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arXiv:2605.22919 · detector doi_compliance · incontrovertible · 2026-05-25 05:50:04.043700+00:00

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Step 5: Identification with the Unparticle Form The unparticle spectral density [1] isρU(ω)∝ω 2dU −3. Matching exponents with Eq. (14): 2dU −3 = 2∆−d−1⇒d U = ∆− d−2 2 ,(15) which is Eq. (2). The dynamical exponents in Table I follow by Fourier transformation of the kernels; full derivations are given in Sec. III. The identification with the unparticle form has a transparent interpretation in the KL language. The spectral weightρ(σ)∝σ ∆−(d+3)/2 is a pure power law, meaning the bath has acontinuous spectrum of states at every invariant mass scale, with no preferred mass and no particle poles. This is precisely the defining feature of an unparticle sector [1]. The KL representation makes this identification automatic: scale invariance does not merely suggest the unparticle picture, it forces it. Remark.The Källén–Lehmann proof derives conclusions (2)–(4) of Theorem 1 directly from Lorentz invariance and scale invariance, without invoking the CFT identification of 10 conclusion (1). Conclusion (1)—that continuous scale invariance implies full conformal invariance—is established independently: ind= 2it follows rigorously from Polchinski’s completion of Zamolodchikov’sc-theorem argument [4]; ind≥3it holds under the additional assumption of unitarity and absence of a virial current, for which strong evidence exists in d= 4[5, 6] but no general proof is available. The logical structure of the theorem is therefore: conclusion (1) is an input from the conformal field theory literature;

Evidence payload

{
  "doi_as_printed": "10.13039/501100011033/",
  "raw_excerpt": "Step 5: Identification with the Unparticle Form The unparticle spectral density [1] is\u03c1U(\u03c9)\u221d\u03c9 2dU \u22123. Matching exponents with Eq. (14): 2dU \u22123 = 2\u2206\u2212d\u22121\u21d2d U = \u2206\u2212 d\u22122 2 ,(15) which is Eq. (2). The dynamical exponents in Table I follow by Fourier transformation of the kernels; full derivations are given in Sec. III. The identification with the unparticle form has a transparent interpretation in the K",
  "ref_index": 13,
  "verdict_class": "incontrovertible"
}