Gauge invariance of the one-loop effective potential in $M^d$${\ge }^1$ Kaluza-Klein theory

By adjusting the Weyl factor in (d+1)-dimensional Kaluza-Klein theory, we obtain a one-parameter family of gauges which give rise to propagating, massive ghosts in the compactified theory on $M^d$×$S^1$. We calculate the one-loop Casimir energy of this vacuum for all values of the gauge parameter S and arbitrary noncompact ‘‘space-time’’ dimension d. As expected, the result is independent of S and coincides with those previously obtained in other gauges. Our calculation confirms recent formal proofs due to Cohler and Chodos and Yasuda concerning the gauge independence of the graviton one-loop effective potential.