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All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

On infinite tensor networks, complementary recovery and type II factors

We initiate a study of local operator algebras at the boundary of infinite tensor networks, using the mathematical theory of inductive limits. In particular, we consider tensor networks in which each layer acts as a quantum code with complementary recovery, a property that features prominently in the bulk-to-boundary maps intrinsic to holographic quantum error-correcting codes. In this case, we decompose the limiting Hilbert space and the algebras of observables in a way that keeps track of the entanglement in the network. As a specific example, we describe this inductive limit for the holographic Harlow-Pastawski-Preskill-Yoshida code model and relate its algebraic and error-correction features. We find that the local algebras in this model are given by the hyperfinite type II$_\infty$ factor. Next, we discuss other networks that build upon this framework and comment on a connection between type II factors and stabilizer circuits. We conclude with a discussion of multiscale entanglement renormalization ansatz networks in which complementary recovery is broken. We argue that this breaking possibly permits a limiting type III von Neumann algebra, making them more suitable ansätze for approximating subregions of quantum field theories.

holographic dualities↗
DOE OSTI2025

Four no-go theorems on the existence of spin and orbital angular momentum of massless bosons

The past decades have seen substantial interest in the so-called orbital angular momentum (OAM) of light, driven largely by its diverse range of applications. However, there are fundamental theoretical issues with decomposing the angular momentum of massless particles, such as photons, into spin (SAM) and orbital angular momentum parts. While the angular momentum of massive particles has a natural splitting into the Wigner SAM and OAM, there are numerous proposed splittings for photons and no consensus about which is correct. Moreover, it has been shown that most of the proposed SAM and OAM operators do not satisfy the defining commutation relations of angular momentum operators and are thus not legitimate splittings. Here, we prove that it is generally impossible to split the total angular momentum operator of massless bosons, such as photons and gravitons, into spin and orbital parts. We prove two further generalizations of this result, showing that there are no SAM-OAM splittings even if (1) the SAM operator generates non-internal symmetries or (2) if one allows the SAM and OAM operators to generate non-SO(3) symmetries.

Chern numbers↗
NASA NTRS1972

On the Poincare noninvariance of a recent alternative to the Dirac equation.

Explicit construction of the infinitesimal generators of the Poincare group, demonstrating that the algebra of commutators closes only for the case of zero mass. Hence, the so-called Stigma equation proposed by Biedenharn et al. (1971) as an alternative to the Dirac equation (1928) for spin one-half, finite mass particles is not Poincare invariant except when the leptonic mass is zero.

Madan, R. N.↗
NASA NTRS1996

Coherent States for Kronecker Products of Non Compact Groups: Formulation and Applications

We introduce and study the properties of a class of coherent states for the group SU(1,1) X SU(1,1) and derive explicit expressions for these using the Clebsch-Gordan algebra for the SU(1,1) group. We restrict ourselves to the discrete series representations of SU(1,1). These are the generalization of the 'Barut Girardello' coherent states to the Kronecker Product of two non-compact groups. The resolution of the identity and the analytic phase space representation of these states is presented. This phase space representation is based on the basis of products of 'pair coherent states' rather than the standard number state canonical basis. We discuss the utility of the resulting 'bi-pair coherent states' in the context of four-mode interactions in quantum optics.

Bambah, Bindu A.↗
DOE OSTI2024

Quantum reference frames from top-down crossed products

All physical observations are made relative to a reference frame, which is a system in its own right. If the system of interest admits a group symmetry, the reference frame observing it must transform commensurately under the group to ensure the covariance of the combined system. We point out that the crossed product is a way to realize quantum reference frames from the bottom-up; adjoining a quantum reference frame and imposing constraints generates a crossed product algebra. We provide a top-down specification of crossed product algebras and show that one cannot obtain inequivalent quantum reference frames using this approach. As a remedy, we define an abstract algebra associated to the system and symmetry group built out of relational crossed product algebras associated with different choices of quantum reference frames. We term this object the G -framed algebra, and show how potentially inequivalent frames are realized within this object. We comment on this algebra’s analog of the classical Gribov problem in gauge theory, its importance in gravity where we show that it is relevant for semiclassical de Sitter and potentially beyond the semiclassical limit, and its utility for understanding the frame dependence of physical notions like observables, density states, and entropies. Published by the American Physical Society 2024

Astronomy & Astrophysics↗
NASA NTRS1977

Linear systems with structure group and their feedback invariants

A general method described by Hermann and Martin (1976) for the study of the feedback invariants of linear systems is considered. It is shown that this method, which makes use of ideas of topology and algebraic geometry, is very useful in the investigation of feedback problems for which the classical methods are not suitable. The transfer function as a curve in the Grassmanian is examined. The general concepts studied in the context of specific systems and applications are organized in terms of the theory of Lie groups and algebraic geometry. Attention is given to linear systems which have a structure group, linear mechanical systems, and feedback invariants. The investigation shows that Lie group techniques are powerful and useful tools for analysis of the feedback structure of linear systems.

Martin, C.↗
NASA NTRS1995

Evaluation of Turbulence Models for Unsteady Flows of an Oscillating Airfoil

Unsteady flowfields of a two-dimensional oscillating airfoil are calculated using an implicit, finite-difference, Navier Stokes numerical scheme. Five widely used turbulence models are used with the numerical scheme to assess the accuracy and suitability of the models for simulating the retreating blade stall of helicopter rotor in forward flight. Three unsteady flow conditions corresponding to an essentially attached flow, light-stall, and deep-stall cases of an oscillating NACA 0015 wing experiment were chosen as test cases for computations. Results of unsteady airloads hysteresis curves, harmonics of unsteady pressures, and instantaneous flowfield patterns are presented. Some effects of grid density, time-step size, and numerical dissipation on the unsteady solutions relevant to the evaluation of turbulence models are examined. Comparison of unsteady airloads with experimental data show that all models tested are deficient in some sense and no single model predicts airloads consistently and in agreement with experiment for the three flow regimes. The chief findings are that the simple algebraic model based on the renormalization group theory (RNG) offers some improvement over the Baldwin Lomax model in all flow regimes with nearly same computational cost. The one-equation models provide significant improvement over the algebraic and the half-equation models but have their own limitations. The Baldwin-Barth model overpredicts separation and underpredicts reattachment. In contrast, the Spalart-Allmaras model underpredicts separation and overpredicts reattachment.

Srinivasan, G. R.↗
NASA NTRS1988

Linear canonical transformations of coherent and squeezed states in the Wigner phase space

It is shown that classical linear canonical transformations are possible in the Wigner phase space. Coherent and squeezed states are shown to be linear canonical transforms of the ground-state harmonic oscillator. It is therefore possible to evaluate the Wigner functions for coherent and squeezed states from that for the harmonic oscillator. Since the group of linear canonical transformations has a subgroup whose algebraic property is the same as that of the (2+1)-dimensional Lorentz group, it may be possible to test certain properties of the Lorentz group using optical devices. A possible experiment to measure the Wigner rotation angle is discussed.

Han, D.↗
NASA NTRS1990

Trees, bialgebras and intrinsic numerical algorithms

Preliminary work about intrinsic numerical integrators evolving on groups is described. Fix a finite dimensional Lie group G; let g denote its Lie algebra, and let Y(sub 1),...,Y(sub N) denote a basis of g. A class of numerical algorithms is presented that approximate solutions to differential equations evolving on G of the form: dot-x(t) = F(x(t)), x(0) = p is an element of G. The algorithms depend upon constants c(sub i) and c(sub ij), for i = 1,...,k and j is less than i. The algorithms have the property that if the algorithm starts on the group, then it remains on the group. In addition, they also have the property that if G is the abelian group R(N), then the algorithm becomes the classical Runge-Kutta algorithm. The Cayley algebra generated by labeled, ordered trees is used to generate the equations that the coefficients c(sub i) and c(sub ij) must satisfy in order for the algorithm to yield an rth order numerical integrator and to analyze the resulting algorithms.

Crouch, Peter↗
DOE OSTI2025

Commuting Line Defects At $q^N$ = 1

Here, we explain the physical origin of a curious property of algebras $\mathcal{A}_{\mathfrak{q}}$ which encode the rotation-equivariant fusion ring of half-BPS line defects in four-dimensional $\mathcal{N}$ = 2 supersymmetric quantum field theories. These algebras are a quantization of the algebras of holomorphic functions on the three-dimensional Coulomb branch of the SQFTs, with deformation parameter log $\mathfrak{q}$. They are known to acquire a large center, canonically isomorphic to the undeformed algebra, whenever $\mathfrak{q}$ is a root of unity. We give a physical explanation of this fact. We also generalize the construction to characterize the action of this center in the $\mathcal{A}_{\mathfrak{q}}$-modules associated to three-dimensional $\mathcal{N}$ = 2 boundary conditions. Finally, we use dualities to relate this construction to a construction in the Kapustin–Witten twist of four-dimensional $\mathcal{N}$ = 4 gauge theory. These considerations give simple physical explanations of certain properties of quantized skein algebras and cluster varieties, and quantum groups, when the deformation parameter is a root of unity.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2024

On a class of selection rules without group actions in field theory and string theory

We discuss a class of selection rules which i) do not come from group actions on fields, ii) are exact at tree level in perturbation theory, iii) are increasingly violated as the loop order is raised, and iv) eventually reduce to selection rules associated with an ordinary group symmetry. We start from basic field-theoretical examples in which fields are labeled by conjugacy classes rather than representations of a group, and discuss generalizations using fusion algebras or hypergroups. We also discuss how such selection rules arise naturally in string theory, such as for non-Abelian orbifolds or other cases with non-invertible worldsheet symmetries.

Kaidi, Justin (ORCID:0000000161440729)↗
NASA NTRS1972

Control systems on Lie groups.

The controllability properties of systems which are described by an evolution equation in a Lie group are studied. The revelant Lie algebras induced by a right invariant system are singled out, and the basic properties of attainable sets are derived. The homogeneous case and the general case are studied, and results are interpreted in terms of controllability. Five examples are given.

Jurdjevic, V.↗
NASA NTRS1990

Nonlinear Reynolds stress models and the renormalization group

The renormalization group is applied to derive a nonlinear algebraic Reynolds stress model of anisotropic turbulence in which the Reynolds stresses are quadratic functions of the mean velocity gradients. The model results from a perturbation expansion that is truncated systematically at second order with subsequent terms contributing no further information. The resulting turbulence model applied to both low and high Reynolds number flows without requiring wall functions or ad hoc modifications of the equations. All constants are derived from the renormalization group procedure; no adjustable constants arise. The model permits inequality of the Reynolds normal stresses, a necessary condition for calculating turbulence-driven secondary flows in noncircular ducts.

Rubinstein, Robert↗
NASA NTRS1991

An algebraic RNG-based turbulence model for three-dimensional turbomachinery flows

An algebraic eddy viscosity turbulence model based on Renormalization Group (RNG) theory for complex three-dimensional turbomachinery flows is presented. Modifications are made to the baseline RNG model for wakes and separated flows. The model has several advantages over popular algebraic models most notably its lack of empirically determined coefficients. The model is used to compute the mean flow in a low speed axial compressor rotor. The agreement with blade boundary layer and radial flow experimental data is very good and shows improvement over the Baldwin-Lomax model. The development of the tip leakage vortex is also well predicted. The computed wake decay also compares favorably with recent experimental data.

Kirtley, K. R.↗
DOE OSTI2024

Algebraic Hastatic Order in One-Dimensional Two-Channel Kondo Lattice

The two-channel Kondo lattice likely hosts a rich array of phases, including hastatic order, a channel symmetry breaking heavy Fermi liquid. In this work, we revisit its one-dimensional phase diagram using density matrix renormalization group and, in contrast to previous work, find algebraic hastatic orders generically for stronger couplings. These are heavy Tomonaga-Luttinger liquids with nonanalyticities at Fermi vectors captured by hastatic density waves. We also find a predicted additional nonlocal order parameter due to interference between hastatic spinors, not present at large N, and residual repulsive interactions at strong coupling suggesting non-Fermi-liquid physics in higher dimensions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Enhancing Lattice Kinetic Schemes for Fluid Dynamics with Lattice-Equivariant Neural Networks

A new class of equivariant neural networks is presented, hereby dubbed lattice-equivariant neural networks (LENNs), designed to satisfy local symmetries of a lattice structure. The approach develops within a recently introduced framework aimed at learning neural network-based surrogate models’ lattice Boltzmann collision operators. Whenever neural networks are employed to model physical systems, respecting symmetries and equivariance properties has been shown to be key for accuracy, numerical stability, and performance. Here, hinging on ideas from group representation theory, trainable layers are defined whose algebraic structure is equivariant with respect to the symmetries of the lattice cell. In this work, the presented method naturally allows for efficient implementations, in terms of both memory usage and computational costs, supporting scalable training/testing for lattices in two spatial dimensions and higher (in which the size of symmetry group grows). The approach is validated and tested considering 2D and 3D flowing dynamics, both in laminar and turbulent regimes. It is compared with group-averaged-based symmetric networks and with plain, nonsymmetric, networks, showing how the presented approach unlocks the (a posteriori) accuracy and training stability of the former models and the train/inference speed of the latter networks. (LENNs are about one order of magnitude faster than group-averaged networks in 3D.) The work in this paper opens toward practical use of machine learning-augmented lattice Boltzmann CFD in real-world simulations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Coupling and recoupling coefficients for Wigner’s U(4) supermultiplet symmetry

A novel procedure for evaluating Wigner coupling coefficients and Racah recoupling coefficients for U(4) in two group–subgroup chains is presented. The canonical U(4) > U(3) > U(2) > U(1) coupling and recoupling coefficients are applicable to any system that possesses U(4) symmetry, while the physical U(4) coupling coefficients are more specific to nuclear structure studies that utilize Wigner’s supermultiplet symmetry concept. The procedure that is proposed sidesteps the use of binomial coefficients and alternating sum series and consequently enables fast and accurate computation of any and all U(4)-underpinned features. The inner multiplicity of a (S, T) pair within a single U(4) > SU S (2) Ⓧ SU T (2) irreducible representation is obtained from the dimension of the null space of the SU(2) raising generators, while the resolution for the outer multiplicity follows from the work of Alex et al. on U(N) . It is anticipated that a C++ library will ultimately be available for determining generic coupling and recoupling coefficients associated with both the canonical and the physical group–subgroup chains of U(4).

Cross-Coupling Reaction↗