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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2014

Matrix elements for type 1 unitary irreducible representations of the Lie superalgebra gl(m|n)

Using our recent results on eigenvalues of invariants associated to the Lie superalgebra gl(m|n), we use characteristic identities to derive explicit matrix element formulae for all gl(m|n) generators, particularly non-elementary generators, on finite dimensional type 1 unitary irreducible representations. We compare our results with existing works that deal with only subsets of the class of type 1 unitary representations, all of which only present explicit matrix elements for elementary generators. Our work therefore provides an important extension to existing methods, and thus highlights the strength of our techniques which exploit the characteristic identities.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

Irreducible representations of finitely generated nilpotent groups

We prove that irreducible complex representations of finitely generated nilpotent groups are monomial if and only if they have finite weight, which was conjectured by Parshin. Note that we consider (possibly infinite-dimensional) representations without any topological structure. In addition, we prove that for certain induced representations, irreducibility is implied by Schur irreducibility. Both results are obtained in a more general form for representations over an arbitrary field. Bibliography: 21 titles.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

IrRep: Symmetry eigenvalues and irreducible representations of ab initio band structures

Here, we present IrRep – a Python code that calculates the symmetry eigenvalues of electronic Bloch states in crystalline solids and the irreducible representations under which they transform. As input it receives bandstructures computed with state-of-the-art Density Functional Theory codes such as VASP, Quantum Espresso, or Abinit, as well as any other code that has an interface to Wannier90. Our code is applicable to materials in any of the 230 space groups and double groups preserving time-reversal symmetry with or without spin-orbit coupling included, for primitive or conventional unit cells. This makes IrRep a powerful tool to systematically analyze the connectivity and topological classification of bands, as well as to detect insulators with non-trivial topology, following the Topological Quantum Chemistry formalism: IrRep can generate the input files needed to calculate the (physical) elementary band representations and the symmetry-based indicators using the [CheckTopologicalMat: https://www.cryst.ehu.es/cgi-bin/cryst/programs/magnetictopo.pl] routine of the Bilbao Crystallographic Server. It is also particularly suitable for interfaces with other plane-waves based codes, due to its flexible structure.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

A simple procedure for construction of the orthonormal basis vectors of irreducible representations of O(5) in the O T ( 3 ) Ⓧ O N ( 2 ) basis

We report a simple and effective algebraic isospin projection procedure for constructing orthonormal basis vectors of irreducible representations of O(5) ⊃ OT(3) Ⓧ O$_\mathscr{N}$ (2) from those in the canonical O(5) ⊃ SU Λ (2) Ⓧ SU I (2) basis is outlined. The expansion coefficients are components of null space vectors of the projection matrix with four nonzero elements in each row in general. Explicit formulae for evaluating O T (3)-reduced matrix elements of O(5) generators are derived.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2024

Stealth dark matter spectrum using Laplacian Heaviside smearing and irreducible representations

We present nonperturbative lattice calculations in the quenched approximation of the low-lying meson and baryon spectrum of the SU(4) gauge theory with fundamental fermion constituents. This theory is one instance of stealth dark matter, a class of strongly coupled theories, where the lowest mass stable baryon is the dark matter candidate. This work constitutes the first milestone in the program to study stealth dark matter self-interactions. Here, we focus on reducing excited state contamination in the single-baryon channel by applying the Laplacian Heaviside method, as well as projecting our baryon operators onto the irreducible representations of the octahedral group. We compare our resulting spectrum to previous work involving Gaussian smeared nonprojected operators and find good agreement with reduced statistical uncertainties. We also present the spectrum of the low-lying odd-parity baryons for the first time.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2014

Oscillators in a (2+1)-dimensional noncommutative space

We study the Harmonic and Dirac Oscillator problem extended to a three-dimensional noncommutative space where the noncommutativity is induced by the shift of the dynamical variables with generators of SL(2,R) in a unitary irreducible representation considered in Falomir et al. [Phys. Rev. D 86, 105035 (2012)]. This redefinition is interpreted in the framework of the Levi's decomposition of the deformed algebra satisfied by the noncommutative variables. The Hilbert space gets the structure of a direct product with the representation space as a factor, where there exist operators which realize the algebra of Lorentz transformations. The spectrum of these models are considered in perturbation theory, both for small and large noncommutativity parameters, finding no constraints between coordinates and momenta noncommutativity parameters. Since the representation space of the unitary irreducible representations SL(2,R) can be realized in terms of spaces of square-integrable functions, we conclude that these models are equivalent to quantum mechanical models of particles living in a space with an additional compact dimension.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2019

Representations of relativistic particles of arbitrary spin in Poincaré, Lorentz, and Euclidean covariant formulations of relativistic quantum mechanics

Background: Relativistic treatments of quantum mechanical systems are important for understanding hadronic structure and dynamics at subnucleon scales. Relativistic invariance of a quantum system means that there is an underlying unitary representation of the Poincaré group. This is equivalent to the requirement that the quantum observables (probabilities, expectation values, and ensemble averages) for equivalent measurements performed in different inertial reference frames are identical. Different representations are used in practice, including Poincaré covariant forms of dynamics, representations based on Lorentz covariant wave functions, Euclidean covariant representations, and representations generated by Lorentz covariant fields. Purpose: The purpose of this work is to illustrate the relation between the different equivalent representations of states in relativistic quantum mechanics. Method: The starting point is a description of a particle of mass m and spin j using irreducible representations of the Poincaré group. Since any unitary representation of the Poincaré group can be decomposed into a direct integral of irreducible representations, these are the basic building blocks of any relativistically invariant quantum theory. The equivalence is established by constructing equivalent Lorentz covariant irreducible representations from Poincaré covariant irreducible representations and constructing equivalent Euclidean covariant irreducible representations from Lorentz covariant irreducible representations. Results: Equivalent descriptions for positive mass representations of arbitrary spin are presented in each of these frameworks. Dynamical realizations of the different representations are briefly discussed. Conclusion: Poincaré covariant, Lorentz covariant, and Euclidean covariant realizations of relativistic dynamics are shown to be equivalent by explicitly relating the positive-mass positive-energy irreducible representations of the Poincaré group that appear in the direct integral.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2014

Structure and properties of the algebra of partially transposed permutation operators

We consider the structure of algebra of operators, acting in n-fold tensor product space, which are partially transposed on the last term. Using purely algebraical methods we show that this algebra is semi-simple and then, considering its regular representation, we derive basic properties of the algebra. In particular, we describe all irreducible representations of the algebra of partially transposed operators and derive expressions for matrix elements of the representations. It appears that there are two kinds of irreducible representations of the algebra. The first one is strictly connected with the representations of the group S(n − 1) induced by irreducible representations of the group S(n − 2). The second kind is structurally connected with irreducible representations of the group S(n − 1)

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

LieART 2.0 – A Mathematica application for Lie Algebras and Representation Theory

In this paper, we present LieART 2.0 which contains substantial extensions to the Mathematica application LieART (Lie Algebras and Representation Theory) for computations frequently encountered in Lie algebras and representation theory, such as tensor product decomposition and subalgebra branching of irreducible representations. The basic procedure is unchanged—it computes root systems of Lie algebras, weight systems and several other properties of irreducible representations, but new features and procedures have been included to allow the extensions to be seamless. The new version of LieART continues to be user friendly. New extended tables of properties, tensor products and branching rules of irreducible representations are included in the supplementary material for use without Mathematica software. LieART 2.0 now includes the branching rules to special subalgebras for all classical and exceptional Lie algebras up to and including rank 15.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Digitization and subduction of S U ( N ) gauge theories

The simulation of lattice gauge theories on quantum computers necessitates digitizing gauge fields. One approach involves substituting the continuous gauge group with a discrete subgroup, but the implications of this approximation still need to be clarified. To gain insights, we investigate the subduction of S U ( 2 ) and S U ( 3 ) to discrete crystal-like subgroups. Using classical lattice calculations, we show that subduction offers valuable information based on subduced direct sums, helping us identify additional terms to incorporate into the lattice action that can mitigate the effects of digitization. Furthermore, we compute the static potentials of all irreducible representations of Σ ( 360 × 3 ) at a fixed lattice spacing. Our results reveal a percent-level agreement with the Casimir scaling of S U ( 3 ) for irreducible representations that subduce to a single Σ ( 360 × 3 ) irreducible representation. This provides a diagnostic measure of approximation quality, as some irreducible representations closely match the expected results while others exhibit significant deviations. Published by the American Physical Society 2024

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2017

Double crystallographic groups and their representations on the Bilbao Crystallographic Server

A new section of databases and programs devoted to double crystallographic groups (point and space groups) has been implemented in the Bilbao Crystallographic Server (http://www.cryst.ehu.es). The double crystallographic groups are required in the study of physical systems whose Hamiltonian includes spin-dependent terms. In the symmetry analysis of such systems, instead of the irreducible representations of the space groups, it is necessary to consider the single- and double-valued irreducible representations of the double space groups. The new section includes databases of symmetry operations ( DGENPOS ) and of irreducible representations of the double (point and space) groups ( REPRESENTATIONS DPG and REPRESENTATIONS DSG ). The tool DCOMPREL provides compatibility relations between the irreducible representations of double space groups at different k vectors of the Brillouin zone when there is a group–subgroup relation between the corresponding little groups. The program DSITESYM implements the so-called site-symmetry approach, which establishes symmetry relations between localized and extended crystal states, using representations of the double groups. As an application of this approach, the program BANDREP calculates the band representations and the elementary band representations induced from any Wyckoff position of any of the 230 double space groups, giving information about the properties of these bands. Recently, the results of BANDREP have been extensively applied in the description of and the search for topological insulators.

Chemistry↗
DOE OSTI2017

Invariants of the Cox rings of double flag varieties of low complexity for exceptional groups

We find the algebras of unipotent invariants of the Cox rings for all double flag varieties of complexity 0 and 1 for the exceptional simple algebraic groups; namely, we obtain presentations of these algebras in terms of generators and relations. It is well known that such an algebra is free in the case of complexity 0. In this paper, we show that, in the case of complexity 1, the algebra in question is either a free algebra or a hypersurface. A similar result for classical groups was previously obtained by the author. Knowing the structure of this algebra enables one to decompose tensor products of some irreducible representations effectively into irreducible summands and to obtain some branching rules. Bibliography: 10 titles. (paper)

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

The total angular momentum algebra related to the S{sub 3} Dunkl Dirac equation

We consider the symmetry algebra generated by the total angular momentum operators, appearing as constants of motion of the S{sub 3} Dunkl Dirac equation. The latter is a deformation of the Dirac equation by means of Dunkl operators, in our case associated to the root system A{sub 2}, with corresponding Weyl group S{sub 3}, the symmetric group on three elements. The explicit form of the symmetry algebra in this case is a one-parameter deformation of the classical total angular momentum algebra so(3), incorporating elements of S{sub 3}. This was obtained using recent results on the symmetry algebra for a class of Dirac operators, containing in particular the Dirac–Dunkl operator for arbitrary root system. For this symmetry algebra, we classify all finite-dimensional, irreducible representations and determine the conditions for the representations to be unitarizable. The class of unitary irreducible representations admits a natural realization acting on a representation space of eigenfunctions of the Dirac Hamiltonian. Using a Cauchy–Kowalevski extension theorem we obtain explicit expressions for these eigenfunctions in terms of Jacobi polynomials.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

BMS symmetry, soft particles and memory

Here, in this work, we revisit unitary irreducible representations of the Bondi–Metzner–Sachs (BMS) group discovered by McCarthy. Representations are labelled by an infinite number of supermomenta in addition to 4-momentum. Tensor products of these irreducible representations lead to particle-like states dressed by soft gravitational modes. Conservation of 4-momentum and supermomentum in the scattering of such states leads to a memory effect encoded in the outgoing soft modes. We note there exist irreducible representations corresponding to soft states with strictly vanishing 4-momentum, which may nevertheless be produced by scattering of particle-like states. This fact has interesting implications for the S-matrix in gravitational theories.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2015

Classification of Arnold-Beltrami flows and their hidden symmetries

In the context of mathematical hydrodynamics, we consider the group theory structure which underlies the so named ABC flows introduced by Beltrami, Arnold and Childress. Main reference points are Arnold’s theorem stating that, for flows taking place on compact three manifolds ℳ{sub 3}, the only velocity fields able to produce chaotic streamlines are those satisfying Beltrami equation and the modern topological conception of contact structures, each of which admits a representative contact one-form also satisfying Beltrami equation. We advocate that Beltrami equation is nothing else but the eigenstate equation for the first order Laplace-Beltrami operator ★{sub g }d, which can be solved by using time-honored harmonic analysis. Taking for ℳ{sub 3}, a torus T {sup 3} constructed as ℝ{sup 3}/Λ, where Λ is a crystallographic lattice, we present a general algorithm to construct solutions of the Beltrami equation which utilizes as main ingredient the orbits under the action of the point group B {sub A }of three-vectors in the momentum lattice *Λ. Inspired by the crystallographic construction of space groups, we introduce the new notion of a Universal Classifying Group GA{sub Λ} which contains all space groups as proper subgroups. We show that the ★{sub g }d eigenfunctions are naturally arranged into irreducible representations of GA{sub Λ} and by means of a systematic use of the branching rules with respect to various possible subgroups H{sub i}⊂GA{sub Λ} we search and find Beltrami fields with non trivial hidden symmetries. In the case of the cubic lattice the point group is the proper octahedral group O{sub 24} and the Universal Classifying Group GA{sub cubic} is a finite group G{sub 1536} of order |G{sub 1536}| = 1536 which we study in full detail deriving all of its 37 irreducible representations and the associated character table. We show that the O{sub 24} orbits in the cubic lattice are arranged into 48 equivalence classes, the parameters of the corresponding Beltrami vector fields filling all the 37 irreducible representations of G{sub 1536}. In this way we obtain an exhaustive classification of all generalized ABC-flows and of their hidden symmetries. We make several conceptual comments about the need of a field-theory yielding Beltrami equation as a field equation and/or an instanton equation and on the possible relation of Arnold-Beltrami flows with (supersymmetric) Chern-Simons gauge theories. We also suggest linear generalizations of Beltrami equation to higher odd-dimensions that are different from the non-linear one proposed by Arnold and possibly make contact with M-theory and the geometry of flux-compactifications.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2017

Baryon-baryon interactions and spin-flavor symmetry from lattice quantum chromodynamics

Lattice quantum chromodynamics is used to constrain the interactions of two octet baryons at the SU(3) flavor-symmetric point, with quark masses that are heavier than those in nature (equal to that of the physical strange quark mass and corresponding to a pion mass of $\approx 806~\tt{MeV}$). Specifically, the S-wave scattering phase shifts of two-baryon systems at low energies are obtained with the application of L\"uscher's formalism, mapping the energy eigenvalues of two interacting baryons in a finite volume to the two-particle scattering amplitudes below the relevant inelastic thresholds. The values of the leading-order low-energy scattering parameters in the irreducible representations of SU(3) are consistent with an approximate SU(6) spin-flavor symmetry in the nuclear and hypernuclear forces that is predicted in the large-$N_c$ limit of QCD. The two distinct SU(6)-invariant interactions between two baryons are constrained at this value of the quark masses, and their values indicate an approximate accidental SU(16) symmetry. The SU(3) irreducible representations containing the $NN~({^1}S_0)$, $NN~({^3}S_1)$ and $\frac{1}{\sqrt{2}}(\Xi^0n+\Xi^-p)~({^3}S_1)$ channels unambiguously exhibit a single bound state, while the irreducible representation containing the $\Sigma^+ p~({^3}S_1)$ channel exhibits a state that is consistent with either a bound state or a scattering state close to threshold. These results are in agreement with the previous conclusions of the NPLQCD collaboration regarding the existence of two-nucleon bound states at this value of the quark masses.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗