%%==============================================================================
\begin{lecture}
\item Ontology of quantum field theory in high-energy physics and condensed matter physics 
\item Basics of classical field theory (Hamiltonian, Lagrangian)
\item Symmetries and conservation laws
\end{lecture}
%%==============================================================================

%%==============================================================================
\begin{problemset}
\item Functional derivatives
\item Lorentz covariance
\item Maxwell equations
\end{problemset}
%%==============================================================================

% %%==============================================================================
\reference{PS:xi--xvi}
% %%==============================================================================

\imp{%
Before we start our journey, here a few general remarks:
%
\begin{itemize}
    %
    \item Quantum field theory (QFT) is concerned with the quantization of
    \emph{fields} that live on smooth manifolds 
    (e.g. Euclidean space-time, Minkowski space-time).
    %
    \item The most prominent example (which you probably already encountered in
    one form or the other) is the \emph{quantum theory of the electromagnetic field},
    which also initiated the field at the beginning of the 20th century.
    %
    \note{
    \item As Maxwell theory has special relativity ``built in,'' its QFT
    must be \emph{relativistic} as well (i.e., the Lorentz group must be a
    global symmetry). Relativistic QFTs are standard in high-energy physics
    and will be the focus of this course.
    %
    \item QFTs are riddled with \emph{infinities} in their expressions,
    which makes it hard to define them rigorously as mathematical objects
    (this is still true for most of them).
    %
    \item In the mid of the 20th century, the technique of
    \emph{renormalization} was developed to systematically deal with these
    infinities and extract physical predictions. This was a crucial step
    to make quantum electrodynamics (and the standard model afterwards) a
    useful and accepted QFT.
    %
    \item At the same time, the development of \emph{Feynman diagrams}
    as a systematic approach to perturbation theory paved the way to
    successful applications of QFTs, in particular quantum electrodynamics.
    %
    \item In the second half of the 20th century, the toolbox of QFT was
    imported from high-energy physics into \emph{condensed matter physics}
    for effective, large scale \& low energy descriptions of many-body systems
    (such as magnets and superconductors); in particular, it proved useful
    for the description of \emph{phase transitions} (\uref \emph{Conformal
    Field Theories}). These QFTs are typically not relativistic as the
    Lorentz group is not a symmetry of condensed matter systems.
    %
    \item In the context of condensed matter physics, the method of
    renormalization is less opaque and has a physical interpretation.
    These insights led to a better understanding of renormalization in
    high-energy physics as well.
    }
    % 
    \item While the methods of QFT in high-energy physics and condensed
    matter physics are very similar, their ontology is very different:
    %
    \begin{center}
        \includegraphics[width=1.0\linewidth]{ontology}
    \end{center}
    %
    \begin{itemize}
        \item In high-energy physics, fields are \emph{elementary} and particles are \emph{emergent}
        \item In condensed matter physics, particles are \emph{elementary} and fields are \emph{emergent} 
    \end{itemize}
    %
    \therefore In this course, fields are the \emph{fundamental entities
    of the world}; particles are emergent, effective models for localized
    excitations of these fields.
\end{itemize}
}

%%==============================================================================
\reference{PS:15--19}
%%==============================================================================

\chapternpb{Elements of Classical Field Theory}

\section{Lagrangian and Hamiltonian Formalism}

\subsubsection{Recap: Classical mechanics of ``points''}

\imp{With ``points'' we mean a discrete set of degrees of freedom.}
%
\begin{lot}
    
    \item \consider Degrees of freedom $q_i$ labeled by $i=1,\dots,N$
   
    \item Lagrangian $L(\{q_i\},\{\dot q_i\},t)=T-V$\\
    \imp{We write $q$ for $\{q_i\}=\{q_1,\dots,q_N\}$.}\\
    \imp{$T$ is the kinetic, $V$ the potential energy.}
  
    \item Action $S[q]=\intdt\,L(q(t),\dot q(t),t)\,\in\mathbb{R}$\\
    \imp{This is a \emph{functional} of trajectories $q=q(t)$.}
 
    \item \Emph{Hamilton's principle} of least action:
    %
    \begin{align}
        \frac{\delta S[q]}{\delta q}\stackrel{!}{=}0
        \quad\Leftrightarrow\quad
        \delta S=\intdt \delta L\stackrel{!}{=}0
    \end{align}
    %
    \imp{$\delta$ denotes functional derivatives/variations (\seepset{1}).}

    \item \Emph{Euler-Lagrange equations} ($i=1,\dots,N$):
    %
    \begin{align}
        \pdiff{L}{q_i}-\ddt\pdiff{L}{\dot q_i}=0
    \end{align}

\end{lot}

\subsubsection{Analogous: Lagrangian Field Theory}

\imp{Now we consider a \emph{continuous} set of degrees of freedom:}
%
\begin{lot}
        
    \item \consider One or more fields $\phi(x)$ on spacetime $x\in\reals^{1,3}/\reals^4$ with derivatives $\pmu\phi(x)$\\
        \imp{where $\partial_0=\partial_t$ and $\partial_{i=1,2,3}=\partial_{x,y,z}$}\\
        \imp{($\reals^{1,3}$: Minkowski space, $\reals^4$: Euclidean space; in the following, we focus on $\reals^{1,3}$)}
       
    \item Lagrangian \Emph{density} $\L(\phi,\partial\phi,x)$\\
        \imp{Most general form: $\L(\{\phi_k\},\{\partial_\mu\phi_k\},\{x^\mu\})$. No explicit $x^\mu$-dependence in the following!}
        \leadsto Lagrangian $L=\int\dd{3}{x}\,\L(\phi,\partial\phi)$\\
        \imp{(We omit the ``density'' in the following.)}
      
    \item Action:
        %
        \begin{align}
            S[\phi]=\intdt L
            =\int\d{t}\dd{3}{x}\,\L(\phi,\partial\phi)
            =\int\dd{4}{x}\,\L(\phi,\partial\phi)
        \end{align}
        \imp{$S[\phi]$ is a functional of ``field trajectories'' in $\reals^{1,3}$.}
     
    \item Action principle:
        %
        \begin{subalign}
            0\stackrel{!}{=}\delta S[\phi]
            &=\int\dd{4}{x}\delta\L
            \\
            &=\int\dd{4}{x}\left\{\pdiff{\L}{\phi}\delta\phi+\pdiff{\L}{(\pmu\phi)}\delta(\pmu\phi)\right\}
            \\
            &\mimp{Add zero and use $\delta(\pmu\phi)=\pmu(\delta\phi)$}
            \\
            &=\int\dd{4}{x}\left\{\pdiff{\L}{\phi}\delta\phi-
            \pmu\left(\pdiff{\L}{(\pmu\phi)}\right)\delta\phi
            +\pmu\left(\pdiff{\L}{(\pmu\phi)}\delta\phi\right)\right\}
            \\
            &\mimp{Gauss theorem}
            \\
            &=\int_\partial\d{\sigma_\mu}\pdiff{\L}{(\pmu\phi)}\underbrace{\delta\phi}_{=0}
            +\int\dd{4}{x}
            \underbrace{\left\{\pdiff{\L}{\phi}-\pmu\left(\pdiff{\L}{(\pmu\phi)}\right)\right\}}_{=0}\delta\phi
        \end{subalign}
        %
        \imp{Note that $\phi$ is fixed on the boundary $\partial$ and therefore $\delta\phi=0$.}\\
        \imp{The second term vanishes because the integral must vanish for arbitrary variations $\delta \phi$.}
    
    \item Euler-Lagrange equations (one for each field $\phi$):
        %
        \begin{empheq}[box=\highlight]{align}
            \pdiff{\L}{\phi}-\pmu\left(\pdiff{\L}{(\pmu\phi)}\right)=0
        \end{empheq}
        %
        \imp{Note the \emph{Einstein summation} over repeated indices.\\This
        expression is manifestly Lorentz invariant if $\L$ is a Lorentz scalar.}

\end{lot}


\subsubsection{Recap: Hamiltonian Mechanics}
%
\begin{align}
    \begin{array}{r}
        \text{Lagrangian}\\ 
        L(q,\dot q,t)
    \end{array}
    \xrightarrow[%
        \begin{array}{c}
        \text{\small Conjugate momentum}\\
        p\equiv\pdiff{L}{\dot q}\,\Leftrightarrow\,\dot q=\dot q(p)
        \end{array}]{\text{Legendre transformation}}
    \begin{array}{l}
        \text{Hamiltonian}\\ 
        H(q,p,t)=p\dot q-L(q,\dot q,t)
    \end{array}
\end{align}

\subsubsection{Analogous: Hamiltonian Field Theory}

\begin{lot}

    \item \consider $\vec x=\vec x_i\hateq i$ discrete \Emph{spatial} coordinates:\\
        %
        \imp{We omit the time dependence of the fields to simplify the notation.}\\
        %
        \begin{subalign}
            \pdiff{L}{\dot q_i}=p_i\hateq p(\vec x)=\pdiff{L}{\dot\phi(\vec x)}
            &=\frac{\partial}{\partial\dot\phi(\vec x)}\int\dd{3}{y}\L(\phi(\vec y),\dot\phi(\vec y))
            \\
            &\sim\sum_{\vec y}\dd{3}{y}\underbrace{\frac{\partial}{%
            \partial\dot\phi(\vec x)}\L(\phi(\vec y),\dot\phi(\vec y))}_{%
                \delta_{\vec x,\vec y}\left.\pdiff{\L}{\dot\phi}\right|_{\vec y=\vec x}}
            =\underbrace{\pdiff{\L}{\dot\phi(\vec x)}}_{\equiv\pi(\vec x)}\dd{3}{x}
        \end{subalign}
        %
        \imp{Spatial derivatives of the fields are represented by finite
        difference quotients and covered by the dependence on
        the (undifferentiated) fields.}\\[10pt]
        %
        \leadsto Momentum \Emph{density} conjugate to $\phi$ is $\pi=\pdiff{\L}{\dot\phi}$

    \item Hamiltonian:
        %
        \begin{align}
            H=\sum_{\vec x}\,\overbrace{p(\vec x)}^{\pi(\vec x)\dd{3}{x}}\dot\phi(\vec x)
            -\overbrace{L}^{\sum_{\vec x}\L(\phi(\vec x),\dot\phi(\vec x))\dd{3}{x}}
        \end{align}
        %
        Therefore
        %
        \begin{empheq}[box=\highlight]{align}
            H=\int\dd{3}{x}\underbrace{\left\{\pi(x)\dot\phi(x)-\L(\phi,\dot\phi)\right\}}_{%
                \text{\Emph{Hamiltonian density}}\,\H(\phi,\pi)}
        \end{empheq}
        %
        \imp{Note that $\dot\phi=\dot\phi(\pi)$. Here we restored the time
        dependence of the fields: $\vec x\mapsto x$.}

\end{lot}

\begin{example}[Free scalar field]

    \begin{lot}

        \item Real field $\phi\,:\,\reals^3\times\reals\rightarrow\reals$ with $(\vec x,t)\mapsto\phi(\vec x,t)=\phi(x)$

        \item Lagrangian (density): $\L=\hlf(\pt\phi)^2-\hlf(\nabla\phi)^2-\hlf m^2\phi^2=\hlf(\pmu\phi)^2-\hlf m^2\phi^2$\\
            \imp{It is $(\pmu\phi)^2\equiv\pmu\phi\Pmu\phi=(\pt\phi)^2-(\px\phi)^2-(\py\phi)^2-(\pz\phi)^2$
            with signature $g_{\mu\nu}=\diag{1,-1,-1,-1}$. Note that then $\pmu\Pmu=\pt^2-\nabla^2$.}

        \item Interpretation:
            %
            \begin{center}
                \includegraphics[width=0.95\linewidth]{tikz/free_boson}
            \end{center}
            %
            \imp{In $\L$, $m$ is refered to as \emph{mass}.
            This is not the inertial mass of the pendula but the stiffness of
            the harmonic potential!}\\
            %
            \note{Continuum of spring-coupled pendula for $m=0$ $\Leftrightarrow$ 1D rubber band}

        \item Equation of motion (``field equation''):
            %
            \begin{align}
                -m^2\phi-\pmu(\Pmu\phi)=0
                \quad\Leftrightarrow\quad
                (\pmu\Pmu+m^2)\phi=0
            \end{align}
            %
            \imp{This is the classical (!) \termdef{Klein-Gordon equation}.}
            %
        \item Conjugate momentum field: $\pi=\pdiff{\L}{\dot\phi}=\dot\phi$

        \item Hamiltonian (density):
            %
            \begin{subalign}
                \H&=\pi\dot\phi-\hlf\dot\phi^2+\hlf(\nabla\phi)^2+\hlf m^2\phi^2\\
                &=\hlf\pi^2+\hlf(\nabla\phi)^2+\hlf m^2\phi^2
            \end{subalign}
            %
            \imp{The Hamiltonian is $H=\int\dd{3}{x}\H(\phi,\pi)$.}

    \end{lot}
\end{example}


\section{Symmetries and Conservation Laws}

\aside{What follows is based on Sénéchal \textit{``Conformal Field Theory''}
(pp. 36--42,45--46) \cite{Francesco2012}.}

\begin{lot}

    \item \consider General transformation of field $\phi\mapsto\phi'$:
    %
    \begin{align}
        x\mapsto x'=x'(x) \quad\text{and}\quad \phi(x)\mapsto \phi'(x')=\F(\phi(x))
    \end{align}
    %
    \Emph{Two} effects: coordinates \Emph{and} (values of the) field transformed\\
    %
    \imp{These are \emph{active transformations} that change
    physics. $x'=x'(x)$ is \emph{not} a (passive) coordinate
    transformation; the frame of reference remains fixed in the following!}

\begin{example}[Rotation of a vector field $\vec\phi$]
    \begin{lot}

    \item \consider 3-component field $\vec\phi=(\phi_1,\phi_2,\phi_3)$ and $R\in\mathrm{SO}(3)$ rotation:
    %
    \begin{center}
        \includegraphics[width=0.5\linewidth]{tikz/symmetry_transformation}
    \end{center}

    \item $\vec x'=R \vec x$ and $\vec\phi'(x')=R\vec\phi(x)=R\vec\phi(R^{-1}x')$\\
    \imp{This defines a \termdef{vector field}.}

    \end{lot}
\end{example}

\item Change of the action under $\phi\mapsto\phi'$:
    %
    \begin{subalign}
        S'\equiv S[\phi']
        &=\int\dd{d}{x}\L(\phi'(x),\pmu\phi'(x))\\
        &\mimp{Rename integration variables $x\to x'$}\\
        &=\int\dd{d}{x'}\L(\phi'(x'),\pmu'\phi'(x'))\\
        &\mimp{Definition}\\
        &=\int\dd{d}{x'}\L(\F(\phi(x)),\pmu'\F(\phi(x)))\\
        &\mimp{Substitution}\\
        &=\int\dd{d}{x}\left|\pdiff{x'}{x}\right|\L\left(\F(\phi(x)),\pdiff{x^\nu}{x'^\mu}\,\partial_\nu\F(\phi(x))\right)
        \label{eq:symtrafo}
    \end{subalign}
    \aside{Skip first step, use colors for primes.}

\end{lot}

\begin{example}[Translations]
    \begin{lot}

        \item $x':=x+a$ and $\phi'(x'):=\phi(x)=\phi(x'-a)$\\
        \imp{This defines a \termdef{scalar field}.}

        \item $\F=\Id$ trivial, $\phi'(x')=\F(\phi(x))=\phi(x(x'))$, 
        and $\pdiff{x^\nu}{x'^\mu}=\delta_\mu^\nu$

        \item Action:
        %
        \begin{align}
            S[\phi']
            =\int\dd{d}{x}\L(\phi'(x),\pmu\phi'(x))
            =\int\dd{d}{x}\L(\phi(x),\pmu\phi(x))
            =S[\phi]
        \end{align}
        %
        \imp{The action is \emph{translation invariant}: $S=S'$!}\\
        %
        \imp{This follows generally from the missing $x$-dependence of $\L$ for scalar fields.}

    \end{lot}
\end{example}

\begin{example}[Scale transformations]
    \begin{lot}

        \item $x':=\lambda x$ and $\phi'(x'):=\lambda^{-\Delta}\phi(x)=\lambda^{-\Delta}\phi(\lambda^{-1}x')$\\
        \imp{$\Delta$ is the \termdef{scaling dimension} of the field $\phi$}

        \item $\F(\phi)=\lambda^{-\Delta}\phi$ and $\pdiff{x^\nu}{x'^\mu}=\lambda^{-1}\delta_\mu^\nu$ and
        $\left|\pdiff{x'}{x}\right|=\lambda^d$

        \item Action:
        %
        \begin{subalign}
            S[\phi']
            &=\lambda^d\int\dd{d}{x}\L(\lambda^{-\Delta}\phi(x),\lambda^{-1-\Delta}\pmu\phi(x))
            \\
            &\mimp{\consider Massless scalar field: $S[\phi]=\textstyle\hlf\int\dd{d}{x}(\pmu\phi)^2$}
            \\
            &=\lambda^{d-2-2\Delta}\int\dd{d}{x}\L(\phi(x),\pmu\phi(x))
            =\lambda^{d-2-2\Delta}S[\phi]
        \end{subalign}
        %
        \leadsto $S'=S$ iff $\Delta=\frac{d}{2}-1$\\
        %
        \imp{This is an example of a \emph{\uref Conformal Field Theory (CFT)}.}

    \end{lot}
\end{example}

\begin{example}[Phase rotation]
    \begin{lot}

        \item $x':=x$ and $\phi'(x'):=e^{i\theta}\phi(x)$\\
        %
        \imp{\therefore There are symmetries that only transform the fields but not the coordinates.}

        \item $\F(\phi)=e^{i\theta}\phi$ and $\pdiff{x^\nu}{x'^\mu}=\delta_\mu^\nu$ and
        $\left|\pdiff{x'}{x}\right|=1$

    \end{lot}
\end{example}


%%==============================================================================
\begin{lecture}
\item Infinitesimal transformations and continuous symmetries
\item Noether's theorem and conserved quantities
\item Application to the energy-momentum tensor 
\end{lecture}
%%==============================================================================

%%==============================================================================
\reference{PS:15--19}
%%==============================================================================

\subsubsection{Infinitesimal Transformations}

\imp{We are interested in \emph{continuous} symmetries (\dref Lie groups).}

\begin{lot}

    \item \consider \Emph{Infinitesimal} transformations (IT):
        %
        \begin{align}
            x'^\mu=x^\mu+w_a\vdiff{x^\mu}{w_a}(x)
            \quad\text{and}\quad
            \phi'(x')=\phi(x)+w_a\vdiff{\F}{w_a}(x)
            \label{eq:inftrafo}
        \end{align}
        %
        \imp{Here, $w_a$ denotes infinitesimal parameters of the transformation (sum over $a$ implied!).}\\
        %
        \imp{They may vary from point to point: $w_a=w_a(x)$ (see below).}

    \item \Emph{Generator} of IT:
        %
        \begin{align}
            &\delta_w\phi(x):=\phi'(x)-\phi(x)\equiv -iw_a\,G_a\phi(x)    
        \end{align}
        %
        With \aside{(omit first line and refer to previous equation)}
        %
        \begin{subalign}
            \phi'(x')
            &=\phi(x)+w_a\vdiff{\F}{w_a}(x)\\
            &=\phi(x')-w_a\vdiff{x^\mu}{w_a}\pmu\phi(x')+w_a\vdiff{\F}{w_a}(x')+\O(w^2)
        \end{subalign}
        %
        it follows \imp{(replace $x'$ by $x$; these are just labels!)}
        %
        \begin{empheq}[box=\highlight]{align}
            iG_a\phi=\vdiff{x^\mu}{w_a}\pmu\phi-\vdiff{\F}{w_a}    
        \end{empheq}
        %
        \imp{This function describes the infinitesimal change of the field at the same point.}

\end{lot}

\begin{example}[Translations]
    \begin{lot}

        \item $x'^\mu:=x^\mu+w^\mu\equiv x^\mu+w^\nu\vdiff{x^\mu}{w^\nu}$ 
            with $\vdiff{x^\mu}{w^\nu}=\delta_\nu^\mu$

        \item $\vdiff{\F}{w^\nu}=0$ \imp{(For a scalar or a vector field.)}

        \item $iG_\mu\phi=\delta_\mu^\nu\partial_\nu\phi-0$ and therefore
            %
            \begin{align}
                G_\mu=-i\partial_\mu\equiv P_\mu    
            \end{align}
            %
            \imp{\therefore The ``momentum operator'' generates translations.}

    \end{lot}
\end{example}

\begin{example}[Scale Transformations]

    $G=-ix^\mu\pmu\equiv D$ \imp{\therefore Generates ``dilations'' in spacetime.}\\
    %
    \imp{This simple form is valid for a scalar field with scaling dimension
    $\Delta=0$ so that $\vdiff{\F}{\lambda}=0$.}

\end{example}

\begin{example}[Spatial Rotations]

        $G_{\mu\nu}=i(x_\mu\partial_\nu-x_\nu\partial_\mu)+S_{\mu\nu}$ for $\mu,\nu=1,2,3$\\
        %
        \imp{The first term generates coordinate rotations (\dref \emph{orbital angular momentum operator}).}\\
        %
        \imp{$S_{\mu\nu}$ are spin matrices that generate field transformations (for non-scalar fields).}\\
        %
        \aside{Question: What generates $G_{\mu\nu}$ if either $\mu=0$ or $\nu=0$? Answer: Boosts.}

\end{example}

\subsubsection{Noether's Theorem}

\begin{lot}

    \item \consider Transformation \cref{eq:inftrafo} which is a
        %
        \begin{empheq}[box=\highlight]{align}
            \text{Symmetry of the action}
            \quad
            :\Leftrightarrow
            \quad
            S[\phi]=S[\phi']
        \end{empheq}
        %
        for $w_a$ independent of $x$ (\termdef{rigid transformation}).

    \item \Emph{Assume} that \cref{eq:inftrafo} is \Emph{not} rigid: $w_a=w_a(x)$\\
        %
        \imp{We assume that $w_a$ is sufficiently smooth so that $\pmu w_a$
        is infinitesimal as well, i.e., $\O(w_a)=\O(\pmu w_a)$.}

    \item Jacobian: $\pdiff{x'^\nu}{x^\mu}=\delta_\mu^\nu+\pmu\left(w_a\vdiff{x^\nu}{w_a}\right)$
        \therefore $\left|\pdiff{x'}{x}\right|=1+\pmu\left(w_a\vdiff{x^\mu}{w_a}\right)$\\
        %
        \imp{Use $\det(\Id+A)=1+\tr A+\O(A^2)$.}

    \item Inverse Jacobian matrix: $\pdiff{x^\nu}{x'^\mu}=\delta_\mu^\nu-\pmu\left(w_a\vdiff{x^\nu}{w_a}\right)$\\
        %
        \imp{This is true in linear order of $w_a$ and $\pmu w_a$.}

    \item Use \cref{eq:symtrafo}:
        %
        \begin{align}
            S'=
            \begin{aligned}[t]
                &\int\dd{d}{x}
                \left[1+\pmu\left(w_a\vdiff{x^\mu}{w_a}\right)\right]
                \\
                \times\,&\L\left(
                \phi+w_a\vdiff{\F}{w_a}
                ,
                \left[\delta_\mu^\nu-\pmu\left(w_a\vdiff{x^\nu}{w_a}\right)\right]
                \times
                \left[\partial_\nu\phi+\partial_\nu\left(w_a\vdiff{\F}{w_a}\right)\right]
                \right)
            \end{aligned}
        \end{align}

    \item Expand in 1st order of $w_a$ and $\pdiff{w_a}{x^\mu}$

    \item \consider $\delta S\equiv S'-S$ \therefore Only terms $\propto\pdiff{w_a}{x^\mu}$ remain\\
        \imp{Because the transformation is a symmetry of the action by assumption, 
        i.e., for $w_a=\const$ (a rigid transformation) it is $S'=S$!}\\
        \note{This is equivalent to the \emph{definition} of a symmetry (of the action).}

    \item For generic, non-rigid symmetry transformation we find
        %
        \begin{align}
            \delta S=-\int\dd{d}{x} j_a^\mu\pmu w_a
        \end{align}
        %
        with the \Emph{current}
        %
        \begin{empheq}[box=\highlight]{align}
            j_a^\mu\nte\left\{\pdiff{\L}{(\pmu\phi)}\partial_\nu\phi-\delta_\nu^\mu\L\right\}\vdiff{x^\nu}{w_a}
            -\pdiff{\L}{(\pmu\phi)}\vdiff{\F}{w_a}
            \label{eq:noethercurrent}
        \end{empheq}
        %
        associated to the IT $\vdiff{x^\nu}{w_a}$ and $\vdiff{\F}{w_a}$.\\
        %
        \imp{This is only true for transformations that are symmetries of the action!}

        \item Integration by parts \therefore $\delta S=\int\dd{d}{x} w_a\,\pmu j_a^\mu$\\
        %
        \imp{Here we assume that the variations $w_a(x)$ vanish on the boundaries (possibly at infinity).}

        \item \consider $\phi$ that \Emph{obeys the equations of motion} \therefore $\delta S=0$ 
            for \Emph{arbitrary} variations $\phi'=\phi+\delta\phi$\\
            %
            In particular, for arbitrary non-rigid transformations $w_a(x)$!
            \\[10pt]
            %
            It follows \Emph{Noether's (first) theorem:}
            %
            \begin{empheq}[box=\highlight]{align}
                \pmu j_a^\mu=0\qquad\forall_{x,a}
            \end{empheq}
            %
            \imp{This is a \emph{conservation law} with \emph{conserved current} $j_a^\mu$.}

        \item \Emph{Conserved charge:}
            %
            \begin{empheq}[box=\highlight]{align}
                Q_a := \int_\text{Space}\dd{d-1}{x} j_a^0
            \end{empheq}
            %
            Indeed:
            %
            \begin{align}
                \diff{Q_a}{t}= \int_\text{Space}\dd{d-1}{x} \partial_0 j_a^0
                \stackrel{\text{Noether}}{=}
                -\int_\text{Space}\dd{d-1}{x} \partial_k j_a^k
                \stackrel{\text{Gauss}}{=}
                -\int_\text{Surface}\d{\sigma_k} j_a^k=0
            \end{align}
            %
            \imp{Here we assume that $j_a^k\equiv 0$ on the
            boundaries---typically at spatial infinity, i.e., the universe
            is closed.} \imp{$k=1,2,3$ denotes the spatial coordinates.}

\end{lot}

\begin{blocknote}

    The current \cref{eq:noethercurrent} is called \Emph{canonical current}
    as there is an ambiguity:
    %
    \begin{align}
    \tilde j_a^\mu:=j_a^\mu+\partial_\nu B_a^{\mu\nu}
    \quad\text{with}\quad B_a^{\mu\nu}=-B_a^{\nu\mu}\quad\text{arbitrary}
    \quad\Rightarrow\quad
    \pmu\tilde j_a^\mu=0
    \end{align}

\end{blocknote}

\begin{blocknote}
    \vspace{-15pt} 
    \begin{align}
        \underbrace{%
            \text{Symmetric Lagrangian}\quad\Rightarrow\quad
            \text{Symmetric action}
        }_{\rightarrow\,\text{Conserved currents}}
        \quad\Rightarrow\quad\text{Symmetric EOMs}
    \end{align}
    %
    \imp{Continuous symmetries of the EOMs do \emph{not} imply conserved currents!}\\[5pt]
    %
    \note{\emph{Example:} 
    %
    $\L=\hlf(\partial\phi)^2-\hlf m^2\phi^2$ yields the EOM $(\partial^2+m^2)\phi=0$
    which clearly is symmetric under rescaling of the field: $\phi'(x)=\lambda\phi(x)$.
    However, the Lagrangian density $\L$ is \emph{not} invariant under this
    transformation (neither does it change by a total derivative), so that
    Noether's theorem does not apply!}

\end{blocknote}

\subsubsection{Application: The Energy-Momentum-Tensor (EMT)}

\note{\emph{Special relativity:}\\
Global spacetime symmetries (Lorentz transformations + \Emph{Translations} = Poincaré group)}\\[5pt]
%
\note{\emph{General relativity:}\\
Local spacetime symmetries (\uref Diffeomorphisms \therefore Gauge symmetries)}

\begin{lot}

    \item \consider Infinitesimal \Emph{spacetime translations}: $x'^\mu=x^\mu+\vep^\mu$
        \therefore $\vdiff{x^\mu}{\vep^\nu}=\delta^\mu_\nu$, $\vdiff{\F}{\vep^\nu}=0$

    \item \consider Translation-invariant action: $S'=S$\\
        %
        \imp{This includes translations in time!}

    \item \Emph{Conserved currents}:
        %
        \begin{align}
            \tensor{T}{^\mu_\nu}
            =\left\{\pdiff{\L}{(\pmu\phi)}\partial_\rho\phi
            -\delta^\mu_\rho\L\right\}\underbrace{%
                \vdiff{x^\rho}{\vep^\nu}
            }_{\delta^\rho_\nu}
            =\pdiff{\L}{(\pmu\phi)}\partial_\nu\phi-\delta^\mu_\nu\L
        \end{align}
        %
        \begin{emphalign}
            T^{\mu\nu}=g^{\nu\rho}\tensor{T}{^\mu_\rho}=\pdiff{\L}{(\pmu\phi)}\partial^\nu\phi-g^{\mu\nu}\L
            \qquad\text{(Energy-Momentum Tensor)}
            \label{eq:emt}
        \end{emphalign}
        %
        with $\pmu T^{\mu\nu}=0$ and four conserved charges
        %
        \begin{emphalign}
            P^\nu=\int\dd{3}{x}T^{0\nu} 
            \label{eq:Pnu}
        \end{emphalign}
        %
        \imp{Note that these quantities are only conserved for classical
        \emph{solutions} of the EOMs.}

    \item \Emph{Energy} ($\nu=0$) \aside{(skip first step)}:
        %
        \begin{align}
            P^0=\int\dd{3}{x}T^{00}=\int\dd{3}{x}\left\{\pdiff{\L}{\dot\phi}\dot\phi-\L\right\}=\int\dd{3}{x}\H(\phi,\pi)=H
            \label{eq:P0}
        \end{align}
        %
        \imp{\therefore The Hamiltonian is the component of a 4-vector and not Lorentz invariant!}\\
        %
        \imp{By contrast, the Lagrangian \emph{is} Lorentz invariant (for relativistic field theories).}\\

    \item \Emph{Kinetic momentum} ($\nu=i$):
        %
        \begin{align}
            P^i=\int\dd{3}{x}T^{0i}=\int\dd{3}{x}\pdiff{\L}{\dot\phi}(-\partial_i\phi)=-\int\dd{3}{x}\pi\partial_i\phi
        \end{align}
        %
        \imp{$\pi$ is the \emph{canonical} momentum.}

\end{lot}

\begin{blocknote}

    In general $T^{\mu\nu}\neq T^{\nu\mu}$ for the canonical EMT. But:
    %
    \begin{align}
        \tilde T^{\mu\nu}:=T^{\mu\nu}+\partial_\rho K^{\rho\mu\nu}
        \quad\text{with}\quad 
        K^{\rho\mu\nu}=-K^{\mu\rho\nu}
    \end{align}
    %
    Choose $K^{\rho\mu\nu}$ such that $\tilde T^{\mu\nu}=\tilde T^{\nu\mu}$
    (\uref \termdef{Belinfante(-Rosenfeld) EMT})\\[5pt]
    %
    \imp{Using the EMT as source of the gravitational field in \emph{general
    relativity} requires a \emph{symmetric} EMT because the Einstein field
    equations read $R_{\mu\nu}-\hlf g_{\mu\nu}R=8\pi G/c^4\,T_{\mu\nu}$
    with the (symmetric) Ricci tensor $R_{\mu\nu}$ and the (symmetric)
    metric $g_{\mu\nu}$.}

\end{blocknote}

\begin{example}[Electromagnetism (EM) in vacuum]
    
    \imp{Details \seepset{1}}

    \begin{lot}

        \item Four-component gauge field: $A^\mu=(\phi,A^1,A^2,A^3)$

        \item EM field tensor: $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$\\
            \imp{Contains $E$- and $B$-field components.}

        \item Lagrangian: $\L_\mathrm{em}(A,\partial A)=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$

        \item Action: $S_\mathrm{em}=\int\dd{4}{x}\L_\mathrm{em}$

        \item Euler-Lagrange equations: $\pmu F^{\mu\nu}=0$ (inhomogeneous Maxwell equations)

        \item $S_\mathrm{em}$ is \Emph{Lorentz invariant} and \Emph{translation invariant} (= Poincaré invariant)\\
            %
            \therefore EMT = conserved currents\\
            %
            \aside{Why is this obvious?}

        \item Canonical EMT: $T_\mathrm{em}^{\mu\nu}=\pdiff{\L_\mathrm{em}}{(\pmu A_\lambda)}\partial^\nu A_\lambda-g^{\mu\nu}\L_\mathrm{em}$

        \item Symmetric EMT using $K^{\lambda\mu\nu}:=F^{\mu\lambda}A^\nu$:
            %
            \begin{emphalign}
                \tilde T_\mathrm{em}^{\mu\nu}
                \nte\frac{1}{4}g^{\mu\nu}F_{\rho\lambda}F^{\rho\lambda}-F^{\mu\rho}\tensor{F}{^\nu_\rho}
            \end{emphalign}
            %
            \begin{itemize}
                \item $\tilde T^{00}=\frac{1}{2}(E^2+B^2)$ \imp{(\dref Energy density)}
                \item $\tilde T^{0i}=(\vec E\times \vec B)_i$ \imp{(\dref Pointing vector)}
                \item $\tilde T^{ij}=\sigma_{ij}$ \imp{(\uref Maxwell stress tensor)}
            \end{itemize}

    \end{lot}

\end{example}