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A general theory of quantum Clifford algebras is presented, based on a quantum generalization of the Cartan theory of spinors. We concentrate on the case when it is possible to apply the quantum‐group formalism of bicovariant bimodules. The general theory is then singularized to the quantum SL(n,C) group case, to generate explicit forms for the whole class of braidings required. The corresponding spinor representations are introduced and investigated. Starting from our Clifford algebras we introduce the quantum‐Euclidean underlying spaces compatible with different choices of *‐structures from where the analogues of Dirac and Laplace operators are built. Using the formalism developed, quantum Spin(n) groups are defined.

Inequivalent spin structures on Lorentzian manifolds are studied and in particular the possibility of defining inequivalent Majorana spinors associated with real irreducible representations of the Clifford algebra corresponding to a metric of signature (3,1) is analyzed. Such exotic complex (Dirac) spinors were first discussed in 1979. It is shown that exotic real (Majorana) spinors can be defined in a consistent way, too. The main idea is the use of a ‘‘chiral’’ U(1) group instead of a ‘‘complex’’ U(1) which contains all the relevant topological information. This result may be interesting in the context of the geometry of supergravity–matter coupling.

A concept of a real index associated with any maximal totally null subspace in a complexified vector space endowed with a scalar product, and also with any complex simple (pure) spinor, is introduced and elaborated. It is shown that the real Pin group acts transitively on the projective space of simple spinors with any given real index. The connection between the real index and multivectors associated with a simple spinor and its charge conjugate is established. The simple spinors with extreme values of the real index deserve a special attention: The generic simple spinors for which the real index is minimal and simple‐r spinors for which it is maximal.

We developed an algorithm to obtain the natural orbitals (natural spinors) from the two-step spin-orbit coupled wave functions. These natural spinors are generally complex-valued, mixing two spin components, and they can have similar symmetry properties as the j-j spinors from the one-step spin-orbit coupling calculations, if the reduced density equally averages all the components of a multi-dimensional irreducible representation. Therefore, the natural spinors can serve as an approximation to the j-j spinors and any wave function analysis based on the j-j spinors can also be performed based on them. The comparison between the natural spinors and the j-j spinors of three representative atoms, Tl, At, and Lu, shows their close similarity and demonstrates the ability of the natural spinors to approximate the j-j spinors.

Choquet-Bruhat was the first to give a proper physical definition of covariant spinors, taking into account the reference system and treating them as equivalence classes defined from the transformation laws of the representatives when the reference system is changed. Recently, Rodriguez et al.[ Int. J. Theor. Phys. 35, 1849 (1996)] have adapted this procedure from covariant spinors to the case of algebraic and operator spinors. These approaches are restrained in the sense that the type of spinor is chosen from the beginning, and it does not admit a general formulation. In this paper, we present a unified definition that is valid for any type of the space of representation, being independent of its particular properties. In our formulation the three types of spinors appear as particular cases of the general definition. Moreover, we stick out the importance of the bilinear covariants in the definition of spinors. From this, we recognize a completely different kind of spinor, characterized by the different n...

We present a new formulation of the maximum clique problem of a graph in complex space. We start observing that the adjacency matrix A of a graph can always be written in the form A = B B where B is a complex, symmetric matrix formed by vectors of zero length (null vectors) and the maximum clique problem can be transformed in a geometrical problem for these vectors. This problem, in turn, is translated in spinorial language and we show that each graph uniquely identifies a set of pure spinors, that is vectors of the endomorphism space of Clifford algebras, and the maximum clique problem is formalized in this setting so that, this much studied problem, may take advantage from recent progresses of pure spinor geometry.

The issue of the meanings of ‘‘spinors’’ in two‐dimensional Minkowski spacetime is investigated. Spinors in two dimensions are defined and shown to behave like the familiar spinors in four dimensions. Both of them can be interpreted as square roots of vectors, and both of them describe fermions in a quantum field theory. We also show that an analogous version of the spin‐statistics theorem holds in two‐dimensional Minkowski spacetime.

The different geometrical properties of Majorana, ‘‘even,’’ Dirac, and Chevalley spinors in the Clifford algebra approach are investigated.

A classification scheme is presented for the finite multiplicative group generated by the gamma matrices associated with a given Clifford algebra. This group reflects the periodicities observed in n‐dimensional spinors, and its representations and other properties are studied, thus highlighting the dependence on the number of spacelike and timelike vectors. The reality of the representations is examined and tabulated; application is made to the imposition of Majorana and Weyl conditions.

Key words of the present paper are visualization and concrete geometry. We develop ordinary two‐component spinor algebra from a concrete geometrical model of spinor space. The null flag picture may be said to constitute such a model as far as topological and differential properties of spinors go. In our model it is also possible to visualize the algebraic properties of spinors by straightforward geometrical constructions. Notably, an interpretation of spinor addition is given in terms of a geometrical procedure, analogous to the addition of real 3‐vectors via the parallelogram rule. By this procedure the relation between the projection of spinors on the 2‐sphere and the projection of their sum can directly be read off. The connection between null flags and our presentation of spinors is touched upon. It is planned to discuss the connection to Minkowski space more closely in a forthcoming paper.

Dirac spinors for vector space‐times Rs,t are considered as real spinors of appropriate extended vector spaces Rs’,t’. This extension is determined by the condition Rs’,t’≂RCs,t. The catalog of reality, chirality conditions allowed for Dirac spinors by the analysis of corresponding Clifford algebras and their left minimal ideals are completed.

Cartan’s simple—often called pure—spinors corresponding to even‐dimensional complex vector spaces are defined in terms of the associated maximal totally null planes. Their geometrical properties are derived and described using notions familiar to physicists: Dirac and Weyl spinors, gamma matrices, tensors formed bilinearly from pairs of spinors, and creation and annihilation operators of Fermi states. A new theorem characterizes a simple spinor φ by the properties of the vector tψBγμφ, where ψ is an arbitrary spinor and B is the matrix connecting the gamma matrices with their transposes. The Cartan constraint equations on the components of simple spinors are given a new, geometrically transparent derivation based on the action on simple spinors of a maximal Abelian subgroup of the group Spin.

The concept of Killing spinor is analyzed in a general way by using the spinorial formalism. It is shown, among other things, that higher derivatives of Killing spinors can be expressed in terms of lower order derivatives. Conformal Killing vectors are studied in some detail in the light of spinorial analysis: Classical results are formulated in terms of spinors. A theorem on Lie derivatives of Debever–Penrose vectors is proved, and it is shown that conformal motion in vacuum with zero cosmological constant must be homothetic, unless the conformal tensor vanishes or is of type N. Our results are valid for either real or complex space–time manifolds.

It is shown that an unambiguous quantum theory of spinors in positively curved deSitter space, based on distinguished coordinates in a Hamiltonian framework, leads to a set of spinors corresponding to unsharp energy but sharp mass defined in a family of novel eigenvalue problems. An example is given in which partly real and partly complex discrete mass spectra come forth.