deletions | additions       diff --git a/Quantize K-G Field.tex b/Quantize K-G Field.tex  index f95d1fc..a314ccb 100644  --- a/Quantize K-G Field.tex  +++ b/Quantize K-G Field.tex 
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\subsection{Quantize Klein-Gordon}  Fourier Transformation: \subsubsection{Fourier Transformation}  $$\hat{\phi}(\mathbf{x}) = \int \frac{d^3 k}{(2\pi)^3 } \hat{\tilde{\phi}}(\mathbf{k})e^{i\mathbf{k}\cdot \mathbf{x}}$$  $$\hat{\pi}(\mathbf{y}) = \int \frac{d^3 k}{(2\pi)^3 } \hat{\tilde{\pi}}(\mathbf{k})e^{i\mathbf{k}\cdot \mathbf{y}}$$ 
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+\frac{1}{2}\omega_\mathbf{k}^2 \hat{\tilde{\phi}}(\mathbf{k})^\dagger \hat{\tilde{\phi}}(\mathbf{k})  \right]$$  Note: These are complex harmonic oscillators. $\tilde{\phi}(-\mathbf{k})=\tilde{\phi}(\mathbf{k})^*$ $\tilde{\phi}(-\mathbf{k})=\tilde{\phi}(\mathbf{k})^*$. All $\mathbf{k}$'s are not independent.     \subsubsection{Commutation Relations}   $$[\hat{\tiled{\pi}}(\mathbf{k'}), \hat{\tilde{\phi}}(\mathbf{k})] = - i (2\pi)^3 \delta^{(3)} (\mathbf{k} + \mathbf{k'})$$