Quantum gravity model with fundamental spinor fields

We discuss the possibility that gravitational potentials (metric, coframe and connection) may emerge as composite fields from more fundamental spinor constituents. We use the formalism of Poincaré gauge gravity as an appropriate theoretical scheme for the rigorous development of such an approach. We postulate the constitutive relations of an elastic Cosserat type continuum that models spacetime. These generalized Hooke and MacCullagh type laws consistently take into account the translational and Lorentz rotational deformations, respectively. The resulting theory extends the recently proposed Diakonov model. An intriguing feature of our theory is that in the lowest approximation it reproduces Heisenberg’s nonlinear spinor model.     

Gravitational fields,spinor fields,and groups of motions

The problem of finding the most general spinor field possessing the same symmetry as a given gravitational field is solved for every group of motions. Its connection with the resolution of Einstein-Dirac equations is briefly pointed out.Aspirant du Fonds National Belge de la Recherche Scientifique.     

Global spinor fields in space-time

This paper is concerned with space-time manifolds that are space- and time-oriented, causal, and possess spinor structures. Five propositions are proven: (1) If a connected, space- and time-oriented manifold is simply con-nected, then it is non-compact; (2) If such a manifold is simply connected, it admits a spinor structure, which, moreover, is unique; (3) If the space-like section of M is compact, then there exists a global system of orthonormal tetrads on M; (4) The necessary and sufficient condition for every space-time M whose space-like section is compact to admit a spinor structure is that M have a global system of orthonormal tetrads; (5) Every space-time M which can be imbedded in R⁶ admits a spinor structure. It is further suggested that in view of the fact that the existence of a spinor structure is related to homotopy properties, space-time manifolds may be classified in terms of their homotopy groups _i (M), i=1,2, 3,4. In a concluding section, some avenues for future research are discussed.Supported by the National Science Foundation.     

Geometry and symmetries in lattice spinor gravity

Lattice spinor gravity is a proposal for regularized quantum gravity based on fermionic degrees of freedom. In our lattice model the local Lorentz symmetry is generalized to complex transformation parameters. The difference between space and time is not put in a priori, and the euclidean and the Minkowski quantum field theory are unified in one functional integral. The metric and its signature arise as a result of the dynamics, corresponding to a given ground state or cosmological solution. Geometrical objects as the vierbein, spin connection or the metric are expectation values of collective fields built from an even number of fermions. The quantum effective action for the metric is invariant under general coordinate transformations in the continuum limit. The action of our model is found to be also invariant under gauge transformations. We observe a “geometrical entanglement” of gauge- and Lorentz-transformations due to geometrical objects transforming non-trivially under both types of symmetry transformations.     

Scalar fields in a seven-dimensional manifold behaving as Lorentz-covariant spinor fields in space-time

A Lorentz-invariant model of vacuum is given in the form of a 7-dimensional manifold endowed with a statistical metrical tensor. Certain scalar fields on this manifold behave then as spinor fields when viewed from their space-time projection. This paper generalizes previous work fromSO(3)-covariance to Lorentz-covariance.     

Random walk construction of spinor fields on a three-dimensional lattice

Euclidean-invariant Klein-Gordon, Dirac and massive Chern-Simons field theories are constructed in terms of a random walk with a spin factor on a three-dimensional lattice. We exactly calculate the free energy and the correlation functions which allow us to obtain the critical diffusion constant and associated critical exponents. It is pointed out that these critical exponents do not satisfy the hyper-scaling relation but the scaling inequalities. We take the continuum limit of this theory on the basis of these analyses. We check the universality of the obtained results on other lattice structures such as the triclinic lattice and the body-centered lattice.     

On the spinor rank of Fermi fields

We show that any Wightman field satisfying equal-time anti-commutation relations involving space derivatives of degree at mostr must have spinor rankr + 1.     

Nonlocal interaction of spinor and pseudoscalar fields

The nonlocal interaction of a spinor and a pseudoscalar field is considered in the framework of gradient coupling. By the introduction of a relativistically invariant form factor in the neutrino propagator, renormalizability of the theory is achieved. It is shown that the number of primitive diagrams is just four, one of which is the neutrino self-energy diagram. The Pauli-Villars regularization procedure is carried out for this diagram.This paper was presented at the session of the Nuclear Physics Section of the USSR Academy of Sciences in Moscow (February 1–4, 1978).Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 84–88, July, 1980.We are grateful to G. V. Efimov and N. Atakishiev for discussing the results.     

Monopole solution in a Lorentz-violating field theory

I present a topological defect solution that arises in a theory where Lorentz symmetry is spontaneously broken by a rank-two antisymmetric tensor field, and I discuss its observational signatures.     

Scalar and spinor fields in the very early universe

Here it is shown how the vacuum energy may dominate the energy density of the very early universe even when the Higg's field in the Coleman-Weinberg potential is confined near the origin at extremely high temperature and the inflationary scenario may start. Also it is shown that supersymmetry breaking may be responsible for this phenomenon. Thus it provides another support for the hypothesis of primordial inflation proposed by Ellis et al. [4],     

Unfolding physics from the algebraic classification of spinor fields

After reviewing the Lounesto spinor field classification, according to the bilinear covariants associated to a spinor field, we call attention and unravel some prominent features involving unexpected properties about spinor fields under such classification. In particular, we pithily focus on the new aspects — as well as current concrete possibilities. They mainly arise when we deal with some non-standard spinor fields concerning, in particular, their applications in physics.     

Group fields,gravity, and angular momentum

The angular momentum properties of different homotopic sectors of a group field are examined. The results are applied to gravity, which is essentially a group field, and it is shown that the usual gravity kinks do not have angular momentum 1/2.Work supported in part by the National Science Foundation Grant No. PHY800-7921.     

Gauge fields,space-time geometry and gravity

A general framework for the gauge theory of the affine group and its various subgroups in terms of connections on the bundle of affine frames and its subbundles is given, with emphasis on the correct gauging of groups including space-time translations. For consistency of interpretation, the appropriate objects to be identified with gravitational vierbeins in such theories are not the translational gauge fields themselves, but their pull backs,via appropriate bundle homomorphisms, to the bundle of frames. This automatically solves the problems usually encountered in constructing a gauge theory of the conventional sort for groups containing translations. We give a consistent formulation of the Poincare gauge theory and also of the theory based on translational gauge invariance which, in the absence of matter fields with intrinsic spin, gives a local Lorentz invariant theory equivalent to Einstein gravity.     

Nonlinear spinor fields in Bianchi-I space: Exact self-consistent solutions

Calculations are performed to obtain exact self-consistent solutions of nonlinear spinor-field equations with self-action terms in Bianchi-I space. The latter terms are arbitrary functions of the invariant . A detailed examination is made of equations with exponential nonlinearity, when the nonlinear term in the Lagrangian of the spinor field L_n=sⁿ. Here, is the nonlinearity parameter, n>1. It is shown that these equations have finite solutions and solutions that are singular at the initial moment of time. The singularity is absent in the case of solutions that describe systems of fields for which the energy dominance condition is violated. It is further shown that if the mass parameter m0 in the spinor-field equation, expansion of Bianchi-I space becomes isotropic as t . This does not occur when m=0. Specific examples of solutions of linear and nonlinear spinor-field equations are presented.Russian University of International Fellowship. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 40–45, July, 1994.     

Matter fields in lattice gravity

The formulations of the scalar, spinor and gauge fields on a curved random lattice are discussed. A new fermion scheme — simplicial fermions — which is more natural in the presence of gravity, is introduced. Some of the propagators in D = 2 are evaluated numerically and are compared with the continuum limit.     

Spinor fields in the theory of relativity and a generalization of Heisenberg's nonlinear spinor equation

Some years ago it was shown that the nonlinear term of Heisenberg's spinor equation can be derived by torsion of the Minkowski space (Cartan space). This result is applied in the investigations of this paper. As the Heisenberg equation does not show any connection with recent phenomenological theories in high energy physics, like the parton or quark model, the problems of the metric of space-time are discussed from the aspect of fundamental axioms of topology (Hausdorff space). It will be shown that Feynman's relativistic parton theory can be derived by means of a quantised de Sitter space, where the constant curvature can assume only discrete values. It is also possible to derive the Dirac equation from the same mathematical considerations. A nonlinear spinor equation will be formulated which contains the parton theory and the nonlinear term of the Heisenberg equation as different approaches in the theory of elementary particles.     

Gravitational interaction in a model of nonlinear spinor theory with derivatives of the fundamental fields

To eliminate the difficulties that arise when particles of high spin (J 2) are treated in nonlinear spinor field theory, it is proposed to include terms with derivatives of the fundamental fields in the self-interaction Lagrangian of the subparticles. It is shown that with this generalization the four-fermion model of nonlinear spinor theory describes the gravitational interaction of subparticles in the approximation of single-graviton exchange. A relation between the gravitational constant and the parameters of the model is found.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 79–85, August, 1977.I thank D. D. Ivanenko for a helpful discussion of this work.