Derivation of the Finslerian Gauge Field Equations

Abstract

AbstractAs is well known the simplest way of formulating the equations for the Yang‐Mills gauge fields consists in taking the Lagrangian to be quadratic in the gauge tensor [1 ‐ 5], whereas the application of such an approach to the gravitational field yields equations which are of essentially more complicated structure than the Einstein equations. On the other hand, in the gravitational field theory the Lagrangian can be constructed to be of forms which may be both quadratic and linear in the curvature tensor, whereas the latter possibility is absent in the current gauge field theories. In previous work [6] it has been shown that the Finslerian structure of the space‐time gives rise to certain gauge fields provided that the internal symmetries may be regarded as symmetries of a three‐dimensional Riemannian space. Continuing this work we show that appropriate equations for these gauge fields can be formulated in both ways, namely on the basis of the quadratic Lagrangian or, if a relevant generalization of the Palatini method is applied, on the basis of a Lagrangian linear in the gauge field strength tensor. The latter possibility proves to result in equations which are similar to the Einstein equations, a distinction being that the Finslerian Cartan curvature tensor rather than the Riemann curvature tensor enters the equations.