Clebsch-Gordan Coefficients Research Articles

An infinite–rank Lie algebra associated to SL(2,R) and SL(2,R)/U(1)

We construct a generalised notion of Kac-Moody algebras using smooth maps from the non-compact manifolds M= SL(2,R) and M= SL(2,R)/U(1) to a finite-dimensional simple Lie group G. This construction is achieved through two equivalent ways: by means of the Plancherel Theorem and by identifying a Hilbert basis within L2(M). We analyze the existence of central extensions and identify those in duality with Hermitean operators on M. By inspecting the Clebsch–Gordan coefficients of sl(2,R), we derive the Lie brackets characterising the corresponding generalised Kac-Moody algebras. The root structure of these algebras is identified, and it is shown that an infinite number of simultaneously commuting operators can be defined. Furthermore, we briefly touch upon applications of these algebras within the realm of supergravity, particularly in scenarios where the scalar fields coordinatize the non-compact manifold SL(2,R)/U(1).

‘The ‘Vector-Model’ wavefunction: spatial description and wavepacket formation of quantum-mechanical angular momenta’

We introduce the concept of an asymptotic spatial angular-momentum wavefunction, Xmjφ,θ,χ=eimφδθ−θm eij+12χ, which treats j in the jm state as a three-dimensional entity; φ, θ, χ are the Euler angles, and θm is the Vector-Model polar angle, given by cosθm=m/j. A wealth of geometric information about j can be deduced from the eigenvalues and spatial transformation properties of the Xmjφ,θ,χ. Specifically, the Xmjφ,θ,χ wavefunction gives a computationally simple description of the sizes of the particle and orbital-angular-momentum wavepackets (constructed from Gaussian distributions in j and m ), given by effective wavepacket angular uncertainty relations for ΔmΔφ, ΔjΔχ, and ΔφΔθ. The Xmjφ,θ,χ also predicts the position of the particle-wavepacket angular motion in the orbital plane, so that the particle-wavepacket rotation can be experimentally probed through continuous and non-destructive j -rotation measurements. Finally, we use the Xmj(φ,θ,χ) to determine geometrically well-known asymptotic expressions for Clebsch–Gordan coefficients, Wigner d-functions, the gyromagnetic ratio of elementary particles, g=2, and the m -state-correlation matrix elements. Interestingly, for low j, even down to j=1/2, these expressions are either exact (the last two) or excellent approximations (the first two), showing that the Xmj(φ,θ,χ) give a useful spatial description of quantum-mechanical angular momentum, and provide a smooth connection with classical angular momentum.

Quark-lepton mass relations from modular flavor symmetry

The so-called Golden Mass Relation provides a testable correlation between charged-lepton and down-type quark masses, that arises in certain flavor models that do not rely on Grand Unification. Such models typically involve broken family symmetries. In this work, we demonstrate that realistic fermion mass relations can emerge naturally in modular invariant models, without relying on ad hoc flavon alignments. We provide a model-independent derivation of a class of mass relations that are experimentally testable. These relations are determined by both the Clebsch-Gordan coefficients of the specific finite modular group and the expansion coefficients of its modular forms, thus offering potential probes of modular invariant models. As a detailed example, we present a set of viable mass relations based on the Γ4 ≅ S4 symmetry, which have calculable deviations from the usual Golden Mass Relation.

New orthogonality relationships of the Gaunt coefficients

A new analytical formula for the Gaunt coefficients is derived using expressions of Clebsch Gordan coefficients in terms of the generalized hypergeometric functions of unit argument and binomial coefficients. In addition, new sum expressions and orthogonality relations containing the Gaunt coefficients are written. These expressions are used to test the accuracy of the numerical calculation for the Gaunt coefficients. These formulae are coded in the Mathematica programming language, and the Gaunt coefficients are calculated numerically and compared with the values from the literature.

Insights into dioxygen binding on metal centers: an ab initio multireference electronic structure analysis.

Why does binding of dioxygen (O2) to metal centers, the initial step of O2 storage, transportation, and activation, almost inevitably induce metal-to-O2 single-electron transfer and generate superoxo (O2-˙) species, instead of genuine O02 adducts? To address this question, this study describes highly correlated wavefunction-based ab initio calculations using CASSCF/NEVPT2 (CASSCF = complete active space self-consistent field, and NEVPT2 = N-electron valence state second-order perturbation theory) approaches to explore the electronic-structure evolution of O2 association on Fe(II)(BDPP) (H2BDPP = 2,6-bis((2-(S)-diphenylhydroxylmethyl-1-pyrrolidinyl)methyl)pyridine) and Co(II)(BDPP) to produce S = 3 Fe(III)(BDPP)(O2-˙) (1) and Co(III)(BDPP)(O2-˙) (2). CASSCF/NEVPT2 calculations suggest that the processes furnishing 1 and 2 feature an avoided crossing resulting from interactions of two diabatic curves, of which one is characterized as Co(II) and Fe(II) centers interacting with a triplet O2 ligand and the other as Co(III) and Fe(III) centers bound to a superoxo ligand. In both cases, the avoided crossing induces a one-electron transfer from the divalent metal center to the incoming O2 ligand and leads to formation of trivalent metal-O2-˙ complexes. To facilitate the interpretation of complicated multireference wavefunctions, we formulated two-fragment spin eigenfunctions utilizing Clebsch-Gordan coefficients (CGCs) to rationalize computed spin populations on the metal centers and the O2 ligand and compared these results with usual valence bonding (VB) analyses. It turns out that both methods give the same results and are complementary to each other. Finally, the limitation of DFT approaches in describing complex electronic structures involving metal-ligand magnetic couplings is delineated.

3j-symbols for tensor product of symmetric representations

We propose an algebraic procedure to obtain quantum 3j-symbols (quantum Clebsch–Gordan coefficients) appearing in the decomposition of tensor product of symmetric representations. In fact, these symbols will be useful to write the spectral parameter dependent R-matrix elements for any bi-partite vertex model whose edges carry states of the symmetric representations.

Elementary coupling coefficients for the Wigner supermultiplet symmetry

An algorithm for evaluating Wigner coefficients of U(4) ⊃ SUS(2) ⊗ SUT(2) for the coupling [n14,n24,n34,n44]⊗[1,0,0,0]↓[m14,m24,m34,m44] with arbitrary irreducible representation [n14,n24,n34,n44] is provided based on the algebraic expressions with the expansion coefficients of the U(4)⊃SUS(2)⊗SUT(2) states in terms of those in the U(4) canonical basis and the Clebsch-Gordan coefficients of U(4) in the canonical basis. The state expansion coefficients are evaluated as the components of the null space vectors of the spin-isospin projection matrices. Program summaryProgram Title: Wigner-U4-SU2xSU2CPC Library link to program files:https://doi.org/10.17632/4b7gf2phh2.1Licensing provisions: GPLv3Programming language: Wolfram Mathematica version 9.0.1.0 and beyondNature of problem: Numerical evaluation of the elementary Wigner coefficients (EWCs) of U(4)⊃SUS(2)⊗SUT(2).Solution method: Mathematica implementation of the EWCs of U(4)⊃SUS(2)⊗SUT(2) [1] using the expansion coefficients of U(4)⊃SUS(2)⊗SUT(2) states in terms of those of the U(4) canonical basis determined as components of the null space vectors of the spin-isospin projection matrices [2].

Asymptotic behavior of Clebsch–Gordan coefficients

Asymptotic behavior of Clebsch–Gordan coefficients

Magnetic-field-induced splitting of Rydberg Electromagnetically Induced Transparency and Autler-Townes spectra in 87Rb vapor cell.

We theoretically and experimentally investigate the Rydberg electromagnetically induced transparency (EIT) and Autler-Townes (AT) splitting of 87Rb vapor under the combined influence of a magnetic field and a microwave field. In the presence of static magnetic field, the effect of the microwave field leads to the dressing and splitting of each mF state, resulting in multiple spectral peaks in the EIT-AT spectrum. A simplified analytical formula was developed to explain the EIT-AT spectrum in a static magnetic field, and the theoretical calculations agree qualitatively with experimental results. The Rydberg atom microwave electric field sensor performance was enhanced by making use of the splitting interval between the two maximum absolute mF states separated by the static magnetic field, which was attributed to the stronger Clebsch-Gordon coefficients between the extreme mF states and the frequency detuning of the microwave electric field under the static magnetic field. The traceable measurement limit of weak electric field by EIT-AT splitting method was extended by an order of magnitude, which is promising for precise microwave electric field measurement.

A closer look at the Yukawa interaction from a symmetry group perspective

Abstract I investigate the use of the SU(3) Clebsch–Gordan coefficients in light of the relations of completeness and closure. I show that in the case of αV = F/(F + D) ≠ 1, there is an additional interaction: the exchange of a ρ meson between a Λ and a Σ0 hyperon that only affects the symmetric coupling. I then calculate these additional coupling constants and show that this recovers the completeness and closure of the SU(3) Clebsch–Gordan coefficients for all values of αV. Besides, it increases the symmetry of the theory, now we can group the baryon octet into four doublets. Finally, I add the new coupling constants to study numerical results in the hyperon onset in dense nuclear matter assuming αV as a free parameter.

Quantum double structure in cold atom superfluids

The theory of topological quantum computation is underpinned by two important classes of models. One is based on non-abelian Chern–Simons theory, which yields the so-called SU(2)k anyon models that often appear in the context of electrically charged quantum fluids. The physics of the other is captured by symmetry broken Yang–Mills theory in the absence of a Chern–Simons term and results in the so-called quantum double models. Extensive resources have been invested into the search for SU(2)k anyon quasi-particles, in particular, the so-called Ising anyons (k = 2) of which Majorana zero modes are believed to be an incarnation. In contrast to the SU(2)k models, quantum doubles have attracted little attention in experiments despite their pivotal role in the theory of error correction. Beyond topological error correcting codes, the appearance of quantum doubles has been limited to contexts primarily within mathematical physics, and as such, they are of seemingly little relevance for the study of experimentally tangible systems. However, recent works suggest that quantum double anyons may be found in spinor Bose–Einstein condensates. In light of this, the core purpose of this article is to provide a self-contained exposition of the quantum double structure, framed in the context of spinor condensates, by constructing explicitly the quantum doubles for various ground state symmetry groups and discuss their experimental realisability. We also derive analytically an equation for the quantum double Clebsch–Gordan coefficients from which the relevant braid matrices can be worked out. Finally, the existence of a particle-vortex duality is exposed and illuminated upon in this context.

LEX-EFT: the Light Exotics Effective Field Theory

We propose the creation of a Light Exotics Effective Field Theory (LEX-EFT) catalog. LEX-EFT is a generic framework to capture all interactions between the Standard Model (SM) and all (or at least a large class of) theoretically allowed exotic states beyond the Standard Model (bSM), indexed by their SM and bSM charges. These states are light enough to be on or nearly on shell in some collider processes. This framework, which subsumes beyond the Standard Model paradigms as generally as possible, is meant to extend recent successful implementations of bSM EFTs and complement e.g. the Standard Model Effective Field Theory (SMEFT), which can capture the off-shell effects of exotic fields. In this work, we review a general method for the construction of a complete list of gauge-invariant operators involving SM interactions with light exotics via iterative tensor product decomposition, up to the desired order in mass dimension. Each operator is characterized by specific Clebsch-Gordan coefficients determined by the charge flow; we show how this charge flow affects the range of EFT validity and cross sections associated with an effective operator. We create an example catalog of exotic scalars coupling to SM gauge boson pairs, and we highlight some operators with exotic weak SU(2)L charges that can produce spectacular LHC phenomenology. We further demonstrate the utility of the LEX-EFT approach with several examples of effects on kinematic distributions and cross sections that would not be captured by EFTs agnostic to the exotic degrees of freedom and may evade the main inclusive collider searches tailored to the existing preferred set of standard bSM theories.

Isotropic N-point basis functions and their properties

Isotropic functions of positions , i.e. functions invariant under simultaneous rotations of all the coordinates, are conveniently formed using spherical harmonics and Clebsch–Gordan coefficients. An orthonormal basis of such functions provides a formalism suitable for analyzing isotropic distributions such as those that arise in cosmology, for instance in the clustering of galaxies as revealed by large-scale structure surveys. The algebraic properties of the basis functions are conveniently expressed in terms of 6-j and 9-j symbols. The calculation of relations among the basis functions is facilitated by ‘Yutsis’ diagrams for the addition and recoupling of angular momenta.

Geometric deep learning and equivariant neural networks

We survey the mathematical foundations of geometric deep learning, focusing on group equivariant and gauge equivariant neural networks. We develop gauge equivariant convolutional neural networks on arbitrary manifolds mathcal {M} using principal bundles with structure group K and equivariant maps between sections of associated vector bundles. We also discuss group equivariant neural networks for homogeneous spaces mathcal {M}=G/K, which are instead equivariant with respect to the global symmetry G on mathcal {M}. Group equivariant layers can be interpreted as intertwiners between induced representations of G, and we show their relation to gauge equivariant convolutional layers. We analyze several applications of this formalism, including semantic segmentation and object detection networks. We also discuss the case of spherical networks in great detail, corresponding to the case mathcal {M}=S^2=textrm{SO}(3)/textrm{SO}(2). Here we emphasize the use of Fourier analysis involving Wigner matrices, spherical harmonics and Clebsch–Gordan coefficients for G=textrm{SO}(3), illustrating the power of representation theory for deep learning.

Constructing the general partial wave and renormalization in effective field theory

We construct the general partial wave amplitude basis for the $N\ensuremath{\rightarrow}M$ scattering, which consists of Poincar\'e Clebsch-Gordan coefficients, in a Lorentz invariant form given in terms of the spinor-helicity variables. The inner product of the Clebsch-Gordan coefficients is defined, which converts on-shell phase space integration into an algebraic problem. We also develop the technique of partial wave expansions of arbitrary amplitudes, including those with infrared divergence. These are applied to the computation of anomalous dimension matrix for general effective operators, where unitarity cuts for the loop amplitudes, with an arbitrary number of external particles, are obtained via partial wave expansion.

Ghost center and representations of the diagonal reduction algebra of osp(1|2)

Reduction algebras are known by many names in the literature, including step algebras, Mickelsson algebras, Zhelobenko algebras, and transvector algebras, to name a few. These algebras, realized by raising and lowering operators, allow for the calculation of Clebsch-Gordan coefficients, branching rules, and intertwining operators; and have connections to extremal equations and dynamical R-matrices in integrable face models. In this paper we continue the study of the diagonal reduction superalgebra A of the orthosymplectic Lie superalgebra osp(1|2). We construct a Harish-Chandra homomorphism, Verma modules, and study the Shapovalov form on each Verma module. Using these results, we prove that the ghost center (center plus anti-center) of A is generated by two central elements and one anti-central element (analogous to the Scasimir due to Leśniewski for osp(1|2)). As another application, we classify all finite-dimensional irreducible representations of A. Lastly, we calculate an infinite-dimensional tensor product decomposition explicitly.

General spherical harmonic bra-ket overlap integrals of trigonometric functions

Closed formulas in terms of double sums of Clebsch–Gordan coefficients are computed for the evaluation of bra-ket spherical harmonic overlap integrals of a wide class of trigonometric functions. These analytical expressions can find useful application in problems involving non-separable wave equations, e.g. general-relativistic perturbation theory, electromagnetism, quantum theory, etc, wherein the overlap integrals arise from the coupling among different angular modes. We provide some examples related to linear perturbations of spinning black holes in general relativity and modified gravity, in which the analytical formulas for the overlap integrals are particularly useful to compute the black-hole spectrum.

The Missing Label of $$\mathfrak {su}_3$$ and Its Symmetry

We present explicit formulas for the operators providing missing labels for the tensor product of two irreducible representations of $$\mathfrak {su}_3$$ . The result is seen as a particular representation of the diagonal centraliser of $$\mathfrak {su}_3$$ through a pair of tridiagonal matrices. Using these explicit formulas, we investigate the symmetry of this missing label problem and we find a symmetry group of order 144 larger than what can be expected from the natural symmetries. Several realisations of this symmetry group are given, including an interpretation as a subgroup of the Weyl group of type $$E_6$$ , which appeared in an earlier work as the symmetry group of the diagonal centraliser. Using the combinatorics of the root system of type $$E_6$$ , we provide a family of representations of the diagonal centraliser by infinite tridiagonal matrices, from which all the finite-dimensional representations affording the missing label can be extracted. Besides, some connections with the Hahn algebra, Heun–Hahn operators and Bethe ansatz are discussed along with some similarities with the well-known symmetries of the Clebsch–Gordan coefficients.

An alternative formalism for modeling spin

We present an alternative formalism for modeling spin. The ontological elements of this formalism are base-2 sequences of length n. The machinery necessary to model physics is then developed by considering correlations between base-2 sequences. Upon choosing a reference base-2 sequence, a relational system of numbers can be defined, which we interpret as quantum numbers. Based on the properties of these relational quantum numbers, the selection rules governing interacting spin systems are derived from first principles. A tool for calculating the associated probabilities, which are the squared Clebsch–Gordan coefficients in quantum mechanics, is also presented. The resulting model offers a vivid information theoretic picture of spin and interacting spin systems. Importantly, this model is developed without making any assumptions about the nature of space-time, which presents an interesting opportunity to study emergent space-time models.

Fully linear band crossings at high symmetry points in layers: classification and role of spin–orbit coupling and time reversal

Symmetry imposed restrictions to the Hamiltonian are systematized and applied to all of 80 clusters of single/double ordinary/gray groups (320 groups in total), to single out linear (in all directions) band crossings and corresponding effective Hamiltonians in high-symmetry Brillouin zone points of layered materials. The resulting dispersion types are isotropic or anisotropic forms of: single cone (with double degenerate crossing point and non-degenerate branches, or four-fold degenerate crossing point with double degenerate conical branches), poppy-flower (four-fold degenerate crossing point with two pairs of non-degenerate mutually rotated conical branches), and fortune teller (with nodal lines). Further, we describe the nontrivial patterns of dispersions’ behavior in high symmetry points when symmetry is varied within a cluster. Namely, Clebsch–Gordan series of the products of spin representation with the integer ones are relevant when spin–orbit coupling is included, and clarify observed scenarios (gap closing, gap opening, cone preserving, cone splitting etc). Analogously, analysis of behavior of dispersions in transition from ordinary to gray group enlightens the role of time reversal symmetry. The results refine and expand data existing in literature, and interesting or even unexpected cases are singled out in discussion.