Opening the Pandora’s box of quantum spinor fields

Lounesto’s classification of spinors is a comprehensive and exhaustive algorithm that, based on the bilinears covariants, discloses the possibility of a large variety of spinors, comprising regular and singular spinors and their unexpected applications in physics and including the cases of Dirac, Weyl, and Majorana as very particular spinor fields. In this paper we pose the problem of an analogous classification in the framework of second quantization. We first discuss in general the nature of the problem. Then we start the analysis of two basic bilinear covariants, the scalar and pseudoscalar, in the second quantized setup, with expressions applicable to the quantum field theory extended to all types of spinors. One can see that an ampler set of possibilities opens up with respect to the classical case. A quantum reconstruction algorithm is also proposed. The Feynman propagator is extended for spinors in all classes.     

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.     

Global existence of generalized solutions of the spherically symmetric Einstein-scalar equations in the large

In this paper we study the global initial value problem for the spherically symmetric Einstein-scalar field equations in the large. We introduce the concept of a generalized solution of our problem, and, taking as initial hypersurface a future light cone with vertex at the center of symmetry, we prove, without any restriction on the size of the initial data, the global, in retarded time, existence of generalized solutions.Research supported in part by National Science Foundation grants MCS-8201599 to the Courant Institute and PHY-8318350 to Syracuse University     

The problem of a self-gravitating scalar field

In this paper we begin the study of the global initial value problem for Einstein's equations in the spherically symmetric case with a massless scalar field as the material model. We reduce the problem to a single nonlinear evolution equation. Taking as initial hypersurface a future light cone with vertex at the center of symmetry, we prove, the local, in retarded time, existence and global uniqueness of classical solutions. We also prove that if the initial data is sufficiently small there exists a global classical solution which disperses in the infinite future.Research supported in part by National Science Foundation grants MCS-8201599 to the Courant Institute and PHY-8318350 to Syracuse University     

Spinor Relativity

Spinor relativity is a unified field theory, which derives gravitational and electromagnetic fields as well as a spinor field from the geometry of an eight-dimensional complex and ‘chiral’ manifold. The structure of the theory is analogous to that of general relativity: it is based on a metric with invariance group GL(ℂ²), which combines the Lorentz group with electromagnetic U(1), and the dynamics is determined by an action, which is an integral of a curvature scalar and does not contain coupling constants. The theory is related to physics on spacetime by the assumption of a symmetry-breaking ground state such that a four-dimensional submanifold with classical properties arises. In the vicinity of the ground state, the scale of which is of Planck order, the equation system of spinor relativity reduces to the usual Einstein and Maxwell equations describing gravitational and electromagnetic fields coupled to a Dirac spinor field, which satisfies a non-linear equation; an additional equation relates the electromagnetic field to the polarization of the ground state condensate.     

On the global “two-sided” characteristic Cauchy problem for linear wave equations on manifolds

The global characteristic Cauchy problem for linear wave equations on globally hyperbolic Lorentzian manifolds is examined, for a class of smooth initial value hypersurfaces satisfying favourable global properties. First it is shown that, if geometrically well-motivated restrictions are placed on the supports of the (smooth) initial datum and of the (smooth) inhomogeneous term, then there exists a continuous global solution which is smooth “on each side” of the initial value hypersurface. A uniqueness result in Sobolev regularity \(H^{1/2+\varepsilon }_{\mathrm {loc}}\) is proved among solutions supported in the union of the causal past and future of the initial value hypersurface, and whose product with the indicator function of the causal future (resp. past) of the hypersurface is past compact (resp. future compact). An explicit representation formula for solutions is obtained, which prominently features an invariantly defined, densitised version of the null expansion of the hypersurface. Finally, applications to quantum field theory on curved spacetimes are briefly discussed.     

Necessity of acceleration‐induced nonlocality

The purpose of this paper is to explain clearly why nonlocality must be an essential part of the theory of relativity. In the standard local version of this theory, Lorentz invariance is extended to accelerated observers by assuming that they are pointwise inertial. This locality postulate is exact when dealing with phenomena involving classical point particles and rays of radiation, but breaks down for electromagnetic fields, as field properties in general cannot be measured instantaneously. The problem is corrected in nonlocal relativity by supplementing the locality postulate with a certain average over the past world line of the observer.     

Nonmetricity and torsion induced by dilaton gravity in two dimension

We develop a theory in which there are couplings amongst Dirac spinor, dilaton and non-Riemannian gravity and explore the nature of connection-induced dilaton couplings to gravity and Dirac spinor when the theory is reformulated in terms of the Levi-Civita connection. After presenting some exact solutions without spinors, we investigate the minimal spinor couplings to the model and in conclusion we cannot find any nontrivial dilaton couplings to spinor.     

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.     

Reformulation of general relativity as a gauge theory

The starting point is a spinor affine space-time. At each point, two-component spinors and a basis in spinor space, called “spin frame,” are introduced. Spinor affine connections are assumed to exist, but their values need not be known. A metric tensor is not introduced. Global and local gauge transformations of spin frames are defined with GL(2) as the gauge group. Gauge potentials B_μ are introduced and corresponding fields F_μν are defined in analogy with the Yang-Mills case. Gravitational field equations are derived from an action principle. Incases of physical interest SL(2, C) is taken as the gauge group, instead of GL(2). In the special case of metric space-times the theory is identical with general relativity in the Newman-Penrose formalism. Linear combinations of B_μ are generalized spin coefficients, and linear combinations of F_μν are generalized Weyl and Ricci tensors and Ricci scalar. The present approach is compared with other formulations of gravitation as a gauge field.     

Constant scalar curvature and warped product globally null manifolds

This paper deals with the curvature properties of a class of globally null manifolds (M,g) which admit a global null vector field and a complete Riemannian hypersurface. Using the warped product technique we study the fundamental problem of finding a warped function such that the degenerate metric g admits a constant scalar curvature on M. Our work has an interplay with the static vacuum solutions of the Einstein equations of general relativity.     

Supersymmetry in Lorentzian Curved Spaces and Holography

We consider superconformal and supersymmetric field theories on four-dimensional Lorentzian curved space-times, and their five-dimensional holographic duals. As in the Euclidean signature case, preserved supersymmetry for a superconformal theory is equivalent to the existence of a charged conformal Killing spinor. Differently from the Euclidean case, we show that the existence of such spinors is equivalent to the existence of a null conformal Killing vector. For a supersymmetric field theory with an R-symmetry, this vector field is further restricted to be Killing. We demonstrate how these results agree with the existing classification of supersymmetric solutions of minimal gauged supergravity in five dimensions.     

Some remarks on the energy-momentum tensor in general relativity

In general relativity, the energy-momentum tensor of a classical tensor field can be constructed by varying the action of the field with respect to the background metric. This paper suggests an alternative interpretation of the construction which also makes sense for spinor fields, and which gives some insight into the locality of energy-momentum operators in generally covariant quantum field theory.     

Twistor theory: An approach to the quantisation of fields and space-time

Twistor theory offers a new approach, starting with conformally-invariant concepts, to the synthesis of quantum theory and relativity. Twistors for flat space-time are the SU(2,2) spinors of the twofold covering group O(2,4) of the conformal group. They describe the momentum and angular momentum structre of zero-rest-mass particles. Space-time points arise as secondary concepts corresponding to linear sets in twistor space. They, rather than the null cones, should become “smeared out” on passage to a quantised gravitational theory. Twistors are represented here in two-component spinor terms. Zero-rest-mass fields are described by holomorphic functions on twistor space, on which there is a natural canonical structure leading to a natural choice of canonical quantum operators. The generalisation to curved space can be accomplished in three ways; i) local twistors, a conformally invariant calculus, ii) global twistors, and iii) asymptotic twistors which provide the framework for an S-matrix approach in asymptotically flat space-times. A Hamiltonian scattering theory of global twistors is used to calculate scattering cross-sections. This leads to twistor analogues of Feynman graphs for the treatment of massless quantum electrodynamics. The recent development of methods for dealing with massive (conformal symmetry breaking) sources and fields is briefly reviewed.     

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.     

General relativity and general Lorentz-covariance

The principle of general relativity means the principle of generalLorentz-covariance of the physical equations in the language of tetrads and metrical spinors. A generalLorentz-Covariant calculus and the generalLorentz-covariant generalisations of the Ricci calculus and of the spinor calculus are given. The generalLorentz-covariant representation implies theEinstein principle of space-time covariance and allows the geometrisation of gravitational fields according toEinstein's principle of equivalence.     

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.     

7-Dimensional compact Riemannian manifolds with Killing spinors

Using a link between Einstein-Sasakian structures and Killing spinors we prove a general construction principle of odd-dimensional Riemannian manifolds with real Killing spinors. In dimensionn=7 we classify all compact Riemannian manifolds with two or three Killing spinors. Finally we classify nonflat 7-dimensional Riemannian manifolds with parallel spinor fields.