Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

===== Start from the shell kinetic contribution =====

Skin=ℏ ⁣∫ ⁣dt  r02 ⁣∫dΩ  12c2 ∣∂tϕ∣2.S_{\text{kin}}=\hbar\!\int\!dt\; r_{0}^{2}\!\int d\Omega\; \frac{1}{2c^{2}}\,|\partial_t\phi|^{2}.Skin=ℏ∫dtr02∫dΩ2c21∣∂tϕ∣2.
Insert the expansion and use the orthonormality of YℓmY_{\ell m}Yℓm. The kinetic term becomes (sum over modes)

Skin=ℏ ⁣∫ ⁣dt  ∑ℓm  r022c2  ∣Φ˙ℓm∣2.S_{\text{kin}} = \hbar\!\int\!dt\;\sum_{\ell m}\; \frac{r_0^{2}}{2c^{2}}\;|\dot\Phi_{\ell m}|^{2}.Skin=ℏ∫dtℓm∑2c2r02∣Φ˙ℓm∣2.
Thus define the effective modal mass:

Meff  =  ℏ r02c2.M_{\rm eff} \;=\; \hbar\,\frac{r_0^{2}}{c^{2}}.Meff=ℏc2r02.
Canonical momentum for each mode:

πℓm(t)  =  ∂L∂Φ˙ℓm  =  Meff  Φ˙ℓm∗(t).\pi_{\ell m}(t) \;=\; \frac{\partial \mathcal{L}}{\partial \dot\Phi_{\ell m}}
\;=\; M_{\rm eff}\; \dot\Phi_{\ell m}^{*}(t).πℓm(t)=∂Φ˙ℓm∂L=MeffΦ˙ℓm∗(t).
(We keep ℏ\hbarℏ explicit in MeffM_{\rm eff}Meff.)