and momentum correspond to components of the angular momentum algebra, which in turn define the coordinate and momentum representation bases. This remarkable result allows the Hamiltonian matrix representationto be expressed in terms of a diagonal matrix characterizing the potential we are considering. The solutions are obtained as an expansion of harmonic oscillator functions by purely algebraic means. As an example of our approach the Morse and asymmetric double Morse potentials are considered.