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+---
+title: "Feynman diagram"
+date: 2021-11-18
+categories:
+- Physics
+- Quantum mechanics
+layout: "concept"
+---
+
+When doing calculations in the context of condensed matter physics and quantum field theory,
+**Feynman diagrams** graphically represent expressions
+that would be tedious or error-prone to work with directly.
+This article is about condensed matter physics.
+
+Suppose we have a many-particle Hamiltonian $\hat{H} = \hat{H}_0 + \hat{H}_1$,
+consisting of an "easy" term $\hat{H}_0$,
+and then a "difficult" term $\hat{H}_1$
+with time-dependent and/or interacting parts.
+Let $\Ket{\Phi_0}$ be a known eigenstate (or superposition thereof)
+of the easily solvable part $\hat{H}_0$,
+with respect to which we will take expectation values $\Expval{}$.
+
+Below, we go through the most notable components of Feynman diagrams
+and how to translate them into a mathematical expression.
+
+
+## Real space
+
+The most common component is a **fermion line**, which represents
+a [Green's function](/know/concept/greens-functions/) $G^0$
+for the simple Hamiltonian $\hat{H}_0$.
+Any type of Green's function is possible in theory (e.g. a retarded),
+but usually the *causal* function is used.
+Let the subscript $I$ refer to the
+[interaction picture](/know/concept/interaction-picture/),
+and $\mathcal{T}\{\}$ denote the
+[time-ordered product](/know/concept/time-ordered-product/):
+
+<a href="freegf.png">
+<img src="freegf.png" style="width:60%">
+</a>
+$$\begin{aligned}
+    = i \hbar G_{s_2 s_1}^0(\vb{r}_2, t_2; \vb{r}_1, t_1)
+    = \Expval{\mathcal{T} \Big\{ \hat{\Psi}_{s_2 I}(\vb{r}_2, t_2) \hat{\Psi}_{s_1 I}^\dagger(\vb{r}_1, t_1) \Big\}}
+\end{aligned}$$
+
+The arrow points in the direction of time, or more generally,
+from the point of creation $\hat{\Psi}{}^\dagger$
+to the point of annihilation $\hat{\Psi}$.
+The dots at the ends are called **vertices**,
+which represent points in space and time with a spin.
+Vertices can be
+**internal** (one Green's function entering AND one leaving)
+or **external** (either one Green's function entering OR one leaving).
+
+Less common is a **heavy fermion line**, representing
+a causal Green's function $G$ for the entire Hamiltonian $\hat{H}$,
+where the subscript $H$ refers to the [Heisenberg picture](/know/concept/heisenberg-picture/):
+
+<a href="fullgf.png">
+<img src="fullgf.png" style="width:60%">
+</a>
+$$\begin{aligned}
+    = i \hbar G_{s_2 s_1}(\vb{r}_2, t_2; \vb{r}_1, t_1)
+    = \Expval{\mathcal{T} \Big\{ \hat{\Psi}_{s_2 H}(\vb{r}_2, t_2) \hat{\Psi}_{s_1 H}^\dagger(\vb{r}_1, t_1) \Big\}}
+\end{aligned}$$
+
+Next, an **interaction line** or **boson line** represents
+a two-body interaction operator $\hat{W}$ (in $\hat{H}_1$),
+which we assume to be instantaneous, i.e. time-independent
+(in quantum field theory this is *not* assumed),
+hence it starts and ends at the same time,
+and no arrow is drawn:
+
+<a href="interaction.png">
+<img src="interaction.png" style="width:60%">
+</a>
+$$\begin{aligned}
+    = \frac{1}{i \hbar} W_{s_2 s_1}(\vb{r}_2, t_2; \vb{r}_1, t_1)
+    = \frac{1}{i \hbar} W(\vb{r}_2, \vb{r}_1; t_1) \: \delta(t_2 - t_1)
+\end{aligned}$$
+
+We have chosen to disallow spin flipping,
+so $W$ does not depend on $s_1$ or $s_2$.
+For reference, this function $W$
+has a time-dependence coming only from the interaction picture,
+and is to be used as follows to get the full two-body operator $\hat{W}$:
+
+$$\begin{aligned}
+    \hat{W}
+    = \frac{1}{2} \sum_{s_1 s_2} \iint_{-\infty}^\infty \hat{\Psi}_{s_1}^\dagger(\vb{r}_1) \hat{\Psi}_{s_2}^\dagger(\vb{r}_2)
+    W(\vb{r}_1, \vb{r}_2) \hat{\Psi}_{s_2}(\vb{r}_2) \hat{\Psi}_{s_1}(\vb{r}_1) \dd{\vb{r}_1} \dd{\vb{r}_2}
+\end{aligned}$$
+
+One-body (time-dependent) operators $\hat{V}$ in $\hat{H}_1$
+are instead represented by a special vertex:
+
+<a href="perturbation.png">
+<img src="perturbation.png" style="width:35%">
+</a>
+$$\begin{aligned}
+    = \frac{1}{i \hbar} V_s(\vb{r}, t)
+\end{aligned}$$
+
+Other graphical components exist representing
+more complicated operators and quantities,
+but these deserve their own articles.
+
+In order for a given Feynman diagram to be valid,
+it must satisfy the following criteria:
+
+a.  Each vertex must be connected to one or two fermion lines,
+    at most one of which leaves,
+    and at most one of which enters.
+b.  Each internal vertex contains at most one "event";
+    which could be $V$ or $W$.
+
+Finally, we need some additional rules to convert
+diagrams into mathematical expressions:
+
+1.  Disallow spin flipping by multiplying
+    each internal vertex by $\delta_{s_\mathrm{in} s_\mathrm{out}}$.
+2.  If both ends of a line are at the same time (always the case for $W$),
+    an infinitesimal $\eta \to 0^+$ must be added
+    to the time of all creation operators,
+    so e.g. $G(t, t) \to G(t, t\!+\!\eta)$.
+3.  Integrate over spacetime coordinates $(\vb{r}, t)$
+    and sum over the spin $s$ of all internal vertices,
+    but not external ones.
+4.  Multiply the result by $(-1)^F$,
+    where $F$ is the number of closed fermion loops.
+5.  Depending on the context, additional constant factors may be required;
+    sometimes they are changed on-the-fly during a calculation.
+
+Note that rules 4 and 5 are convention,
+just like the factors $i \hbar$ in $G^0$, $G$, $V$ and $W$;
+it simply turns out to be nicer to do it this way
+when using Feynman diagrams in the wild.
+
+The combination of rules 2 and 3 means that spin
+belongs to lines rather than vertices,
+so that a particle with a given spin propagates
+from vertex to vertex without getting flipped.
+
+
+## Fourier space
+
+If the system is time-independent and spatially uniform,
+meaning it has continuous translational symmetry in time and space,
+then it is useful to work in [Fourier space](/know/concept/fourier-transform/):
+
+$$\begin{aligned}
+    G_{s_2 s_1}^0(\vb{r}_2, t_2; \vb{r}_1, t_1)
+    &= G_{s_1}^0(\vb{r}_2 - \vb{r}_1, t_2 - t_1) \: \delta_{s_2 s_1}
+    \\
+    &= \frac{\delta_{s_2 s_1}}{(2 \pi)^4} \iint_{-\infty}^\infty G_{s_1}^0(\vb{k}, \omega) \:
+    e^{i \vb{k} \cdot (\vb{r}_2 - \vb{r}_1)} e^{- i \omega (t_2 - t_1)} \dd{\vb{k}} \dd{\omega}
+    \\
+    W_{s_2 s_1}(\vb{r}_2, t_2; \vb{r}_1, t_1)
+    &= W(\vb{r}_2 - \vb{r}_1) \: \delta(t_2 - t_1)
+    \\
+    &= \frac{1}{(2 \pi)^4} \iint_{\infty}^\infty W(\vb{k}) \:
+    e^{i \vb{k} \cdot (\vb{r}_2 - \vb{r}_1)} e^{- i \omega (t_2 - t_1)} \dd{\vb{k}} \dd{\omega}
+\end{aligned}$$
+
+Where we have used an integral representation of
+the [Dirac delta function](/know/concept/dirac-delta-function/).
+Note the inconsistent sign of the exponent
+in the Fourier transform definitions for space and time.
+
+Working in Fourier space allows us to simplify calculations.
+Consider the following diagram and the resulting expression,
+where $\tilde{\vb{r}} = (\vb{r}, t)$, and $\tilde{\vb{k}} = (\vb{k}, \omega)$:
+
+<a href="conservation.png">
+<img src="conservation.png" style="width:40%">
+</a>
+$$\begin{aligned}
+    &= (i \hbar)^3 \sum_{s s'} \!\!\iint \dd{\tilde{\vb{r}}} \dd{\tilde{\vb{r}}'}
+    G_{s_1's}^0(\tilde{\vb{r}}_1', \tilde{\vb{r}}) G_{s s_1}^0(\tilde{\vb{r}}, \tilde{\vb{r}}_1) \delta_{s_1 s_1'}
+    W(\tilde{\vb{r}}, \tilde{\vb{r}}')
+    G_{s_2' s'}^0(\tilde{\vb{r}}_2', \tilde{\vb{r}}') G_{s' s_2}^0(\tilde{\vb{r}}', \tilde{\vb{r}}_2) \delta_{s_2 s_2'}
+    \\
+    &= \frac{-i \hbar^3}{(2 \pi)^{20}}
+    \sum_{s_1 s_2} \!\!\iint \dd{\tilde{\vb{r}}} \dd{\tilde{\vb{r}}'}
+    \bigg(\! \int \dd{\tilde{\vb{k}}_2} G_{s_1}^0(\tilde{\vb{k}}_2) e^{i \tilde{\vb{k}}_2 \cdot (\tilde{\vb{r}}_1' - \tilde{\vb{r}})} \!\bigg)
+    \bigg(\! \int \dd{\tilde{\vb{k}}_1} G_{s_1}^0(\tilde{\vb{k}}_1) e^{i \tilde{\vb{k}}_1 \cdot (\tilde{\vb{r}} - \tilde{\vb{r}}_1)} \!\bigg)
+    \\
+    &\qquad\times \bigg(\! \int \dd{\tilde{\vb{p}}} W(\tilde{\vb{p}}) e^{i \tilde{\vb{p}} \cdot (\tilde{\vb{r}}' - \tilde{\vb{r}})} \!\bigg)
+    \bigg(\! \int \dd{\tilde{\vb{q}}_2} G_{s_2}^0(\tilde{\vb{q}}_2) e^{i \tilde{\vb{q}}_2 \cdot (\tilde{\vb{r}}_2' - \tilde{\vb{r}}')} \!\bigg)
+    \bigg(\! \int \dd{\tilde{\vb{q}}_1} G_{s_2}^0(\tilde{\vb{q}}_1) e^{i \tilde{\vb{q}}_1 \cdot (\tilde{\vb{r}}' - \tilde{\vb{r}}_2)} \!\bigg)
+    \\
+    &= \frac{-i \hbar^3}{(2 \pi)^{12}}
+    \sum_{s_1 s_2} \!\!\iint \dd{\tilde{\vb{k}}_1} \dd{\tilde{\vb{k}}_2}
+    G_{s_1}^0(\tilde{\vb{k}}_2) G_{s_1}^0(\tilde{\vb{k}}_1)
+    \iint \dd{\tilde{\vb{q}}_1} \dd{\tilde{\vb{q}}_2}
+    G_{s_2}^0(\tilde{\vb{q}}_2) G_{s_2}^0(\tilde{\vb{q}}_1)
+    \\
+    &\qquad\times
+    e^{i \tilde{\vb{k}}_2 \cdot \tilde{\vb{r}}_1' - i \tilde{\vb{k}}_1 \cdot \tilde{\vb{r}}_1
+    + i \tilde{\vb{q}}_2 \cdot \tilde{\vb{r}}_2' - i \tilde{\vb{q}}_1 \cdot \tilde{\vb{r}}_2}
+    \!\!\int \dd{\tilde{\vb{p}}} W(\tilde{\vb{p}})
+    \bigg( \frac{1}{(2 \pi)^8} \!\!\iint \dd{\tilde{\vb{r}}} \dd{\tilde{\vb{r}}'}
+    e^{i (\tilde{\vb{k}}_1 - \tilde{\vb{k}}_2 - \tilde{\vb{p}}) \cdot \tilde{\vb{r}}}
+    e^{i (\tilde{\vb{q}}_1 - \tilde{\vb{q}}_2 + \tilde{\vb{p}}) \cdot \tilde{\vb{r}}'} \bigg)
+    \\
+    &= \frac{-i \hbar^3}{(2 \pi)^{12}}
+    \sum_{s_1 s_2} \!\!\iint \dd{\tilde{\vb{k}}_1} \dd{\tilde{\vb{k}}_2}
+    G_{s_1}^0(\tilde{\vb{k}}_2) G_{s_1}^0(\tilde{\vb{k}}_1)
+    \iint \dd{\tilde{\vb{q}}_1} \dd{\tilde{\vb{q}}_2}
+    G_{s_2}^0(\tilde{\vb{q}}_2) G_{s_2}^0(\tilde{\vb{q}}_1)
+    \\
+    &\qquad\times
+    e^{i \tilde{\vb{k}}_2 \cdot \tilde{\vb{r}}_1' - i \tilde{\vb{k}}_1 \cdot \tilde{\vb{r}}_1
+    + i \tilde{\vb{q}}_2 \cdot \tilde{\vb{r}}_2' - i \tilde{\vb{q}}_1 \cdot \tilde{\vb{r}}_2}
+    \!\!\int \dd{\tilde{\vb{p}}} W(\tilde{\vb{p}})
+    \: \delta(\tilde{\vb{k}}_1 \!-\! \tilde{\vb{k}}_2 \!-\! \tilde{\vb{p}})
+    \: \delta(\tilde{\vb{q}}_1 \!-\! \tilde{\vb{q}}_2 \!+\! \tilde{\vb{p}})
+    \\
+    &= \frac{-i \hbar^3}{(2 \pi)^{12}}
+    \sum_{s_1 s_2} \!\!\int \dd{\tilde{\vb{p}}} W(\tilde{\vb{p}})
+    \int \dd{\tilde{\vb{k}}_1} G_{s_1}^0(\tilde{\vb{k}}_1 \!-\! \tilde{\vb{p}}) G_{s_1}^0(\tilde{\vb{k}}_1)
+    \int \dd{\tilde{\vb{q}}_1} G_{s_2}^0(\tilde{\vb{q}}_1 \!+\! \tilde{\vb{p}}) G_{s_2}^0(\tilde{\vb{q}}_1)
+    \\
+    &\qquad\times
+    e^{i \tilde{\vb{k}}_1 \cdot (\tilde{\vb{r}}_1' - \tilde{\vb{r}}_1)}
+    e^{i \tilde{\vb{q}}_1 \cdot (\tilde{\vb{r}}_2' - \tilde{\vb{r}}_2)}
+    e^{i \tilde{\vb{p}} \cdot (\tilde{\vb{r}}_2' - \tilde{\vb{r}}_1')}
+\end{aligned}$$
+
+Conveniently, the Dirac delta functions that appear from the integrals
+represent conservation of wavevector $\vb{k}$ (momentum $\hbar \vb{k}$)
+and angular frequency $\omega$ (energy $\hbar \omega$).
+
+In Fourier space, it makes more sense
+to regard the incoming energies and momenta and spins as given,
+and only integrate over the internal quantities.
+We thus modify the Feynman diagram rules
+such that we end up with the following result:
+
+$$\begin{aligned}
+    \equiv \frac{-i \hbar^3}{(2 \pi)^4}
+    \sum_{s} \!\!\int \dd{\tilde{\vb{p}}} W(\tilde{\vb{p}})
+    \: G_{s_1}^0(\tilde{\vb{k}}_1 \!-\! \tilde{\vb{p}}) \: G_{s_1}^0(\tilde{\vb{k}}_1)
+    \: G_{s_2}^0(\tilde{\vb{q}}_1 \!+\! \tilde{\vb{p}}) \: G_{s_2}^0(\tilde{\vb{q}}_1)
+\end{aligned}$$
+
+Therefore, we say that fermion lines represent $i \hbar G_{s}^0(\vb{k}, \omega)$,
+interaction lines $W(\vb{k}) / i \hbar$, etc.,
+and the other interpretation rules are modified to the following:
+
+1.  Each line has a momentum $\vb{k}$ and energy $\omega$,
+    and each fermion line has a spin $s$;
+    these must all be conserved at each vertex.
+2.  If both ends of a *fermion* line would be at the same time,
+    multiply it by $e^{i \omega \eta}$,
+    where $\eta \to 0^+$ is a positive infinitesimal,
+    so e.g. $G(\tau, \tau) \to e^{i \omega \eta} G(\tau, \tau)$.
+3.  Integrate over all internal $(\vb{k}, \omega)$,
+    and sum over all internal spins $s$.
+    Let each $(\vb{k}, \omega)$ integral contribute a factor $1 / (2 \pi)^4$.
+4.  Multiply the end result by $(-1)^F$, where $F$ is the number of closed fermion loops.
+5.  Depending on the context, additional constant factors may be required;
+    sometimes they are changed on-the-fly during a calculation.
+
+Note that if the diagram is linear (i.e. does not contain interactions),
+then conservation removes all internal variables,
+so no integrals would be needed.
+
+
+## Imaginary time
+
+Feynman diagrams are also useful when working with
+[imaginary time](/know/concept/imaginary-time/).
+In that case, the meaning of fermion lines is changed as follows,
+involving the [Matsubara Green's function](/know/concept/matsubara-greens-function/):
+
+$$\begin{aligned}
+    i \hbar G_{s_2 s_1}^0(\vb{r}_2, t_2; \vb{r}_1, t_1)
+    \:\: &\longrightarrow \:\:
+    \hbar G_{s_2 s_1}^0(\vb{r}_2, \tau_2; \vb{r}_1, \tau_1)
+    = \Expval{\mathcal{T} \Big\{ \hat{\Psi}_I(\vb{r}_2, \tau_2) \hat{\Psi}_I^\dagger(\vb{r}_1, \tau_1) \Big\}}
+    \\
+    i \hbar G_{s_2 s_1}(\vb{r}_2, t_2; \vb{r}_1, t_1)
+    \:\: &\longrightarrow \:\:
+    \hbar G_{s_2 s_1}(\vb{r}_2, \tau_2; \vb{r}_1, \tau_1)
+    = \Expval{\mathcal{T} \Big\{ \hat{\Psi}_H(\vb{r}_2, \tau_2) \hat{\Psi}_H^\dagger(\vb{r}_1, \tau_1) \Big\}}
+\end{aligned}$$
+
+Where the time-ordering is with respect to $\tau$.
+Interaction lines are modified like so:
+
+$$\begin{aligned}
+    \frac{1}{i \hbar} W_{s_2 s_1}(\vb{r}_2, t_2; \vb{r}_1, t_1)
+    \:\: &\longrightarrow \:\:
+    -\frac{1}{\hbar} W_{s_2 s_1}(\vb{r}_2, \tau_2; \vb{r}_1, \tau_1)
+    = -\frac{1}{\hbar} W(\vb{r}_2, \vb{r}_1; \tau_1) \delta(\tau_2 \!-\! \tau_1)
+\end{aligned}$$
+
+One-body $V$-vertices are usually not used,
+because they are intended for real-time-dependent operators,
+but in theory they would get a factor $-1/\hbar$ too.
+
+For imaginary time, the Fourier transform is defined differently,
+and a distinction must be made between
+fermionic Matsubara frequencies $i \omega_n^f$ (for $G$ and $G^0$)
+and bosonic Matsubara ones $i \omega_n^b$ (for $W$).
+This distinction is compatible with frequency conservation,
+since a sum of two fermionic frequencies is always bosonic.
+We have:
+
+$$\begin{aligned}
+    G_{s_2 s_1}^0(\vb{r}_2, \tau_2; \vb{r}_1, \tau_1)
+    &= \frac{\delta_{s_2 s_1}}{(2 \pi)^3} \int_{-\infty}^\infty \frac{1}{\hbar \beta} \sum_{n = -\infty}^\infty
+    G_{s_1}^0(\vb{k}, i \omega_n^f) e^{i \vb{k} \cdot (\vb{r}_2 - \vb{r}_1)} e^{- i \omega_n^f (\tau_2 - \tau_1)} \dd{\vb{k}}
+    \\
+    W_{s_2 s_1}(\vb{r}_2, \tau_2; \vb{r}_1, \tau_1)
+    &= \frac{1}{(2 \pi)^3} \int_{-\infty}^\infty \frac{1}{\hbar \beta} \sum_{n = -\infty}^\infty
+    W(\vb{k}) e^{i \vb{k} \cdot (\vb{r}_2 - \vb{r}_1)} e^{- i \omega_n^b (\tau_2 - \tau_1)} \dd{\vb{k}}
+\end{aligned}$$
+
+The interpretation in Fourier space is the same,
+except that each internal integral/sum
+instead gives a constant $1 / \big(\hbar \beta (2 \pi)^3\big)$,
+and same-time fermion lines need a factor of $e^{i \omega_n^f \eta}$.
+
+
+
+## References
+1.  H. Bruus, K. Flensberg,
+    *Many-body quantum theory in condensed matter physics*,
+    2016, Oxford.
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