\chapter{Discretization of Quantum Field Theories}
\label{cha:QFT}

Quantum Field Theory (QFT) is the standard framework for describing modern particle physics. Compared to ordinary quantum mechanics, where operators and states depend only on time, in QFT they depend on all spacetime coordinates. In this sense, quantum mechanics can be interpreted as a $(0+1)$ dimensional QFT. Unlike classical field theory, which deals with real- or complex-valued fields, QFT involves operator-valued fields. Most QFTs require Lorentz invariance, which is especially necessary when describing particles whose momenta are large compared to their masses~\cite{Klümper2022lecture,Hölbling2019lecture,Risch2024lecture}.

\section{Classical Scalar and Vector Fields}
\label{sec:classical_fields}

Classical fields $\varphi(\vec x,t)$ are typically real or complex-valued functions defined over spacetime, with wide-ranging applications in cosmology~\cite{Hentschke:2020bla}, classical electrodynamics~\cite{Hentschke:2017}, and continuum mechanics~\cite{Landau:345895}. The most fundamental classical field theories are often used in their quantized form in modern particle physics~\cite{Schwartz_2013,Peskin,Weinberg}.

\subsection{Action and Lagrangian}

The Hamiltonian principle states that a classical field $\varphi^j$ extremizes the action $S[\varphi^j(x), \partial_\mu \varphi^j(x)]$. The index $j$ denotes different field components, such as spin, flavor, or vector direction, and $x$ represents the spacetime coordinates. The action is a functional given by the integral of the Lagrangian density $\mathcal{L}(\varphi^j, \partial_\mu \varphi^j)$ over all spacetime. A necessary condition for minimizing the action is:

\begin{align}
    \frac{\delta S[\varphi^j, \partial_\mu \varphi^j]}{\delta \varphi^i(x)} = 0,
\end{align}

where $\delta/\delta \varphi(x)$ denotes the functional derivative. Assuming that $\varphi$ vanishes at the boundary of $\mathbb{R}^{3+1}$, one can derive the Euler-Lagrange equations:

\begin{align}
    \partial_\mu \frac{\partial \mathcal{L}}{\partial(\partial_\mu \varphi^j)} - \frac{\partial \mathcal{L}}{\partial \varphi^j} = 0.
\end{align}

\subsubsection*{Noether's Theorem}
Noether's theorem describes the relationship between a symmetry transformation and a conservation law~\cite{Peskin,Risch2024lecture}. The infinitesimal version of a continuous transformation is given by
\begin{align}
    \varphi \to \varphi' = \varphi + \varepsilon \delta \varphi,
\end{align}
where $\epsilon$ is infinitesimal and $\delta \varphi$ is some deformation of the field. This transformation leaves a theory invariant whenever the corresponding Lagrange density only changes by the four gradient of some current $J^\mu$. The corresponding conservation law states that the vector current is conserved:
\begin{align}
     \partial_\mu j^\mu = 0 \quad \textrm{with} \quad  j^\mu = \frac{\partial \mathcal{L}}{\partial(\partial_\mu \varphi)}\delta\varphi - J^\mu.
\end{align}

\subsection{The Klein-Gordon theory}
The simplest scalar field theory that is consistent with special relativity is the Klein-Gordon theory~\cite{Peskin,Risch2024lecture,Hölbling2019lecture}. It is motivated by the relativistic energy-momentum-mass relation:
\begin{align}
    E^2 = \vec p^2 + m^2
\end{align}
Using the well-known quantum mechanics substitutions
\begin{align}
    E \to i \partial_t,\\
    \vec p \to -i \vec \nabla,
\end{align}
and the application on a real- or complex-valued wave function $\phi(t,\vec x)$ we get the Lorentz-covariant wave-equation 
\begin{align}
    \left(\partial_\mu\partial^\mu + m^2\right)\phi(x^\mu) = 0.
\end{align}
Elementary solutions of this equation are given by
\begin{align}
    \phi_{\vec k}^{\pm}(x^\mu)  = A_{\vec k}^\pm e^{\pm ik_\mu x^\mu}\bigg\vert_{k_0 = \sqrt{\vec k^2 + m^2}},
\end{align}
and using Fourier transformation the most-general solution can be represented in the following way:
\begin{align}
    \phi(x^\mu) = \sum_{\sigma \in \{+,-\}} \int \mathrm{d}^4k\,  A_{\vec k}^\sigma e^{\mathrm{i}\sigma k_\mu x^\mu} \delta\left(k_0 +\sigma \sqrt{m^2 + \vec k^2}\right)
\end{align}
\subsection{The Dirac theory}
\label{sec:Dirac}
While the Klein-Gordon theory uses a second order differential equation to describe fields that fulfill the relativistic energy-momentum relation, the Dirac theory is a first order differential equation to describe fields that fulfill the relativistic energy-momentum relation~\cite{Klümper2022lecture,Peskin}. The Dirac equation is given by
\begin{align}
    \left(\mathrm{i} \slashed \partial - m\right) \psi = 0,
\end{align}
where we use the Feynman-slash notation
\begin{align}
    \slashed q = \gamma^\mu q_\mu
\end{align}
and $\gamma^\mu$ represents a set of matrices fulfilling the condition
\begin{align}
    \{\gamma^\mu,\gamma^\nu\} = 2\eta^{\mu\nu},
\end{align}
with the metric tensor $\eta^{\mu\nu} = {\diag\{1,-1,-1,-1\}^{\mu}}_{\nu}$. It can be shown that these $\gamma$-matrices are at least four dimensional~\cite{Risch2024lecture}. The Dirac equation is often called the root of the Klein-Gordon equation due to the identity
\begin{align}
    (-\mathrm{i}\slashed \partial -m)(\mathrm{i}\slashed \partial + m) = -(\partial_\mu \partial^\mu + m^2).
\end{align}
From this, it can be concluded that each component of every solution of the Dirac equation is a solution of the Klein-Gordon equation. It can be shown that real- or complex-valued Dirac fields $\psi$ and $\bar \psi$ can lead to contradictions like the issue that the Hamiltonian is not bound from below~\cite{Risch2024lecture}. This can be solved by taking Grassmann-valued fields instead~\cite{GattringerLang} (see Appendix~\ref{sec:Grassmann}).  
\section{Path Integral Quantization}
\label{sec:PI}
There are several ways to quantize field theories but Path Integral Quantization is particularly suited for quantizing fields on the lattice.
\subsection{Hamiltonian version of the path integral}
To perform the Path Integral Quantization we start with the canonical commutation relations that are known from quantum mechanics~\cite{Klümper2023lecture}:
\begin{align}
\label{eq:can_com_rel}
    [Q_a,P_b] = \mathrm{i}\delta_{ab},\quad  [Q_a,Q_b] = [P_a,P_b] =0
\end{align}
for a set of operators $\{Q_a\}_a$ with the corresponding conjugate momentum operators $\{P_a\}_a$. Here, the index $a$ describes the position\footnote{Since we work in the Schrödinger picture, there is no time-dependence in the operators.} and internal (spinor-, flavor-, color- etc.) degrees of freedom of the quantum field~\cite{Weinberg}. As the spatial degrees of freedom are continuous and integrated instead of summed over, the Kronecker delta in Equation~\eqref{eq:can_com_rel} has to be replaced by the Dirac distribution.

All operators $Q_a$ commute with each other so there has to be a system of common eigenstates $\ket{q}$ with the corresponding eigenvalues $q_a$. The eigenvectors can be chosen orthogonally and normalized, such that they fulfill the conditions
\begin{align}
\label{eq:orhtonormal}
    \braket{q'}{q} = \delta(q'-q),\quad  \mathbbm{1} = \sumint\,\mathrm{d}q_a\ket{q}\bra{q}
\end{align}
where $\sumint$ denotes integration over spatial coordinates and summation over discrete indices. The same construction can be done for the canonical momenta.
The scalar product of the eigenstates can be computed similar to how it is done in quantum mechanics as
\begin{align}
\label{eq:scalarprod}
    \braket{q}{p} = \prod_a \frac{\exp(\mathrm{i}p_a q_a) }{\sqrt{2\pi}}.
\end{align}
Up to this point we worked at a fixed time-slice. Next we want to describe the time-evolution of the operators in the Heisenberg picture:
\begin{align}
    Q_a(t) = \exp(\mathrm{i}Ht)Q_a\exp(-\mathrm{i}Ht)\\
    P_a(t) = \exp(\mathrm{i}Ht)P_a\exp(-\mathrm{i}Ht)
\end{align}
Here, $H$ is the Hamilton operator of the system. The new operators $Q_a(t)$ and $P_a(t)$ have the eigenvectors $\ket{q,t}$ resp. $\ket{p,t}$ with the eigenvalues $q_a$ and $p_a$. These eigenstates are related to the eigenstates of $Q_a$ and $P_a$ by
\begin{align}
    \ket{q,t} =\exp(\mathrm{i}Ht) \ket{q} \text{ and }  \ket{p,t} =\exp(\mathrm{i}Ht) \ket{p}.
\end{align}
These time-dependent states are not the time-evolution of the eigenstates $\ket{q}$ and $\ket{p}$ (in this case there would be a minus-sign in the exponent of the time-evolution operator), but the eigenvectors of the time-evolved operators. At a given time-slice $t$, these states fulfill the relations from Equation~\eqref{eq:orhtonormal} and Equation~\eqref{eq:scalarprod}. The expression $\braket{q',t'}{q,t}$ describes the probability amplitude of the process going from state $\ket{q}$ to state $\ket{q'}$ in the time from $t$ to $t'$. To compute this expression we start by assuming that the time difference of $t$ and $t'$ is infinitesimal:
\begin{align}
\begin{aligned}
      \braket{q',t+\mathrm{d}t}{q,t}  = \bra{q',t}\exp(-\mathrm{i}H\mathrm{d}t)\ket{q,t}\\
      =\int\left( \prod_a\,\mathrm{d}p_a\right) \bra{q',t}\exp(-\mathrm{i}H(Q(t),P(t))\mathrm{d}t) \ket{p,t}\braket{p,t}{q,t}\\
      =\int\left( \prod_a\,\mathrm{d}p_a\right)\bra{q',t}\exp(-\mathrm{i}H(q(t),p(t))\mathrm{d}t) \ket{p,t}\braket{p,t}{q,t}\\
      =\int\left( \prod_a\,\mathrm{d}p_a\right) \exp\left[-\mathrm{i}H(q',p)\mathrm{d}t + \mathrm{i}\sumint_a (q'_a-q_a)p_a\right].
\end{aligned}
\end{align}
In the third step it is assumed that all momentum operators are moved to the right and all $Q$-operators are moved to the left\footnote{This argument is taken from~\cite{Weinberg}, nonetheless it is not taken care of the fact that exponentiating a sorted operator can destroy the ordering. Luckily it can be derived from Trotter's formula that unordered terms are of order $\mathrm{d}t^2$~\cite{Varnhorst2019lecture}. In fact, this is the reason why infinitesimal time intervals are considered.}. The momenta in the above equation are integrated over the whole range $(-\infty,\infty)$. Now let us split up the interval $(t,t')$ into $N+1$ sub-intervals $[t,t_1],\,[t_1,t_2],\,...,\,[t_n,t']$ of length $\mathrm{d}t$. Now at each $t_k\,(k = 1,\,...,\,N)$ a complete set of states $\ket{q,t_k}$ is included.
\begin{align}
\label{eq:PI}
    \begin{aligned}
        \braket{q',t'}{q,t} = \int \left( \prod_{k = 1}^N \mathrm{d}q_k \right)\, \braket{q',t'}{q_N,t_N}\,...\,\braket{q_1,t_1}{q,t}\\
        =\int \left( \prod_{k = 1}^N \frac{\mathrm{d}q_k\,\mathrm{d}p_k}{2\pi} \right)\exp\left[\mathrm{i}\sum_{k = 1}^N \left((q_k-q_{k-1})\cdot p_{k-1} - H(q_k,p_{k-1}) \right)\mathrm{d}t \right]
    \end{aligned}
\end{align}
Equation~\eqref{eq:PI} is called the \textit{Path Integral Quantization} of the transition amplitude. Here the indices $a$ are included implicitly. That means in particular that in the scalar products spatial variables are integrated out.\\
Not only transition amplitudes can be computed using the path integral but also matrix elements of generalized operators $F(P,Q)$ at times $\tau$ in the time-interval $(t,t')$:
\begin{align}
    \begin{aligned}
\label{eq:PI_O}
         \bra{q',t'}F(Q(\tau),P(\tau))\ket{q,t}    =\int \left( \prod_{k = 1}^N \frac{\mathrm{d}q_k\,\mathrm{d}p_k}{2\pi} \right)F(q(\tau),p(\tau))\exp\left[\mathrm{i}\sum_{k = 1}^N \left((q_k-q_{k-1})\cdot p_{k-1} - H(q_k,p_{k-1}) \right)\mathrm{d}t \right],
    \end{aligned}
\end{align}
which results in the limit $N \to \infty$ in
\begin{align}
\label{eq:PI_cont}
    \int \left( \prod_{k = 1}^N \frac{\mathrm{d}q_k\,\mathrm{d}p_k}{2\pi} \right)F(q(\tau),p(\tau))\exp\left[\mathrm{i}\int_t^{t'}\mathrm{d}\Tilde{t} \left(\dot q(\Tilde{t})\cdot p_(\Tilde{t}) - H(q(\Tilde{t}),p(\Tilde{t})) \right) \right].
\end{align}
Finally we got an general expression for matrix elements in QFTs where we got rid of all operators~\cite{Weinberg}.
\subsection{Lagrangian version of the path integral}
Since most commonly field theories are defined using their Lagrangian density, one wants to replace the Hamilton function in Equation~\eqref{eq:PI_O} by the corresponding Lagrange function. The following derivation is similar to the one given in Section 9.3 of the QFT book by Weinberg~\cite{Weinberg}.\\
In general, it can be assumed that the Hamilton function is polynomial of order two in the momenta and can therefore be written as
\begin{align}
    H[Q,P] = \frac{1}{2}A^{ab}[Q]P_aP_b + B^a[Q]P_a + C[Q].
\end{align}
So the exponent from Equation~\eqref{eq:PI_cont} is transformed to
\begin{align}
  -\frac{1}{2}  \int \mathrm{d}\tau\,\mathrm{d}\tau'\,\mathcal{A}_{\tau\tau'}^{ab}[q]\,p_a(\tau)p_b(\tau') - \int\mathrm{d}\tau\, \mathcal{B}^a_\tau\,p_a[\tau] - \mathcal{C}[q],\\
  \mathcal{A}_{\tau\tau'}^{ab}[q] = A^{ab}[q(\tau)]\delta(\tau-\tau'),\\
  \mathcal{B}^a_\tau[q] = B^a[q(\tau)]- \dot q^a[\tau],\\
  \mathcal{C}[q] = \int \mathrm{d}\tau \, C[q(\tau)].
\end{align}
With the assumption that the operators whose matrix elements we want to compute do not depend on the canonical momenta (which has shown to be the case in general), the momentum integration for the path integral can be solved with common techniques to get the result 
\begin{align}
    \left( \det \frac{\mathrm{i}\mathcal{A}}{2\pi}\right)^{-1/2}\exp\left[-\frac{\mathrm{i}}{2}\mathcal{A}^{ab}\xi_a\xi_b - \mathrm{i}\mathcal{B}^a\xi_a - \mathrm{i}\mathcal{C} \right],
\end{align}
where the $\xi$ denote the stationary point of the exponent. This is given by the condition
\begin{align}
    \dot q = \frac{\delta H}{\delta p}\bigg\vert_{p = \xi},
\end{align}
that is known for being one of the Hamilton equations of motion. This can be used to interpret the integrand in the exponent as a Legendre transformation to write the matrix elements in terms of the Lagrange function
\begin{align}
\label{eq:PI_full}
             \bra{q',t'}F(Q(\tau),P(\tau))\ket{q'',t''}    =\int \left( \prod_{k = 1}^N \mathrm{d}q_k \right)\bigg\vert_{q(t'' = q''),\, q(t') = q'}\,F(q(\tau),p(\tau))\exp\left[\mathrm{i}\int\mathrm{d}t\, L[q(t),\dot q(t)] \right].
\end{align}
The integral over the Lagrange function is called the action of the theory~\cite{Weinberg}.
\section{Quantum electrodynamics}
\label{sec:QED}
Quantum electrodynamics (QED) is one of the most common QFTs. The fields that occur in this equation are $2\times N_f$ sets of Dirac spinor fields $\bar \psi_f$ and $\psi_f$ and a vector potential $A_\mu$. $N_f$ denotes the number of flavors used in this theory, whose masses are given by $m_f$. If one wants to describe only photons and electrons, $N_f$ can be set to one. Usually, it is defined via its Lagrange density
\begin{align}
\label{eq:L_QED}
    \mathcal{L}^\mathrm{QED} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \sum_{f = 1}^{N_f}\bar \psi_f (\mathrm{i}\slashed D - m_f)\psi_f,\\
    F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu,\\
    D_\mu = \partial_\mu - ieA_\mu,
\end{align}
where $e$ denotes the coupling constant of this theory. The first term in this expression describes the propagation of photons while the second term describes the propagation of fermions (zeroth order in $e$) and the electron photon interaction. These are the three fundamental building blocks of QED, their respective Feynman diagrams are shown in Figure~\ref{fig:QED_diagrams}.
\begin{figure}
    \centering
    \begin{tikzpicture}
          \begin{feynman}
        \vertex (a) ;
        \vertex [right=of a] (b);
        
        % Quarks im Pion mit gekrümmten Linien und manuellen Positionierungen der Labels
        \diagram* {
            (a) -- [boson, edge label=\(\gamma\)] (b),
        };
    \end{feynman}
    \end{tikzpicture}
    \hspace{3cm}
    \begin{tikzpicture}
          \begin{feynman}
        \vertex (a) ;
        \vertex [right=of a] (b);
        
        % Quarks im Pion mit gekrümmten Linien und manuellen Positionierungen der Labels
        \diagram* {
            (a) -- [fermion, edge label=\(e^-\)] (b),
        };
    \end{feynman}
    \end{tikzpicture}
    \hspace{3cm}
    \begin{tikzpicture}
          \begin{feynman}
        \vertex (a) ;
        \vertex [above=of a] (b);
        \vertex [below left=of a] (c);
        \vertex [below right=of a] (d);
        
        % Quarks im Pion mit gekrümmten Linien und manuellen Positionierungen der Labels
        \diagram* {
            (a) -- [boson, edge label=\(\gamma\)] (b),
            (c) -- [fermion, edge label=\(e^-\)] (a),
            (d) -- [anti fermion, edge label=\(e^-\)] (a),
        };
    \end{feynman}
    \end{tikzpicture}
    \caption[Fundamental diagrams of QED]{The fundamental diagrams, which can be used to construct every interaction of QED. The photon propagator is on the left, the fermion propagator is in the middle and on the right there is the only interaction vertex of QED.}
    \label{fig:QED_diagrams}
\end{figure}
Setting up the Euler-Lagrange equations for the photon field $A_\mu$, one immediately gets the inhomogeneous Maxwell equation, while the equations of motion of the fermion fields lead to the Dirac equation from Section~\ref{sec:Dirac}.\\
Furthermore, QED is one of the most fundamental gauge theories. A gauge theory has a local symmetry transformation $\Omega(x)$ that leaves the Lagrangian invariant. The co-domain of the gauge transformations is a Lie group, called the gauge group. In QED, the gauge group is the one dimensional compact Lie group $U(1)$. The fields are transformed according to
\begin{align}
    \psi(x) \to \psi'(x) = e^{\mathrm{-ie\alpha(x)}}\psi(x),\\
    \bar \psi(x) \to \bar \psi'(x) = \bar \psi(x) e^{\mathrm{i}e\alpha(x)},\\
    A_\mu(x) \to A'_\mu(x) = A_\mu(x) + e\partial_\mu \alpha(x).
\end{align}
where $\alpha(x)$ is a real-valued function that is at least $C^1$ (differentiable). As for most functions in physics, it is often assumed to be $C^\infty$ (infinitely many times differentiable).
\section{Quantum chromodynamics}
\label{sec:QCD}
Quantum chromodynamics (QCD) is the standard theory used to describe the strong interaction within the framework of the Standard Model, although other theoretical approaches to strong processes also exist, especially in contexts beyond or effective to QCD~\cite{scherer2002introductionchiralperturbationtheory}. Its overall structure is very similar to the one of QED, but there are some crucial difficulties. First the fermion fields (called quark fields in this context) have an additional color-index that can take values from one to three. The term ``color''\ in this context does not denote actual visual color, but is instead a conventional quantum label --- introduced by analogy with ``flavor'' --- to distinguish between different types of fermions. The three components of the quark field are related to the symmetry group, which has to be a subgroup of $U(3)$ in order to leave the mass term invariant. From experimental observations, it can be concluded that the subgroup in question must be $SU(3)$~\cite{Griffiths}. The transformation of the vector potential $A_\mu$ takes care of the invariance of the dynamical part of the quarks under $SU(3)$-transformations. Every gauge-transformation $\Omega$ can be written as the exponential of an anti-Hermitian $3\times 3$-matrix:
\begin{align}
    \Omega(x) = \exp \left(-\mathrm{i}\alpha^a(x)\frac{\lambda_a}{2} \right).
\end{align}
The index $a$ takes values from 1 to 8 and  $\frac{\lambda_a}{2}$ is a normalized orthogonal basis of the vector space of Hermitian and trace-less $3\times 3$-matrices $\mathfrak{su}(3)$. $\lambda_a$ are the Gell-Mann-matrices~\cite{halzen1984quarks}. The gauge transformations for all the fields appearing in QCD are given by
\begin{align}
    \psi(x) \to \psi'(x) = \Omega(x)\psi(x),\\
    \bar \psi(x) \to \bar \psi'(x) = \bar \psi(x) \Omega^\dagger(x),\\
    A_\mu(x) \to A'_\mu(x) = \Omega(x)A_\mu(x)\Omega^\dagger(x) + \left(\partial_\mu \Omega(x)\right)\Omega^\dagger(x).
\end{align}
So there are three components of the quarks and an eight dimensional symmetry group. The boson fields (that are called gluons in QCD). The Lagrangian of QCD is defined via
\begin{align}
     \mathcal{L}^\mathrm{QCD} = -\frac{1}{4}F_{\mu\nu}^aF^{\mu\nu}_a + \sum_{f = 1}^{N_f}\bar \psi_f^c (\mathrm{i}\slashed D_{cd} - m_f\delta_{cd})\psi_f^d,\\
    F_{\mu\nu}^a = \partial_\mu A^a_\nu - \partial_\nu A^a_\mu + g\sqrt{2}f^{abc}A_\mu^bA_\nu^c,\\
    D_\mu^{cd} = \partial_\mu \delta^{cd} - \frac{\mathrm{i}e}{2}A_\mu^{a}\lambda_a^{cd},
\end{align}
One of the most fundamental differences of the QCD and QED Lagrangian is the non linear term in the field strength tensor $F_{\mu\nu}$. It has to be included to take care of terms arising from the non-abelian structure of $SU(3)$. This quadratic term generates contributions of third and fourth powers of the gauge field $A$ in the Lagrangian which can be translated directly into gluon-gluon-interactions as shown in Figure~\ref{fig:QCD_diagrams}~\cite{Weinberg2}.
\begin{figure}
    \centering
    \begin{tikzpicture}
          \begin{feynman}
        \vertex (a) ;
        \vertex [right=of a] (b);
        
        % Quarks im Pion mit gekrümmten Linien und manuellen Positionierungen der Labels
        \diagram* {
            (a) -- [gluon, edge label=\(g\)] (b),
        };
    \end{feynman}
    \end{tikzpicture}
    \hspace{1cm}
    \begin{tikzpicture}
          \begin{feynman}
        \vertex (a) ;
        \vertex [right=of a] (b);
        
        % Quarks im Pion mit gekrümmten Linien und manuellen Positionierungen der Labels
        \diagram* {
            (a) -- [fermion, edge label=\(q\)] (b),
        };
    \end{feynman}
    \end{tikzpicture}
    \hspace{1cm}
    \begin{tikzpicture}
          \begin{feynman}
        \vertex (a) ;
        \vertex [above=of a] (b);
        \vertex [below left=of a] (c);
        \vertex [below right=of a] (d);
        
        % Quarks im Pion mit gekrümmten Linien und manuellen Positionierungen der Labels
        \diagram* {
            (a) -- [gluon, edge label=\(g\)] (b),
            (c) -- [fermion, edge label=\(q\)] (a),
            (d) -- [anti fermion, edge label=\(q\)] (a),
        };
    \end{feynman}
    \end{tikzpicture}
    \hspace{1cm}
    \begin{tikzpicture}
          \begin{feynman}
        \vertex (a) ;
        \vertex [above=of a] (b);
        \vertex [below left=of a] (c);
        \vertex [below right=of a] (d);
        
        % Quarks im Pion mit gekrümmten Linien und manuellen Positionierungen der Labels
        \diagram* {
            (a) -- [gluon, edge label=\(g\)] (b),
            (c) -- [gluon, edge label=\(g\)] (a),
            (d) -- [gluon, edge label=\(g\)] (a),
        };
    \end{feynman}
    \end{tikzpicture}
    \hspace{1cm}
    \begin{tikzpicture}
          \begin{feynman}
        \vertex (a) ;
        \vertex [above right =of a] (b);
        \vertex [below left=of a] (c);
        \vertex [below right=of a] (d);
      \vertex [above left =of a] (e);

        % Quarks im Pion mit gekrümmten Linien und manuellen Positionierungen der Labels
        \diagram* {
            (a) -- [gluon, edge label=\(g\)] (b),
            (c) -- [gluon, edge label=\(g\)] (a),
            (d) -- [gluon, edge label=\(g\)] (a),
            (e) -- [gluon, edge label=\(g\)] (a),
        };
    \end{feynman}
    \end{tikzpicture}
    \caption[Fundamental diagrams of QCD]{The fundamental diagrams, which can be used to construct every interaction of QCD. Compared to QED, photons are replaced by gluons and the fermions are limited to quarks. In addition, there are self-interactions of three or four gluons.}
    \label{fig:QCD_diagrams}
\end{figure}
\begin{figure}
    \centering
    \includegraphics[width=0.5\linewidth]{Figures/2/asymptotic_freedom.pdf}
    \caption[Experimental confirmation of asymptotic freedom]{Experimental confirmation of the running of the strong coupling with respect to the energy scale~\cite{BETHKE2007351}. The black lines are theoretical predictions for different values of the QCD energy scale $\Lambda_{\overline{\textrm{MS}}}^{(5)}$. This scale parameter, defined within the modified minimal subtraction scheme ($\overline{\textrm{MS}}$), sets the reference point at which the strong coupling constant $\alpha_s$ is matched to experimental data. Typical values lie in the range of a few hundred MeV~\cite{ParticleDataGroup:2024cfk,Bethke:2012jm}.}
    \label{fig:asymptotic_freedom}
\end{figure}
The general approach for computing matrix elements and transition amplitudes in most QFTs and especially QED is perturbation theory. 
The general idea behind perturbation theory (in the context of QED) is that each of the fundamental vertices (see Figure~\ref{fig:QED_diagrams}) corresponds to a mathematical expression. Each vertex includes another factor of $\sqrt{\alpha}$ ($\sim e$) in the transition amplitude. Summing up all diagrams describing the process will give an asymptotic series in powers of $e$ that converges to the correct QED result.\\ 
The self-interaction of the gluons on the other hand leads to an interesting feature, called asymptotic freedom, that is shown in Figure~\ref{fig:asymptotic_freedom}. The coupling strength $\alpha_s$ of QCD increases at low energies, or equivalently, at large distances~\cite{PhysRevLett.30.1343,PhysRevLett.30.1346}. Therefore the perturbative series only converges at sufficient high energy scales. So another method has to be used to compute the matrix element in QCD at low energies. In 1974, Kenneth Wilson introduced the idea of regularizing QFTs by a spacetime lattice for non-perturbative studies, such as gauge theories with confinement~\cite{PhysRevD.10.2445}.
\section{Wick rotation}
\label{sec:wick}
The exponent in the path integral in Equation~\eqref{eq:PI_full} is given by $\mathrm{i}S$, where $S$ is the action of the corresponding theory in Minkowski spacetime. The action is a real quantity, so the exponential has an absolute value of one along all the field configurations. However, there are exceptions to this rule, e.g. when a finite chemical potential is introduced. So when considering the transition amplitude ($F = 1$) of the process, one finds an integral that is not convergent in a classical sense, because the domain is unbound and the absolute value of the integrand is bound from below. Of course, at configurations with large action, the fluctuation of the exponential is huge and therefore destructive interference suppresses the contribution in these action regimes, but a mathematical well-defined solution of this problem is needed.\\
In Minkowski spacetime, the action is defined with an integration of the Lagrange-function with respect to the time along the real axis. Now the time can be re-interpreted as a complex quantity. So we integrate along a path in the complex plane, and this path is given by the real axis. Now this path can be modified. If the integrand is a holomorphic function (which can be assumed if the action is analytic on the real axis) and the path does not cross any poles during the modification, the result of the integral remains unchanged. We only have to accept the assumption that the integrand converges to zero fast enough if $\vert t \vert \to \infty$. In case of the Wick rotation, the modification of the path is given by
\begin{align}
    \gamma_\alpha(t) = \exp(\mathrm{i}\alpha)t \quad (t \in \mathbb{R},\, \alpha \in [0,\pi/2]).
\end{align}
For $\alpha = 0$ we have the integration along the real axis and for $\alpha = \frac{\pi}{2}$ it is along the imaginary axis. Now we can replace the time in the action according to
\begin{align}
    t \to t_E = \mathrm{i}t.
\end{align}
This affects the Lagrangian and the action such that they are replaced in the following way by their Euclidean counterparts:
\begin{align}
    L \to -L_E\\
    S \to \mathrm{i}S_E
\end{align}
The metric tensor $\eta^{\mu\nu}$ of the Lorentz covariant theories is also replaced:
\begin{align}
    \eta^{\mu\nu} \to \delta^{\mu\nu}.
\end{align}
Usually, the index zero is used for Minkowski time, while Euclidean time is denoted with the index 4.
With the aforementioned substitutions, the exponential in the path integral has the form $\exp(-S_E)$ with $S_E$ bounded from below, so the integrals converge properly~\cite{PhysRev.96.1124}.
The Euclidean actions for the Klein-Gordon and the Dirac theory are
\begin{align}
    S_E^{\mathrm{KG}}[\phi] = \frac{1}{2}\int \mathrm{d}^4x \,\left[(\partial_t\phi)^2 + (\partial_x\phi)^2+(\partial_y\phi)^2+(\partial_z\phi)^2 + m^2 \phi^2 \right] \eqqcolon \frac{1}{2}\int \mathrm{d}^4x\, \left( \partial_\mu \psi \partial_\mu \phi + m^2\phi^2\right)\\
    \label{eq:Dirac_euclidean}
    S_E^\mathrm{D}[\bar \psi,\psi] = \int\mathrm{d}^4x \, \bar \psi(\gamma_E^4 \partial_t + \gamma_E^1 \partial_x + \gamma^E_2 \partial_y + \gamma^3_E \partial_z + m)\psi  \eqqcolon \int \mathrm{d}^4x\, \bar \psi \left( \slashed \partial^E + m \right)\psi,
\end{align}
where the $\gamma^E$ matrices are defined according to
\begin{align}
    \gamma_E^4 = \gamma^0,\quad \gamma_E^i = \mathrm{i}\gamma_E^i.
\end{align}
It follows that the Euclidean $\gamma$-matrices are Hermitian.
The Yang-Mills action for photons and gluons changes to~\cite{Varnhorst2019lecture,GattringerLang}
\begin{align}
\label{eq:YM_Euclidean}
    S^{\mathrm{YM}}_E = \frac{1}{4}F_{\mu\nu}F_{\mu\nu}.
\end{align}
\section{Lattice discretization}
\label{sec:lattice}
To compute observables in a QFT in a mathematically correct way one has to introduce an ultraviolet regulator. In case of lattice gauge theory a hard momentum cutoff is used. This is done by discretizing spacetime and only considering field values on a lattice. The spacetime variable $x^\mu$ is replaced by $a\cdot n^\mu$ with $n \in \mathbb{Z}^4$ and $a$ being the lattice spacing. In principle it is possible to use anisotropic lattice spacings, but in the studies that are discussed here, a globally fixed spacing is used. Of course the definition of operations like derivatives and integrals (which are fundamental objects in the formulation of field theories) require a continuous spacetime. So they have to be replaced by discrete operators that reproduce them in the limit $a \to 0$. A possible (but not unique) discretization of these operators could be
\begin{align}
    \partial_\mu \phi(x) \to \frac{\phi(a(n+\hat\mu)) - \phi(a(n-\hat\mu))}{2a},\\
    \int\mathrm{d}^4x\, \phi(x) \to a^4 \sum_{n \in \mathbb{Z}^4}\phi(an),
\end{align}
where $\hat{\mu}$ is a short notation for the unit vector in $\mu$-direction.
Applying this discretization on the free Dirac action for fermions that is given in Equation~\eqref{eq:Dirac_euclidean} yields
\begin{align}
    S_F^0[\psi,\bar\psi] = a^4 \sum_n \bar \psi(an) \left(\sum_\mu \gamma_\mu \frac{\psi(a(n+\hat\mu)) - \psi(a(n-\hat\mu))}{2a}  + m\psi(an)\right).
\end{align}
The lattice spacing $a$ is set to one (lattice units) often.
As it was discussed before there are gauge transformations that can act on the quark fields, on the lattice they are given by $\Omega(an)$. So if the transformation of the quark fields is given by
\begin{align}
    \psi(n) \to \Omega(n)\psi(n), \\
    \bar\psi(n) \to\bar\psi(n) \Omega^\dagger(n),
\end{align}
the product of quark fields at neighboring lattice sites transforms as
\begin{align}
    \bar\psi(n)\psi(n+\hat \mu) \to \bar\psi(n)\Omega^\dagger(n)\Omega(n+\hat \mu)\psi(n+\hat \mu).
\end{align}
So the discretized version of the fermion action is not gauge invariant anymore, which is a necessary condition for the regularization. To restore gauge invariance, the gauge links $U_\mu(n)\in SU(3)$ are included. They are given by $N_c\times N_c$-matrices ($N_c$ is the number of colors in the corresponding theory, three in case of QCD and 1 in case of QED). They are located on the links between the lattices sites $n$ and $n + \hat \mu$ as it is shown in Figure~\ref{fig:gaugelink}. Whenever two quark fields $\bar \psi(m)$ and $\psi(n)$ are multiplied, one has to include the product of the gauge links along a path that connects the sites $m$ and $n$ between them. The gauge links transform according to 
\begin{align}
    U_\mu(n) \to \Omega(n)U_\mu(n) \Omega^\dagger(n+\mu), 
\end{align}
which guarantees that the product
\begin{align}
    \bar \psi(n) U_\mu(n)\psi(n+\mu)
\end{align}
is gauge invariant. A gauge link connecting the sites $n$ and  $n-\hat\mu$ is given by $U_\mu^\dagger(n-\mu)$. There is another way to construct gauge invariant quantities on the lattice and that is to take the trace of the product of gauge links along a closed path on the lattice. The most fundamental object is the plaquette
\begin{align}
\label{eq:Plaquette}
    P_{\mu\nu}(n) =\Tr{ U_\mu(n) U_\mu(n+\hat \mu)U_\mu^\dagger(n+\hat \nu)U^\dagger_\nu(n)},
\end{align}
which is a closed loop along the smallest possible square in the $\mu\nu$-plane starting at $n$\cite{PhysRevD.10.2445,Varnhorst2019lecture,GattringerLang}. An example is shown in Figure~\ref{fig:Plaquette}. 

\begin{figure}[H]
\centering
\begin{tikzpicture}
    \tikzset{EmptyDotShadow/.style={draw, circle, circular drop shadow = {on layer = back}, fill = black!25, minimum size = 6pt, inner sep=0pt, fill = gray!50}}
    \draw[very thick] (0,0) -- node {\midarrow} (4,0);
    \node[EmptyDotShadow] at (0,0) {};
    \node[EmptyDotShadow] at (4,0) {};
    \node                 at (0, -0.5) {$n$};
    \node                 at (2, -0.5) {$U_\mu(n)$};
    \node                 at (4, -0.5) {$n + \hat{\mu}$};
    \node[anchor=east]    at (-0.5, 0) {$\psi(n)$};
    \node[anchor=west]    at (4.5, 0) {$\psi(n + \hat{\mu})$};
\end{tikzpicture}
\caption[Location of quark fields and gauge links]{The quark fields are located at the sites on the lattice, while the gauge fields is at the links that connect adjacent lattice sites. This picture is taken from the Master Thesis of Timo Eichhorn~\cite{TimoMaster}.}
\label{fig:gaugelink}
\end{figure}
\begin{figure}[H]
	\centering
	\begin{tikzpicture}[scale=0.9]
		\tikzset{EmptyDotShadow/.style={draw, circle, circular drop shadow = {on layer = back}, fill = black!25, minimum size = 6pt, inner sep=0pt, fill = gray!50}}
		\foreach \x in {0,4}
		{
			\draw[thick] ({\x},-0.4) -- ({\x},4.4);
		}
		\foreach \y in {0,4}
		{
			\draw[thick] (-0.4,{\y}) -- (4.4,{\y});
		}
		\begin{scope}[every node/.style={sloped,allow upside down}]
			\draw[very thick] (0.4,0.4) -- node {\midarrow} (3.6,0.4) -- node {\midarrow} (3.6,3.6) -- node {\midarrow} (0.4,3.6) -- node {\midarrow} cycle;
		\end{scope}
		\foreach \x in {0,4}
		{
			\foreach \y in {0,4}
			{
				\node[EmptyDotShadow] at (\x,\y) {};
			}
		}
		\node at (-0.3, -0.3) {$n$};
		\node at (4.6, -0.3) {$n + \hat{\mu}$};
		\node at (-0.6, 4.3) {$n + \hat{\nu}$};
		\node at (4.95, 4.3) {$n + \hat{\mu} + \hat{\nu}$};
		\node at (2, -0.3) {$U_{\mu}(n)$};
		\node at (-0.6, 2) {$U^{\dagger}_{\nu}(n)$};
		\node at (2, 4.3) {$U^{\dagger}_{\mu}(n + \hat{\nu})$};
		\node at (4.95, 2) {$U_{\nu}(n + \hat{\mu})$};
	\end{tikzpicture}
	\caption[Illustration of a plaquette]{The plaquette is in general the most simple observable that can be constructed from gauge links only. It is the smallest contractible loop. This picture is taken from the Master Thesis of Timo Eichhorn~\cite{TimoMaster}.}
	\label{fig:Plaquette}
\end{figure}
\section{The Wilson gauge action and improvements}
\label{sec:improvements}
It is evident how the lattice fermion action is related to the continuum one. The relation between gauge links on the lattice and the gauge fields in the continuum theory is a bit more involved. The gauge links are group-valued while the gauge fields takes values in the corresponding algebra. So they might be related by an exponential function. This is indeed true, but the gauge field is defined at a single point, while the gauge links are obviously not, so the gauge field is integrated along the link before the exponentiation:
\begin{align}
\label{eq:exp_gauge_link}
    U_\mu(an) = \exp\left(ig \int_{an}^{a(n+\hat\mu)}A^\nu\mathrm{d}s^\nu \right) \eqqcolon \exp \left(\mathrm{i}ag \tilde A_\mu(n)\right).
\end{align}
Here we introduced an algebra valued lattice quantity $\tilde A_\mu(n)$ that differs to the gauge links by terms of $\mathcal{O}(a)$.\\
To get a lattice version of the Yang-Mills-Lagrangian, let us consider again the most simple gauge-invariant quantity on the lattice, the plaquette (see Figure~\ref{fig:Plaquette}
and Equation~\eqref{eq:Plaquette}). If we use the exponential representation in Equation~\eqref{eq:exp_gauge_link} and apply a Taylor expansion of the continuous gauge fields
\begin{align}
    A_\nu(an + a\hat \mu) = A_\nu(an) + a\partial_\mu A_\nu(n) + \mathcal{O}(a^2),
\end{align}
we find that
\begin{align}
    P_{\mu\nu}(an) = \exp \left(\mathrm{i}a^2g^2 F_{\mu\nu}(an) + \mathcal{O}(a^3)\right),
\end{align}
which can be expanded and reformulated to
\begin{align}
    S_G[U] = \underbrace{\frac{2N_c}{g^2}}_{\beta} \sum_n \sum_{\mu <\nu}\Re \left(1-\frac{1}{N_c}\Tr P_{\mu\nu}(an) \right) = \frac{a^4}{2g^2}\sum_n\sum_{\mu,\nu}\Tr \left( F_{\mu\nu}^2\right) + \mathcal{O}(a^2).
\end{align}
So we got a lattice version of the gauge action that matches with the continuum up to terms proportional to $a^2$. However, it is possible to remove some of those artifacts~\cite{SYMANZIK1983187}. By including loops of size $2\times 1$, the action is modified to
\begin{align}
	S_G^\textit{rect}[U] = \frac{\beta}{\mathrm{N_c}} \sum\limits_{n \in \Lambda} \Bigl( \sum\limits_{\mu < \nu} c_0 \bigl( \mathrm{N_c} - \Re\tr[P_{\mu\nu}(n)] \bigr)
	+ \sum\limits_{\mu \neq \nu} c_1 \bigl( \mathrm{N_c} - \Re\tr[P_{\mu, 2\nu}(n)] \bigr) \Bigr),
\end{align}
where $P_{\mu,2\nu}$ is the $1\times 2$ closed loop in the $\mu\nu$-plane. The coefficients have to be chosen according to
\begin{align}
    c_0 + 8c_1 = 1, \quad c_0 > 0.
\end{align}
The condition ensures that the continuum limit is correct and the tuning of the parameters can exclude higher-order contributions. For example, the Lüscher-Weisz (tree-level) action uses $c_0 = \frac{5}{3}$ and $c_1 = -\frac{1}{12}$ and has classically leading discretization effects of $\mathcal{O}(a^4)$~\cite{Luscher:1984xn}. Other commonly used improved actions are the DBW2 and the Iwasaki action~\cite{Takaishi:1996xj,Iwasaki:1983iya}.
