Noncommutative Field Research Articles

Braided Scalar Quantum Electrodynamics

AbstractWe formulate scalar electrodynamics in the braided ‐algebra formalism and study its perturbative expansion in the algebraic framework of Batalin–Vilkovisky quantization. We also confirm that UV/IR mixing is absent at one‐loop order in this noncommutative field theory, and that the non‐anomalous Ward‐Takahashi identities for the braided gauge symmetry are satisfied.

Inverses and Determinants of Arrowhead and Diagonal-Plus-Rank-One Matrices over Associative Algebras

This article considers arrowhead and diagonal-plus-rank-one matrices in Fn×n where F∈{R,C,H} and where H is a noncommutative algebra of quaternions. We provide unified formulas for fast determinants and inverses for considered matrices. The formulas are unified in the sense that the same formula holds in both commutative and noncommutative associative fields or algebras, with noncommutative examples being matrices of quaternions and block matrices. Each formula requires O(n) arithmetic operations, as does multiplication of such matrices with a vector. The formulas are efficiently implemented using the polymorphism or multiple-dispatch feature of the Julia programming language.

Noncommutative field theory of the Tkachenko mode: Symmetries and decay rate

We construct an effective field theory describing the collective Tkachenko oscillation mode of a vortex lattice in a two-dimensional rotating Bose-Einstein condensate in the long-wavelength regime. The theory has the form of a noncommutative field theory of a Nambu-Goldstone boson, which exhibits a noncommutative version of dipole symmetry. From the effective field theory, we show that, at zero temperature, the decay width Γ of the Tkachenko mode scales with its energy E as Γ∼E3 in the low-energy limit. We also discuss the width of the Tkachenko mode at a small temperature. Published by the American Physical Society 2024

Gauge theory on twist-noncommutative spaces

We construct actions for four dimensional noncommutative Yang-Mills theory with star-gauge symmetry, with non-constant noncommutativity, to all orders in the noncommutativity. Our construction covers all noncommutative spaces corresponding to Drinfel’d twists based on the Poincaré algebra, including nonabelian ones, whose r matrices are unimodular. This includes particular Lie-algebraic and quadratic noncommutative structures. We prove a planar equivalence theorem for all such noncommutative field theories, and discuss how our actions realize twisted Poincaré symmetry, as well as twisted conformal and twisted supersymmetry, when applicable. Finally, we consider noncommutative versions of maximally supersymmetric Yang-Mills theory, conjectured to be AdS/CFT dual to certain integrable deformations of the AdS5 × S5 superstring.

Noncommutative scalar field theory in a curved background: Duality between noncommutative and effective commutative description

We study a noncommutative (NC) deformation of a charged scalar field, minimally coupled to a classical (commutative) Reissner–Nordström-like background. The deformation is performed via a particularly chosen Killing twist to ensure that the geometry remains undeformed (commutative). An action describing a NC scalar field minimally coupled to the RN geometry is manifestly invariant under the deformed [Formula: see text] gauge symmetry. We find the equation of motion and conclude that the same equation is obtained from the commutative theory in a modified geometrical background described by an effective metric. This correspondence we call “duality between formal and effective approach”. We also show that a NC deformation via semi-Killing twist operator cannot be rewritten in terms of an effective metric. There is dual description for those particular deformations.

The algebraic structure of the non-commutative nonlinear Schrödinger and modified Korteweg–de Vries hierarchy

We prove that each member of the non-commutative nonlinear Schrödinger and modified Korteweg–de Vries hierarchy is a Fredholm Grassmannian flow, and for the given linear dispersion relation and corresponding equivalencing group of Fredholm transformations, is unique in the class of odd-polynomial partial differential fields. Thus each member is linearisable and integrable in the sense that time-evolving solutions can be generated by solving a linear Fredholm Marchenko equation, with the scattering data solving the corresponding linear dispersion equation. At each order, each member matches the corresponding non-commutative Lax hierarchy field which thus represent odd-polynomial partial differential fields. We also show that the cubic form for the non-commutative sine–Gordon equation corresponds to the first negative order case in the hierarchy, and establish the rest of the negative order non-commutative hierarchy. To achieve this, we construct an abstract combinatorial algebra, the Pöppe skew-algebra, that underlies the hierarchy. This algebra is the non-commutative polynomial algebra over the real line generated by compositions, endowed with the Pöppe product—the product rule for Hankel operators pioneered by Ch. Pöppe for classical integrable systems. Establishing the hierarchy members at non-negative orders, involves proving the existence of a ‘Pöppe polynomial’ expansion for basic compositions in terms of ‘linear signature expansions’ representing the derivatives of the underlying non-commutative field. The problem boils down to solving a linear algebraic equation for the polynomial expansion coefficients, at each order.

Braided quantum electrodynamics

The homotopy algebraic formalism of braided noncommutative field theory is used to define the explicit example of braided electrodynamics, that is, U(1) gauge theory minimally coupled to a Dirac fermion. We construct the braided L∞-algebra of this field theory and obtain the braided equations of motion, action functional and conserved matter current. The modifications of the electric charge conservation law due to the braided noncommutative deformation are described. We develop a braided generalization of Wick’s theorem, and use it to compute correlation functions of the braided quantum field theory using homological perturbation theory. Our putative calculations indicate that the braided theory does not contain the non-planar Feynman diagrams of conventional noncommutative quantum field theory, and that correlators do not exhibit UV/IR mixing.

Homotopy Double Copy of Noncommutative Gauge Theories

We discuss the double-copy formulation of Moyal–Weyl-type noncommutative gauge theories from the homotopy algebraic perspective of factorisations of L∞-algebras. We define new noncommutative scalar field theories with rigid colour symmetries taking the role of the zeroth copy, where the deformed colour algebra plays the role of a kinematic algebra; some of these theories have a trivial classical limit but exhibit colour–kinematics duality, from which we construct the double copy theory explicitly. We show that noncommutative gauge theories exhibit a twisted form of colour–kinematics duality, which we use to show that their double copies match with the commutative case. We illustrate this explicitly for Chern–Simons theory, and for Yang–Mills theory where we obtain a modified Kawai–Lewellen–Tye relationship whose momentum kernel is linked to a binoncommutative biadjoint scalar theory. We reinterpret rank-one noncommutative gauge theories as double copy theories and discuss how our findings tie in with recent discussions of Moyal–Weyl deformations of self–dual Yang–Mills theory and gravity.

Quantum simulation of quantum mechanical system with spatial noncommutativity

Quantum simulation has become a promising avenue of research that allows one to simulate and gain insight into the models of high energy physics whose experimental realizations are either complicated or inaccessible with current technology. We demonstrate the quantum simulation of such a model, a quantum mechanical system with spatial noncommutativity, which is inspired by the works in noncommutative geometry and noncommutative field theory for a universal quantum computer. We use the novel group theoretical formalism to map the Hamiltonian of such a noncommutative quantum system into the ordinary quantum mechanical Hamiltonian and then carry out the quantum simulation using the Trotter–Suzuki product formula. Furthermore, we distinguish the impact of the noncommutativity parameter on the quantum simulation, especially on the Trotter error, and point out how its sizable value affects the simulation.

Quantum geodesic flows and curvature

We study geodesics flows on curved quantum Riemannian geometries using a recent formulation in terms of bimodule connections and completely positive maps. We complete this formalism with a canonical * operation on noncommutative vector fields. We show on a classical manifold how the Ricci tensor arises naturally in our approach as a term in the convective derivative of the divergence of the geodesic velocity field and use this to propose a similar object in the noncommutative case. Examples include quantum geodesic flows on the algebra of 2 times 2 matrices, fuzzy spheres and the q-sphere.

Total variation bound for Milstein scheme without iterated integrals

Abstract The paper gives new results for the Milstein scheme of stochastic differential equations. We show that (i) the Milstein scheme holds as a weak approximation in total variation sense and is given by second-order polynomials of Brownian motion without using iterated integrals under non-commutative vector fields; (ii) the accuracy of the Milstein scheme is better than that of the Euler–Maruyama scheme in an asymptotic sense. In particular, we prove d TV ⁢ ( X T ε , X ¯ T ε , Mil , ( n ) ) ≤ C ⁢ ε 2 / n d_{\mathrm{TV}}(X_{T}^{\varepsilon},\bar{X}_{T}^{\varepsilon,\mathrm{Mil},(n)})\leq C\varepsilon^{2}/n and d TV ⁢ ( X T ε , X ¯ T ε , EM , ( n ) ) ≤ C ⁢ ε / n d_{\mathrm{TV}}(X_{T}^{\varepsilon},\bar{X}_{T}^{\varepsilon,\mathrm{EM},(n)})\leq C\varepsilon/n , where d TV d_{\mathrm{TV}} is the total variation distance, X ε X^{\varepsilon} is a solution of a stochastic differential equation with a small parameter 𝜀, and X ¯ ε , Mil , ( n ) \bar{X}^{\varepsilon,\mathrm{Mil},(n)} and X ¯ ε , EM , ( n ) \bar{X}^{\varepsilon,\mathrm{EM},(n)} are the Milstein scheme without iterated integrals and the Euler–Maruyama scheme, respectively. In computational aspect, the scheme is useful to estimate probability distribution functions by a simple simulation without Lévy area computation. Numerical examples demonstrate the validity of the method.

Topological quantum field theories from Hecke algebras

We construct one-dimensional non-commutative topological quantum field theories (TQFTs), one for each Hecke algebra corresponding to a finite Coxeter system. These TQFTs associate an invariant to each ciliated surface, which is a Laurent polynomial for punctured surfaces. There is a graphical way to compute the invariant using minimal colored graphs. We give explicit formulas in terms of the Schur elements of the Hecke algebra and prove positivity properties for the invariants when the Coxeter group is of classical type, or one of the exceptional typesH3H_3,E6E_6andE7E_7.

Lifting Bratteli diagrams between Krajewski diagrams: Spectral triples, spectral actions, and AF algebras

In this paper, we present a framework to construct sequences of spectral triples on top of an inductive sequence defining an $AF$-algebra. One aim of this paper is to lift arrows of a Bratteli diagram to arrows between Krajewski diagrams. The spectral actions defining Non-commutative Gauge Field Theories associated to two spectral triples related by these arrows are compared (tensored by a commutative spectral triple to put us in the context of Almost Commutative manifolds). This paper is a follow up of a previous one in which this program was defined and physically illustrated in the framework of the derivation-based differential calculus, but the present paper focuses more on the mathematical structure without trying to study the physical implications.

Division algebras and MRD codes from skew polynomials

AbstractLet $D$ be a division algebra, finite-dimensional over its center, and $R=D[t;\;\sigma,\delta ]$ a skew polynomial ring.Using skew polynomials $f\in R$ , we construct division algebras and maximum rank distance codes consisting of matrices with entries in a noncommutative division algebra or field. These include Jha Johnson semifields, and the classes of classical and twisted Gabidulin codes constructed by Sheekey.

Quantum Hall valley ferromagnets as a platform for topologically protected quantum memory

Materials hosting topologically protected non-Abelian zero modes offer the exciting possibility of storing and manipulating quantum information in a manner that is protected from decoherence at the hardware level. In this paper, we study the possibility of realizing such excitations along line defects in certain fractional quantum Hall states in multivalley systems. Such line defects have been recently observed experimentally between valley polarized Hall states on the surface of Bi(111), and excitations near these defects appear to be gapped (gapless) depending on the presence (absence) of interaction-induced gapping perturbations constrained by momentum selection rules, while the position of defects is determined by strain. In this paper, we use these selection rules to show that a hybrid structure involving a superlattice imposed on such a multivalley quantum Hall surface realizes non-Abelian anyons, which can then be braided by modulating strain locally to move line defects. Specifically, we explore such defects in Abelian fractional quantum Hall states of the form $\ensuremath{\nu}=2/m$ using a K-matrix approach, and identify relevant gapping perturbations. Charged modes on these line defects remain gapped, while charge neutral valley pseudospin modes may be gapped with the aid of two (mutually orthogonal) superlattices, which pin noncommuting fields. When these superlattices are alternated along the line defect, non-Abelian zero modes result at points where the gapping perturbation changes. Given that these pseudospin modes carry no net physical charge or spin, the setup eschews utilizing superconducting and magnetic elements to engineer gapping perturbations. We provide a scheme to braid these modes using strain modulation, and confirm that the resulting unitaries satisfy a representation of the braid group.

Approximate treatment of noncommutative curvature in quartic matrix model

We study a Hermitian matrix model with the standard quartic potential amended by a tr(RΦ2) term for fixed external matrix R. This is motivated by a curvature term in the truncated Heisenberg algebra formulation of the Grosse-Wulkenhaar model — a renormalizable noncommutative field theory. The extra term breaks the unitary symmetry of the action and leads, after perturbative calculation of the unitary integral, to an effective multitrace matrix model. Accompanying the analytical treatment of this multitrace approximation, we also study the model numerically by Monte Carlo simulations. The phase structure of the model is investigated, and a modified phase diagram is identified. We observe a shift of the transition line between the 1-cut and 2-cut phases of the theory that is consistent with the previous numerical simulations and also with the removal of the noncommutative phase in the Grosse-Wulkenhaar model.

A Laplacian to Compute Intersection Numbers on overline{{{mathcal {M}}}}_{g,n} and Correlation Functions in NCQFT

Let F_g(t) be the generating function of intersection numbers of psi -classes on the moduli spaces overline{{{mathcal {M}}}}_{g,n} of stable complex curves of genus g. As by-product of a complete solution of all non-planar correlation functions of the renormalised Phi ^3-matrical QFT model, we explicitly construct a Laplacian Delta _t on a space of formal parameters t_i which satisfies exp (sum _{gge 2} N^{2-2g}F_g(t))=exp ((-Delta _t+F_2(t))/N^2)1 as formal power series in 1/N^2. The result is achieved via Dyson-Schwinger equations from noncommutative quantum field theory combined with residue techniques from topological recursion. The genus-g correlation functions of the Phi ^3-matricial QFT model are obtained by repeated application of another differential operator to F_g(t) and taking for t_i the renormalised moments of a measure constructed from the covariance of the model.

Deformable Protein Shape Classification Based on Deep Learning, and the Fractional Fokker-Planck and Kähler-Dirac Equations.

The classification of deformable protein shapes, based solely on their macromolecular surfaces, is a challenging problem in protein-protein interaction prediction and protein design. Shape classification is made difficult by the fact that proteins are dynamic, flexible entities with high geometrical complexity. In this paper, we introduce a novel description for such deformable shapes. This description is based on the bifractional Fokker-Planck and Dirac-Kähler equations. These equations analyse and probe protein shapes in terms of a scalar, vectorial and non-commuting quaternionic field, allowing for a more comprehensive description of the protein shapes. An underlying non-Markovian Lévy random walk establishes geometrical relationships between distant regions while recalling previous analyses. Classification is performed with a multiobjective deep hierarchical pyramidal neural network, thus performing a multilevel analysis of the description. Our approach is applied to the SHREC'19 dataset for deformable protein shapes classification and to the SHREC'16 dataset for deformable partial shapes classification, demonstrating the effectiveness and generality of our approach.

The Non-Commutative Korteweg—De Vries Hierarchy and Combinatorial Poppe Algebra

We give a constructive proof, to all orders, that each member of the non-commutative potential Korteweg-de Vries hierarchy is a Fredholm Grassmannian flow and is therefore linearisable. Indeed we prove this for any linear combination of fields from this hierarchy. That each member of the hierarchy is linearisable, and integrable in this sense, means that the time evolving solution can be generated from the solution to the corresponding linear dispersion equation in the hierarchy, combined with solving an associated linear Fredholm equation representing the Marchenko equation. Further, we show that within the class of polynomial partial differential fields, at every order, each member of the non-commutative potential Korteweg--de Vries hierarchy is unique. Indeed, we prove to all orders, that each such member matches the non-commutative Lax hierarchy field, which is therefore a polynomial partial differential field. We achieve this by constructing the abstract combinatorial algebra that underlies the non-commutative potential Korteweg-de Vries hierarchy. This algebra is the non-commutative polynomial algebra over the real line generated by the set of all compositions endowed with the Poppe product. This product is the abstract representation of the product rule for Hankel operators pioneered by Ch. Poppe for integrable equations such as the Sine-Gordon and Korteweg-de Vries equations. Integrability of the hierarchy members translates, in the combinatorial algebra, to proving the existence of a `Poppe polynomial' expansion for basic compositions in terms of `linear signature expansions'. Proving the existence of such Poppe polynomial expansions boils down to solving a linear algebraic problem for the expansion coefficients, which we solve constructively to all orders.

Implications of Seiberg–Witten map on Type-I superconductors

Guided by Pippard superconductivity, we incorporate the Seiberg–Witten map in the classical London theory of Type-I superconductors when an external magnetic field is applied. After defining the noncommutative Maxwell potentials, we derive the London equation for the supercurrent as a function of the noncommutative parameter, up to second order in gauge fields expansions. Keeping track of the effects of noncommutative gauge fields, we argue that noncommutative magnetic field effects can be cast in the London penetration length similar to nonlocal Pippard superconductivity. Also, we show that the flux quantization remains consistent relative to the commutative case. Our effective London penetration length reduces to the standard one in the commutative limit. These results allow us to argue that the framework of noncommutative electrodynamics can give some insights into the anisotropic and nonlocal structure of superconductivity and Condensed Matter Systems, perhaps out of the ultra-microscopic scale regime.