THE PROPER TIME EQUATION AND THE ZAMOLODCHIKOV METRIC

The connection between the proper time equation and the Zamolodchikov metric is discussed. The connection is twofold. First, as already know n, the proper time equation is the product of the Zamolodchikov metric and the renormalization group beta function. Second, the condition th at the two-point function is the Zamolodchikov metric implies the prop er time equation. We study the massless vector of the open string in d etail. In the exactly calculable case of a uniform electromagnetic fie ld strength we recover the Born-Infeld equation. We describe the syste matics of the perturbative evaluation of the gauge-invariant proper ti me equation for the massless vector field. The method is valid for non uniform fields and gives results that are exact to all orders in deriv atives. As a nontrivial check, we show that in the limit of uniform fi elds it reproduces the lowest order Born-Infeld equation.