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

Abstract

We show irreversibility of the renormalization group flow in non-unitary but $\mathcal{P}\mathcal{T}$-invariant quantum field theory in two space-time dimensions. In addition to unbroken $\mathcal{P}\mathcal{T}$-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 $\mathcal{P}\mathcal{T}$-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.

26 pages, 1 figure

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