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% Quantum Vacuum Energy Extraction Framework

% 1. Vacuum State and Field Operators
\begin{align*}
& \text{Vacuum state: } |0\rangle \text{ (unbounded)} \\
& \text{Field operator: } \hat{\phi}(x,t) = \sum_k \left(\hat{a}_k \phi_k(x,t) + \hat{a}_k^\dagger \phi_k^*(x,t)\right) \\
& \text{Where: } \phi_k(x,t) = \frac{1}{\sqrt{2\omega_k V}} e^{i(k\cdot x - \omega_k t)}
\end{align*}

% 2. Boundary Condition Introduction
\begin{align*}
& \text{Bounded region: } \Omega \text{ with boundary } \partial\Omega \\
& \text{Boundary condition: } \left.\hat{\phi}(x,t)\right|_{\partial\Omega} = f(x,t) \\
& \text{Mode expansion with boundary: } \phi_k^B(x,t) = \sum_n c_n \psi_n(x) e^{-i\omega_n t}
\end{align*}

% 3. Energy Operator and Expectation Values
\begin{align*}
& \hat{H} = \frac{1}{2} \int_\Omega d^3x \left[(\partial_t\hat{\phi})^2 + (\nabla\hat{\phi})^2 + m^2\hat{\phi}^2\right] \\
& \text{Vacuum energy (unbounded): } E_0 = \langle 0|\hat{H}|0\rangle = \infty \\
& \text{Bounded vacuum energy: } E_B = \langle 0_B|\hat{H}|0_B\rangle
\end{align*}

% 4. Energy Extraction Process
\begin{align*}
& \text{Interaction Hamiltonian: } \hat{H}_I = g\int_\Omega d^3x \hat{\phi}(x)\hat{A}(x) \\
& \text{Where } \hat{A}(x) \text{ is the photon field} \\
& \text{Energy difference: } \Delta E = E_B - E_0 = -\frac{\pi^2\hbar c}{720d^3} \text{ (Casimir effect)}
\end{align*}

% 5. Photon-Induced Boundary Effect
\begin{align*}
& \text{Photon state: } |\gamma\rangle = \hat{a}_\gamma^\dagger|0\rangle \\
& \text{Energy of boundary system: } E_{\gamma B} = \langle\gamma|\hat{H}|0_B\rangle \\
& \text{Extractable energy: } E_{ext} = |E_{\gamma B}| = E_\gamma
\end{align*}