Extended Theories of Gravity

A generalization of Einstein's gravitational theory is discussed in which the spin of matter as well as its mass plays a dynamical role. The spin of matter couples to a non-Riemannian structure in space-time, Cartan's torsion tensor. The theory which emerges from taking this coupling into account, the ${U}_{4}$ theory of gravitation, predicts, in addition to the usual infinite-range gravitational interaction medicated by the metric field, a new, very weak, spin contact interaction of gravitational origin. We summarize here all the available theoretical evidence that argues for admitting spin and torsion into a relativistic gravitational theory. Not least among this evidence is the demonstration that the ${U}_{4}$ theory arises as a local gauge theory for the Poincar\'e group in space-time. The deviations of the ${U}_{4}$ theory from standard general relativity are estimated, and the prospects for further theoretical development are assessed.

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General Relativity with Spin and Torsion: Foundations and Prospects

Feynman Diagrams for the Yang-Mills Field

Gravitation, Gauge Theories and Differential Geometry

An argument leading from the Lorentz invariance of the Lagrangian to the introduction of the gravitational field is presented. Utiyama's discussion is extended by considering the 10‐parameter group of inhomogeneous Lorentz transformations, involving variation of the coordinates as well as the field variables. It is then unnecessary to introduce a priori curvilinear coordinates or a Riemannian metric, and the new field variables introduced as a consequence of the argument include the vierbein components hk μ as well as the ``local affine connection'' Ai jμ . The extended transformations for which the 10 parameters become arbitrary functions of position may be interpreted as general coordinate transformations and rotations of the vierbein system. The free Lagrangian for the new fields is shown to be a function of two covariant quantities analogous to Fμν for the electromagnetic field, and the simplest possible form is just the usual curvature scalar density expressed in terms of hk μ and Ai jμ . This Lagrangian is of first order in the derivatives, and is the analog for the vierbein formalism of Palatini's Lagrangian. In the absence of matter, it yields the familiar equationsRμν =0 for empty space, but when matter is present there is a difference from the usual theory (first pointed out by Weyl) which arises from the fact that Ai jμ appears in the matter field Lagrangian, so that the equation of motion relating Ai jμ to hk μ is changed. In particular, this means that, although the covariant derivative of the metric vanishes, the affine connection Γλ μν is nonsymmetric. The theory may be reexpressed in terms of the Christoffel connection, and in that case additional terms quadratic in the ``spin density'' Sk ij appear in the Lagrangian. These terms are almost certainly too small to make any experimentally detectable difference to the predictions of the usual metric theory.

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Lorentz Invariance and the Gravitational Field

Some systems of fields have been considered which are invariant under a certain group of transformations depending on $n$ parameters. A general rule is obtained for introducing a new field in a definite way with a definite type of interaction with the original fields by postulating the invariance of these systems under a wider group derived by replacing the parameters of the original group with a set of arbitrary functions. The transformation character of this new field under the wider group is determined from the invariance postulate. The possible types of the equations of the new fields can be also derived, giving rise to a certain conservation law owing to the invariance. As examples, the electromagnetic, the gravitational and the Yang-Mills fields are reconsidered following this line of approach.

Invariant theoretical interpretation of interaction

: The singularities of Green's functions of electrons and photons in a classical gravitational field are investigated. These singularities can be removed by the introduction of some suitable counter terms into the equations of Green's functions. These counter terms can be given by rewriting the conventional renormalization technique in a generally covari nt way. The infinite renormalization const nts appearing in t e conventional quantum electrodyn mics ar shown to be sufficient for the removal of the singularities of the present system; any other renormalization is not necessary. The segregation of these singularities needs an exclusive use of the transfor atio properties of Gre n's functio s under general coordinate transformations, Vieb in rotatio s, and gaug transformatons. (Author)

Renormalization of quantum electrodynamics in a classical gravitational field

A tentative scheme is developed to formulate the behavior of the new unstable particles. This scheme is a straightforward generalization of the usual charge-independent meson theory. The selection rules for isotopic spin are identical with those suggested by Gell-Mann. Owing to the particular form of the interaction assumed in this scheme, we can derive a new selection rule which seems to be of some use in interpreting the metastability of the new particles.

Ryoyu Utiyama

Papers

Invariant theoretical interpretation of interaction

Renormalization of quantum electrodynamics in a classical gravitational field

A theory of new particles