From dc3498fd50121eadbdd3ddac5bf950a16e2b50cb Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 18 Nov 2021 20:07:12 +0100
Subject: Expand knowledge base
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content/know/concept/greens-functions/index.pdc | 68 +++--
content/know/concept/imaginary-time/index.pdc | 4 +-
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diff --git a/content/know/concept/dyson-equation/index.pdc b/content/know/concept/dyson-equation/index.pdc
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--- a/content/know/concept/dyson-equation/index.pdc
+++ b/content/know/concept/dyson-equation/index.pdc
@@ -63,8 +63,8 @@ Let us assume that $\hat{H}_0$ is simple,
such that $G_0$ and $\hat{G}{}_0^{-1}$ can be found without issues
by solving the defining equation above.
-Suppose we now perturb this Hamiltonian with
-a possibly time-dependent operator $\hat{H}_1(\vb{r}, t)$,
+Suppose we now add a more complicated and
+possibly time-dependent term $\hat{H}_1(\vb{r}, t)$,
in which case the corresponding fundamental solution
$G(\vb{r}, \vb{r}', t, t')$ satisfies:
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diff --git a/content/know/concept/feynman-diagram/index.pdc b/content/know/concept/feynman-diagram/index.pdc
new file mode 100644
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--- /dev/null
+++ b/content/know/concept/feynman-diagram/index.pdc
@@ -0,0 +1,337 @@
+---
+title: "Feynman diagram"
+firstLetter: "F"
+publishDate: 2021-11-15
+categories:
+- Physics
+- Quantum mechanics
+
+date: 2021-11-15T21:01:46+01:00
+draft: false
+markup: pandoc
+---
+
+# Feynman diagram
+
+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/):
+
+
+
+
+$$\begin{aligned}
+ = i \hbar G_{s_2 s_1}^0(\vb{r}_2, t_2; \vb{r}_1, t_1)
+ = \expval{\mathcal{T} \Big\{ \hat{\Psi}_{s_2 I}(\vb{r}_2, t_2) \hat{\Psi}_{s_1 I}^\dagger(\vb{r}_1, t_1) \Big\}}
+\end{aligned}$$
+
+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/):
+
+
+
+
+$$\begin{aligned}
+ = i \hbar G_{s_2 s_1}(\vb{r}_2, t_2; \vb{r}_1, t_1)
+ = \expval{\mathcal{T} \Big\{ \hat{\Psi}_{s_2 H}(\vb{r}_2, t_2) \hat{\Psi}_{s_1 H}^\dagger(\vb{r}_1, t_1) \Big\}}
+\end{aligned}$$
+
+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:
+
+
+
+
+$$\begin{aligned}
+ = \frac{1}{i \hbar} W_{s_2 s_1}(\vb{r}_2, t_2; \vb{r}_1, t_1)
+ = \frac{1}{i \hbar} W(\vb{r}_2, \vb{r}_1; t_1) \: \delta(t_2 - t_1)
+\end{aligned}$$
+
+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:
+
+
+
+
+$$\begin{aligned}
+ = \frac{1}{i \hbar} V_I(\vb{r}, t, \sigma)
+\end{aligned}$$
+
+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 $\tilde{\vb{k}} = (\vb{k}, \omega)$:
+
+
+
+
+$$\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:
+
+$$\begin{aligned}
+ G_{s_2 s_1}^0(\vb{r}_2, \tau_2; \vb{r}_1, \tau_1)
+ &= \frac{\delta_{s_2 s_1}}{(2 \pi)^3} \int_{-\infty}^\infty \frac{1}{\hbar \beta} \sum_{n = -\infty}^\infty
+ G_{s_1}^0(\vb{k}, i \omega_n^f) e^{i \vb{k} \cdot (\vb{r}_2 - \vb{r}_1)} e^{- i \omega_n^f (\tau_2 - \tau_1)} \dd{\vb{k}}
+ \\
+ W_{s_2 s_1}(\vb{r}_2, \tau_2; \vb{r}_1, \tau_1)
+ &= \frac{1}{(2 \pi)^3} \int_{-\infty}^\infty \frac{1}{\hbar \beta} \sum_{n = -\infty}^\infty
+ W(\vb{k}) e^{i \vb{k} \cdot (\vb{r}_2 - \vb{r}_1)} e^{- i \omega_n^b (\tau_2 - \tau_1)} \dd{\vb{k}}
+\end{aligned}$$
+
+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.
diff --git a/content/know/concept/feynman-diagram/interaction.png b/content/know/concept/feynman-diagram/interaction.png
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diff --git a/content/know/concept/feynman-diagram/perturbation.png b/content/know/concept/feynman-diagram/perturbation.png
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diff --git a/content/know/concept/greens-functions/index.pdc b/content/know/concept/greens-functions/index.pdc
index 2f86e63..b3c9ede 100644
--- a/content/know/concept/greens-functions/index.pdc
+++ b/content/know/concept/greens-functions/index.pdc
@@ -32,14 +32,26 @@ If the two operators are single-particle creation/annihilation operators,
then we get the **single-particle Green's functions**,
for which the symbol $G$ is used.
-The **retarded Green's function** $G_{\nu \nu'}^R$
-and the **advanced Green's function** $G_{\nu \nu'}^A$
-are defined like so,
-where the expectation value $\expval{}$ is
+The **time-ordered** or **causal Green's function** $G_{\nu \nu'}$
+is defined as follows,
+where $\mathcal{T}$ is the [time-ordered product](/know/concept/time-ordered-product/),
+the expectation value $\expval{}$ is
with respect to thermodynamic equilibrium,
$\nu$ and $\nu'$ are labels of single-particle states,
and $\hat{c}_\nu$ annihilates a particle from $\nu$, etc.:
+$$\begin{aligned}
+ \boxed{
+ G_{\nu \nu'}(t, t')
+ \equiv -\frac{i}{\hbar} \expval{\mathcal{T} \Big\{ \hat{c}_{\nu}(t) \: \hat{c}_{\nu'}^\dagger(t') \Big\}}
+ }
+\end{aligned}$$
+
+Arguably more prevalent are
+the **retarded Green's function** $G_{\nu \nu'}^R$
+and the **advanced Green's function** $G_{\nu \nu'}^A$
+which are defined like so:
+
$$\begin{aligned}
\boxed{
\begin{aligned}
@@ -75,15 +87,19 @@ $$\begin{aligned}
}
\end{aligned}$$
-Where $-$ is for bosons, and $+$ is for fermions.
-The retarded and advanced Green's functions can thus be expressed as follows:
+Where $-$ is for bosons, and $+$ for fermions.
+With this, the causal, retarded and advanced Green's functions
+can thus be expressed as follows:
$$\begin{aligned}
+ G_{\nu \nu'}(t, t')
+ &= \Theta(t - t') \: G_{\nu \nu'}^>(t, t') + \Theta(t' - t) \: G_{\nu \nu'}^<(t, t')
+ \\
G_{\nu \nu'}^R(t, t')
- &= \Theta(t - t') \Big( G_{\nu \nu'}^>(t, t') - G_{\nu \nu'}^<(t, t') \Big)
+ &= \Theta(t - t') \big( G_{\nu \nu'}^>(t, t') - G_{\nu \nu'}^<(t, t') \big)
\\
G_{\nu \nu'}^A(t, t')
- &= \Theta(t' - t) \Big( G_{\nu \nu'}^<(t, t') - G_{\nu \nu'}^>(t, t') \Big)
+ &= \Theta(t' - t) \big( G_{\nu \nu'}^<(t, t') - G_{\nu \nu'}^>(t, t') \big)
\end{aligned}$$
If the Hamiltonian involves interactions,
@@ -93,14 +109,14 @@ In that case, instead of a label $\nu$,
we use the spin $s$ and position $\vb{r}$, leading to:
$$\begin{aligned}
- G_{ss'}^R(\vb{r}, t; \vb{r}', t')
- &= -\frac{i}{\hbar} \Theta(t - t') \expval{\comm*{\hat{\Psi}_{s}(\vb{r}, t)}{\hat{\Psi}_{s'}^\dagger(\vb{r}', t')}_{\mp}}
+ G_{ss'}(\vb{r}, t; \vb{r}', t')
+ &= -\frac{i}{\hbar} \Theta(t - t') \expval{\mathcal{T}\Big\{ \hat{\Psi}_{s}(\vb{r}, t) \hat{\Psi}_{s'}^\dagger(\vb{r}', t') \Big\}}
\\
- &= \sum_{\nu \nu'} \psi_\nu(\vb{r}) \: \psi^*_{\nu'}(\vb{r}') \: G_{\nu \nu'}^R(t, t')
+ &= \sum_{\nu \nu'} \psi_\nu(\vb{r}) \: \psi^*_{\nu'}(\vb{r}') \: G_{\nu \nu'}(t, t')
\end{aligned}$$
-And analogously for $G_{ss'}^A$, $G_{ss'}^>$ and $G_{ss'}^<$.
-Note that the time-dependence is given to the old $G_{\nu \nu'}^R$,
+And analogously for $G_{ss'}^R$, $G_{ss'}^A$, $G_{ss'}^>$ and $G_{ss'}^<$.
+Note that the time-dependence is given to the old $G_{\nu \nu'}$,
i.e. to $\hat{c}_\nu$ and $\hat{c}_{\nu'}^\dagger$,
because we are in the Heisenberg picture.
@@ -108,7 +124,9 @@ If the Hamiltonian is time-independent,
then it can be shown that all the Green's functions
only depend on the time-difference $t - t'$:
-$$\begin{aligned}
+$$\begin{gathered}
+ G_{\nu \nu'}(t, t') = G_{\nu \nu'}(t - t')
+ \\
G_{\nu \nu'}^R(t, t') = G_{\nu \nu'}^R(t - t')
\qquad \quad
G_{\nu \nu'}^A(t, t') = G_{\nu \nu'}^A(t - t')
@@ -116,7 +134,7 @@ $$\begin{aligned}
G_{\nu \nu'}^>(t, t') = G_{\nu \nu'}^>(t - t')
\qquad \quad
G_{\nu \nu'}^<(t, t') = G_{\nu \nu'}^<(t - t')
-\end{aligned}$$
+\end{gathered}$$
@@ -324,16 +342,20 @@ i.e. the Hamiltonian only contains kinetic energy.
## Two-particle functions
-The above can be generalized to two arbitrary operators $\hat{A}$ and $\hat{B}$,
+We generalize the above to two arbitrary operators $\hat{A}$ and $\hat{B}$,
giving us the **two-particle Green's functions**,
or just **correlation functions**.
-The **retarded correlation function** $C_{AB}^R$
-and the **advanced correlation function** $C_{AB}^A$ are defined as
+The **causal correlation function** $C_{AB}$,
+the **retarded correlation function** $C_{AB}^R$
+and the **advanced correlation function** $C_{AB}^A$ are defined as follows
(in the Heisenberg picture):
$$\begin{aligned}
\boxed{
\begin{aligned}
+ C_{AB}(t, t')
+ &\equiv -\frac{i}{\hbar} \expval{\mathcal{T}\Big\{\hat{A}(t) \hat{B}(t')\Big\}}
+ \\
C_{AB}^R(t, t')
&\equiv -\frac{i}{\hbar} \Theta(t - t') \expval{\comm*{\hat{A}(t)}{\hat{B}(t')}_{\mp}}
\\
@@ -350,13 +372,15 @@ of two single-particle creation/annihilation operators.
Like for the single-particle Green's functions,
if the Hamiltonian is time-independent,
-then it can be shown that $C_{AB}^R$ and $C_{AB}^A$
+then it can be shown that the two-particle functions
only depend on the time-difference $t - t'$:
$$\begin{aligned}
- G_{\nu \nu'}^>(t, t') = G_{\nu \nu'}^>(t - t')
- \qquad \quad
- G_{\nu \nu'}^<(t, t') = G_{\nu \nu'}^<(t - t')
+ G_{\nu \nu'}(t, t') = G_{\nu \nu'}(t \!-\! t')
+ \qquad
+ G_{\nu \nu'}^R(t, t') = G_{\nu \nu'}^>(t \!-\! t')
+ \qquad
+ G_{\nu \nu'}^A(t, t') = G_{\nu \nu'}^<(t \!-\! t')
\end{aligned}$$
diff --git a/content/know/concept/imaginary-time/index.pdc b/content/know/concept/imaginary-time/index.pdc
index 68e4e02..55f163a 100644
--- a/content/know/concept/imaginary-time/index.pdc
+++ b/content/know/concept/imaginary-time/index.pdc
@@ -25,8 +25,8 @@ $$\begin{aligned}
&= \frac{1}{Z} \Tr\!\Big( \exp\!(-\beta \hat{H}_{0,S}(t)) \: \hat{A}_H(t) \: \hat{B}_H(t') \Big)
\end{aligned}$$
-Where the Hamiltonian $\hat{H}_{0,S}$ is time-independent.
-Suppose a time-dependent $\hat{H}_{1,S}$ is added,
+Where the "simple" Hamiltonian $\hat{H}_{0,S}$ is time-independent.
+Suppose a (maybe time-dependent) "difficult" $\hat{H}_{1,S}$ is added,
so that the total Hamiltonian is $\hat{H}_S = \hat{H}_{0,S} + \hat{H}_{1,S}$.
Then it is easier to consider the expectation value
in the [interaction picture](/know/concept/interaction-picture/):
--
cgit v1.2.3
+
+
+$$\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}$$
+
+Next, an **interaction line** or **boson line** represents
+a two-body interaction operator $\hat{W}$ (in $\hat{H}_1$),
+which we assume to be instantaneous, i.e. time-independent
+(in quantum field theory this is *not* assumed),
+hence it starts and ends at the same time,
+and no arrow is drawn:
+
+
+
+
+$$\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}$$
+
+One-body (time-dependent) operators $\hat{V}$ in $\hat{H}_1$
+are instead represented by a special vertex:
+
+
+
+
+$$\begin{aligned}
+ = \frac{1}{i \hbar} V_I(\vb{r}, t, \sigma)
+\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}$$
+
+Where we have used an integral representation of
+the [Dirac delta function](/know/concept/dirac-delta-function/).
+Note the inconsistent sign of the exponent
+in the Fourier transform definitions for space and time.
+
+Working in Fourier space allows us to simplify calculations.
+Consider the following diagram and the resulting expression,
+where $\tilde{\vb{r}} = (\vb{r}, t)$, and $\tilde{\vb{k}} = (\vb{k}, \omega)$:
+
+
+
+
+$$\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}$$
+
+One-body $V$-vertices are usually not used,
+because they are intended for real-time-dependent operators,
+but in theory they would get a factor $-1/\hbar$ too.
+
+For imaginary time, the Fourier transform is defined differently,
+and a distinction must be made between
+fermionic Matsubara frequencies $i \omega_n^f$ (for $G$ and $G^0$)
+and bosonic Matsubara ones $i \omega_n^b$ (for $W$).
+This distinction is compatible with frequency conservation,
+since a sum of two fermionic frequencies is always bosonic:
+
+$$\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}$$
+
+The interpretation in Fourier space is the same,
+except that each internal integral/sum
+instead gives a constant $1 / \big(\hbar \beta (2 \pi)^3\big)$,
+and same-time fermion lines need a factor of $e^{i \omega_n^f \eta}$.
+
+
+
+## References
+1. H. Bruus, K. Flensberg,
+ *Many-body quantum theory in condensed matter physics*,
+ 2016, Oxford.
diff --git a/content/know/concept/feynman-diagram/interaction.png b/content/know/concept/feynman-diagram/interaction.png
new file mode 100644
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diff --git a/content/know/concept/feynman-diagram/perturbation.png b/content/know/concept/feynman-diagram/perturbation.png
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diff --git a/content/know/concept/greens-functions/index.pdc b/content/know/concept/greens-functions/index.pdc
index 2f86e63..b3c9ede 100644
--- a/content/know/concept/greens-functions/index.pdc
+++ b/content/know/concept/greens-functions/index.pdc
@@ -32,14 +32,26 @@ If the two operators are single-particle creation/annihilation operators,
then we get the **single-particle Green's functions**,
for which the symbol $G$ is used.
-The **retarded Green's function** $G_{\nu \nu'}^R$
-and the **advanced Green's function** $G_{\nu \nu'}^A$
-are defined like so,
-where the expectation value $\expval{}$ is
+The **time-ordered** or **causal Green's function** $G_{\nu \nu'}$
+is defined as follows,
+where $\mathcal{T}$ is the [time-ordered product](/know/concept/time-ordered-product/),
+the expectation value $\expval{}$ is
with respect to thermodynamic equilibrium,
$\nu$ and $\nu'$ are labels of single-particle states,
and $\hat{c}_\nu$ annihilates a particle from $\nu$, etc.:
+$$\begin{aligned}
+ \boxed{
+ G_{\nu \nu'}(t, t')
+ \equiv -\frac{i}{\hbar} \expval{\mathcal{T} \Big\{ \hat{c}_{\nu}(t) \: \hat{c}_{\nu'}^\dagger(t') \Big\}}
+ }
+\end{aligned}$$
+
+Arguably more prevalent are
+the **retarded Green's function** $G_{\nu \nu'}^R$
+and the **advanced Green's function** $G_{\nu \nu'}^A$
+which are defined like so:
+
$$\begin{aligned}
\boxed{
\begin{aligned}
@@ -75,15 +87,19 @@ $$\begin{aligned}
}
\end{aligned}$$
-Where $-$ is for bosons, and $+$ is for fermions.
-The retarded and advanced Green's functions can thus be expressed as follows:
+Where $-$ is for bosons, and $+$ for fermions.
+With this, the causal, retarded and advanced Green's functions
+can thus be expressed as follows:
$$\begin{aligned}
+ G_{\nu \nu'}(t, t')
+ &= \Theta(t - t') \: G_{\nu \nu'}^>(t, t') + \Theta(t' - t) \: G_{\nu \nu'}^<(t, t')
+ \\
G_{\nu \nu'}^R(t, t')
- &= \Theta(t - t') \Big( G_{\nu \nu'}^>(t, t') - G_{\nu \nu'}^<(t, t') \Big)
+ &= \Theta(t - t') \big( G_{\nu \nu'}^>(t, t') - G_{\nu \nu'}^<(t, t') \big)
\\
G_{\nu \nu'}^A(t, t')
- &= \Theta(t' - t) \Big( G_{\nu \nu'}^<(t, t') - G_{\nu \nu'}^>(t, t') \Big)
+ &= \Theta(t' - t) \big( G_{\nu \nu'}^<(t, t') - G_{\nu \nu'}^>(t, t') \big)
\end{aligned}$$
If the Hamiltonian involves interactions,
@@ -93,14 +109,14 @@ In that case, instead of a label $\nu$,
we use the spin $s$ and position $\vb{r}$, leading to:
$$\begin{aligned}
- G_{ss'}^R(\vb{r}, t; \vb{r}', t')
- &= -\frac{i}{\hbar} \Theta(t - t') \expval{\comm*{\hat{\Psi}_{s}(\vb{r}, t)}{\hat{\Psi}_{s'}^\dagger(\vb{r}', t')}_{\mp}}
+ G_{ss'}(\vb{r}, t; \vb{r}', t')
+ &= -\frac{i}{\hbar} \Theta(t - t') \expval{\mathcal{T}\Big\{ \hat{\Psi}_{s}(\vb{r}, t) \hat{\Psi}_{s'}^\dagger(\vb{r}', t') \Big\}}
\\
- &= \sum_{\nu \nu'} \psi_\nu(\vb{r}) \: \psi^*_{\nu'}(\vb{r}') \: G_{\nu \nu'}^R(t, t')
+ &= \sum_{\nu \nu'} \psi_\nu(\vb{r}) \: \psi^*_{\nu'}(\vb{r}') \: G_{\nu \nu'}(t, t')
\end{aligned}$$
-And analogously for $G_{ss'}^A$, $G_{ss'}^>$ and $G_{ss'}^<$.
-Note that the time-dependence is given to the old $G_{\nu \nu'}^R$,
+And analogously for $G_{ss'}^R$, $G_{ss'}^A$, $G_{ss'}^>$ and $G_{ss'}^<$.
+Note that the time-dependence is given to the old $G_{\nu \nu'}$,
i.e. to $\hat{c}_\nu$ and $\hat{c}_{\nu'}^\dagger$,
because we are in the Heisenberg picture.
@@ -108,7 +124,9 @@ If the Hamiltonian is time-independent,
then it can be shown that all the Green's functions
only depend on the time-difference $t - t'$:
-$$\begin{aligned}
+$$\begin{gathered}
+ G_{\nu \nu'}(t, t') = G_{\nu \nu'}(t - t')
+ \\
G_{\nu \nu'}^R(t, t') = G_{\nu \nu'}^R(t - t')
\qquad \quad
G_{\nu \nu'}^A(t, t') = G_{\nu \nu'}^A(t - t')
@@ -116,7 +134,7 @@ $$\begin{aligned}
G_{\nu \nu'}^>(t, t') = G_{\nu \nu'}^>(t - t')
\qquad \quad
G_{\nu \nu'}^<(t, t') = G_{\nu \nu'}^<(t - t')
-\end{aligned}$$
+\end{gathered}$$