Observer, Window, Mellin--Heisenberg\\and Reparametrization Covariance

Observer, Window, Mellin–Heisenberg and Reparametrization Covariance Auric Version 1.0 November 24, 2025 Abstract Establish observer-dependent window framework with Mellin–Heisenberg phase– scale symmetry and reparametrization covariance. Core results: (i) Window choice encodes observer resolution; (ii) Mellin transform M[f](s) = R∞ 0f(x)xs−1dx provides phase–scale representation via Weyl–Heisenberg commutation relations; (iii) Energy reparametrization E7→ ϕ(E) induces window pushforward preserving windowed observables; (iv) Wexler–Raz biorthogonality ensures frame completeness. 1 Mellin–Heisenberg Framework On Ha=L2(R+, xa−1dx), define modulation/dilation: (Uτf)(x) = xiτ f(x),(Vσf)(x) = eσa/2f(eσx) satisfying Weyl relation VσUτ=eiτσUτVσ. Via x=etisometry with L2(R) Gabor frame. 2 Window Reparametrization Theorem 2.1 (Reparametrization Covariance).For monotone ϕ:R→Rand window w, windowed observable Ow[ρ] = Zw(E)ρ(E)dE under E7→ ϕ(E)becomes Owϕ[ρϕ] = Zw(ϕ−1(E′))ρ(ϕ−1(E′))|ϕ′(ϕ−1(E′))|−1dE′ preserving integral value. 3 Observer Resolution Window bandwidth ∆Eencodes observer energy resolution. Wexler–Raz density condition ∆E·∆t≤2πℏensures frame completeness. Different observers with windows w1, w2relate via Wexler–Raz transform, preserving quantum information up to resolution limits. 1 4 Discussion Framework provides:  Observer-dependent but covariant measurement theory  Mellin–Heisenberg phase–scale symmetry  Reparametrization invariance  Frame-theoretic foundations 2