  We show irreversibility of the renormalization group flow in non-unitary but
${\cal PT}$-invariant quantum field theory in two space-time dimensions. In
addition to unbroken $\mathcal{PT}$-symmetry and a positive energy spectrum, we
assume standard properties of quantum field theory including a local
energy-momentum tensor and relativistic invariance. This generalizes
Zamolodchikov's $c$-theorem to ${\cal PT}$-symmetric hamiltonians. Our proof
follows closely Zamolodchikov's arguments. We show that a function
$c_{\mathrm{eff}}(s)$ of the renormalization group parameter $s$ exists which
is non-negative and monotonically decreasing along renormalization group flows.
Its value at a critical point is the "effective central charge" entering the
specific free energy. At least in rational models, this equals
$c_{\mathrm{eff}}=c-24\Delta$, where $c$ is the central charge and $\Delta$ is
the lowest primary field dimension in the conformal field theory which
describes the critical point.
