Classical probabilities for Majorana and Weyl spinors

We construct a map between the quantum field theory of free Weyl or Majorana fermions and the probability distribution of a classical statistical ensemble for Ising spins or discrete bits. More precisely, a Grassmann functional integral based on a real Grassmann algebra specifies the time evolution of the real wave function q_τ(t) for the Ising states τ. The time dependent probability distribution of a generalized Ising model obtains as . The functional integral employs a lattice regularization for single Weyl or Majorana spinors. We further introduce the complex structure characteristic for quantum mechanics. Probability distributions of the Ising model which correspond to one or many propagating fermions are discussed explicitly. Expectation values of observables can be computed equivalently in the classical statistical Ising model or in the quantum field theory for fermions.     

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.     

Beta decays with momentum space Majorana spinors

We construct and apply to decays a truly neutral local quantum field that is entirely based upon momentum space Majorana spinors. We make the observation that theory with momentum space Majorana spinors of real C parities is equivalent to Diracs theory. For imaginary C parities, the neutrino mass can drop out from the single decay trace and reappear in , a curious and in principle experimentally testable signature for a non-trivial impact of Majorana framework in polarization experiments.Received: 11 November 2003, Revised: 20 April 2004, Published online: 12 October 2004PACS: 11.30.Er Charge conjugation, parity, time reversal, and other discrete symmetries - 14.60.St Non-standard-model neutrinos, right-handed neutrinos, etc.     

Dirac and Majorana spinors on non-orientable Riemann surfaces

It is well known that (Weyl) spinors cannot be consistently defined on nonorientable manifolds. We prove that Dirac spinors can be defined on nonorientable Riemann surfaces. It is also shown that Majorana spinors cannot be defined consistently on closed nonorientable Riemann surfaces with odd Euler number, but can be consistently defined in all other cases.Work supported in part by the U.S. Department of Energy under contract DEAC 03-81-ER40050Weingart Research Fellow     

Vertex operator construction of the SO(9,1) Majorana-Weyl spinors in the Neveu-Schwarz-Ramond model

By paying a careful attention to the hermiticity property, the SO(9,1) left-handed Majorana-Weyl spinors are obtained as vertex operators. The hermiticity requires an extra condition on the two-cocycle involved in the vertex operators, whose existence is proved by making an explicit construction.     

Classification of the Weyl curvature spinors of neutral metrics in four dimensions

Neutral geometry is of increasing interest. As with Riemannian and Lorentzian geometry, spinors can be expected to provide a valuable tool in neutral geometry. For a neutral metric in four dimensions, the classification of the Weyl curvature spinors by the pattern of principal spinors each admits is given. For each Weyl curvature spinor, there are nine nontrivial types. This classification is then related to the classification, given previously by the author, of a Weyl curvature spinor when regarded as a curvature endomorphism (four types). These results are the neutral analogues of well known and fundamental results in Lorentzian geometry, but display the peculiarities of neutral geometry. One can expect these results to be an essential ingredient in a full understanding of neutral geometry in four dimensions.     

Pseudo-supersymmetry, consistent sphere reduction and Killing spinors for the bosonic string

Certain supergravity theories admit a remarkable consistent dimensional reduction in which the internal space is a sphere. Examples include type IIB supergravity reduced on S⁵, and eleven-dimensional supergravity reduced on S⁴ or S⁷. Consistency means that any solution of the dimensionally-reduced theory lifts to give a solution in the higher dimension. Although supersymmetry seems to play a role in the consistency of these reductions, it cannot be the whole story since consistent sphere reductions of non-supersymmetric theories are also known, such as the reduction of the effective action of the bosonic string in any dimension D on either a 3-sphere or a (D−3)-sphere, retaining the gauge bosons of SO(4) or SO(D−2) respectively. We show that although there is no supersymmetry, there is nevertheless a natural Killing spinor equation for the D-dimensional bosonic string. A projection of the full integrability condition for these Killing spinors gives rise to the bosonic equations of motion (just as happens in the supergravity examples). Thus it appears that by extending the notion of supersymmetry to “pseudo-supersymmetry” in this way, one may be able to obtain a broader understanding of a relation between Killing spinors and consistent sphere reductions.     

Overlap formulation of Majorana-Weyl fermions

An overlap method for regularizing Majorana-Weyl fermions interacting with gauge fields is presented. A mod(2) index is introduced in relation to the anomalous violation of a discrete global chiral symmetry. Most of the paper is restricted to 2 dimensions but generalizations to 2 + 8k dimensions should be straightforward.     

Dimensional reduction of symmetries

It has been shown that ordinary (d−2)-dimensional quantum field theories are equivalent to corresponding quantum field theories defined on a (d+2)-dimensional superspace with two anticommuting variables. This dimensional reduction is a consequence of superrotation invariance in the superspace. In this letter we study general conformal transformations in the superspace and the properties of field theories which are invariant under such transformations. We show that the symmetries are dimensionally reduced.     

Dimensional reduction and flavor chirality

We show that phenomenologically realistic flavor-chiral Yang-Mills-Higgs theories in 4 dimensions can be derived by dimensional reduction of 10-dimensional vectorlike and gauge theories, where the extra 6 dimensions form a compact coset space with scale size r. The dimensional reduction often implies a symmetry breaking pattern like that of the electroweak theory, in which case it is natural to propose $\text{r ? G}{}_{\mathrm{<mtext>F</mtext>}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. Quantum effects then determine the short-distance behavior of the theory, including any additional symmetry breaking.     

Null vectors,spinors, and strings

It is shown how, in the frame of the Cartan-conception of spinors, the old theorems onminimal surfaces, as generated from null-curves, formulated by Enneper-Weierstrass (1864–1866) for 3-dimensional ordinary space, and by Eisenhart (1911) for 4-dimensional space-time, may be reformulated in terms ofcomplex 2- and 4-component projective spinors respectively. For the correspondingreal (Majorana) spinors instead the same procedure naturally leads tostrings in 3-dimensional and 4-dimensional space-time (?^{2, 1} and ?^{3, 1}). It is suggested that this close connection with Cartan-spinors, and the corresponding (projective) null-geometry, may be the clue for understanding the fundamental nature of strings.     

Covariant,algebraic, and operator spinors

We deal with three different definitions for spinors: (I) thecovariant definition, where a particular kind ofcovariant spinor (c-spinor) is a set of complex variables defined by its transformations under a particular spin group; (II) theideal definition, where a particular kind of algebraic spinor (e-spinor) is defined as an element of a lateral ideal defined by the idempotente in an appropriated real Clifford algebra _p,q (whene is primitive we writea-spinor instead ofe-spinor); (III) the operator definition where a particular kind of operator spinor (o-spinor) is a Clifford number in an appropriate Clifford algebra _p,q determining a set of tensors by bilinear mappings. By introducing the concept of spinorial metric in the space of minimal ideals ofa-spinors, we prove that forp+q5 there exists an equivalence from the group-theoretic point of view among covariant and algebraic spinors. We also study in which senseo-spinors are equivalent toc-spinors. Our approach contain the following important physical cases: Pauli, Dirac, Majorana, dotted, and undotted two-component spinors (Weyl spinors). Moreover, the explicit representation of thesec-spinors asa-spinors permits us to obtain a new approach for the spinor structure of space-time and to represent Dirac and Maxwell equations in the Clifford and spin-Clifford bundles over space-time.     

Dimensional reduction of superstring models

Compactification of ten-dimensional supergravity on Calabi-Yau manifolds (as recently proposed by Candelas, Horowitz, Strominger, and the author) gives n=1 supergravity theories in four dimensions. This paper is devoted to working out the Kähler potential and superpotential which arise.     

Gauge group of gravity,spinors, and anomalies

A discussion is given of the gravitational anomalies that arise from coupling Weyl spinors to gravity, treating the metric, the soldering form, and the connection as independent dynamical variables. This system is strictly analogous to Weyl spinors coupled to Yang-Mills fields and a nonlinear sigma model. The larger gauge group of this formulation is seen to lie at the root of the equivalence between Einstein and Lorentz anomalies.On leave of absence from SISSA, Trieste, Italy.