  Recent work has shown that complex quantum field theory emerges as a
statistical mechanical approximation to an underlying noncommutative operator
dynamics based on a total trace action. In this dynamics, scale invariance of
the trace action becomes the statement $0=Re Tr T_{\mu}^{\mu}$, with $T_{\mu
\nu}$ the operator stress energy tensor, and with $Tr$ the trace over the
underlying Hilbert space. We show that this condition implies the vanishing of
the cosmological constant and vacuum energy in the emergent quantum field
theory. However, since the scale invariance condition does not require the
operator $T_{\mu}^{\mu}$ to vanish, the spontaneous breakdown of scale
invariance is still permitted.
