Geometrical interpretation of Benenti systems

We show that the following two separately developed theories, the theory of Benenti systems in mathematical physics and the theory of projectively equivalent metrics in classical differential geometry, study essentially the same object. Combining methods and results from these two theories, one can prove the commutative integrability of projectively equivalent pseudo-Riemannian metrics and construct infinitely many new Hamiltonian systems, integrable in the classical and in the quantum sense.     

Geometrical interpretation of extended supergravity

SO₂ extended supergravity is shown to be a geometrical theory, whose underlying gauge group is OSp(4, 2). The couplings which gauge the SO₂ symmetry as well as the accompanying cosmological and masslike terms are directly obtained, and the usual SO₂ model is obtained after a Wigner-Inönü group contraction.     

Geometrical interpretation of Inönü-Wigner contractions

The Inönü-Wigner contractions which interrelate the Lie algebras of the isometry groups of metric spaces are discussed with reference to deformations of the absolutes of the spaces. A general formula is derived for the Lie algebra commutation relations of the isometry group for anyN-dimensional metric space. These ideas are illustrated by a discussion of important particular cases, which interrelate the four-dimensional de Sitter, Poincaré, and Galilean groups.     

0(4,2) Operator replacements: Geometrical interpretation

The method ofo(4,2) operator replacements relies upon a particular realization of theo(4,2) Lie algebra in terms of position and momentum operators, involving a free dimensionless parameter . The geometrical significance of the operator replacements is given. The momentum space becomes a three-dimensional sphere of radius [exp()]/2 (atomic units) embedded in a four-dimensional Euclidean space. A much simpler realization of the replaced operators is obtained.     

Geometrical interpretation of the bilinear equations for the KP hierarchy

The Grassmann manifold approach to the KP hierarchy, in the spirit of Segal and Wilson, is used to define the wavefunction _W and its adjoint _W. From the fact that _W and _W are the orthogonal, we derive the bilinear equations. The modified equations are treated at the same time.     

Fermions without spinors

The classical Kähler equation for an inhomogeneous differential form is analysed in some detail with respect to the physical properties of its Minkowski space solutions. Although the components of the field contain only integer representations of the Lorentz group for a physical interpretation of the quantum theory, we impose fermionic commutators. The electromagnetic interactions are identical to those of a Dirac spinor field with an extra fourfold degeneracy. Possibilities for the interpretation of the extra degrees of freedom are discussed.     

Algebraic spinors and SUSY

Using the Dirac-Kaehler formalism, we formulate the supersymmetric Wess-Zumino model. Special attention is paid to the proper definition of a two-dimensional spinor, its adjoint, and its generalization to other dimensions.     

Supergauges and goldstone spinors

An example of a local conserved spinor current density is given, such that the corresponding symmetry is spontaneously broken and Goldstone states with spin one half occur.     

Twisted spinors on black holes

It is noted that the standard black hole topology admits twisted configurations of the spinor field owing to the existence of twisted spinor bundles, and they are analyzed using the Schwarzschild black hole as an example. This is physically linked with the natural presence of Dirac monopoles on black holes and entails marked modification of the Hawking radiation for spinor particles. Pis’ma Zh. éksp. Teor. Fiz. 69, No. 9, 619–625 (10 May 1999) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Killing spinors on Lorentzian manifolds

The aim of this paper is to describe some results concerning the geometry of Lorentzian manifolds admitting Killing spinors. We prove that there are imaginary Killing spinors on simply connected Lorentzian Einstein–Sasaki manifolds. In the Riemannian case, an odd-dimensional complete simply connected manifold (of dimension n≠7) is Einstein–Sasaki if and only if it admits a non-trivial Killing spinor to . The analogous result does not hold in the Lorentzian case. We give an example of a non-Einstein Lorentzian manifold admitting an imaginary Killing spinor. A Lorentzian manifold admitting a real Killing spinor is at least locally a codimension one warped product with a special warping function. The fiber of the warped product is either a Riemannian manifold with a real or imaginary Killing spinor or with a parallel spinor, or it again is a Lorentzian manifold with a real Killing spinor. Conversely, all warped products of that form admit real Killing spinors.     

On factorization properties of algebraic spinors

We discuss algebraic spinors as elements of a minimal left ideal of the Clifford algebra and observe their factorization property as a column times row in an arbitrary irreducible faithful matrix representation. This allows us to parametrize the set of primitive idempotents and the set of minimal left ideals, and find the dimensionality of the intersection of any minimal left with a minimal right ideal. We propose to generalize Cartan's concept of pure spinors to arbitrary linear spaceR ^{p, q} orC ⁿ .Institute of Theoretical Physics, University of Wrocaw Poland     

Dirac spinors and flavor oscillations

In the standard treatment of particle oscillations the mass eigenstates are implicitly assumed to be scalars and, consequently, the spinorial form of the neutrino wave functions is not included in the calculations. To analyze this additional effect, we discuss the oscillation probability formula obtained by using the Dirac equation as evolution equation for the neutrino mass eigenstates. The initial localization of the spinor state also implies an interference between positive and negative energy components of mass eigenstate wave packets which modifies the standard oscillation probability.Received: 8 April 2004, Revised: 12 July 2004, Published online: 20 October 2004     

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 and later 1967/75 he recognized the importance of hisoperator spinors Cl _1,3 ⁺ 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 ₀=0, ₀₀₁₂₃=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_±, _± 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 i.     

Third-order equation for two-component spinors

We present the properties of a two-component spinor field that obeys a third-order equation. It is separated into a massive part that corresponds closely to a Dirac field, and a massless part that obeys the Weyl equation. We discuss the interaction of such a field with an external electromagnetic field and the (weak) interactions of two such fields. They can be considered both in terms of relativistic quantum mechanics and quantum field theory. We conclude that this formulation has some attractive features, such as a unified treatment of electrons and muons with their neutrinos, a special role of thePC transformation, a more convergent propagator and a new approach to interactions. It also has some serious difficulties, aside from those generally associated with higher-order equations. These are mainly related to inconsistencies in the simultaneous considerations of electromagnetic and weak interactions. The approach also suggests a further unification of the electron and muon fields into a single bispinor field.     

Covariant inertial forces for spinors

In this paper we consider the Dirac spinor field in interaction with a background of electrodynamics and torsion-gravity; by performing the polar reduction we acquire the possibility to introduce a new set of objects that have the geometrical status of non-vanishing tensors but which seem to contain the same information of the connection: thus they appear to be describing something that seems like an inertial force but which is also essentially covariant. After a general introduction, we exemplify these tensors in the very well known instance of the orbital of minimal energy for an electron in the case of the Hydrogen atom: we will see that the invariants built with these tensors remain different from zero even for free field configurations. An outlook regarding possible interpretations of such a set of tensors will be sketched. A few final comments will eventually be given.