Representations Of SU Research Articles

Yang-Mills field from fuzzy sphere quantum Kaluza-Klein model

Using the framework of quantum Riemannian geometry, we show that gravity on the product of spacetime and a fuzzy sphere is equivalent under minimal assumptions to gravity on spacetime, an su2-valued Yang-Mills field Aμi and a real-symmetric-matrix valued Liouville-sigma model field hij for gravity on the fuzzy sphere. Moreover, a massless real scalar field on the product appears as a tower of scalar fields on spacetime, with one for each internal integer spin l representation of SU(2), minimally coupled to Aμi and with mass depending on l and the fuzzy sphere size. For discrete values of the deformation parameter, the fuzzy spheres can be reduced to matrix algebras M2j+1(ℂ) for j any non-negative half-integer, and in this case only integer spins 0 ≤ l ≤ 2j appear in the multiplet. Thus, for j = 1 a massless field on the product appears as a massless SU(2) internal spin 0 field, a massive internal spin 1 field and a massive internal spin 2 field, in mass ratio 0, 1,3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{3} $$\\end{document} respectively, which we conjecture could arise in connection with an approximate SU(2) flavour symmetry.

A, B, C of three-qubit entanglement: three vectors to control it all

In this paper we are focusing on entanglement control problem in a three-qubit system. We demonstrate that vector representation of entanglement, associated with SO(6) representation of SU(4) two-qubit group, can be used to solve various control problems analytically including (i) the transformation between a W- type states and GHZ state, and (ii) manipulating bipartite concurrences and three-tangle under a restricted access to only two qubits, and (iii) designing USp(4)- type quaternionic operations and quantum states.

Bosonic Representation of Matrices and Angular Momentum Probabilistic Representation of Cyclic States.

The Jordan-Schwinger map allows us to go from a matrix representation of any arbitrary Lie algebra to an oscillator (bosonic) representation. We show that any Lie algebra can be considered for this map by expressing the algebra generators in terms of the oscillator creation and annihilation operators acting in the Hilbert space of quantum oscillator states. Then, to describe quantum states in the probability representation of quantum oscillator states, we express their density operators in terms of conditional probability distributions (symplectic tomograms) or Husimi-like probability distributions. We illustrate this general scheme by examples of qubit states (spin-1/2 su(2)-group states) and even and odd Schrödinger cat states related to the other representation of su(2)-algebra (spin-j representation). The two-mode coherent-state superpositions associated with cyclic groups are studied, using the Jordan-Schwinger map. This map allows us to visualize and compare different properties of the mentioned states. For this, the su(2) coherent states for different angular momenta j are used to define a Husimi-like Q representation. Some properties of these states are explicitly presented for the cyclic groups C2 and C3. Also, their use in quantum information and computing is mentioned.

A Differential-Geometric Approach to Quantum Ignorance Consistent with Entropic Properties of Statistical Mechanics.

In this paper, we construct the metric tensor and volume for the manifold of purifications associated with an arbitrary reduced density operator ρS. We also define a quantum coarse-graining (CG) to study the volume where macrostates are the manifolds of purifications, which we call surfaces of ignorance (SOI), and microstates are the purifications of ρS. In this context, the volume functions as a multiplicity of the macrostates that quantifies the amount of information missing from ρS. Using examples where the SOI are generated using representations of SU(2), SO(3), and SO(N), we show two features of the CG: (1) A system beginning in an atypical macrostate of smaller volume evolves to macrostates of greater volume until it reaches the equilibrium macrostate in a process in which the system and environment become strictly more entangled, and (2) the equilibrium macrostate takes up the vast majority of the coarse-grained space especially as the dimension of the total system becomes large. Here, the equilibrium macrostate corresponds to a maximum entanglement between the system and the environment. To demonstrate feature (1) for the examples considered, we show that the volume behaves like the von Neumann entropy in that it is zero for pure states, maximal for maximally mixed states, and is a concave function with respect to the purity of ρS. These two features are essential to typicality arguments regarding thermalization and Boltzmann's original CG.

Sterile neutrinos, 0nu beta beta decay and the W-boson mass anomaly in a flipped SU(5) from F-theory

We investigate the low energy properties of an effective local model with flipped SU(5)times U(1)_{chi } gauge group, constructed within the framework of F-theory. Its origin is traced back to the SO(10) symmetry – associated with a geometric singularity of the compactification manifold – broken by an internal flux which is turned on along the seven-brane in the U(1)_{chi } direction. Topological properties and the choice of flux parameters determine the massless spectrum of the model to be that of the minimal flipped SU(5) supplemented with an extra right-handed electron-type state and its complex conjugate, E^c+{{bar{E}}}^c, as well as neutral singlet fields. The subsequent symmetry breaking to the SU(3)times SU(2)times U(1)_Y gauge group occurs with a Higgs pair in 10+{overline{10}} representations of SU(5). Next we proceed to the phenomenological analysis of the resulting effective model and the salient outcomes are: The E^c+{{bar{E}}}^c pair acquires a mass of few TeV and as such could solve the g_{mu }-2 discrepancy. Neutrino couplings to extra neutral singlets lead to an inverse seesaw mechanism where an extra light state could be a suitable dark matter candidate. The predictions of the model for the {0nu }beta beta decay rate could be tested in near future experiments. There are non-unitarity deviations from the lepton mixing matrix (U_{PMNS}), which could in principle explain the new precision measurement of the W-boson mass recently reported by the CDF II collaboration.

A representation of SU(2, 2) which can be interpreted as describing chronometric fermions (proton, neutrino, and electron) in terms of a single composition series

In 1991 I. E. Segal stated (on the basis of a certain induced representation of the conformal group, but without proof) that there are four chronometric elementary particles of spin 1/2. In terms of the same representation (but using a different parallelization), we discuss a model where there are just three of those particles. They are interpreted as the proton, the (electronic) neutrino , and the electron, respectively. Mathematically, the three particles correspond to factors of the composition series. Each of the three representation spaces ( F p , F v , and F e , respectively) is provided with an invariant unitary structure and, in each of the three cases, energy positivity holds. These findings may shed some light on why there are lepton generations at all. The conclusions are mathematically based on the specific formula for the action of our representation in the full representation space F. This formula is presented in terms of the so-called reproducing kernel functions which are related to coherent states .

Solving gauge anomaly equations in the standard model using the method of chords

In a recent paper, Allanach et al (2020 J. High Energy Phys. JHEP05(2020)065) introduced a geometric method to solve the anomaly cancellation equations for a U(1) gauge theory with an arbitrary number of charges—the method of chords known in Diophantine analysis. We extend their result to non-abelian gauge groups, and show that this method can be used to find the general solution to the anomaly cancellation equations for a theory with the standard model gauge group on a curved background. Given K charges in (2, 3), L charges in (2, 1), M charges in (1, 3), and N charges in (1, 1) representations of SU(2) × SU(3), the equations reduce to a homogeneous cubic Diophantine equation in K + L + M + N − 4 variables.

Index-like theorem for massless fermions in spherically symmetric monopole backgrounds

In this paper we study massless fermions coupled to spherically symmetric SU(N) monopoles without Yukawa couplings between the Higgs and fermion fields. The corresponding Dirac operator is not Fredholm and the associated eigenfunctions are not L2-normalizable. Here we derive a formula for the dimension of the plane-wave normalizable kernel of such a Dirac operator for fermions of any representation of SU(N) in the presence of any spherically symmetric monopole background. Notably, our results also apply to fermions coupled to monopoles that preserve non-abelian gauge symmetry.

Standard model gauge fields localized on non-Abelian vortices in six dimensions

Abstract A brane-world SU(5) grand unified theory model with global non-Abelian vortices is constructed in six-dimensional spacetime. We find a solution with a vortex associated to SU(3) separated from another vortex associated to SU(2). This 3–2 split configuration achieves a geometric Higgs mechanism for SU(5) → SU(3) × SU(2) × U(1) symmetry breaking. A simple deformation potential induces a domain wall between non-Abelian vortices, leading to a linear confining potential. The confinement stabilizes the vortex separation moduli, and ensures the vorticities of the SU(3) and SU(2) groups are identical. This dictates the equality of the numbers of fermion zero modes in the fundamental representation of SU(3) (quarks) and SU(2) (leptons), leading to quark/lepton generations. The standard model massless gauge fields are localized on the non-Abelian vortices thanks to a field-dependent gauge kinetic function. We perform fluctuation analysis with an appropriate gauge fixing and obtain a four-dimensional effective Lagrangian of unbroken and broken gauge fields at quadratic order. We find that SU(3) × SU(2) × U(1) gauge fields are localized on the vortices and exactly massless. Complications in analyzing the spectra of gauge fields with the nontrivial gauge kinetic function are neatly worked out by a vector-analysis-like method.

A self-consistent framework of topological amplitude and its SU(N) decomposition

We propose a systematic theoretical framework for the topological amplitudes of the heavy meson decays and their SU(N) decomposition. In the framework, the topologies are expressed in invariant tensors and classified into tree- and penguin-operator-induced diagrams according to which four-quark operators, tree or penguin, being inserted into their effective weak vertexes. The number of possible topologies contributing to one type of decay can be counted by permutations and combinations. The Wigner-Eckhart theorem ensures the topological amplitudes under flavor symmetry are the same for different decay channels. By decomposing the four-quark operators into irreducible representations of SU(N) group, one can get the SU(N) irreducible amplitudes. Taking the D → PP decay (P denoting a pseudoscalar meson) with SU(3)F symmetry as an example, we present our framework in detail. The linear correlation of topologies in the SU(3)F limit is clarified in group theory. It is found there are only nine independent topologies in all tree- and penguin-operator-induced diagrams contributing to the D → PP decays in the Standard Model. If a large quark-loop diagram, named TLP, is assumed, the large ∆ACP and the very different D0→ K+K− and D0→ π+π− branching fractions can be explained with a normal U-spin breaking. Moreover, our framework provides a simple way to analyze the SU(N) breaking effects. The linear SU(3)F breaking and the high order U-spin breaking in charm decays are re-investigated in our framework, which are consistent with literature. Analogous to the degeneracy and splitting of energy levels, we propose the concepts of degeneracy and splitting of topologies to describe the flavor symmetry breaking effects in decay. As applications, we analyze the strange-less D decays in SU(3)F symmetry breaking into Isospin symmetry and the charm-less B decays in SU(4)F symmetry breaking into SU(3)F symmetry.

New universality classes of the non-Hermitian Dirac operator in QCD-like theories

In non-Hermitian random matrix theory there are three universality classes for local spectral correlations: the Ginibre class and the nonstandard classes ${\mathrm{AI}}^{\ifmmode\dagger\else\textdagger\fi{}}$ and ${\mathrm{AII}}^{\ifmmode\dagger\else\textdagger\fi{}}$. We show that the continuum Dirac operator in two-color QCD coupled to a chiral U(1) gauge field or an imaginary chiral chemical potential falls in class ${\mathrm{AI}}^{\ifmmode\dagger\else\textdagger\fi{}}$ (${\mathrm{AII}}^{\ifmmode\dagger\else\textdagger\fi{}}$) for fermions in pseudoreal (real) representations of SU(2). We introduce the corresponding chiral random matrix theories and verify our predictions in lattice simulations with staggered fermions, for which the correspondence between representation and universality class is reversed. Specifically, we compute the complex eigenvalue spacing ratios introduced recently. We also derive novel spectral sum rules.

Designing locally maximally entangled quantum states with arbitrary local symmetries

One of the key ingredients of many LOCC protocols in quantum information is a multiparticle (locally) maximally entangled quantum state, aka a critical state, that possesses local symmetries. We show how to design critical states with arbitrarily large local unitary symmetry. We explain that such states can be realised in a quantum system of distinguishable traps with bosons or fermions occupying a finite number of modes. Then, local symmetries of the designed quantum state are equal to the unitary group of local mode operations acting diagonally on all traps. Therefore, such a group of symmetries is naturally protected against errors that occur in a physical realisation of mode operators. We also link our results with the existence of so-called strictly semistable states with particular asymptotic diagonal symmetries. Our main technical result states that the Nth tensor power of any irreducible representation of SU(N) contains a copy of the trivial representation. This is established via a direct combinatorial analysis of Littlewood-Richardson rules utilising certain combinatorial objects which we call telescopes.

How Does the Planck Scale Affect Qubits?

Gedanken experiments in quantum gravity motivate generalised uncertainty relations (GURs) implying deviations from the standard quantum statistics close to the Planck scale. These deviations have been extensively investigated for the non-spin part of the wave function, but existing models tacitly assume that spin states remain unaffected by the quantisation of the background in which the quantum matter propagates. Here, we explore a new model of nonlocal geometry in which the Planck-scale smearing of classical points generates GURs for angular momentum. These, in turn, imply an analogous generalisation of the spin uncertainty relations. The new relations correspond to a novel representation of SU(2) that acts nontrivially on both subspaces of the composite state describing matter-geometry interactions. For single particles, each spin matrix has four independent eigenvectors, corresponding to two 2-fold degenerate eigenvalues ħ±(ħ+β)/2, where β is a small correction to the effective Planck’s constant. These represent the spin states of a quantum particle immersed in a quantum background geometry and the correction by β emerges as a direct result of the interaction terms. In addition to the canonical qubits states, |0⟩=|↑⟩ and |1⟩=|↓⟩, there exist two new eigenstates in which the spin of the particle becomes entangled with the spin sector of the fluctuating spacetime. We explore ways to empirically distinguish the resulting "geometric" qubits, |0′⟩ and |1′⟩, from their canonical counterparts.

Scalar anomaly cancellation reveals the hidden superalgebraic structure of the quantum chiral SU(2/1) model of leptons and quarks

At the classical level, the SU(2/1) superalgebra offers a natural description of the elementary particles: leptons and quarks massless states, graded by their chirality, fit the smallest irreducible representations of SU(2/1). Our new proposition is to pair the left/right space-time chirality with the superalgebra chirality and to study the model at the one-loop quantum level. If, despite the fact that they are non-Hermitian, we use the odd matrices of SU(2/1) to minimally couple an oriented complex Higgs scalar field to the chiral Fermions, novel anomalies occur. They affect the scalar propagators and vertices. However, these undesired new terms cancel out, together with the Adler-Bell-Jackiw vector anomalies, because the quarks compensate the leptons. The unexpected and striking consequence is that the scalar propagator must be normalized using the anti-symmetric super-Killing metric and the scalar-vector vertex must use the symmetric d_aij structure constants of the superalgebra. Despite this extraordinary structure, the resulting Lagrangian is actually Hermitian.

Special unitary particle pusher for extreme fields

An exact momentum update useful for particle codes was previously given. The expressions involved were unwieldy. By treating the momentum as a representation of SU(2)⊗SU(2) instead of SO(3,1), a more compact expression for the exact momentum update is obtained. An expansion in powers of the timestep can be formulated such that invariance properties are exactly preserved, and push rates comparable to the standard Boris pusher are obtained.

Two-loop beta function for complex scalar electroweak multiplets

We present the general form of the renormalizable four-point interactions of a complex scalar field furnishing an irreducible representation of SU(2), and derive a set of algebraic identities that facilitates the calculation of higher-order radiative corrections. As an application, we calculate the two-loop beta function for the SM extended by a scalar multiplet, and provide the result explicitly in terms of the group invariants. Our results include the evolution of the Higgs-portal couplings, as well as scalar “minimal dark matter”. We present numerical results for the two-loop evolution of the various couplings.

Generalized Lie Algebraic Geometry in R3x SO(3) Configuration Space for SU(3) of Elementary Particles and for Wave-packing of Atomic Structure

In this paper, we show, by extending Lie’s original 1871 thesis “on philosophical reflections upon the nature of Cartesian geometry” based on “transformation by which surfaces that touch each other are turned into similar surfaces … between the Plucker line geometry and a geometry whose elements are the space’s spheres” to include toroidal deformation of the sphere, how an algebraic geometric principle of duality between points, lines and planes of 3..dimesional space provides a sufficiently general framework for realizations of Lie-algebra and its Lie-isotopic and Lie-admissible generalizations in solid state configuration space R3×SO(3) compatible with translational periodicity of 3–dimensional space lattice. The generalization provide not only representation of SU(3) symmetry of extended (string-like) elementary particles with complementary duality of leptons and baryons, but also dual wave-packet representation of atomic structure and the periodic table, highlighting the significance of the fact that Mendeleev originally moulded his two-dimensional rendering of the periodic system on the dual Sanskrit grammar/phonetics.

Scattering amplitudes in mathcal{N} = 3 supersymmetric SU(N) Chern-Simons-matter theory at large N

We study mathcal{N} = 3 supersymmetric Chern-Simons-matter theory coupled to matter in the fundamental representation of SU(N). In the ’t Hooft large N limit, we compute the exact 2 → 2 scattering amplitudes of the fundamental scalar superfields to all orders in the ’t Hooft coupling λ. Our computations are presented in mathcal{N} = 1 superspace and make significant use of the residual SO(2)R symmetry in order to solve for the exact four-point correlator of the scalar superfields. By taking the on-shell limit, we are able to extract the exact 2 → 2 scattering amplitudes of bosons/fermions in the symmetric, anti-symmetric and adjoint channels of scattering. We find that the scattering amplitude of the mathcal{N} = 3 theory in the planar limit is tree-level exact to all orders in the ’t Hooft coupling λ. The result is consistent with the conjectured bosonization duality and is expected to have enhanced symmetry structures such as dual superconformal symmetry and Yangian symmetry.

Temperley–Lieb, Brauer and Racah algebras and other centralizers of 𝔰𝔲(2)

In the spirit of the Schur–Weyl duality, we study the connections between the Racah algebra and the centralizers of tensor products of three (possibly different) irreducible representations of s u ( 2 ) \mathfrak {su}(2) . As a first step we show that the Racah algebra always surjects onto the centralizer. We then offer a conjecture regarding the description of the kernel of the map, which depends on the irreducible representations. If true, this conjecture would provide a presentation of the centralizer as a quotient of the Racah algebra. We prove this conjecture in several cases. In particular, while doing so, we explicitly obtain the Temperley–Lieb algebra, the Brauer algebra and the one-boundary Temperley–Lieb algebra as quotients of the Racah algebra.

TetrahedralA4symmetry in anti-SU(5) GUT

We construct a flavor model in an anti-SU(5) GUT with a tetrahedral symmetry $A_4$. We choose a basis where $Q_{text{em}}=-\frac13$ quarks and charged leptons are already mass eigenstates. This choice is possible from the $A_4$ symmetry. Then, matter representation $\overline{10}_{-1}^{\rm\, matter}$ contains both a quark doublet and a heavy neutrino $N$, which enables us to use the $A_4$ symmetry to both $Q_{text{em}}=+\frac23$ quark masses and neutrino masses (through the see-saw via $N$). This is made possible because the anti-SU(5) breaking is achieved by the Higgs fields transforming as anti-symmetric representations of SU(5), $\overline{10}_{-1}^H\oplus 10_{+1}^H$, reducing the rank-5 anti-SU(5) group down to the rank-4 standard model group \smg. For possible mass matrices, the $A_4$ symmetry predictions on mass matrices at field theory level are derived. Finally, an illustration from string compactification is presented.