Obse e , Window, Mellin–Heisenbe g
and Repa ame iza ion Co a iance
Au ic
Ve sion 1.0
No embe 24, 2025
Abs ac
Es ablish obse e -dependen window amewo k wi h Mellin–Heisenbe g phase–
scale symme y and epa ame iza ion co a iance. Co e esul s: (i) Window choice
encodes obse e esolu ion; (ii) Mellin ans o m M[ ](s) = R∞
0 (x)xs−1dx p o ides
phase–scale ep esen a ion ia Weyl–Heisenbe g commu a ion ela ions; (iii) Ene gy
epa ame iza ion E7→ ϕ(E) induces window push o wa d p ese ing windowed ob-
se ables; (i ) Wexle –Raz bio hogonali y ensu es ame comple eness.
1 Mellin–Heisenbe g F amewo k
On Ha=L2(R+, xa−1dx), de ine modula ion/dila ion:
(Uτ )(x) = xiτ (x),(Vσ )(x) = eσa/2 (eσx)
sa is ying Weyl ela ion VσUτ=eiτσUτVσ.
Via x=e isome y wi h L2(R) Gabo ame.
2 Window Repa ame iza ion
Theo em 2.1 (Repa ame iza ion Co a iance).Fo mono one ϕ:R→Rand window w,
windowed obse able
Ow[ρ] = Zw(E)ρ(E)dE
unde E7→ ϕ(E)becomes
Owϕ[ρϕ] = Zw(ϕ−1(E′))ρ(ϕ−1(E′))|ϕ′(ϕ−1(E′))|−1dE′
p ese ing in eg al alue.
3 Obse e Resolu ion
Window bandwid h ∆Eencodes obse e ene gy esolu ion. Wexle –Raz densi y condi ion
∆E·∆ ≤2πℏensu es ame comple eness.
Di e en obse e s wi h windows w1, w2 ela e ia Wexle –Raz ans o m, p ese ing quan-
um in o ma ion up o esolu ion limi s.
1
4 Discussion
F amewo k p o ides:
Obse e -dependen bu co a ian measu emen heo y
Mellin–Heisenbe g phase–scale symme y
Repa ame iza ion in a iance
F ame- heo e ic ounda ions
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