The problem of solving the combined gravitational and Yang–Mills field systems is regarded as a purely geometrical problem of determining a linear connection on the principal frame bundle L(M) from a connection on a SU(2) principal bundle over a space‐time M. It is suggested that mapping theorems of connections on bundles may provide a means of actually solving ‘‘field equations.’’

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Mapping of connections on bundles and gauge field theories

Foundations of 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

A new integral formalism for gauge fields is described. Further developments are presented, including gravitation equations related to, but not identical with, Einstein's equations.

Integral Formalism for Gauge Fields

These Lecture Notes are an expanded version of the Fermi Lectures I gave at Scuola Normale Superiore in Pisa, the Loeb Lectures at Harvard and the Whittemore Lectures at Yale, in 1978. In all cases I was addressing a mixed audience of mathematicians and physicists and the presentation had to be tailored accordingly. Throughout, I presented the mathematical material in a somewhat unorthodox order, following a pattern which I felt would relate the new techniques to familiar ground for physicists. The main new results presented in the lectures, namely the construction of all multi-istanton solutions of Yang-Mills fields, is the culmination of several years of fruitful interaction between many physicists and mathematicians.The major breakthrough came with the observation by Ward that the complex methods developed by Penrose in his 'twistor programme' were ideally suited to the study of the Yang-Mills equations. The instanton problem was then seen to be equivalent to a problem in complex analysis and to one in algebraic geometry. Using the powerful methods of modern algebraic geometry it was not long before the problem was finally solved.

Mapping of connections on bundles and gauge field theories

References

Foundations of Differential Geometry

Lorentz Invariance and the Gravitational Field

Integral Formalism for Gauge Fields

Geometry of Yang-Mills fields

Fibre bundles associated with space-time☆

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