Linear Response in the $\mathcal{PT}$-Broken Phase: Conductivity of Fermions with Complex Conjugate Masses

Abstract

We examine the linear response of the $\PT$-broken phase of a BRST invariant model whose fermion masses form the complex conjugate pair $m_{\pm}=N\pm iav$. It splits into two channels, kept apart by an exact $\mathbb{Z}_{2}$, and one Euclidean correlator decides both. In the diagonal channel it oscillates in Euclidean time, so that channel admits no positive-norm realization: we exhibit a test function of negative norm and close every repair, the choice of boundary value among them. In the mixed channel the same correlator is completely monotone, which proves the spectral density positive; there we obtain the vacuum conductivity in closed form, with a real threshold at $ω=2N$, and positivity survives at finite temperature below a bound whose onset we give exactly. The longitudinal density of that channel is negative by the square of the mass splitting; an antisymmetric derivative removes it, and the resulting field-strength composite has a positive one-particle space and a causal commutator --- at the level of the two-point function only, since a four-point reflection form on the original composite comes out negative. The positive realization is therefore not a sector of the fermion model.