Commutative Field Research Articles

Quantum scalar field on fuzzy de Sitter space. Part I. Field modes and vacua

We study a scalar field on a noncommutative model of spacetime, the fuzzy de Sitter space, which is based on the algebra of the de Sitter group SO(1, d) and its unitary irreducible representations. We solve the Klein-Gordon equation in d = 2, 4 and show, using a specific choice of coordinates and operator ordering, that all commutative field modes can be promoted to solutions of the fuzzy Klein-Gordon equation. To explore completeness of this set of modes, we specify a Hilbert space representation and study the matrix elements (integral kernels) of a scalar field: in this way the complete set of solutions of the fuzzy Klein-Gordon equation is found. The space of noncommutative solutions has more degrees of freedom than the commutative one, whenever spacetime dimension is d > 2. In four dimensions, the new non-geometric, internal modes are parametrised by S2 × W, where W is a discrete matrix space. Our results pave the way to analysis of quantum field theory on the fuzzy de Sitter space.

Pseudo-algebraically commutative fields of quasi-fourier–smale curves and hyperbolic topological spaces

Let us suppose we are given a category Σ. Is it possible to construct commutative vectors? We show that I < ∞. It is essential to consider that C may be geometric. A useful survey of the subject can be found in [1].

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.

Non-integrability of the restricted three-body problem

Abstract The problem of non-integrability of the circular restricted three-body problem is very classical and important in the theory of dynamical systems. It was partially solved by Poincaré in the nineteenth century: he showed that there exists no real-analytic first integral which depends analytically on the mass ratio of the second body to the total and is functionally independent of the Hamiltonian. When the mass of the second body becomes zero, the restricted three-body problem reduces to the two-body Kepler problem. We prove the non-integrability of the restricted three-body problem both in the planar and spatial cases for any non-zero mass of the second body. Our basic tool of the proofs is a technique developed here for determining whether perturbations of integrable systems which may be non-Hamiltonian are not meromorphically integrable near resonant periodic orbits such that the first integrals and commutative vector fields also depend meromorphically on the perturbation parameter. The technique is based on generalized versions due to Ayoul and Zung of the Morales–Ramis and Morales–Ramis–Simó theories. We emphasize that our results are not just applications of the theories.

Nonintegrability of dissipative planar systems

We consider dissipative autonomous perturbations of planar Hamiltonian systems and give a sufficient condition for them not to be complex-meromorphically Bogoyavlenskij-integrable such that the first integral or commutative vector field also depends complex-meromorphically on the small parameter. We illustrate the theoretical result for three examples including systems with the Morse potential and effective potential of the Kepler problem.

Complete description of invariant, associative pseudo-Euclidean metrics on left Leibniz algebras via quadratic Lie algebras

A pseudo-Euclidean non-associative algebra (g,•) is a finite dimensional algebra over a field K that has a metric, i.e., a bilinear, symmetric, and non-degenerate form 〈,〉. The metric is considered L-invariant (resp. R-invariant) if all left multiplications (resp. right multiplications) are skew-symmetric. The metric is called associative if 〈u•v,w〉=〈u,v•w〉 for all u,v,w∈g. These three notions coincide when g is a Lie algebra and in this case g endowed with the metric is known as a quadratic Lie algebra.This paper provides a complete description of L-invariant, R-invariant, or associative pseudo-Euclidean metrics on left Leibniz algebras over a commutative field of characteristic zero. It shows that a left Leibniz algebra with an associative metric is also right Leibniz and can be obtained easily from its underlying Lie algebra, which is a quadratic Lie algebra. Additionally, it shows that at the core of a left Leibniz algebra endowed with a L-invariant or R-invariant metric, there are two Lie algebras with one quadratic and the left Leibniz algebra can be built from these Lie algebras. We derive many important results from this complete description. Finally, the paper provides a list of left Leibniz algebras with an associative metric up to dimension 6, as well as a list of left Leibniz algebras with an L-invariant metric, up to dimension 4, and R-invariant metric up to dimension 5.

Nonintegrability of the SEIR epidemic model

We prove the nonintegrability of the susceptible–exposed–infected–removed (SEIR) epidemic model in the Bogoyavlenskij sense. This property of the SEIR model is different from the more fundamental susceptible–infected–removed (SIR) model, which is Bogoyavlenskij-integrable. Our basic tool for the proof is an extension of the Morales–Ramis theory due to Ayoul and Zung. Moreover, we introduce three new state variables and extend the system to a six-dimensional system to treat transcendental first integrals and commutative vector fields. We also use the fact that the incomplete gamma function Γ(α,x) is not elementary for α⁄∈N, of which a proof is included.

Rational symbolic cubature rules over the first quadrant in a Cartesian plane

In this paper we introduce a new symbolic Gaussian formula for theevaluation of an integral over the first quadrant in a Cartesianplane, in particular with respect to the weight function$w(x)=\exp(-x^T x-1/x^T x)$, where $x=(x_1,x_2)^T\in \mathbb{R}^2_+$. Itintegrates exactly a class of homogeneous Laurent polynomials withcoefficients in the commutative field of rational functions in twovariables. It is derived using the connection between orthogonalpolynomials, two-point Padé approximants, and Gaussian cubatures.We also discuss the connection to two-point Padé-typeapproximants in order to establish symbolic cubature formulas ofinterpolatory type. Numerical examples are presented toillustrate the different formulas developed in the paper.

Skew Field on the Binomial Coefficients in Combinatorial Geometric Series

This paper discusses a commutative group, ring, and field under addition and multiplication of the binomial coefficients in combinatorial geometric series. The combinatorial geometric series is derived from the multiple summations of geometric series. The coefficient for each term in combinatorial geometric series refers to a binomial coefficient. This idea can enable the scientific researchers to solve the real life problems.

Nonintegrability of Nearly Integrable Dynamical Systems Near Resonant Periodic Orbits

In a recent paper by the author (K. Yagasaki, Nonintegrability of the restricted three-body problem, submitted for publication), a technique was developed for determining whether nearly integrable systems are not meromorphically Bogoyavlenskij-integrable such that the first integrals and commutative vector fields also depend meromorphically on the small parameter. Here we continue to demonstrate the technique for some classes of dynamical systems. In particular, we consider time-periodic perturbations of single-degree-of-freedom Hamiltonian systems and discuss a relationship of the technique with the subharmonic Melnikov method, which enables us to detect the existence of periodic orbits and their stability. We illustrate the theory for the periodically forced Duffing oscillator and two more additional examples: second-order coupled oscillators and a two-dimensional system of pendulum-type subjected to a constant torque.

Cubic Action of a Rank one Group

We consider a rank one group G = ⟨ A , B ⟩ G = \langle A,B \rangle acting cubically on a module V V , this means [ V , A , A , A ] = 0 [V,A,A,A] =0 but [ V , G , G , G ] ≠ 0 [V,G,G,G] \ne 0 . We have to distinguish whether the group A 0 := C A ( [ V , A ] ) ∩ C A ( V / C V ( A ) ) A_0 :=C_A([V,A]) \cap C_A(V/C_V(A)) is trivial or not. We show that if A 0 A_0 is trivial, G G is a rank one group associated to a quadratic Jordan division algebra. If A 0 A_0 is not trivial (which is always the case if A A is not abelian), then A 0 A_0 defines a subgroup G 0 G_0 of G G acting quadratically on V V . We will call G 0 G_0 the quadratic kernel of G G . By a result of Timmesfeld we have G 0 ≅ S L 2 ( J , R ) G_0 \cong \mathrm {SL}_2(J,R) for a ring R R and a special quadratic Jordan division algebra J ⊆ R J \subseteq R . We show that J J is either a Jordan algebra contained in a commutative field or a Hermitian Jordan algebra. In the second case G G is the special unitary group of a pseudo-quadratic form π \pi of Witt index 1 1 , in the first case G G is the rank one group for a Freudenthal triple system. These results imply that if ( V , G ) (V,G) is a quadratic pair such that no two distinct root groups commute and c h a r V ≠ 2 , 3 \mathrm {char} V\ne 2,3 , then G G is a unitary group or an exceptional algebraic group.

Quantitative theory of composite fermions in Bose-Fermi mixtures at ν=1

Composite fermions provide a simple and unified picture to understand a vast amount of phenomenology in the quantum Hall regime. However it has remained challenging to formulate this concept properly within a single Landau level. Recently a low-energy noncommutative field theory for bosons at Landau-level filling factor $\nu=1$ has been formulated by Dong and Senthil. In the limit of long-wavelength and small-amplitude gauge fluctuation, they found it reduces to the celebrated Halperin-Lee-Read theory of composite fermion liquid. In this work we consider a Bose-Fermi mixture at total filling factor $\nu=1$. Different from previous work, the number density of composite fermions in the mixture and corresponding Fermi momentum can be tuned by changing the filling factor of bosons, $\nu_b = 1 -\nu_f$. This tunability enables us to study the dilute limit $\nu_b\ll 1$, which allows for a controlled and asymptotically exact calculation of the energy dispersion and effective mass of composite fermions. Furthermore, the approximation of the low-energy description by a commutative field theory is manifestly justified. Most importantly, we demonstrate gauge fluctuations acquire a Higgs mass due to the presence of a composite boson condensate, as a result of which the system behaves like a genuine Landau Fermi liquid. Combined with the irrelevance of four-fermion interaction in the dilute limit, we are able to obtain asymptotically exact properties of this composite fermion Fermi liquid. In the opposite limit of $\nu_f\ll 1$, the Higgs mass goes to zero and we find crossover between Fermi liquid and non-Fermi liquid as temperature increases. Observing these properties either experimentally or numerically provides unambiguous evidence of not only the composite fermions and the Fermi surface they form, but also the presence of emergent gauge fields and their fluctuations due to strong correlation.

Persistence of periodic and homoclinic orbits, first integrals and commutative vector fields in dynamical systems

We study persistence of periodic and homoclinic orbits, first integrals and commutative vector fields in dynamical systems depending on a small parameter ɛ > 0 and give several necessary conditions for their persistence. Here we treat homoclinic orbits not only to equilibria but also to periodic orbits. We also discuss some relationships of these results with the standard subharmonic and homoclinic Melnikov methods for time-periodic perturbations of single-degree-of-freedom Hamiltonian systems, and with another version of the homoclinic Melnikov method for autonomous perturbations of multi-degree-of-freedom Hamiltonian systems. In particular, we show that a first integral which converges to the Hamiltonian or another first integral as the perturbation tends to zero does not exist near the unperturbed periodic or homoclinic orbits in the perturbed systems if the subharmonic or homoclinic Melnikov functions are not identically zero on connected open sets. We illustrate our theory for four examples: the periodically forced Duffing oscillator, two identical pendula coupled with a harmonic oscillator, a periodically forced rigid body and a three-mode truncation of a buckled beam.

Quaternion rings over local rings

In 1843, Hamilton (1805–1865) discovered the 4-dimensional division algebra [Formula: see text] over the field [Formula: see text] of real numbers. Hamilton’s big discovery is the following beautiful multiplications for the basis [Formula: see text] :[Formula: see text] [Formula: see text] contains the field [Formula: see text] of complex numbers; therefore [Formula: see text]. Frobenius (1849–1917) showed the following outstanding theorem. Theorem (Frobenius). Up to isomorphism, the only finite-dimensional non-commutative division algebra over [Formula: see text] is [Formula: see text]. Starting from given any ring [Formula: see text] and a free right [Formula: see text]-module [Formula: see text], the quaternion ring [Formula: see text] is canonically defined by the multiplications [Formula: see text]. For a commutative field [Formula: see text] with [Formula: see text], [Formula: see text] is a division ring or isomorphic to the ring of [Formula: see text] matrices over [Formula: see text]. This is a classical theorem. For nonzero [Formula: see text] in the center of a ring [Formula: see text], the generalized quaternion ring [Formula: see text] is defined. [Formula: see text] is [Formula: see text]. In [I. Kikumasa, K. Koike and K. Oshiro, Complex rings and quaternion rings, East-West J. Math. 21 (2019) 1–19; I. Kikumasa, G. Lee and K. Oshiro, Complex Rings, Quaternion Rings and Octonion Rings (Lambert Academic Publishing, 2020); G. Lee and K. Oshiro, Quaternion rings and octonion rings, Front. Math. China 12(1) (2017) 143–155], quaternion rings and generalized quaternion rings over division rings or other rings are studied. Now, in this paper, from ring theoretic viewpoints, we study quaternion rings [Formula: see text] and generalized quaternion rings [Formula: see text] over local rings [Formula: see text]. From our results, we can clearly look over several classical results on [Formula: see text] and [Formula: see text] over commutative fields [Formula: see text].

A general class of Danielewski algebras

Given a commutative field K of characteristic zero we study in this paper a class of K-algebras which contains the coordinate rings of the classical Danielewski surfaces. We describe their locally nilpotent K-derivations and compute their Makar-Limanov invariant. We also determine the K-algebras in our class that have a trivial Makar-Limanov invariant.

Bases for skew derivations

Let R be a prime ring, Q the Utumi quotient ring of R and Ω the expansion closed set of products of automorphisms and skew derivations of Q. Let Λ⊆Ω∖Ωm, where m≥0, be such that for any w∈Λ(g,h), where g,h are automorphisms of Q, w(xy)−g(x)w(y)−w(x)h(y) is a sum of terms with δ(x) and δ′(y), where δ∈Λ|w|−1orδ′∈Λ|w|−1orδ,δ′∈Ωm (Definition 3). Suppose that no monic linear identities with words in Λ∪Ωm involve words in Λ. We prove the following:(i) Λ extends to a basis of Ω (Theorem 1).(ii) If a generalized polynomial φ with words in Λ∪Ωm is an identity, then so is the generalized polynomial obtained from φ by replacing every occurrence of ν(x) in φ by an arbitrary new variable yν,x for ν∈Λ and variable x (Theorem 2).Dedekind's Lemma asserts that distinct automorphisms σi of a commutative field are independent maps over the field and hence, via linearization, σi(xj) in distinct variables xj, considered as distinct maps for distinct ordered pair (i,j), are transcendental over the field in the case of characteristic 0. (ii) above generalizes this to maps defined by w∈Λ with dependency and transcendency over the prime ring R.

Differential Equations in a Tangent Category I: Complete Vector Fields, Flows, and Exponentials

This paper describes how to define and work with differential equations in the abstract setting of tangent categories. The key notion is that of a curve object which is, for differential geometry, the structural analogue of a natural number object. A curve object is a preinitial object for dynamical systems; dynamical systems may, in turn, be viewed as determining systems of differential equations. The unique map from the curve object to a dynamical system is a solution of the system, and a dynamical system is said to be complete when for all initial conditions there is a solution. A subtle issue concerns the question of when a dynamical system is complete, and the paper provides abstract conditions for this. This abstract formulation also allows new perspectives on topics such as commutative vector fields and flows. In addition, the stronger notion of a differential curve object, which is the centrepiece of the last section of the paper, has exponential maps and forms a differential exponential rig. This rig then, somewhat surprisingly, has an action on every differential object and bundle in the setting. In this manner, in a very strong sense, such a curve object plays the role of the real numbers in standard differential geometry.

⋆-cohomology, third type Chern character and anomalies in general translation-invariant noncommutative Yang–Mills

A representation of general translation-invariant star products ⋆ in the algebra of [Formula: see text] is introduced which results in the Moyal–Weyl–Wigner quantization. It provides a matrix model for general translation-invariant noncommutative quantum field theories in terms of the noncommutative calculus on differential graded algebras. Upon this machinery a cohomology theory, the so-called ⋆-cohomology, with groups [Formula: see text], [Formula: see text], is worked out which provides a cohomological framework to formulate general translation-invariant noncommutative quantum field theories based on the achievements for the commutative fields, and is comparable to the Seiberg–Witten map for the Moyal case. Employing the Chern–Weil theory via the integral classes of [Formula: see text] a noncommutative version of the Chern character is defined as an equivariant form which contains topological information about the corresponding translation-invariant noncommutative Yang–Mills theory. Thereby, we study the mentioned Yang–Mills theories with three types of actions of the gauge fields on the spinors, the ordinary, the inverse, and the adjoint action, and then some exact solutions for their anomalous behaviors are worked out via employing the homotopic correlation on the integral classes of ⋆-cohomology. Finally, the corresponding consistent anomalies are also derived from this topological Chern character in the ⋆-cohomology.

Parallelizability of control systems

This manuscript presents a notion of parallelizability of control systems. Parallelizability is a well-known concept of dynamical systems that associates with complete instability and dispersiveness. The concept of dispersiveness has been successfully interpreted in the setup of control systems. This naturally asks about the meaning of a parallelizable control system. The answer can be given in the setting of control affine systems by evoking their control flows. The main result shows that a parallelizable control flow characterizes a dispersive control affine system. The dispersiveness is then equivalent to the existence of a functional with infinite limit at infinity. The results of the paper contribute to the controllability studies, since dispersive control systems admit no control set. For invariant control systems with commutative vector fields, null trace representative matrices are a necessary condition for the existence of control set.

A standard form in (some) free fields: How to construct minimal linear representations

AbstractWe describe a standard form for the elements in the universal field of fractions of free associative algebras (over acommutativefield). It is a special version of the normal form provided by Cohn and Reutenauer and enables the use of linear algebra techniques for the construction of minimal linear representations (in standard form) for the sum and the product of two elements (given in a standard form). This completes “minimal” arithmetic in free fields since “minimal” constructions for the inverse are already known. The applications are wide: linear algebra (over the free field), rational identities, computing the left gcd of two non-commutative polynomials, etc.