Paraparticles intrinsically exhibit Hardy-space breakdown

Abstract

The memory kernel of an open quantum system obeys Kramers--Kronig (KK) relations if and only if its Laplace transform is analytic in the upper half-plane -- a property known as Hardy-space analyticity. Here we show that non-unitary exchange statistics, the defining property of paraparticles, intrinsically breaks Hardy-space analyticity. The metric $\hat{I}\cdot$ that guarantees a real closed-system spectrum for these particles necessarily differs from the physical Born inner product ($\Vert \hat{I}\cdot -I{\Vert }_F/\Vert I{\Vert }_F=0.51$) -- a mathematical consequence of the R-matrix's non-unitarity, not a parameter choice. This metric is a "shadow metric": Schur's lemma forces it to commute with every bilinear observable, making the distortion physically invisible in the closed system. But when the paraparticle is coupled to a bath, any coupling operator that lies outside the symmetry algebra -- that is, any interaction that sees the internal flavour structure -- exposes the distortion. The memory kernel then develops upper-half-plane poles at coupling $g_c\approx 0.1$, breaking standard dispersion relations before the closed-system spectrum complexifies. Fermions and bosons, whose exchange is unitary ($\hat{I}\cdot =I$ as an analytic fact of the canonical anticommutation algebra), are immune at any coupling, because there is no distortion to expose. The violation is intrinsic: it distinguishes non-unitary exchange statistics from ordinary particle statistics at the level of the memory kernel's analytic structure.

8 pages main + 13 pages Supplemental Material, 3 figures. Data and code: [Liu, K. (2026). Data and code for "Paraparticles intrinsically exhibit Hardy-space breakdown" [Dataset]. Zenodo. https://doi.org/10.5281/zenodo.21342378

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