States and operators in the spacetime algebra

The spacetime algebra (STA) is the natural, representation-free language for Dirac's theory of the electron. Conventional Pauli, Dirac, Weyl, and Majorana spinors are replaced by spacetime multivectors, and the quantum - and -matrices are replaced by two-sided multivector operations. The STA is defined over the reals, and the role of the scalar unit imaginary of quantum mechanics is played by a fixed spacetime bivector. The extension to multiparticle systems involves a separate copy of the STA for each particle, and it is shown that the standard unit imaginary induces correlations between these particle spaces. In the STA, spinors and operators can be manipulated without introducing any matrix representation or coordinate system. Furthermore, the formalism provides simple expressions for the spinor bilinear covariants which dispense with the need for the Fierz identities. A reduction to2+1 dimensions is given, and applications beyond the Dirac theory are discussed.Supported by a SERC studentship.     

Dirac-Hestenes spinor fields on Riemann-Cartan manifolds

In this paper we study Dirac-Hestenes spinor fields (DHSF) on a four-dimensional Riemann-Cartan spacetime (RCST). We prove that these fields must be defined as certain equivalence classes of even sections of the Clifford bundle (over the RCST), thereby being certain particular sections of a new bundle named the spin-Clifford bundle (SCB). The conditions for the existence of the SCB are studied and are shown to be equivalent to Geroch's theorem concerning the existence of spinor structures in a Lorentzian spacetime. We introduce also the covariant and algebraic Dirac spinor fields and compare these with DHSF, showing that all three kinds of spinor fields contain the same mathematical and physical information. We clarify also the notion of (Crumeyrolle's) amorphous spinors (Dirac-Kähler spinor fields are of this type), showing that they cannot be used to describe fermionic fields. We develop a rigorous theory for the covariant derivatives of Clifford fields (sections of the Clifford bundle, CB) and of Dirac-Hestenes spinor fields. We show how to generalize the original Dirac-Hestenes equation in Minkowski spacetime for the case of RCST. Our results are obtained from a variational principle formulated through the multiform derivative approach to Lagrangian field theory in the Clifford bundle.     

On supersymmetric Lorentzian Einstein-Weyl spaces

We consider weighted parallel spinors in Lorentzian Weyl geometry in arbitrary dimensions, choosing the weight such that the integrability condition for the existence of such a spinor implies the geometry to be Einstein-Weyl. We then use techniques developed for the classification of supersymmetric solutions to supergravity theories to characterise those Lorentzian EW geometries that allow for a weighted parallel spinor, calling the resulting geometries supersymmetric. The overall result is that they are either conformally related to ordinary geometries admitting parallel spinors (w.r.t. the Levi-Cività connection), or they are conformally related to certain Kundt spacetime. A full characterisation is obtained for the 4- and 6-dimensional cases.     

The role of singular spinor fields in a torsional gravity,Lorentz-violating,framework

In this work, we consider a generalization of quantum electrodynamics including Lorentz violation and torsional-gravity, in the context of general spinor fields as classified in the Lounesto scheme. Singular spinor fields will be shown to be less sensitive to the Lorentz violation, as far as couplings between the spinor bilinear covariants and torsion are regarded. In addition, we prove that flagpole spinor fields do not admit minimal coupling to the torsion. In general, mass dimension four couplings are deeply affected when singular—flagpoles—spinors are considered, instead of the usual Dirac spinors. We also construct a mapping between spinors in the covariant framework and spinors in Lorentz symmetry breaking scenarios, showing how one may transliterate spinors of different classes between the two cases. Specific examples concerning the mapping of Dirac spinor fields in Lorentz violating scenarios into flagpole and flag-dipole spinors with full Lorentz invariance (including the cases of Weyl and Majorana spinors) are worked out.     

Axial Anomaly and the Triality Symmetry of Octonion

With an assumption that in the Yang-Mills Lagrangian, a left-handed fermion and a right-handed fermion both expressed as the quaternion makes an octonion, and the gauge field can be treated as self-dual, we calculate the axial current and two vector currents triangle diagram of Bardeen, which yields the contribution of the axial anomaly. The octonion possesses the triality symmetry, and there are 5 symmetry operations G _ij and G _ijk (ijk = 123), in which mixing of spinors and vectors occur. G ₂₃ does not mix vectors and spinors, but mismatch of the spinor and vector fields occurs. Hence, electro magnetic (EM) wave emitted from galaxies transformed by the five transformations would not be detected by EM detectors in our galaxy, and the source would be regarded as dark matter. The axial anomaly appears as a reflection of the symmetry of the matter field and not as a reflection of the symmetry of the pure vacuum, which is consistent with recent arguments on condensates and confinement.     

On the spinor representation

A systematic study of the spinor representation by means of the fermionic physical space is accomplished and implemented. The spinor representation space is shown to be constrained by the Fierz–Pauli–Kofink identities among the spinor bilinear covariants. A robust geometric and topological structure can be manifested from the spinor space, wherein the first and second homotopy groups play prominent roles on the underlying physical properties, associated to fermionic fields. The mapping that changes spinor fields classes is then exemplified, in an Einstein–Dirac system that provides the spacetime generated by a fermion.     

The Spin-Statistics Theorem in Arbitrary Dimensions

We investigate the spin-statistics connection in arbitrary dimensions for hermitian spinor or tensor quantum fields with a rotationally invariant bilinear Lagrangian density. We use essentially the same simple method as for space dimension D=3. We find the usual connection (tensors as bosons and spinors as fermions) for D=8n+3,8n+4,8n+5, but only bosons for spinors and tensors in dimensions 8n±1 and 8n. In dimensions 4n+2 the spinors may be chosen as bosons or fermions. The argument hinges on finding the identity representation of the rotation group either on the symmetric or the antisymmetric part of the square of the field representation. Permanent address of L.J. Boya: Departamento de Física Teórica, Universidad de Zaragoza, 50009 Zaragoza, Spain     

A multivector derivative approach to Lagrangian field theory

A new calculus, based upon the multivector derivative, is developed for Lagrangian mechanics and field theory, providing streamlined and rigorous derivations of the Euler-Lagrange equations. A more general form of Noether's theorem is found which is appropriate to both discrete and continuous symmetries. This is used to find the conjugate currents of the Dirac theory, where it improves on techniques previously used for analyses of local observables. General formulas for the canonical stress-energy and angular-momentum tensors are derived, with spinors and vectors treated in a unified way. It is demonstrated that the antisymmetric terms in the stress-energy tensor are crucial to the correct treatment of angular momentum. The multivector derivative is extended to provide a functional calculus for linear functions which is more compact and more powerful than previous formalisms. This is demonstrated in a reformulation of the functional derivative with respect to the metric, which is then used to recover the full canonical stress-energy tensor. Unlike conventional formalisms, which result in a symmetric stress-energy tensor, our reformulation retains the potentially important antisymmetric contribution.Supported by a SERC studentship.     

SO（d,d;Z) Transformation Property for Gauge Fluxes and Ramond-Ramond Fields in Noncommutative Geometry

In this paper we study the spinor constructions of gauge fluxes and Ramond-Ramond fields on noncommutative tori T^d up to d=6. In which the spinor and conjugate spinor are distinguished and dual bases are also introduced.So that we can express the Chern-Simons Lagrangian in toroidal compactification as a product of spinors.     

Gauge Formalism for General Relativity and Fermionic Matter

A new formulation for General Relativity is developed; it is a canonical, global and geometrically well posed formalism in which gravity is described using only variables related to spin structures. It does not require any background metric fixing and it applies to quite general manifolds, i.e. it does not need particular symmetries requirement or global frames. A global Lagrangian framework for Dirac spinors is also provided; conserved quantities and superpotentials are given. The interaction between gravity and spinors is described in a minimal coupling fashion with respect to the new variables and the Hilbert stress tensor of spinor fields is computed, providing the gravitational field generated by spinors. Finally differences and analogies between this formalism and gauge theories are discussed.     

Mass generation in the aether: Neutrino oscillations and massive gauge fields

Neutrino mixing is studied in an absolute spacetime conception based on a dispersive aether. The effect of the frequency-dependent permeability of the aether on the interference phase of neutrino mass eigenstates is analyzed. Neutrinos are treated as massless Dirac spinors, and mass eigenstates are due to the neutrino permeability of spacetime. The aether can also generate effective gauge masses, resulting in massive dispersion relations preserving the gauge symmetry. The propagators of gauge and spinor fields are derived, illustrating mass generation by isotropic permeability tensors in the aether frame, the rest frame of the cosmic background radiation.     

An analysis of Fierz identities,factorization and inversion theorems

We show that the full set of Fierz identities which are used to compute electro-weak interactions reported by Y. Takahashi can be considered as particular cases of the Clifford product between multivector Cartan maps. Moreover, we think that our approach can be generalized to higher-dimensional models.We discuss the factorization and inversion theorems for the recovery of the spinor from its multivectorial Cartan map.A new classification given by P. Lounesto is applied to the recovered spinors for Cl_1,3 space-time symmetry and SU(2)×U(1) isotopic group.Dedicated to David Hestenes on his 60th birthday.     

Geometric significance of the spinor Lie derivative. I

In a previous article, the writer explored the geometric foundation of the generally covariant spinor calculus. This geometric reasoning can be extended quite naturally to include the Lie covariant differentiation of spinors. The formulas for the Lie covariant derivatives of spinors, adjoint spinors, and operators in spin space are deduced, and it is observed that the Lie covariant derivative of an operator in spin space must vanish when taken with respect to a Killing vector. The commutator of two Lie covariant derivatives is calculated; it is noted that the result is consistent with the geometric interpretation of the Jacobi identity for vectors. Lie current conservation is seen to spring from the result that the operator of spinor affine covariant differentiation commutes with the operator of spinor Lie covariant differentiation with respect to a Killing vector. It is shown that differentiations of the spinor field defined geometrically are Lorentz-covariant.     

Triality and Quantization of Singularities in Massive Fermion

It is proved that fermions can acquire the mass through the additional non-integrable exponential factor. For this propose the special vector potential associated with the spinor field was introduced. Such a vector potential has closk relation with the. triality property in Dirac spinors and plays crucial role in the construction of massive term. It is shown that the change in phase of a wavefunction round any closed curve with the possibility of there being singularities in our vector potential will lead to the law of quantization of physical constants including the mass. The triality properties of Dirac's spinors are studied and it leads to a double covering vector representation of Dirac spinor field. It is proved that massive Dirac equation in the bosonic representation is self-dual.     

Clifford algebra geometric-multispinor particles and multivector-current gauge fields

A new class of multivector quantum mechanics is defined in which the theoretical gains over standard formalism are fully illustrated. Multiple generations of particles appear when column spinors are replaced by Clifford multivectors (matrices associated with physical geometry). New gauge fields arise from now-allowable right-side-applied transformations, independent of the usual left-sided ones. The number and group structure of the gauge generators is a function of the dimension and metric of the underlying geometric space, where constraints on a multivector Lagrangian suppress some interactions.     

Flag-dipole and flagpole spinor fluid flows in Kerr spacetimes

Flagpole and flag-dipole spinors are particular classes of spinor fields that has been recently used in different branches of theoretical physics. In this paper, we study the possibility and consequences of these spinor fields to induce an underlying fluid flow structure in the background of Kerr spacetimes. We show that flag-dipole spinor fields are solutions of the equations of motion in this context. To our knowledge, this is the second time that this class of spinor field appears as a physical solution, the first one occurring as a solution of the Dirac equation in ESK gravities.     

On potentials for several classes of spinor and tensor fields in curved spacetimes

Already known results with respect to the existence of a vector potential for the Maxwell field tensor and a tensor potential for Weyl's conformal curvature tensor in four-dimensional spacetimes are generalized. It is shown that there exists a spinor potential of type (n–1,1) for any totally symmetric spinor field of rankn. From this theorem we deduce a series of corollaries, for example, that every antisymmetric tensor of second rank admits a linear representation in terms of the first derivatives of two vector fields. Further, some investigations are made on the existence of potentials for arbitrary symmetric spinors of type (n, m).     

Polarized Spacetime Foam

An approximate model of a spacetime foam is presented. It is supposed that in the spacetime foam each quantum handle is like to an electric dipole and therefore the spacetime foam is similar to a dielectric. If we neglect of linear sizes of the quantum handle then it can be described with an operator containing a Grassman number and either a scalar or a spinor field. For both fields the Lagrangian is presented. For the scalar field it is the dilaton gravity + electrodynamics and the dilaton field is a dielectric permeability. The spherically symmetric solution in this case give us the screening of a bare electric charge surrounded by a polarized spacetime foam and the energy of the electric field becomes finite one. In the case of the spinor field the spherically symmetric solution give us a ball of the polarized spacetime foam filled with the confined electric field. It is shown that the full energy of the electric field in the ball can be very big.     

Boson-Fermion Duality in A2 Toda Field Theory

In this paper, we consider a two-dimensional integrable and conformal invariant field theory with two Dirac spinors and two scalar fields. This model has chiral symmetry and CP-like symmetry. Moreover, this model also has a Neother current depending only on the matter field. At last, we bosonize the spinor fields.