Noncommutative Theory Research Articles

Field Equations and Radial Solutions in a Noncommutative Spherically Symmetric Geometry

We study a noncommutative theory of gravity in the framework of torsional spacetime. This theory is based on a Lagrangian obtained by applying the technique of dimensional reduction of noncommutative gauge theory and that the yielded diffeomorphism invariant field theory can be made equivalent to a teleparallel formulation of gravity. Field equations are derived in the framework of teleparallel gravity through Weitzenbock geometry. We solve these field equations by considering a mass that is distributed spherically symmetrically in a stationary static spacetime in order to obtain a noncommutative line element. This new line element interestingly reaffirms the coherent state theory for a noncommutative Schwarzschild black hole. For the first time, we derive the Newtonian gravitational force equation in the commutative relativity framework, and this result could provide the possibility to investigate examples in various topics in quantum and ordinary theories of gravity.

Canonical approach to the closed string non-commutativity

We consider the closed string moving in the weakly curved background and its totally T-dualized background. Using T-duality transformation laws, we find the structure of the Poisson brackets in the T-dual space corresponding to the fundamental Poisson brackets in the original theory. From this structure we obtain that the commutative original theory is equivalent to the non-commutative T-dual theory, whose Poisson brackets are proportional to the background fluxes times winding and momenta numbers. The non-commutative theory of the present article is more nongeometrical then T-folds and in the case of three space-time dimensions corresponds to the nongeometric space-time with $R$-flux.

REFLEXIVE PROPERTY ON IDEMPOTENTS

The reflexive property for ideals was introduced by Mason and has important roles in noncommutative ring theory. In this note we study the structure of idempotents satisfying the reflexive property and introduce reflexive-idempotents-property (simply, RIP) as a generaliza- tion. It is proved that the RIP can go up to polynomial rings, power series rings, and Dorroh extensions. The structure of non-Abelian RIP rings of minimal order (with or without identity) is completely investi- gated. Throughout this paper all rings are associative with identity unless otherwise stated. Given a ring R the polynomial ring (resp., the power series ring) with an indeterminate x over R is denoted by R(x) (resp., R((x))). For any polynomial f(x) in R(x), let Cf(x) denote the set of all coefficients off(x). Denote the n by n full matrix ring over R by Matn(R) and the n by n upper triangular matrix ring over R by Un(R). Use Eij for the matrix with (i,j)-entry 1 and elsewhere 0. Let Id(R) be the set of all idempotent elements of R. Denote {a ∈ Un(R) | the diagonal entries of a are all equal} by Dn(R). Z and Zn denote the ring of integers and the ring of integers modulo n, respectively. GF(p n ) denotes the Galois field of order p n for a prime p and n ≥ 1. J(R) denotes the Jacobson radical of R. |S | denotes the cardinality of given a set S. Mason (18) introduced the reflexive property for ideals, and then this concept was generalized by Kim and Baik (9, 10) by defining idempotent reflexive right ideals and rings. A right ideal I of a ring R (possibly without identity) is called reflexive (18) if aRb ⊆ I implies bRa ⊆ I for a,b ∈ R. R is called reflexive if 0 is a reflexive ideal (i.e., aRb = 0 implies bRa = 0 for a,b ∈ R.) In (12), Kwak and Lee characterized aspects of the reflexive and one-sided idempotent reflexive properties, showing that the concept of idempotent reflexive ring is not left- right symmetric (12, Example 3.3). For a one-sided ideal I of a ring R, I is

VANISHING DIMENSIONS: A REVIEW

We review a growing theoretical motivation and evidence that the number of dimensions actually reduces at high energies. This reduction can happen near the Planck scale, or much before, the dimensions that are reduced can be effective, spectral, topological or the usual dimensions, but many things point toward the fact that the high energy theories appear to propagate in a lower-dimensional space, rather than a higher-dimensional one. We will concentrate on a particular scenario of "vanishing" or "evolving dimensions" where the dimensions open up as we increase the length scale that we are probing, but will also mention related models that point to the same direction, i.e. the causal dynamical triangulation, asymptotic safety, as well as evidence coming from a noncommutative quantum theories, the Wheeler–DeWitt equation and phenomenon of "asymptotic silence". It is intriguing that experimental evidence for the high energy dimensional reduction may already exist — a statistically significant planar alignment of events with energies higher than TeV has been observed in high altitude cosmic ray experiments. A convincing evidence for dimensional reduction may be found in future in collider experiments and gravity waves observatories.

Homotopy probability theory I

This is the second of two papers that introduce a deformation theoretic framework to explain and broaden a link between homotopy algebra and probability theory. This paper outlines how the framework can assist in the development of homotopy probability theory, where a vector space of random variables is replaced by a chain complex of random variables. This allows the principles of derived mathematics to participate in classical and noncommutative probability theory. A simple example is presented.

Holographic entanglement entropy in nonlocal theories

We compute holographic entanglement entropy in two strongly coupled nonlocal field theories: the dipole and the noncommutative deformations of SYM theory. We find that entanglement entropy in the dipole theory follows a volume law for regions smaller than the length scale of nonlocality and has a smooth cross-over to an area law for larger regions. In contrast, in the noncommutative theory the entanglement entropy follows a volume law for up to a critical length scale at which a phase transition to an area law occurs. The critical length scale increases as the UV cutoff is raised, which is indicative of UV/IR mixing and implies that entanglement entropy in the noncommutative theory follows a volume law for arbitrary large regions when the size of the region is fixed as the UV cutoff is removed to infinity. Comparison of behaviour between these two theories allows us to explain the origin of the volume law. Since our holographic duals are not asymptotically AdS, minimal area surfaces used to compute holographic entanglement entropy have novel behaviours near the boundary of the dual spacetime. We discuss implications of our results on the scrambling (thermalization) behaviour of these nonlocal field theories.

Spectral metric spaces for Gibbs measures

We construct spectral metric spaces for Gibbs measures on a one-sided topologically exact subshift of finite type. That is, for a given Gibbs measure we construct a spectral triple and show that Connesʼ corresponding pseudo-metric is a metric and that its metric topology agrees with the weak-⁎-topology on the state space over the set of continuous functions defined on the subshift. Moreover, we show that each Gibbs measure can be fully recovered from the noncommutative integration theory and that the noncommutative volume constant of the associated spectral triple is equal to the reciprocal of the measure theoretical entropy of the shift invariant Gibbs measure.

P-adic L-functions over the false Tate curve extensions

AbstractLet f be a primitive modular form of CM type of weight k and level Γ0(N). Let p be an odd prime which does not divide N, and for which f is ordinary. Our aim is to p-adically interpolate suitably normalized versions of the critical values L(f, ρχ,n), where n=1,2,. . .,k − 1, ρ is a fixed self-dual Artin representation of M∞ defined by (1.1) below, and χ runs over the irreducible Artin representations of the Galois group of the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. As an application, if k ≥ 4, we will show that there are only finitely many χ such that L(f, ρχ,k/2)=0, generalizing a result of David Rohrlich. Also, we conditionally establish a congruence predicted by non-commutative Iwasawa theory and give numerical evidence for it.

The minimal and the new minimal supersymmetric Grand Unified Theories on noncommutative space-time

We construct noncommutative versions of both the minimal and the new minimal supersymmetric Grand Unified Theories (GUTs). The enveloping-algebra formalism is used to carry out such constructions. The beautiful formulation of the Higgs sector of these noncommutative theories is a consequence of the fact that, in the GUTs at hand, the ordinary Higgs fields can be realized as elements of the Clifford algebra . In the noncommutative supersymmetric GUTs we formulate, supersymmetry is linearly realized by the noncommutative fields; but it is not realized by the ordinary fields that define those noncommutative fields via the Seiberg–Witten map.

ON μ-INVARIANTS OF SELMER GROUPS OF SOME CM ELLIPTIC CURVES

We show that the Selmer groups defined over the cyclotomic ℤ3-extension of the field of 3-torsion points of the CM elliptic curves 256a1, 256a2, 256d1, 256d2, 121b1, 121b2 have μ-invariant equal to zero, verifying a conjecture in non-commutative Iwasawa theory.

Backreacted flavor in non-commutative gauge theories

We construct the gravity dual of N=1 Non-Commutative SYM theory coupled to Nf smeared massless fundamental flavors. Our solution is analytic and non-perturbative. Near the origin the background reduces to the one corresponding to an ordinary SYM theory. At large radial distances, the dilaton diverges signaling the presence of a UV Landau pole. Considering a probe D3-brane we calculate the effective YM coupling and show that it is independent on the parameter of non-commutativity. We calculate the corresponding beta function and show that it remains positive at all energy scales.

Free biholomorphic functions and operator model theory, II

In a companion to this paper, we introduced the class of n-tuples f=(f1,…,fn) of formal power series in noncommutative indeterminates Z1,…,Zn with the model property and developed an operator model theory for pure n-tuples of operators in noncommutative domains Bf(H)⊂B(H)n, where the associated universal model is an n-tuple (MZ1,…,MZn) of multiplication operators on a Hilbert space H2(f) of formal power series. In the present paper, we continue this work by considering the completely non-coisometric (c.n.c) case. In the second part of the paper, several results concerning the noncommutative multivariable operator theory on the unit ball [B(H)n]1− are extended to noncommutative varieties Vf,J(H)⊆Bf(H) defined byVf,J(H):={(T1,…,Tn)∈Bf(H):ψ(T1,…,Tn)=0 for any ψ∈J} for an appropriate evaluation ψ(T1,…,Tn) and associated with WOT-closed two-sided ideals J of the Hardy algebra H∞(Bf). In particular, we obtain commutative versions for all the results.

B-Construction and C-Construction

B-construction is a way of obtaining a graded algebra from the triple consisting of an additive category, an object, and an autoequivalence, while C-construction is a way of obtaining an algebra (without unity) from the pair consisting of an additive category and a set of objects. In this article, we study and compare three important classes of algebras in noncommutative algebraic geometry and representation theory of finite dimensional algebras, namely, quantum polynomial algebras, preprojetive algebras and trivial extensions, via these constructions.

Tensorial Function Theory: From Berezin Transforms to Taylor’s Taylor Series and Back

Let H ∞(E) be the Hardy algebra of a W*-correspondence E over a W*-algebra M. Then the ultraweakly continuous completely contractive representations of H ∞(E) are parametrized by certain sets $${{\mathcal{AC}}(\sigma)}$$ indexed by NRep(M)—the normal *-representations σ of M. Each set $${{\mathcal{AC}}(\sigma)}$$ has analytic structure, and each element $${F \in H^{\infty}(E)}$$ gives rise to an analytic operator-valued function $${\widehat{F}_{\sigma}}$$ on $${{\mathcal{AC}}(\sigma)}$$ that we call the σ-Berezin transform of F. The sets $${\{{\mathcal{AC}}(\sigma)\}_{\sigma\in\Sigma}}$$ and the family of functions $${\{\widehat{F}_{\sigma}\}_{\sigma\in\Sigma}}$$ exhibit “matricial structure” that was introduced by Joeseph Taylor in his work on noncommutative spectral theory in the early 1970s. Such structure has been exploited more recently in other areas of free analysis and in the theory of linear matrix inequalities. Our objective here is to determine the extent to which the matricial structure characterizes the Berezin transforms.

On main conjectures in non-commutative Iwasawa theory and related conjectures

Abstract The main conjecture of non-commutative Iwasawa theory is shown to be equivalent, modulo an appropriate torsion hypothesis, to a family of main conjectures over cyclotomic ℤ p -extensions together with suitable compatibility relations between the associated abelian p-adic L-functions. This result combines with techniques of Ritter and Weiss and of Kakde to give a concrete strategy for deducing main conjectures for (non-abelian) compact p-adic Lie extensions of arbitrarily large rank from main conjectures in the classical sense. As a first application of this approach we combine it with a recent result of Ritter and Weiss to prove, modulo an appropriate vanishing hypothesis on μ-invariants, the main conjecture of non-commutative Iwasawa theory for Tate motives over compact p-adic Lie extensions of totally real number fields of arbitrarily large rank. We then also deduce several interesting consequences of this result concerning a range of related conjectures including special cases of the equivariant Tamagawa number conjecture.

Identification of the delay parameter for nonlinear time-delay systems with unknown inputs

Using the theory of non-commutative rings, this paper studies the delay identification of nonlinear time-delay systems with unknown inputs. A sufficient condition is given in order to deduce an output delay equation from the studied system. Then necessary and sufficient conditions are proposed to judge whether the deduced output delay equation can be used to identify delay involved in this equation. Two different cases are discussed for the dependent and independent outputs, respectively. The presented result is applied to identify delay in a biological system.

SOLUTIONS FOR THE LANDAU PROBLEM USING SYMPLECTIC REPRESENTATIONS OF THE GALILEI GROUP

Symplectic unitary representations for the Galilei group are studied. The formalism is based on the noncommutative structure of the star-product, and using group theory approach as a guide, a consistent physical theory in phase space is constructed. The state of a quantum mechanics system is described by a quasi-probability amplitude that is in association with the Wigner function. As a result, the Schrödinger and Pauli–Schrödinger equations are derived in phase space. As an application, the Landau problem in phase space is studied. This shows how this method of quantum mechanics in phase space is to be brought to the realm of spatial noncommutative theories.

About the Connes embedding conjecture

In his celebrated paper in 1976, A. Connes casually remarked that any finite von Neumann algebra ought to be embedded into an ultraproduct of matrix algebras, which is now known as the Connes embedding conjecture or problem. This conjecture became one of the central open problems in the field of operator algebras since E. Kirchberg’s seminal work in 1993 that proves it is equivalent to a variety of other seemingly totally unrelated but important conjectures in the field. Since then, many more equivalents of the conjecture have been found, also in some other branches of mathematics such as noncommutative real algebraic geometry and quantum information theory. In this note, we present a survey of this conjecture with a focus on the algebraic aspects of it.

Non-commutative Iwasawa theory for modular forms

The aim of the present paper is to give evidence, largely numerical, in support of the non-commutative main conjecture of Iwasawa theory for the motive of a primitive modular form of weight k > 2 over the Galois extension of ℚ obtained by adjoining to ℚ all p-power roots of unity, and all p-power roots of a fixed integer m > 1. The predictions of the main conjecture are rather intricate in this case because there is more than one critical point, and also there is no canonical choice of periods. Nevertheless, our numerical data agree perfectly with all aspects of the main conjecture, including Kato's mysterious congruence between the cyclotomic Manin p-adic L-function, and the cyclotomic p-adic L-function of a twist of the motive by a certain non-abelian Artin character of the Galois group of this extension.

A bi-invariant Einstein–Hilbert action for the non-geometric string

Inspired by recent studies on string theory with non-geometric fluxes, we develop a differential geometry calculus combining usual diffeomorphisms with what we call β-diffeomorphisms. This allows us to construct a manifestly bi-invariant Einstein–Hilbert type action for the graviton, the dilaton and a dynamical (quasi-)symplectic structure. The equations of motion of this symplectic gravity theory, further generalizations and the relation to the usual form of the string effective action are discussed. The Seiberg–Witten limit, known for open strings to relate commutative with non-commutative theories, makes an interesting appearance.