Classes Of Spinors Research Articles

Characterizing nonlocality of pure symmetric three-qubit states

We explore nonlocality of three-qubit pure symmetric states shared between Alice, Bob and Charlie using the Clauser-Horne-Shimony-Holt (CHSH) inequality. We make use of the elegant parametrization in the canonical form of these states, proposed by Meill and Meyer (Phys. Rev. A, 96, 062310 (2017)) based on Majorana geometric representation. The reduced two-qubit states, extracted from an arbitrary pure entangled symmetric three-qubit state do not violate the CHSH inequality and hence they are CHSH-local. However, when Alice and Bob perform a CHSH test, after conditioning over measurement results of Charlie, nonlocality of the state is revealed. We have also shown that two different families of three-qubit pure symmetric states, consisting of two and three distinct spinors (qubits) respectively, can be distinguished based on the strength of violation in the conditional CHSH nonlocality test. Furthermore, we identify six of the 46 classes of tight Bell inequalities in the three-party, two-setting, two-outcome i.e., (3,2,2) scenario (Phys. Rev. A 94, 062121 (2016)). Among the two inequivalent families of three-qubit pure symmetric states, only the states belonging to three distinct spinor class show maximum violations of these six tight Bell inequalities.

Constraints on mapping the Lounesto\u2019s classes

The so-called Lounesto’s classification engenders six distinct classes of spinors, divided into two sectors: one composed by regular spinors (single-helicity spinors) and the other composed by singular spinors (comprising dual-helicity spinors). In the present essay we develop a mechanism to fully define the right class within the Lounesto’s classification a spinor belongs to, without necessity to evaluate the 16 bilinear forms. The analysis lies in the following criteria: a judicious inspection of the phases factor present in both spinor’s components. Thus, the machinery developed here works for both regular and singular spinors. Taking advantage of the present algorithm, we analyse, under certain conditions, the possibility to transmute between the six classes.

Revealing how different spinors can be: The Lounesto spinor classification

This paper aims to give a coordinate-based introduction to the so-called Lounesto spinorial classification scheme. Among other results, it has evinced classes of spinors which fail to satisfy Dirac equation. The underlying idea and the central aspects of such spinorial categorization are introduced in an argumentative basis, after which we delve into a commented account on recent results obtained from (and within) this branch of research.

Representation fields for orders of small rank

A representation field for a non-maximal order H in a central simple algebra is a subfield of the spinor class field of maximal orders which determines the set of spinor genera of maximal orders representing H. In our previous work we have proved the existence of the representation field for several important families of suborders, like commutative orders, while we have also found examples where the representation field fails to exist. To be precise, we have found full-rank orders, in central simple algebras of dimension 9 or larger over a suitable field, whose representation field is undefined. In this article, we prove that the representation field is defined for any order H of rank r < 7. This is done by defining representation fields for arbitrary representations of orders into central simple algebra and showing that the computation of these generalized representation fields can be reduced to the case of irreducible representations. The same technique yields the existence of representation fields for any order in an algebra whose semi-simple reduction is commutative. We also construct a rank-8 order, in a 16-dimensional matrix algebra, whose representation field is not defined.

New spinor fields classes and applications

After revisiting the well-known Lounesto classification of spinor fields, wherefrom the bilinear covariants are considered as the landmark beyond Wigner classification, relevant features of regular and singular spinor fields are presented. Hence, non-standard spinor fields are scrutinised, together with their associated dynamics, paving recently found applications in Physics. The case of the new classes of spinors on 7-manifolds is revisited to provide concrete examples.

The spinorial energy functional: solutions of the gradient flow on Berger spheres

We study the negative gradient flow of the spinorial energy functional (introduced by Ammann, Wei{\ss}, and Witt) on 3-dimensional Berger spheres. For a certain class of spinors we show that the Berger spheres collapse to a 2-dimensional sphere. Moreover, for special cases, we prove that the volume-normalized standard 3-sphere together with a Killing spinor is a stable critical point of the volume-normalized version of the flow. Our results also include an example of a critical point of the volume-normalized flow on the 3-sphere, which is not a Killing spinor.

Spinor class fields for generalized Eichler orders

Spinor class fields for generalized Eichler orders

Questing mass dimension 1 spinor fields

This work deals with new classes of spinors of mass dimension 1 in Minkowski spacetime. In order to accomplish it, Lounesto’s classification scheme and the inversion theorem are going to be used. The algebraic framework shall be revisited by explicating the central point performed by the Fierz aggregate. Then the spinor classification is generalized in order to encompass the new mass dimension 1 spinors. The spinor operator is shown to play a prominent role to engender the new mass dimension 1 spinors, accordingly.

Tau Function and Moduli of Spin Curves

The goal of the paper is to give an analytic proof of the formula of G. Farkas for the divisor class of spinors with multiple zeros in the moduli space of odd spin curves. We make use of the technique developed by Korotkin and Zograf that is based on properties of the Bergman tau function. We also show how the Farkas formula for the {\it theta-null} in the rational Picard group of the moduli space of even spin curves can be derived from classical theory of theta functions.

EICHLER ORDERS, TREES AND REPRESENTATION FIELDS

The spinor class field for a genus of orders of maximal rank in a quaternion algebra 𝔄 over a number field K is an abelian extension Σ/K provided with a distance function associating elements of the corresponding Galois group to pairs of orders in that genus. If ℌ ⊆ 𝔇 are two orders in a quaternion algebra 𝔄 with 𝔇 of maximal rank, the representation field F = F(𝔇 | ℌ) is a subfield of the spinor class field for the genus of 𝔇 such that, the set of spinor genera of orders in that genus representing the order ℌ, coincides with the set of spinor genera of orders whose distance to 𝔇 fixes F pointwise. Previous works have focused on two cases: maximal orders 𝔇 and commutative orders ℌ. In this work, we give a method to compute the representation field F(𝔇|ℌ) when 𝔇 is the intersection of a finite family of maximal orders, e.g., an Eichler order, and ℌ is arbitrary. Examples are provided.

Representation fields for commutative orders

Un corps de représentation pour un ordre non maximal ℌ dans une algèbre centrale simple est un sous-corps du corps de classes spinoriel d’ordres maximaux qui détermine l’ensemble de genres spinoriels d’ordres maximaux qui contiennent un conjugué de ℌ. Un ordre non maximal ne possède pas forcément un corps de représentation. Dans ce travail, nous montrons que chaque ordre commutatif a un corps de représentation F et nous donnons une formule pour F. Le résultat principal est prouvé pour des algèbres simples centrales sur des corps globaux arbitraires.

Representation fields for quaternionic skew-hermitian forms

Classes of indefinite quadratic forms in a genus are in correspondence with the Galois group of an abelian extension called the spinor class field (Estes and Hsia, Japanese J. Math. 16, 341–350 (1990)). Hsia has proved (Hsia et al., J. Reine Angew. Math. 494, 129–140 (1998)) the existence of a representation field F with the property that a lattice in the genus represents a fixed given lattice if and only if the corresponding element of the Galois group is trivial on F. This far, the corresponding result for skew-hermitian forms was known only in some special cases, e.g., when the ideal (2) is square free over the base field. In this work we prove the existence of representation fields for quaternionic skew-hermitian forms in complete generality.

Relative spinor class fields: A counterexample

It is known that classes of indefinite quadratic forms in a genus are classified by the Galois group of a spinor class field [4]. Hsia has proved the existence of a representation field F with the property that a lattice in the genus represents a fixed given lattice if and only if the corresponding element of the Galois group is trivial on F. Spinor class fields can also be used to classify conjugacy classes of maximal orders in a central simple algebra. In [1] we left open the issue of whether for every fixed given non-maximal order $$\mathfrak{H}$$ in a central simple division algebra there exists a representation field L with the property that $$\mathfrak{H}$$ embeds into a given maximal order if and only if the corresponding element of the Galois group is trivial on L. In this work we give a negative answer to this question for central simple division algebras of dimension ≥ 32. The case of non-division algebras is also treated by replacing the phrase embeds into by is contained in a conjugate of. As a byproduct of the techniques used in this paper we compute the representation field of an Eichler order in a quaternion algebra.

Applications of spinor class fields: embeddings of orders and quaternionic lattices

Nous etendons la theorie des corps de classe spinoriels et corps de classes spinoriels relatifs a l'etude des problemes de representation des reseaux par rapport a plusieurs groupes algebriques lineaires classiques sur les corps de nombres. Nous appliquons cette theorie pour etudier l'ensemble des classes d'isomorphismes d'ordres maximaux dans une algebre centrale simple qui contiennent un sous-ordre abelien donne. Nous etudions aussi les isometries injectives d'un reseau anti-hermitien quaternionique dans un autre.

Clifford algebras and Hestenes spinors

This article reviews Hestenes' work on the Dirac theory, where his main achievement is a real formulation of the theory within thereal Clifford algebra Cl 1,3 ≃ M2 (H). Hestenes invented first in 1966 hisideal spinors $$\phi \in Cl_{1,3 _2}^1 (1 - \gamma _{03} )$$ and later 1967/75 he recognized the importance of hisoperator spinors ψ ∈ Cl ≃ M2 (C). This article starts from the conventional Dirac equation as presented with matrices by Bjorken-Drell. Explicit mappings are given for a passage between Hestenes' operator spinors and Dirac's column spinors. Hestenes' operator spinors are seen to be multiples of even parts of real parts of Dirac spinors (real part in the decompositionC ⊗ Cl 1,3 andnot inC ⊗ M4 (R)=M4 (C)). It will become apparent that the standard matrix formulation contains superfluous parts, which ought to be cut out by Occam's razor. Fierz identities of bilinear covariants are known to be sufficient to study the non-null case but are seen to be insufficient for the null case ψ†γ0ψ=0, ψ†γ0γ0123ψ=0. The null case is thoroughly scrutinized for the first time with a new concept calledboomerang. This permits a new intrinsically geometric classification of spinors. This in turn reveals a new class of spinors which has not been discussed before. This class supplements the spinors of Dirac, Weyl, and Majorana; it describes neither the electron nor the neutron; it is awaiting a physical interpretation and a possible observation. Projection operators P±, Σ± are resettled among their new relatives in End(Cl 1,3 ). Finally, a new mapping, calledtilt, is introduced to enable a transition from Cl 1,3 to the (graded) opposite algebra Cl 3,1 without resorting to complex numbers, that is, not using a replacement γμ →iγμ.

Spinor genera under field extensions IV: Spinor class fields

Spinor genera under field extensions IV: Spinor class fields

Spinor class groups of orders

Spinor class groups of orders