The construction of spinors in geometric algebra

The relationship between spinors and Clifford (or geometric) algebra has long been studied, but little consistency may be found between the various approaches. However, when spinors are defined to be elements of the even subalgebra of some real geometric algebra, the gap among algebraic, geometric, and physical methods is closed. Spinors are developed in any number of dimensions from a discussion of spin groups, followed by the specific cases of U (1), SU (2), and spinors. The physical observables in Schrödinger-Pauli theory and Dirac theory are found, and the relationship between Dirac, Lorentz, Weyl, and Majorana spinors is made explicit. The use of a real geometric algebra, as opposed to one defined over the complex numbers, provides a simpler construction and advantages of conceptual and theoretical clarity not available in other approaches.     

Quaternionic Lorentz Group and Dirac Equation

We formulate Lorentz group representations in which ordinary complex numbers are replaced by linear functions of real quaternions and introduce dotted and undotted quaternionic one-dimensional spinors. To extend to parity the space-time transformations, we combine these one-dimensional spinors into bi-dimensional column vectors. From the transformation properties of the two-component spinors, we derive a quaternionic chiral representation for the space-time algebra. Finally, we obtain a quaternionic bi-dimensional version of the Dirac equation.     

Euclidean Majorana and Weyl Spinors

The complex form algebra of Schwinger functions of a Dirac field on a Euclidean R ^d with arbitrary dimension d is decomposed into the form algebras of Majorana spinors and of Weyl spinors. The existence of real form algebras is investigated. The reality condition leads to severe restrictions in the case of Majorana forms which do not agree with the results of classical field theory. For all real form algebras Euclidean spinors are constructed as elements of a measure space.     

A METHOD ON DERIVATION OF GAUGE FIELDS

Usually the study of gauge field is based on the wave function. By discussing thebehaviour of Dirac particles in gravitation, one has a famous difficulty, that is, thewave functions appear as scalars under general coordinate transformations. In thispaper, a method is suggested to constitute the gauge fields directly from algebraicstructures, Lie algebra and Jordan algebra. We introduce a concept called represen-tation group of algebras, the transformations, of wave function are connected with therepresentation group. The global and local representation groups are connected withglobal and local transformations of wave function respectively. According to thismethod we find that it is equivalent to the usual one for all of the problems concernedwith internal freedom as Yang-Mills field etc. For spinors, one can introduce gravi-tation by changing the algebraic structure, one find that the vierbein is unneccessaryand the wave functions transform as spinors corresponding to Dirac theory. Somerelated problems are also discussed.     

Fundamentals of nonassociative classical field theory

A nonassociative classical field theory is constructed. Octonion algebra is studied. The octonion is represented as the sum of a quaternion and an associator. The octonion algebra is expanded and Lorentz group generators are specified in terms of octonion bases in one of the subalgebras. Lorentz vectors and spinors are constructed in the nonassociative algebra. The representation of the Lorentz group in terms of spin and the associator is obtained.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 22–27, November, 1986.     

Graded manifold theory as the geometry of supersymmetry

Building upon Kostant's graded manifold theory, we present a new way of introducing spinors into the spacetime manifold, by expanding the algebra of functions on spacetime to a graded algebra. The elements of differential geometry are generalized to accomodate the expanded algebra of functions and in this enriched geometry we find the elements of supersymmetry and of supergravity theory. The geometrical role of the supergravity fields is discussed and a derivation of their transformation rules is given.Research supported in part by the National Science Foundation under Grant Nos. PHY77-22864 and PHY77-05299     

A Dirac algebraic approach to supersymmetry

The power of the Dirac algebra is illustrated through the Kähler correspondence between a pair of Dirac spinors and a 16-component bosonic field. The SO(5, 1) group acts on both the fermion and boson fields, leading to a supersymmetric equation of the Dirac type involving all these fields.     

Fermi-Bose hypersymmetry

A new algebra, combining supersymmetry and internal symmetry, is presented. A massless vector hypermultiplet contains a vector, an isodoublet of left-handed Dirac spinors, and a complex scalar. These can be used as generalized gauge fields. Abelian as well as non-Abelian gauge theories are studied, and the Higgs mechanism is extended in a hypersymmetric way. We present, also, a (mom-realistic) SU(2)× U(1) model; gauge invariance and hypersymmetry are spontaneously broken; two Goldstone spinors appear. Hypersymmetry allows one to define “electronic” and “muonic” numbers, and suggests that a weakly interacting scalar particle ωγ is associated with the photon and the two neutrinos.     

Charged Representations of the Infinite Fermi and Clifford Algebras

The real and quaternionic charge conjugation operators invariant under the infinite-dimensional Clifford algebra, or compatible with the Fermi algebra, are determined. There results a maze of inequivalent irreducible charged representations, all of which are non-Fock. The representation vectors and their charges admit two interpretations besides those of spinors or states of quantum fields: as wavelets on the circle, with charge conjugations acting via ordinary complex conjugation; and as infinite-dimensional numbers, with charge conjugations acting by automorphisms.     

Geometrical interpretation of spinors

Geometrical properties of elements of the unique representation of the Clifford algebra of quadratic form (+, –, –, –) are investigated. A connection between horospheres on the positive Lobatschevsky space of timelike directions and spinors is established.Partly supported by the Polish Government under the Research Program MR I.7.     

Eleven-dimensional supergravity and octonions

We analyse an aspect of spontaneous compactification of 11-dimensional simple supergravity on the 7-sphere: the geometric meaning of the pseudo-scalar modes on S₇. In particular, we present the geometrical interpretation of the (anti) self-dual equation and of its 35 solutions. This is deduced from the properties of Killing spinors and of the algebra of octonions.     

Unification of Spins and Charges

Polynomials in Grassmann algebra can be used to describe the internal degrees, spins and charges of spinors, scalars, and vectors. It was shown by Manko Bortnik and Nielsen that Kähler spinors can be generalized to describe spins of vectors as well as spins and charges of scalars, vectors, and spinors. In dimensions 14 and higher, the spontaneous breaking of symmetry leads gravity in d dimensions to manifest in 4-dimensional subspace as ordinary gravity and all needed gauge fields as well as the Yukawa couplings. Both approaches, Kähler's one (if generalized) and ours, manifest four generations of massless fermions, which are left-handed SU(2) doublets and right-handed SU(2) singlets. A possible way of spontaneously breaking symmetries is pointed out at the level of canonical momentum.     

Symplectic reduction,BRS cohomology,and infinite-dimensional Clifford algebras

This paper gives the mathematical foundations for the BRS quantization procedure. We first discuss the classical finite dimensional BRS procedure and relate it to Marsden-Weinstein reduction. This leads to interesting relations between Lie algebras and Clifford algebras and a novel way of computing Lie algebra cohomology in terms of the spin representation. We then discuss infinite-dimensional Clifford algebras and their spin representations. We find that in the infinite-dimensional case, the analog of the finite-dimensional construction of Lie algebra cohomology breaks down, the obstruction (anomaly) being the Kac-Peterson class which is the cohomology class associated to the representation of the Lie algebra on spinors which is now only a projective representation. Tensoring by a projective representation of opposite class kills the obstruction and gives rise to a cohomology theory and a quantization procedure. We discuss the gradings and Hermitian structures on the absolute and relative complexes.     

Modified supersymmetries

An attempt is made to modify super-symmetry groups and the algebra of the underlying elements, so that the mappings involved in the realizations are those of an ordinary manifold on itself. The resulting group possesses representations that transform constant tensors and spinors among each other. Applications to fiber bundles lead to fields over general space-time manifolds whose structure group is the modified super-symmetry group.Supported by National Science Foundation Grant No. MPS74-15246.     

Spinors,algebraic geometry,and the classification of second-order symmetric tensors in general relativity

Two approaches to the problem of classifying second-order symmetric tensors in space-time given by Ludwig and Scanlon and by Penrose are discussed. Ludwig and Scanlon use both spinor and tensor algebra in their approach, whereas Penrose uses spinors and the properties of certain curves in complex projective 3-space. These approaches yield essentially identical classifications, and this paper points out the connections between them in detail and tabulates the results.     

Spinorial charges and their role in the fusion of internal and space-time symmetries

The advent of supersymmetry immediately led to speculations that a non-trivial mixing of internal and space-time symmetries could be achieved within its framework. In fact, the well-known no-go theorems do not apply to the supersymmetry algebra due to the presence, in the latter, of (anticommuting) spinorial charges. However, not until the recent work of Haag, Lopuszanski and Sohnius did a clearcut picture emerge as to how the aforementioned nontrivial mixing can take place. Most notably, the presence of the conformal algebra within the supersymmetry algebra turns out to be vital. We solidify the findings of Haag et al. through an explicit construction which uses as underlying space the pseudo-Euclidean space E(4,2), i.e. the space for which the conformal group is the group of rotations, and which employs as main tools the spinors associated with the space E(4,2). We follow the algebro-geometric approach of Cartan in order to understand both the introduction and the properties of these spinors. In this manner, we gain many insights regarding the mathematical foundations of supersymmetry. Thus, we fully understand the emergence of the anticommutator, rather than the commutator, among spinor charges as a natural algebraic consequence and not as an a priori given fact. In addition, we clearly see how an (internal) unitary symmetry group can make its appearance within the supersymmetry scheme and verify, via our explicit construction, the results of Haag et al.     

An introduction to the theory of local twistors

This paper deals with the formalism of local twistors, which has developed from the twistor algebra, and extends some of the basic twistor concepts to curved space-time. Essentially, the central ideas are to define a twistor space at each point of the spacetime, and to define a covariant derivative so that an operation of local twistor transport is possible; this leads to the definition of a conformally invariant curvature twistor. In an appendix, some conformally invariant spinors are discussed.     

Possible origin of isotopic spin from extended conformal symmetry

In conformally covariant lagrangian density for conformal spinors interacting with tensor fields, the terms with Lorentz scalar-pseudoscalar bosons have a maximal internal symmetry su(2)_L⊕ su(2)_R⊕ u(1)_L⊕ u(1)_R and this internal algebra is Poincaré invariant; pion-nucleon Yukawa interaction is a particular case.     

Octonionic representations of Clifford algebras and triality

The theory of representations of Clifford algebras is extended to employ the division algebra of the octonions or Cayley numbers. In particular, questions that arise from the nonassociativity and noncommutativity of this division algebra are answered. Octonionic representations for Clifford algebras lead to a notion of octonionic spinors and are used to give octoninic representations of the respective orthogonal groups. Finally, the triality automorphisms are shown to exhibit a manifest ₃ ×SO(8) structure in this framework.