Spinors

The intention of this article is to propose a generalization to the Black Hole (BH) model, prioritizing an understanding from graphic intuitions. To promote the model presented here (based on the use of quaternions) I will try to convince you that with an incredibly simple change of coordinates of the type (T, X) → (iT + jX, iT − jX), it is possible to see (and therefore also intuit) a much more understandable 4D representation than those offered so far in General Relativity for known BH models. But beyond the possible results that I will defend from my physcicist position (such as: 1) the diagonalization of the Kerr metric, 2) a toroidal interpretation of the "infinite universes" that arise when rotating a BH and 3) a model that simultaneously contemplates BHs with and without angular momentum, in which singularities are interpreted as "a problem of coordinates"), what this trans and self-proclaimed anthropologist-of-science seeks above all is to promote a "modal" metaphysic for science, in which every model exists and can be interpreted as physical with the appropriate correspondence between "coordinates" and "metaphysical parameters" (in particular, with such a position I will try to give "materiality" to the universes that today we call "imaginary", such as of "complex" numbers). Inspired by a theory that I imported from anthropology, called Multinaturalismo (Viveiros de Castro), I will defend the epistemological (that is, political) utility that a primitivist twist of scientific languages would bring to the construction of Science and its importance in relation to "Society". My (not so) secret intention? De-binarize the Academy. Help to finally blur the boundaries between Discover/Create, Reality/Fiction, Nature/Society.

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A group theoretical description of basic discrete symmetries (space inversion P, time reversal T and charge conjugation C) is given. Discrete subgroups of orthogonal groups of multidimensional spaces over the fields of real and complex numbers are considered in terms of fundamental automorphisms of Clifford algebras. In accordance with a division ring structure, a complete classification of automorphism groups is established for the Clifford algebras over the fields of real and complex numbers. Finite-dimensional representations of the proper orthochronous Lorentz group are studied in terms of spinor representations of the Clifford algebras. Real, complex, quaternionic and octonionic representations of the Lorentz group are considered. The Atiayh-Bott-Shapiro periodicity is defined on the Lorentz group. Quotient representations of the Lorentz group are introduced. It is shown that quotient representations are the most suitable for description of massless physical fields. An algebraic construction of basic physical fields is presented.

Spinor representations of surfaces immersed into 4-dimensional pseudo-riemannian manifolds are defined in terms of minimal left ideals and tensor decompositions of Clifford algebras. The classification of spinor fields and Dirac operators on the immersed surfaces is given. The Dirac-Hestenes spinor field on surfaces immersed into Lorentzian manifolds and on surfaces conformally immersed into Minkowski spacetime is defined. Mathematics Subject Classification (1991): 15A66, 53A50, 53A10, 35Q55

A representation of generalized Weierstrass formulae for an immersion of generic surfaces into a 4-dimensional complex space in terms of spinors treated as minimal left ideals of Clifford algebras is proposed. The relation between integrable deformations of surfaces via mVN-hierarchy and integrable deformations of spinor fields on the surface is also discussed.

The mass positivity theorem, first proven by Schoen and Yau [13], [14] using methods of Geometric Analysis, is the most basic and fundamental geometric inequality in General Relativity. An alternative proof, using spinorial methods, has been given by Witten [16], from where the importance of the use of spinors in General Relativity was established. The present research proposal intends to show the relevance of geometric inequalities in General Relativity, introduce the spinorial method, and elaborate on the new approach which will be pursued during the duration of the doctoral program, emphasising the results that can be obtained with this new powerful and elegant vision. The research project will be based on the seminal work developed by the supervisor, Dr. Juan Valiente Kroon. A couple of simple, but fundamental results fruit of this method, such as the characterisation of the Kerr and Minkowski spacetimes [1], [2],[3], are shown, as well as the new approach to the mass possitivity theorem.

Spinor fields on surfaces of revolution conformally immersed into 3-dimensional space are considered in the framework of the spinor representations of surfaces. It is shown that a linear problem (a 2-dimensional Dirac equation) related with a modified Veselov-Novikov hierarchy in the case of the surface of revolution reduces to a well-known Zakharov-Shabat system. In the case of one-soliton solution an explicit form of the spinor fields is given by means of linear Bargmann potentials and is expressed via the Jost functions of the Zakharov-Shabat system. It is shown also that integrable deformations of the spinor fields on the surface of revolution are defined by a modified Korteweg-de Vries hierarchy.

La question de savoir si on peut determiner un spineur connaissant les tenseurs sans derivation de la theorie de Dirac a ete etudiee par plusieurs auteurs. Nous proposons ici une methode algebrique basee sur la representation du spineur comme element de C 2 (dans le cas d'un spineur de Pauli), ou d'un element de C 4 (dans le cas d'un spineur de Dirac). Cette methode conduit de facon naturelle a la construction d'un fibre de Hopf au-dessus de l'algebre des observables.

Abstract. We show that the space of Euclid’s parameters for Pythagorean triples is endowed with a natural symplectic structure and that it emerges as a spinor space of the Clifford algebra R21, whose minimal version may be con-ceptualized as a 4-dimensional real algebra of “kwaternions. ” We observe that this makes Euclid’s parameterization the earliest appearance of the concept of spinors. We present an analogue of the “magic correspondence ” for the spinor representation of Minkowski space and show how the Hall matrices fit into the scheme. The latter obtain an interesting and perhaps unexpected geomet-ric meaning as certain symmetries of an Apollonian gasket. An extension to more variables is proposed and explicit formulae for generating all Pythagorean quadruples, hexads, and decuples are provided.

Spinor representations of surfaces immersed into 4–dimensional pseudo– riemannian manifolds are defined in terms of minimal left ideals and tensor decompositions of Clifford algebras. The classification of spinor fields and Dirac operators on the immersed surfaces is given. The Dirac–Hestenes spinor field on surfaces immersed into Lorentzian manifolds and on surfaces conformally immersed into Minkowski spacetime is defined.

At present time methods of the theory of surfaces penetrate into many areas of theoretical and mathematical physics and become an important and inherent part of the modern physics. The deep relation between the theory of surfaces and soliton theory is well known, there exists a numerous literature devoted to this question (see, for example, [Sym85, Bob94]). Recently, Konopelchenko and Taimanov introduced a new approach relating the methods of integrable systems with a theory of surfaces conformally immersed into 3-and 4-dimensional spaces [Kon96, KT95, KT96, Tai97a, Tai97b]. The main tool of this approach is a generalized Weierstrass representation for a conformal immersion of surfaces into R 3 or R 4. A consideration of the linear problem along with the Weierstrass representation allows one to express integrable deformations of surfaces via such hierarchies of nonlinear differential equations as a modified Veselov-Novikov hierarchy, Davy-Stewartson hierarchy and so on. This approach is also used for the study of the basic quantities related with 2D gravity such as Polyakov extrinsic action, Nambu-Goto action, geometric action and Euler characteristic [CK96, KL98c, KL99].

For each quadratic form Q ∈ Quad(V) over a given vector space over a field K we have the Clifford algebra Cℓ(V, Q) defined as the quotient T (V)/I(Q) of the tensor algebra T (V) over the two-sided ideal generated by expressions of the form x⊗x−Q(x), x ∈ V. In the present paper we consider the whole family {Cℓ(V, Q), Q ∈ Quad(V)} in a geometric way as a Z2-graded vector bundle over the base manifold

An algebraic description of basic discrete symmetries (space inversion P , time reversal T , charge conjugation C and their combinations P T , CP , CT , CPT) is studied. Discrete subgroups {1, P, T, P T } of orthogonal groups of multidimensional spaces over the fields of real and complex numbers are considered in terms of fundamental auto-morphisms of Clifford algebras. The fundamental automorphisms form a finite group of order 4. The charge conjugation is represented by a pseudoautomorphism of the Clifford algebra. Such a description allows one to extend the automorphism group. It is shown that an extended automorphism group (CP T-group) forms a finite group of order 8. The group structure and isomorphisms between the extended automorphism groups and finite groups are studied in detail. It is proved that there exist 64 different realizations of CPT-group. An extension of universal coverings (Clifford-Lipschitz groups) of the orthogonal groups is given in terms of CPT-structures which include well-known Shirokov-Dabrowski PT-structures as a particular case.

We show that the effective theory of long wavelength low energy behavior of a dipolar Bose-Einstein condensate(BEC) with large dipole moments (treated as a classical spin) can be modeled using an extended Non-linear sigma model (NLSM) like energy functional with an additional non-local term that represents long ranged anisotropic dipole-dipole interaction. Minimizing this effective energy functional we calculate the density and spin-profile of the dipolar Bose-Einstein condensate in the mean-field regime for various trapping geometries. The resulting configurations show strong intertwining between the spin and mass density of the condensate, transfer between spin and orbital angular momentum in the form of Einstein-de Hass effect, and novel topological properties. We have also described the theoretical framework in which the collective excitations around these mean field solutions can be studied and discuss some examples qualitatively.

Spinor fields on surfaces of revolution conformally immersed into 3-dimensional space are considered in the framework of the spinor representations of surfaces. It is shown that a linear problem (a 2-dimensional Dirac equation) related with a modified Veselov- Novikov hierarchy in the case of the surface of revolution reduces to a well-known Zakharov-Shabat system. In the case of one-soliton solution an explicit form of the spinor fields is given by means of linear Bargmann potentials and is expressed via the Jost functions of the Zakharov-Shabat system. It is shown also that integrable deformations of the spinor fields on the surface of revolution are defined by a modified Korteweg-de Vries hierarchy.