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

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Expand the scalar field on spherical harmonics (on the shell at r0r_0r0):

ϕ(t,θ,φ)  =  ∑ℓ=0∞∑m=−ℓℓΦℓm(t)  Yℓm(θ,φ).\phi(t,\theta,\varphi) \;=\; \sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell} \Phi_{\ell m}(t)\;Y_{\ell m}(\theta,\varphi).ϕ(t,θ,φ)=ℓ=0∑∞m=−ℓ∑ℓΦℓm(t)Yℓm(θ,φ).
Because ΔΩYℓm=−ℓ(ℓ+1)Yℓm\Delta_{\Omega}Y_{\ell m}=-\ell(\ell+1)Y_{\ell m}ΔΩYℓm=−ℓ(ℓ+1)Yℓm, the shell EOM (no A for the moment) becomes for each mode:

1c2Φ¨ℓm(t)+ℓ(ℓ+1)r02 Φℓm(t)+ms2 Φℓm(t)=0.\frac{1}{c^{2}}\ddot{\Phi}_{\ell m}(t) + \frac{\ell(\ell+1)}{r_0^{2}}\,\Phi_{\ell m}(t) + m_s^{2}\,\Phi_{\ell m}(t)=0.c21Φ¨ℓm(t)+r02ℓ(ℓ+1)Φℓm(t)+ms2Φℓm(t)=0.
So the free (uncoupled) mode frequency is

  ωℓ2  =  c2[ℓ(ℓ+1)r02  +  ms2].  \boxed{\;
\omega_{\ell}^{2}
\;=\; c^{2}\left[\frac{\ell(\ell+1)}{r_0^{2}} \;+\; m_s^{2}\right].
\;}ωℓ2=c2[r02ℓ(ℓ+1)+ms2].
This is a clean and important result: modes are indexed by ℓ\ellℓ, the geometry sets the ℓ(ℓ+1)/r02\ell(\ell+1)/r_0^2ℓ(ℓ+1)/r02 term, and ms2m_s^2ms2 is an effective shell mass.

===== 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.)

===== Each mode is a harmonic oscillator: =====

Hℓm  =  ∣πℓm∣22Meff  +  12Meff ωℓ2 ∣Φℓm∣2.H_{\ell m} \;=\; \frac{|\pi_{\ell m}|^{2}}{2M_{\rm eff}} \;+\; \frac{1}{2}M_{\rm eff}\,\omega_{\ell}^{2}\,|\Phi_{\ell m}|^{2}.Hℓm=2Meff∣πℓm∣2+21Meffωℓ2∣Φℓm∣2.
Quantize by canonical commutators (on the shell):

[Φℓm,πℓ′m′]  =  iℏ δℓℓ′δmm′.[\Phi_{\ell m},\pi_{\ell' m'}] \;=\; i\hbar\,\delta_{\ell\ell'}\delta_{mm'}.[Φℓm,πℓ′m′]=iℏδℓℓ′δmm′.
Define annihilation/creation operators aℓma_{\ell m}aℓm, aℓm†a_{\ell m}^\daggeraℓm† with the usual normalization so the quantized Hamiltonian is

  Hshell  =  ∑ℓ,mℏωℓ(aℓm†aℓm+12).  \boxed{\;
H_{\rm shell} \;=\; \sum_{\ell,m} \hbar\omega_{\ell}\left(a_{\ell m}^\dagger a_{\ell m} + \tfrac12\right).
\;}Hshell=ℓ,m∑ℏωℓ(aℓm†aℓm+21).
Important: zero-point energy per mode is 12ℏωℓ\tfrac12\hbar\omega_\ell21ℏωℓ. Note ℏ\hbarℏ here is h/2πh/2\pih/2π, explicitly.

===== Minimal coupling replaced time/angle derivatives by covariant ones: =====

∂tΦℓm  ⟶  ∂tΦℓm+iqAt(ℓm)(t),\partial_t\Phi_{\ell m}\;\longrightarrow\; \partial_t\Phi_{\ell m} + i q A_t^{(\ell m)}(t),∂tΦℓm⟶∂tΦℓm+iqAt(ℓm)(t),
and similarly for angular derivatives. Expanding to leading order in AμA_\muAμ yields an interaction Hamiltonian of the dipole/multipole form:

Hint  =  q∑ℓm[ Oℓm(t)  At(ℓm)(t)  +  Pℓm(t)⋅AΩ(ℓm)(t)]+O(q2A2).H_{\rm int}
\;=\; q \sum_{\ell m} \left[\,\mathcal{O}_{\ell m}(t)\;A_t^{(\ell m)}(t) \;+\; \mathcal{P}_{\ell m}(t)\cdot A_{\Omega}^{(\ell m)}(t)
\right] + \mathcal{O}(q^2 A^2).Hint=qℓm∑[Oℓm(t)At(ℓm)(t)+Pℓm(t)⋅AΩ(ℓm)(t)]+O(q2A2).
Here Oℓm\mathcal{O}_{\ell m}Oℓm, Pℓm\mathcal{P}_{\ell m}Pℓm are mode operators constructed from Φℓm\Phi_{\ell m}Φℓm and πℓm\pi_{\ell m}πℓm (they generalize charge and multipole moments).

At quadratic order the interaction also produces a shift of the eigenfrequencies (Stark-like or diamagnetic terms) of order q2⟨A2⟩/Meffq^2\langle A^2\rangle/M_{\rm eff}q2⟨A2⟩/Meff.

===== - A shell charge multipole σℓm(t)\sigma_{\ell m}(t)σℓm(t) (equal to the shell JtJ^tJt expanded in YℓmY_{\ell m}Yℓm) produces an external potential outside that scales like r−(ℓ+1)r^{-(\ell+1)}r−(ℓ+1) and inside like rℓr^\ellrℓ. That gives the external EM field that the shell mode radiates into. =====
* Thus radiation and damping of shell modes can be obtained by coupling the shell oscillators to the continuum of EM modes (standard open-system treatment): shell modes will have decay widths Γℓ\Gamma_{\ell}Γℓ into EM radiation determined by the coupling strength qqq and the multipole order ℓ\ellℓ.