  It is argued that the zero-point energies of free quantum fields diverge at
most quadratically and not quartically, as is generally believed. This is a
consequence of the relativistic invariance which requires that the energy
density of the vacuum $\rho$ and its pressure $p$ satisfy $\rho=-p$. The
usually obtained quartic divergence is an artifact of the use of a noninvariant
regularization which violates this relation. One consequence of our results is
that the zero-point energies of free massless fields vanish. Implications for
the cosmological constant problem are briefly discussed.
