  It is known that there is a proportionality factor relating the
$\beta$-function and the equations of motion viz. the Zamolodchikov metric.
Usually this factor has to be obtained by other methods. The proper time
equation, on the other hand, is the full equation of motion. We explain the
reasons for this and illustrate it by calculating corrections to Maxwell's
equation. The corrections are calculated to cubic order in the field strength,
but are exact to all orders in derivatives. We also test the gauge covariance
of the proper time method by calculating higher (covariant) derivative
corrections to the Yang-Mills equation.
