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Haifeng Yang deleted file Conserved P and Noether.tex
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\subsection{Noether Theorem and Conservation}
Starting point: $a_\mathbf{k}^\dagger$ should also increase the momentum of the system by $\hbar \mathrm{k}$.
\subsubsection{Transformation and Symmetry}
\begin{enumerate}
\item Lorentz transform:
$$x^\mu \rightarrow \Lambda^\mu_{\,\,\nu}x^\nu$$
It can be a boost or a rotation or a combination of both. Inverse: $\Lambda^{-1}$. ($\beta \to -\beta$ for example)
\item All Lorentz Transformations satisfy:
$$\eta \Lambda \eta = (\Lambda^{-1})^T \qquad \mathrm{or} \qquad \eta\Lambda = (\Lambda^{-1})^T \eta$$
\item Invariant thing: $X^T \eta Y$ (Ex: Prove this)
\item Notation of covariance and contravariance. $\eta^\mu_{\,\,\nu} \equiv \delta^\mu_\nu$.
$$\eta x \to (\Lambda^{-1})^T\eta x$$
\end{enumerate}
