Lie Algebra Of SO Research Articles

Hamiltonian formulation of gravity as a spontaneously-broken gauge theory of the Lorentz group

A number of approaches to gravitation have much in common with the gauge theories of the standard model of particle physics. In this paper, we develop the Hamiltonian formulation of a class of gravitational theories that may be regarded as spontaneously-broken gauge theories of the complexified Lorentz group SO(1,3)C with the gravitational field described entirely by a gauge field valued in the Lie algebra of SO(1,3)C and a ‘Higgs field’ valued in the group’s fundamental representation. The theories have one free parameter β which appears in a similar role to the inverse of the Barbero–Immirzi parameter of Einstein–Cartan theory. However, contrary to that parameter, it is shown that the number of degrees of freedom (DOFs) crucially depends on the value of β. For non-zero values of β, it is shown the theories possesses three complex DOFs, and for the specific values β=±i an extension to general relativity is recovered in a symmetry-broken regime. For the value β = 0, the theory possesses no local DOFs. A non-zero value of β corresponds to the self-dual and anti-self-dual gauge fields appearing asymmetrically in the action, therefore in these models, the existence of gravitational DOFs is tied to chiral asymmetry in the gravitational sector.

Integrable nonlocal nonlinear Schrödinger equations associated with 𝑠𝑜(3,ℝ)

We construct integrable PT-symmetric nonlocal reductions for an integrable hierarchy associated with the special orthogonal Lie algebra so ⁡ ( 3 , R ) \operatorname {so}(3,\mathbb {R}) . The resulting typical nonlocal integrable equations are integrable PT-symmetric nonlocal reverse-space, reverse-time and reverse-spacetime nonlinear Schrödinger equations associated with so ⁡ ( 3 , R ) \operatorname {so}(3,\mathbb {R}) .

Wilson loops in mathcal{N} = 4 SO(N) SYM and D-branes in AdS5 \xd7 \u211d\u21195

We study the half-BPS circular Wilson loop in mathcal{N} = 4 super Yang-Mills with orthogonal gauge group. By supersymmetric localization, its expectation value can be computed exactly from a matrix integral over the Lie algebra of SO(N). We focus on the large N limit and present some simple quantitative tests of the duality with type IIB string theory in AdS5× ℝℙ5. In particular, we show that the strong coupling limit of the expectation value of the Wilson loop in the spinor representation of the gauge group precisely matches the classical action of the dual string theory object, which is expected to be a D5-brane wrapping a ℝℙ4 subspace of ℝℙ5. We also briefly discuss the large N, large λ limits of the SO(N) Wilson loop in the symmetric/antisymmetric representations and their D3/D5-brane duals. Finally, we use the D5-brane description to extract the leading strong coupling behavior of the “bremsstrahlung function” associated to a spinor probe charge, or equivalently the normalization of the two-point function of the displacement operator on the spinor Wilson loop, and obtain agreement with the localization prediction.

Numerical comparisons based on new NCP-functions

We consider studying numerical comparisons based on variants of new NCP-functions denoted by . The nonlinear programming (NLP) can be converted to nonlinear complementarity problem (NCP) by employing the Karush-Kuhn Tucker (KKT) optimality conditions. One of the most popular ways to solve NCP is Lagrangian globalization (LG) method by transforming NCP as a system of nonsmooth (semismooth) equations. The second one is a novel method named the fictitious time integration method (FTIM). We reformulate NCP as a system of nonlinear algebraic equations (NAEs) and then construct an ordinary differential equation (ODE) by utilizing time-like functions. A group preserving scheme (GPS) is a set of ODEs which is a tool for systematically reformulating into the nonlinear dynamical system (NDS). Afterward, the NDS can be manipulated to numerical equations through activating the Lorentz group SO 0(n, 1) and its Lie algebra SO 0(n, 1). The FTIM will be operated on this numerical equation in numerical simulations for getting approximation solutions. All of the numerical experiments are carried out in performance profile theories. The comparisons of new NCP-functions by utilizing FTIM and LG method will be discussed in numerical experiments. Lastly, an accurate test of both FTIM and LG method is going to be committed by performance profile concepts as well.

Pfaffian particles and strings in SO(2N) gauge theories

We introduce (generalised) Pfaffian operators into our lattice calculations of the mass spectra and confining string tensions of SO(2N) gauge theories, complementing the conventional trace operators used in previous lattice calculations. In SO(6) the corresponding ‘Pfaffian’ particles match the negative charge conjugation particles of SU(4), thus resolving a puzzle arising from the observation that SO(6) and SU(4) have the same Lie algebra. The same holds true (but much more trivially) for SO(2) and U(1). For SO(4) the Pfaffian particles are degenerate with, but orthogonal to, those obtained with the usual single trace operators. That is to say, there is a doubling of the spectrum, as one might expect given that the Lie algebra of SO(4) is the same as that of SU(2) × SU(2). Additional SO(8) and SO(10) calculations of the Pfaffian spectrum confirm the naive expectation that these masses increase with N, so that they cease to play a role in the physics of SO(N) gauge theories as N → ∞. We also calculate the energies of Pfaffian ‘strings’ in these gauge theories. Although all our lattice calculations are for gauge theories in D = 2 + 1, similar conclusions should hold for D = 3 + 1.

Bi-integrable and tri-integrable couplings of a soliton hierarchy associated with SO(4)

Abstract In our paper, the theory of bi-integrable and tri-integrable couplings is generalized to the discrete case. First, based on the six-dimensional real special orthogonal Lie algebra SO(4), we construct bi-integrable and tri-integrable couplings associated with SO(4) for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Moreover, Hamiltonian structures of the obtained bi-integrable and tri-integrable couplings are constructed by the variational identities.

Global robust image rotation from combined weighted averaging

In this paper we present a novel rotation averaging scheme as part of our global image orientation model. This model is based on homologous points in overlapping images and is robust against outliers. It is applicable to various kinds of image data and provides accurate initializations for a subsequent bundle adjustment. The computation of global rotations is a combined optimization scheme: First, rotations are estimated in a convex relaxed semidefinite program. Rotations are required to be in the convex hull of the rotation group SO(3), which in most cases leads to correct rotations. Second, the estimation is improved in an iterative least squares optimization in the Lie algebra of SO(3). In order to deal with outliers in the relative rotations, we developed a sequential graph optimization algorithm that is able to detect and eliminate incorrect rotations. From the beginning, we propagate covariance information which allows for a weighting in the least squares estimation. We evaluate our approach using both synthetic and real image datasets. Compared to recent state-of-the-art rotation averaging and global image orientation algorithms, our proposed scheme reaches a high degree of robustness and accuracy. Moreover, it is also applicable to large Internet datasets, which shows its efficiency.

Bi-Integrable and Tri-Integrable Couplings of a Soliton Hierarchy Associated with SO(3)

Based on the three-dimensional real special orthogonal Lie algebra SO(3), by zero curvature equation, we present bi-integrable and tri-integrable couplings associated with SO(3) for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Moreover, Hamiltonian structures of the obtained bi-integrable and tri-integrable couplings are constructed by applying the variational identities.

A New Soliton Hierarchy Associated withso⁡3,Rand Its Conservation Laws

Based on the three-dimensional real special orthogonal Lie algebraso(3,R), we construct a new hierarchy of soliton equations by zero curvature equations and show that each equation in the resulting hierarchy has a bi-Hamiltonian structure and thus integrable in the Liouville sense. Furthermore, we present the infinitely many conservation laws for the new soliton hierarchy.

GLOBAL ROTATION ESTIMATION USING WEIGHTED ITERATIVE LIE ALGEBRAIC AVERAGING

Abstract. In this paper we present an approach for a weighted rotation averaging to estimate absolute rotations from relative rotations between two images for a set of multiple overlapping images. The solution does not depend on initial values for the unknown parameters and is robust against outliers. Our approach is one part of a solution for a global image orientation. Often relative rotations are not free from outliers, thus we use the redundancy in available pairwise relative rotations and present a novel graph-based algorithm to detect and eliminate inconsistent rotations. The remaining relative rotations are input to a weighted least squares adjustment performed in the Lie algebra of the rotation manifold SO(3) to obtain absolute orientation parameters for each image. Weights are determined using the prior information we derived from the estimation of the relative rotations. Because we use the Lie algebra of SO(3) for averaging no subsequent adaptation of the results has to be performed but the lossless projection to the manifold. We evaluate our approach on synthetic and real data. Our approach often is able to detect and eliminate all outliers from the relative rotations even if very high outlier rates are present. We show that we improve the quality of the estimated absolute rotations by introducing individual weights for the relative rotations based on various indicators. In comparison with the state-of-the-art in recent publications to global image orientation we achieve best results in the examined datasets.

Bi-Integrable Couplings of a New Soliton Hierarchy Associated withSO(4)

Based on the six-dimensional real special orthogonal Lie algebraSO(4), a new Lax integrable hierarchy is obtained by constructing an isospectral problem. Furthermore, we construct bi-integrable couplings for this hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Hamiltonian structures of the obtained bi-integrable couplings are constructed by the variational identity.

Two soliton hierarchies associated with SO(4) and the applications of SU(2) ⊗ SU(2) ≅ SO(4)

Based on the six-dimensional real special orthogonal Lie algebra SO(4), we construct two different isospectral problems. And the two hierarchies of soliton equations from zero curvature equations can be generated. At last, we use the property of the isomorphism SU(2) ⊗ SU(2) ≅ SO(4), and obtain the relations between the hierarchy of SU(2) ⊗ SU(2) and the hierarchy of SO(4).

Euclidean and super Euclidean algebras and Localizations of Uq((2)) and Uq((1|2))

In Ref. 1 we described homomorphisms of Uq((2)) and U((2)) into localizations of U((2)) and Uq((2)), respectively, with U((2)) being the universal enveloping algebra of the Lie algebra of SO(2) × sT2 the Euclidean group in the plane, with T2 denoting translations of the plane and × s semidirect product. In particular, we obtained explicit expressions for the generators of Uq((2)) and U((2)), respectively as irrational functions of U((2)) and Uq((2)). Herewith we further develop our results extending them to Uq((1|2)) and U ) with being the super Euclidean algebra.

Cartan gravity, matter fields, and the gauge principle

Gravity is commonly thought of as one of the four force fields in nature. However, in standard formulations its mathematical structure is rather different from the Yang–Mills fields of particle physics that govern the electromagnetic, weak, and strong interactions. This paper explores this dissonance with particular focus on how gravity couples to matter from the perspective of the Cartan-geometric formulation of gravity. There the gravitational field is represented by a pair of variables: (1) a ‘contact vector’ VA which is geometrically visualized as the contact point between the spacetime manifold and a model spacetime being ‘rolled’ on top of it, and (2) a gauge connection AμAB, here taken to be valued in the Lie algebra of SO(2,3) or SO(1,4), which mathematically determines how much the model spacetime is rotated when rolled. By insisting on two principles, the gauge principle and polynomial simplicity, we shall show how one can reformulate matter field actions in a way that is harmonious with Cartan’s geometric construction. This yields a formulation of all matter fields in terms of first order partial differential equations. We show in detail how the standard second order formulation can be recovered. In particular, the Hodge dual, which characterizes the structure of bosonic field equations, pops up automatically. Furthermore, the energy–momentum and spin-density three-forms are naturally combined into a single object here denoted the spin-energy–momentum three-form. Finally, we highlight a peculiarity in the mathematical structure of our first-order formulation of Yang–Mills fields. This suggests a way to unify a U(1) gauge field with gravity into a SO(1,5)-valued gauge field using a natural generalization of Cartan geometry in which the larger symmetry group is spontaneously broken down to SO(1,3)×U(1). The coupling of this unified theory to matter fields and possible extensions to non-Abelian gauge fields are left as open questions.

A PURE DIRAC'S CANONICAL ANALYSIS FOR FOUR-DIMENSIONAL BF THEORIES

We perform Dirac's canonical analysis for a four-dimensional BF and for a generalized four-dimensional BF theory depending on a connection valued in the Lie algebra of SO(3, 1). This analysis is developed by considering the corresponding complete set of variables that define these theories as dynamical, and we find out the relevant symmetries, the constraints, the extended Hamiltonian, the extended action, gauge transformations and the counting of physical degrees of freedom. The results obtained are compared with other approaches found in the literature.

A Geometric Algebra Perspective on Quantum Computational Gates and Universality in Quantum Computing

We investigate the utility of geometric (Clifford) algebras (GA) methods in two specific applications to quantum information science. First, using the multiparticle spacetime algebra (MSTA, the geometric algebra of a relativistic configuration space), we present an explicit algebraic description of one and two-qubit quantum states together with a MSTA characterization of one and two-qubit quantum computational gates. Second, using the above mentioned characterization and the GA description of the Lie algebras SO(3) and SU(2) based on the rotor group Spin+(3, 0) formalism, we reexamine Boykin's proof of universality of quantum gates. We conclude that the MSTA approach does lead to a useful conceptual unification where the complex qubit space and the complex space of unitary operators acting on them become united, with both being made just by multivectors in real space. Finally, the GA approach to rotations based on the rotor group does bring conceptual and computational advantages compared to standard vectorial and matricial approaches.

Hamiltonian study for Chern-Simons and Pontryagin theories

The Hamiltonian analysis for the Chern-Simons theory and Pontryagin invariant, which depends of a connection valued in the Lie algebra of SO(3, 1), is performed. By applying a pure Dirac’s method we find for both theories the extended Hamiltonian, the extended action, the constraint algebra, the gauge transformations and we carry out the counting of degrees of freedom. From the results obtained in the present analysis, we will conclude that the theories under study have a closed relation among its constraints and defines a topological field theory. In addition, we extends the configuration space for the Pontryagin theory and we develop the Hamiltonian analysis for this modified version, finding a best description of the results previously obtained.

Coupling coefficients for SO(5) with applications to nuclear physics

The Lie algebra SO(5) has several applications in nuclear theory, involving its various subalgebra chains. In this contribution, the calculation of SO(5) coupling coefficients, reduced with respect to these different subalgebras, is discussed. It is shown that "Racah's method" for extracting coupling coefficients can be formulated generically for a subalgebra chain G ⊃ H, provided the reduced matrix elements of the generators of G and the recoupling coefficients of H are known. The calculation of SO(5) ⊃ SO(4) reduced coupling coefficients, for generic irreps, is considered as an example.

Angular momentum adaptation of spinor basis

Fermion operator realization of the Lie-algebra of SCX(N) (N = 2n, 2n + 1) has been used extensively to study the symmetry adaptation of the fundamental spinor space of SO(N) to SO(3). However, the multispinor basis cannot be handled directly using such fermion operators since they generate only the totally antisymmetric irreducible representation (irrep) of U(4l + 2). Thus, the required spin-free states, which belong to two column irrep of U(2l + 1), do not follow readily. Recently, the Lie-Algebra of SO(N) has been realized using the restriction U(2n) ↓ SO(2n + 1) and was further adapted to SO(2n + 1) = ⊃ SO(2n) ⊃ U(n) = ⊃ U(n − 1) ⊃ … ⊃U(1) for the CI studies in molecules. However, for spin-free electronic configurations in atoms, the total angular momentum is a good quantum number and hence the relevant subgroup chain is SO(2n + 1) = ⊃ SO(2n) ⊃ U(n) ⊃ SC(n) = SO(3) (n odd). It is interesting to generate the configuration space in complex atoms using, as the starting point, the symmetric two box Weyl states of U(2n) and then restrict it to the above chain. This has been attempted in the present note in the hope that it will lead to simpler procedures for MC SCF studies in atoms.

The Chern–Simons state for topological invariants

The covariant canonical formalism for the second Chern and Euler topological invariants which depends of a connection valued in the Lie algebra of SO(3,1) is performed. We show that the Chern–Simons state corresponds to an eigenfunction of zero energy for such characteristic classes, in particular, for the Euler class within self-dual (or anti-self-dual) scenario. In addition, to complete our analysis we develop the Hamiltonian analysis for the theories under study, obtaining a best description of the results obtained with the symplectic method.