Irreversibility of the renormalization group flow in non-unitary quantum field theory

We show irreversibility of the renormalization group flow in non-unitary but PT -invariant quantum field theory in two space-time dimensions. In addition to unbroken 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 PT -symmetric Hamiltonians. Our proof follows closely Zamolodchikov’s arguments. We show that a function ceff(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 ceff = c − 24∆, where c is the central charge and Δ is the lowest primary field dimension in the conformal field theory which describes the critical point.