Commutative Field Research Articles

Commutative presemifields and semifields

Strong conditions are derived for when two commutative presemifields are isotopic. It is then shown that any commutative presemifield of odd order can be described by a planar Dembowski–Ostrom polynomial and conversely, any planar Dembowski–Ostrom polynomial describes a commutative presemifield of odd order. These results allow a classification of all planar functions which describe presemifields isotopic to a finite field and of all planar functions which describe presemifields isotopic to Albert's commutative twisted fields. A classification of all planar Dembowski–Ostrom polynomials over any finite field of order p 3 , p an odd prime, is therefore obtained. The general theory developed in the article is then used to show the class of planar polynomials X 10 + a X 6 − a 2 X 2 with a ≠ 0 describes precisely two new commutative presemifields of order 3 e for each odd e ⩾ 5 .

Parametric Representation of Noncommutative Field Theory

In this paper we investigate the Schwinger parametric representation for the Feynman amplitudes of the recently discovered renormalizable \({\phi^{4}_4}\) quantum field theory on the Moyal non commutative \({\mathbb R^{4}}\) space. This representation involves new hyperbolic polynomials which are the non-commutative analogs of the usual “Kirchoff” or “Symanzik” polynomials of commutative field theory, but contain richer topological information.

NONCOMMUTATIVE QUANTIZATION FOR NONCOMMUTATIVE FIELD THEORY

We present a new procedure for quantizing field theory models on a noncommutative space–time. Our new quantization scheme depends on the noncommutative parameter explicitly and reduces to the canonical quantization in the commutative limit. It is shown that a quantum field theory constructed by this quantization yields exactly the same correlation functions as those of the commutative field theory, that is, the noncommutative effects disappear completely after the quantization. This implies, for instance, that the noncommutativity may be incorporated in the process of quantization, rather than in the action as conventionally done.

On the stability of Jensen’s functional equation on groups

In this paper we establish the stability of Jensen’s functional equation on some classes of groups. We prove that Jensen equation is stable on noncommutative groups such as metabelian groups and T (2, K), where K is an arbitrary commutative field with characteristic different from two. We also prove that any group A can be embedded into some group G such that the Jensen functional equation is stable on G.

Noncommutative Quantization for Noncommutative Field Theory

We present a new procedure for quantizing field theory models on a noncommutative space-time. The new quantization depends on the noncommutative parameter explicitly and reduces to the canonical quantization in the commutative limit. It is shown that a QFT constructed by the new quantization yields exactly the same correlation functions as those of the commutative field theory, that is, the noncommutative effects disappear completely after quantization. In addition, it is proved that the Poincare symmetry of the commutative QFT corresponds to the twisted Poincare symmetry for new QFT.

NONCOMMUTATIVITY FROM THE SYMPLECTIC POINT OF VIEW

The great deal in noncommutative (NC) field theories started when it was noted that NC spaces naturally arise in string theory with a constant background magnetic field in the presence of D-branes. In this work we explore how NC geometry can be introduced into a commutative field theory besides the usual introduction of the Moyal product. We propose a nonperturbative systematic new way to introduce NC geometry into commutative systems, based mainly on the symplectic approach. Further, as example, this formalism describes precisely how to obtain a Lagrangian description for the NC version of some systems reproducing well-known theories.

One-loop unitarity of scalar field theories on Poincaré invariant commutative nonassociative spacetimes

We study scalar field theories on Poincare invariant commutative nonassociative spacetimes. We compute the one-loop self-energy diagrams in the ordinary path integral quantization scheme with Feynman's prescription, and find that the Cutkosky rule is satisfied. This property is in contrast with that of noncommutative field theory, since it is known that noncommutative field theory with space/time noncommutativity violates unitarity in the above standard scheme, and the quantization procedure will necessarily become complicated to obtain a sensible Poincare invariant noncommutative field theory. We point out a peculiar feature of the non-locality in our nonassociative field theories, which may explain the property of the unitarity distinct from noncommutative field theories. Thus commutative nonassociative field theories seem to contain physically interesting field theories on deformed spacetimes.

Non-anticommutative supersymmetric field theory and quantum shift

Abstract Non-anticommutative Grassmann coordinates in four-dimensional twist-deformed N = 1 Euclidean superspace are decomposed into geometrical ones and quantum shift operators. This decomposition leads to the mapping from the commutative to the non-anticommutative supersymmetric field theory. We apply this mapping to the Wess–Zumino model in commutative field theory and derive the corresponding non-anticommutative Lagrangian. Based on the theory of twist deformations of Hopf algebras, we comment on the preservation of the (initial) N = 1 super-Poincare algebra and on the consequent super-Poincare invariant interpretation of the discussed model, but also provide a measure for the violation of the super-Poincare symmetry.

The geometry of k-transvection groups

Let k be a (commutative) field and G a group, then a conjugacy class of Abelian subgroups of G is called a class of k-transvection subgroups in G if and only if it generates G and any two elements of the class either commute or are full unipotent subgroups of the group they generate and which is isomorphic to ( P ) SL 2 ( k ) . In this paper we study the geometry of k-transvection groups. Given a class of k-transvection groups Σ, we consider a partial linear space whose points are the elements of Σ, and whose lines correspond to the groups generated by two noncommuting elements from Σ. We derive several properties of this partial linear space. These properties are used to give a characterization of the geometries of k-transvection groups and provide a classification of groups generated by k-transvection subgroups.

Non-constant non-commutativity in 2d field theories and a new look at fuzzy monopoles

We write down scalar field theory and gauge theory on two-dimensional non-commutative spaces M with non-vanishing curvature and non-constant non-commutativity. Usual dynamics results upon taking the limit of M going to (i) a commutative manifold M 0 having non-vanishing curvature and (ii) the non-commutative plane. Our procedure does not require introducing singular algebraic maps or frame fields. Rather, we exploit the Kähler structure in the limit (i) and identify the symplectic two-form with the volume two-form. As an example, we take M to be the stereographically projected fuzzy sphere, and find magnetic monopole solutions to the non-commutative Maxwell equations. Although the magnetic charges are conserved, the classical theory does not require that they be quantized. The non-commutative gauge field strength transforms in the usual manner, but the same is not, in general, true for the associated potentials. We develop a perturbation scheme to obtain the expression for gauge transformations about limits (i) and (ii). We also obtain the lowest order Seiberg–Witten map to write down corrections to the commutative field equations and show that solutions to Maxwell theory on M 0 are stable under inclusion of lowest order non-commutative corrections. The results are applied to the example of non-commutative AdS 2.

Moufang sets and jordan division algebras

We give a simple criterion which determines when a permutation group U and one additional permutation give rise to a Moufang set. We apply this criterion to show that every Jordan division algebra gives rise in a very natural way to a Moufang set, to provide sufficient conditions for a Moufang set to arise from a Jordan division algebra and to give a characterization of the projective Moufang sets over a commutative field of characteristic different from 2.

Automorphisms of Unitals

It is shown that every automorphism of classical unitals over certain (not necessarily commutative) fields is induced by a semi-similitude of a corresponding hermitian form. In particular, this is true if the form uses an involution of the second kind. In [16], J. Tits has studied classical unitals, defined by suitable polarities. Using

An interpretation of noncommutative field theory in terms of a quantum shift

Noncommutative coordinates are decomposed into a sum of geometrical ones and a universal quantum shift operator. With the help of this operator, the mapping of a commutative field theory into a noncommutative field theory (NCFT) is introduced. A general measure for the Lorentz-invariance violation in NCFT is also derived.

Non-planar diagrams and non-commutative superspace in Dijkgraaf–Vafa theory

We consider the field theory on non-commutative superspace and non-commutative spacetime that arises on D-branes in Type II superstring theory with a constant self-dual graviphoton and NS–NS B field background. N=1 supersymmetric field theories on this non-commutative space (such theories are called N=1/2 supersymmetric theories) can be reduced to supermatrix models as in hep-th/0303210. We take an appropriate commutative limit in these theories and show that holomorphic quantities in commutative field theories are equivalent to reduced models, including non-planar diagrams to which the graviphoton contributes. This is a new derivation of Dijkgraaf–Vafa theory including non-planar diagrams.

Orthogonal Cayley-Klein Groups

A. Cayley [5] and F. Klein [12] were the first mathematicians who introduced the notion of a metric in a real projective space, by specializing a set of quadrics (called the absolute). We show in this paper, that from an algebraic point of view these projective spaces can be described by vector spaces, in which a metric structure is given by one or more symmetric bilinear forms. We study these Cayley- Klein vector spaces and their groups of automorphisms (orthogonal Cayley- Klein groups) over arbitrary commutative fields of characteristic ≠ 2.

A characterization of the projective line

Let X X be a set (with at least three different points) and let G G be a group of bijections of X X . If the action of G G on X X satisfies three natural conditions, then X X admits a canonical structure of a projective line over a commutative field, such that G G is the group of all projective transformations of X X .

RANK t ℋ-PRIMES IN QUANTUM MATRICES#

ABSTRACT Let 𝕂 be a (commutative) field and consider a nonzero element q in 𝕂 that is not a root of unity. Goodearl and Lenagan (2002) have shown that the number of ℋ-primes in R = O q (ℳ n (𝕂)) that contain all (t + 1) × (t + 1) quantum minors but not all t × t quantum minors is a perfect square. The aim of this paper is to make precise their result: we prove that this number is equal to (t!) 2 S(n + 1, t + 1)2, where S(n + 1, t + 1) denotes the Stirling number of the second kind associated to n + 1 and t + 1. This result was conjectured by Goodearl, Lenagan, and McCammond. The proof involves some closed formulas for the poly-Bernoulli numbers that were established by Kaneko (1997) and Arakawa and Kaneko (1999).

Isometries and collineations of the Cayley surface

Let F be Cayley’s ruled cubic surface in a projective three-space over any commutative field K. We determine all collineations fixing F , as a set, and all cubic forms defining F . For both problems the cases |K| = 2, 3 turn out to be exceptional. On the other hand, if |K| ≥ 4 then the set of simple points of F can be endowed with a non-symmetric distance function. We describe the corresponding circles, and we establish that each isometry extends to a unique projective collineation of the ambient space.

THE ENERGY–MOMENTUM TENSOR IN NONCOMMUTATIVE GAUGE FIELD MODELS

We discuss the different possibilities of constructing the various energy–momentum tensors for noncommutative gauge field models. We use Jackiw's method in order to get symmetric and gauge invariant stress tensors — at least for commutative gauge field theories. The noncommutative counterparts are analyzed with the same methods. The issues for the noncommutative cases are worked out.

Affine extensions of loops

The most known examples of loops L having strong relations to geometry have classical groups as the groups generated by their left translations ( [5], [8], [7],[4], [6], Chapter 9, [10], Chapters 22 and 25, [2], [3]). These groups G may be seen as subgroups of the stabilizer of 0 in the group of affinities of suitable affine spaces An, and as the elements of the loops L may be often taken certain projective subspaces of the hyperplane at infinity of An. The semidirect products T o G, where T is the translation group of the affine space An have in many cases a geometric interpretation as motion groups of affine metric geometries. In the papers [2], [3] three dimensional connected differentiable loops are constructed which have the connected component of the motion group of the 3-dimensional hyperbolic or pseudo-euclidean geometry as the group topologically generated by the left translations and which are Bol, Bruck or left A-loops. The set of the left translations of these loops induces on the plane at infinity the set of left translations of a loop isotopic to the hyperbolic plane loop (cf. [10], Chapter 22, p. 280, [7], p. 189). This and the fact that, up to our knowledge, there are only few known examples of sharply transitive sections in affine metric motion groups, motivated us to seek a simple geometric procedure for an extension of a loop realized as the image Σ∗ of a sharply transitive section in a subgroup G∗ of the projective linear group PGL(n − 1, F ) to a loop realized as the image of sharply transitive section in a subgroup G of the group of affinities of the n-dimensional space An = F n over the commutative field F . Moreover, we desire that G contains a large normal subgroup of affine translations and that α(G) = G∗ holds for the canonical homomorphism α : GL(n, F ) → PGL(n, F ). We show that this goal can be achived if there exists for G∗ in the (n−1)-dimensional projective space Pn−1(F ) an orbitO ofm-dimensional subspaces such that Σ∗ acts sharply transitively on O, there is a subspace of dimension (n− 1−m) having empty intersection with any element of O and if the restriction of α−1 to Σ∗ defines a bijection from α−1(Σ∗) to Σ∗.