Weyl's Gauge Field and Its Behavior

Abstract

Using a manifestly gauge-invariant Lagrangian density of a system in which a real scalar field (matter field) is interacting with itself and with Weyl's gauge field, we shall study equations of the real scalar field and of Weyl's gauge field, and discuss the self-interacting term of the real scalar field. For a special self-interacting term, we shall obtain an equation of only Weyl's gauge field which plays an important role in solving the equation of Weyl's gauge field interacting with the real scalar field. By making use of the above mentioned equation we shall obtain a rigorous solution for Weyl's gauge field. Next, combining the equation of only Weyl's gauge field with the condition in Weyl's gauge field that the length scale of any vector changes under parallel transfer, we shall obtain a nonlinear equation for the length scale of Weyl's gauge field, which may be important in mathematical physics and is shown to have meron-type solution. By making use of the same techniques being used above, we shall study solution of equation of gradient Weyl's gauge field and as a result, obtain a nonlinear equation of the same type as being found above. Finally we shall study relation between local gauge transformation and symmetric connection in space-time. As a result, we can partly make clear relation between the change in the measure of length scale of a vector due to an infinitesimal parallel transfer and the coefficients of affine connection of Weyl's geometry.