Expand knowledge base

6 files changed, 555 insertions, 11 deletions

diff --git a/content/know/concept/dyson-equation/index.pdc b/content/know/concept/dyson-equation/index.pdc
index 7b94124..be93379 100644
--- a/content/know/concept/dyson-equation/index.pdc
+++ b/content/know/concept/dyson-equation/index.pdc
@@ -21,7 +21,8 @@ $$\begin{aligned}
     = \hat{H}_0(\vb{r}) \: \Psi_0(\vb{r}, t)
 \end{aligned}$$
 
-By definition, this equation's *fundamental solution*
+By definition, this equation's
+[fundamental solution](/know/concept/fundamental-solution/)
 $G_0(\vb{r}, t; \vb{r}', t')$ satisfies the following:
 
 $$\begin{aligned}
@@ -142,7 +143,7 @@ and integrate over $G$'s second argument pair:
 
 $$\begin{aligned}
     \iint \big( \hat{G}{}_0^{-1} \!-\! \hat{H}_1 \big) G(\vb{r}', t') \: f(\vb{r}', t') \dd{\vb{r}'} \dd{t'}
-    = \iint^\infty \delta(\vb{r} \!-\! \vb{r}') \: \delta(t \!-\! t') \: f(\vb{r}, t) \dd{\vb{r}'} \dd{t'}
+    = \iint \delta(\vb{r} \!-\! \vb{r}') \: \delta(t \!-\! t') \: f(\vb{r}', t') \dd{\vb{r}'} \dd{t'}
     = f
 \end{aligned}$$
 
diff --git a/content/know/concept/fundamental-solution/index.pdc b/content/know/concept/fundamental-solution/index.pdc
new file mode 100644
index 0000000..2ae9899
--- /dev/null
+++ b/content/know/concept/fundamental-solution/index.pdc
@@ -0,0 +1,151 @@
+---
+title: "Fundamental solution"
+firstLetter: "F"
+publishDate: 2021-11-02
+categories:
+- Mathematics
+- Physics
+
+date: 2021-11-01T14:57:46+01:00
+draft: false
+markup: pandoc
+---
+
+# Fundamental solution
+
+Given a linear operator $\hat{L}$ acting on $x \in [a, b]$,
+its **fundamental solution** $G(x, x')$ is defined as the response
+of $\hat{L}$ to a [Dirac delta function](/know/concept/dirac-delta-function/)
+$\delta(x - x')$ for $x \in ]a, b[$:
+
+$$\begin{aligned}
+    \boxed{
+        \hat{L}\{ G(x, x') \}
+        = A \delta(x - x')
+    }
+\end{aligned}$$
+
+Where $A$ is a constant, usually $1$.
+Fundamental solutions are often called **Green's functions**,
+but are distinct from the (somewhat related)
+[Green's functions](/know/concept/greens-functions/)
+in many-body quantum theory.
+
+Note that the definition of $G(x, x')$ generalizes that of
+the [impulse response](/know/concept/impulse-response/).
+And likewise, due to the superposition principle,
+once $G$ is known, $\hat{L}$'s response $u(x)$ to
+*any* forcing function $f(x)$ can easily be found as follows:
+
+$$\begin{aligned}
+    \hat{L} \{ u(x) \}
+    = f(x)
+        \quad \implies \quad
+    \boxed{
+        u(x)
+        = \frac{1}{A} \int_a^b f(x') \: G(x, x') \dd{x'}
+    }
+\end{aligned}$$
+
+<div class="accordion">
+<input type="checkbox" id="proof-solution"/>
+<label for="proof-solution">Proof</label>
+<div class="hidden">
+<label for="proof-solution">Proof.</label>
+$\hat{L}$ only acts on $x$, so $x' \in ]a, b[$ is simply a parameter,
+meaning we are free to multiply the definition of $G$
+by the constant $f(x')$ on both sides,
+and exploit $\hat{L}$'s linearity:
+
+$$\begin{aligned}
+    A f(x') \: \delta(x - x')
+    = f(x') \hat{L}\{ G(x, x') \}
+    = \hat{L}\{ f(x') \: G(x, x') \}
+\end{aligned}$$
+
+We then integrate both sides over $x'$ in the interval $[a, b]$,
+allowing us to consume $\delta(x \!-\! x')$.
+Note that $\int \dd{x'}$ commutes with $\hat{L}$ acting on $x$:
+
+$$\begin{aligned}
+    A \int_a^b f(x') \: \delta(x - x') \dd{x'}
+    &= \int_a^b \hat{L}\{ f(x') \: G(x, x') \} \dd{x'}
+    \\
+    A f(x)
+    &= \hat{L} \int_a^b f(x') \: G(x, x') \dd{x'}
+\end{aligned}$$
+
+By definition, $\hat{L}$'s response $u(x)$ to $f(x)$
+satisfies $\hat{L}\{ u(x) \} = f(x)$, recognizable here.
+</div>
+</div>
+
+While the impulse response is typically used for initial value problems,
+the fundamental solution $G$ is used for boundary value problems.
+Suppose those boundary conditions are homogeneous,
+i.e. $u(x)$ or one of its derivatives is zero at the boundaries.
+Then:
+
+$$\begin{aligned}
+    0
+    &= u(a)
+    = \frac{1}{A} \int_a^b f(x') \: G(a, x') \dd{x'}
+    \qquad \implies \quad
+    G(a, x') = 0
+    \\
+    0
+    &= u_x(a)
+    = \frac{1}{A} \int_a^b f(x') \: G_x(a, x') \dd{x'}
+    \quad \implies \quad
+    G_x(a, x') = 0
+\end{aligned}$$
+
+This holds for all $x'$, and analogously for the other boundary $x = b$.
+In other words, the boundary conditions are built into $G$.
+
+What if the boundary conditions are inhomogeneous?
+No problem: thanks to the linearity of $\hat{L}$,
+those conditions can be given to the homogeneous solution $u_h(x)$,
+where $\hat{L}\{ u_h(x) \} = 0$,
+such that the inhomogeneous solution $u_i(x) = u(x) - u_h(x)$
+has homogeneous boundaries again,
+so we can use $G$ as usual to find $u_i(x)$, and then just add $u_h(x)$.
+
+If $\hat{L}$ is self-adjoint
+(see e.g. [Sturm-Liouville theory](/know/concept/sturm-liouville-theory/)),
+then the fundamental solution $G(x, x')$
+has the following **reciprocity** boundary condition:
+
+$$\begin{aligned}
+    \boxed{
+        G(x, x') = G^*(x', x)
+    }
+\end{aligned}$$
+
+<div class="accordion">
+<input type="checkbox" id="proof-reciprocity"/>
+<label for="proof-reciprocity">Proof</label>
+<div class="hidden">
+<label for="proof-reciprocity">Proof.</label>
+Consider two parameters $x_1'$ and $x_2'$.
+The self-adjointness of $\hat{L}$ means that:
+
+$$\begin{aligned}
+    \int_a^b G^*(x, x_1') \Big( \hat{L} \{ G(x, x_2') \} \Big) \dd{x}
+    &= \int_a^b \Big( \hat{L} \{ G(x, x_1') \} \Big)^* G(x, x_2') \dd{x}
+    \\
+    \int_a^b G^*(x, x_1') \: \delta(x - x_2') \dd{x}
+    &= \int_a^b \delta^*(x - x_1') \: G(x, x_2') \dd{x}
+    \\
+    G^*(x_2', x_1')
+    &= G(x_1', x_2')
+\end{aligned}$$
+</div>
+</div>
+
+
+
+## References
+1.  O. Bang,
+    *Applied mathematics for physicists: lecture notes*, 2019,
+    unpublished.
diff --git a/content/know/concept/greens-functions/index.pdc b/content/know/concept/greens-functions/index.pdc
new file mode 100644
index 0000000..10ab09b
--- /dev/null
+++ b/content/know/concept/greens-functions/index.pdc
@@ -0,0 +1,152 @@
+---
+title: "Green's functions"
+firstLetter: "G"
+publishDate: 2021-11-03
+categories:
+- Physics
+- Quantum mechanics
+
+date: 2021-11-01T09:46:27+01:00
+draft: false
+markup: pandoc
+---
+
+# Green's functions
+
+In many-body quantum theory, **Green's functions**
+are correlation functions between particle creation/annihilation operators.
+They are somewhat related to
+[fundamental solution](/know/concept/fundamental-solution/) functions,
+which are also often called *Green's functions*.
+
+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
+with respect to thermal equilibrium,
+$\nu$ and $\nu'$ are labels of single-particle states that may include spin,
+and $\hat{c}_\nu$ and $\hat{c}_{\nu'}^\dagger$ are annihilation/creation operators
+from the [second quantization](/know/concept/second-quantization/):
+
+$$\begin{aligned}
+    \boxed{
+        \begin{aligned}
+            G_{\nu \nu'}^R(t, t')
+            &\equiv -\frac{i}{\hbar} \Theta(t - t') \expval{\comm{\hat{c}_{\nu}(t)}{\hat{c}_{\nu'}^\dagger(t')}}
+            \\
+            G_{\nu \nu'}^A(t, t')
+            &\equiv \frac{i}{\hbar} \Theta(t' - t) \expval{\comm{\hat{c}_{\nu}(t)}{\hat{c}_{\nu'}^\dagger(t')}}
+        \end{aligned}
+    }
+\end{aligned}$$
+
+Where $\Theta$ is the [Heaviside step function](/know/concept/heaviside-step-function/).
+This is for bosons; for fermions the commutator
+must be replaced by an anticommutator, as usual.
+Notice that $G^R_{\nu \nu'}$ has the same form as the correlation function
+from the [Kubo formula](/know/concept/kubo-formula/).
+
+Furthermore, the **greater Green's function** $G_{\nu \nu'}^>$
+and **lesser Green's function** $G_{\nu \nu'}^<$ are:
+
+$$\begin{aligned}
+    \boxed{
+        \begin{aligned}
+            G_{\nu \nu'}^>(t, t')
+            &\equiv -\frac{i}{\hbar} \expval{\hat{c}_{\nu}(t) \hat{c}_{\nu'}^\dagger(t')}
+            \\
+            G_{\nu \nu'}^<(t, t')
+            &\equiv \mp \frac{i}{\hbar} \expval{\hat{c}_{\nu'}^\dagger(t') \hat{c}_{\nu}(t)}
+        \end{aligned}
+    }
+\end{aligned}$$
+
+Where $-$ is for bosons, and $+$ is for fermions.
+The retarded and advanced Green's functions can thus be expressed as follows:
+
+$$\begin{aligned}
+    G_{\nu \nu'}^R(t, t')
+    &= \Theta(t - t') \Big( G_{\nu \nu'}^>(t, t') - G_{\nu \nu'}^<(t, t') \Big)
+    \\
+    G_{\nu \nu'}^A(t, t')
+    &= \Theta(t' - t) \Big( G_{\nu \nu'}^<(t, t') - G_{\nu \nu'}^>(t, t') \Big)
+\end{aligned}$$
+
+If the Hamiltonian involves interactions,
+it might be more natural to use quantum field operators $\hat{\Psi}(\vb{r}, t)$
+instead of choosing a basis of single-particle states $\psi_\nu$.
+In that case, instead of a label $\nu$,
+we use the spin $s$ and position $\vb{r}$, leading to:
+
+$$\begin{aligned}
+    G_{ss'}^R(\vb{r}, t; \vb{r}', t')
+    &= -\frac{i}{\hbar} \Theta(t - t') \expval{\comm{\hat{\Psi}_{s}(\vb{r}, t)}{\hat{\Psi}_{s'}^\dagger(\vb{r}', t')}}
+    \\
+    &= \sum_{\nu \nu'} \psi_\nu(\vb{r}) \: \psi^*_{\nu'}(\vb{r}') \: G_{\nu \nu'}^R(t, t')
+\end{aligned}$$
+
+And analogously for $G_{ss'}^A$, $G_{ss'}^>$ and $G_{ss'}^<$.
+Note that the time-dependence is given to the old $G_{\nu \nu'}^R$,
+i.e. to $\hat{c}_\nu$ and $\hat{c}_{\nu'}^\dagger$.
+In other words, we are using the
+[Heisenberg picture](/know/concept/heisenberg-picture/).
+
+If the Hamiltonian is time-independent,
+then it can be shown that all the Green's functions
+only depend on the time-difference $t - t'$
+(for a proof, see [Kubo formula](/know/concept/kubo-formula/)):
+
+$$\begin{aligned}
+    G_{\nu \nu'}^>(t, t') = G_{\nu \nu'}^>(t - t')
+    \qquad \quad
+    G_{\nu \nu'}^<(t, t') = G_{\nu \nu'}^<(t - t')
+\end{aligned}$$
+
+
+If the Hamiltonian is both time-independent and non-interacting,
+then the time-dependence of $\hat{c}_\nu$
+can simply be factored out as follows:
+
+$$\begin{aligned}
+    \hat{c}_\nu(t)
+    = \hat{c}_\nu \exp\!(- i \varepsilon_\nu t / \hbar)
+\end{aligned}$$
+
+Then the diagonal ($\nu = \nu'$) greater and lesser Green's functions
+can be written in the form below, where $f_\nu$ is either
+the [Fermi-Dirac distribution](/know/concept/fermi-dirac-distribution/)
+or the [Bose-Einstein distribution](/know/concept/bose-einstein-distribution/).
+Note that the off-diagonal ($\nu \neq \nu'$) functions vanish,
+because $\expval*{\hat{c}_{\nu} \hat{c}_{\nu'}^\dagger} = 0$ there,
+since the many-particle states are simply orthogonal
+[Slater determinants](/know/concept/slater-determinant/)/permanents:
+
+$$\begin{aligned}
+    G_{\nu \nu}^>(t, t')
+    &= -\frac{i}{\hbar} \expval{\hat{c}_{\nu} \hat{c}_{\nu}^\dagger} \exp\!\big(\!-\! i \varepsilon_\nu (t \!-\! t') / \hbar \big)
+    \\
+    &= -\frac{i}{\hbar} (1 - f_\nu) \exp\!\big(\!-\! i \varepsilon_\nu (t \!-\! t') / \hbar \big)
+    \\
+    G_{\nu \nu}^<(t, t')
+    &= \mp \frac{i}{\hbar} \expval{\hat{c}_{\nu}^\dagger \hat{c}_{\nu}} \exp\!\big(\!-\! i \varepsilon_\nu (t \!-\! t') / \hbar \big)
+    \\
+    &= \mp \frac{i}{\hbar} f_\nu \exp\!\big(\!-\! i \varepsilon_\nu (t \!-\! t') / \hbar \big)
+\end{aligned}$$
+
+The diagonal retarded and advanced Green's functions then reduce to
+the following, where $+$ applies to fermions, and $-$ to bosons:
+
+$$\begin{aligned}
+    G_{\nu \nu}^R(t, t')
+    &= - \frac{i}{\hbar} \Theta(t - t') \big( 1 - f_\nu \pm f_\nu \big) \exp\!\big(\!-\! i \varepsilon_\nu (t \!-\! t') / \hbar \big)
+    \\
+    G_{\nu \nu}^A(t, t')
+    &= \frac{i}{\hbar} \Theta(t - t') \big( 1 - f_\nu \pm f_\nu \big) \exp\!\big(\!-\! i \varepsilon_\nu (t \!-\! t') / \hbar \big)
+\end{aligned}$$
+
+
+
+## References
+1.  H. Bruus, K. Flensberg,
+    *Many-body quantum theory in condensed matter physics*,
+    2016, Oxford.
diff --git a/content/know/concept/impulse-response/index.pdc b/content/know/concept/impulse-response/index.pdc
index fa921fa..f4c40a8 100644
--- a/content/know/concept/impulse-response/index.pdc
+++ b/content/know/concept/impulse-response/index.pdc
@@ -28,9 +28,9 @@ This can be used to find the response $u(t)$ of $\hat{L}$ to
 by simply taking the convolution with $u_p(t)$:
 
 $$\begin{aligned}
+    \hat{L} \{ u(t) \} = f(t)
+    \quad \implies \quad
     \boxed{
-        \hat{L} \{ u(t) \} = f(t)
-        \quad \implies \quad
         u(t) = (f * u_p)(t)
     }
 \end{aligned}$$
@@ -68,6 +68,17 @@ $$\begin{aligned}
 </div>
 </div>
 
+This is useful for solving initial value problems,
+because any initial condition can be satisfied
+due to the linearity of $\hat{L}$,
+by choosing the initial values of the homogeneous solution $\hat{L}\{ u_h(t) \} = 0$
+such that the total solution $(f * u_p)(t) + u_h(t)$
+has the desired values.
+
+Meanwhile, for boundary value problems,
+the related [fundamental solution](/know/concept/fundamental-solution/)
+is preferable.
+
 
 
 ## References
diff --git a/content/know/concept/lehmann-representation/index.pdc b/content/know/concept/lehmann-representation/index.pdc
new file mode 100644
index 0000000..f38f803
--- /dev/null
+++ b/content/know/concept/lehmann-representation/index.pdc
@@ -0,0 +1,235 @@
+---
+title: "Lehmann representation"
+firstLetter: "L"
+publishDate: 2021-11-03
+categories:
+- Physics
+- Quantum mechanics
+
+date: 2021-11-02T19:04:16+01:00
+draft: false
+markup: pandoc
+---
+
+# Lehmann representation
+
+In many-body quantum theory, the **Lehmann representation**
+is an alternative way to write the [Green's functions](/know/concept/greens-functions/),
+obtained by expanding in the many-particle eigenstates
+under the assumption of a time-independent Hamiltonian $\hat{H}$.
+
+We start by writing out the
+greater Green's function $G_{\nu \nu'}(t, t')$,
+and then expanding its thermal expectation value $\expval{}$
+into a sum of many-particle basis states $\ket{n}$:
+
+$$\begin{aligned}
+    G_{\nu \nu'}^>(t, t')
+    = - \frac{i}{\hbar} \expval{\hat{c}_\nu(t) \hat{c}_{\nu'}^\dagger(t')}
+    &= - \frac{i}{\hbar Z} \sum_{n} \matrixel**{n}{\hat{c}_\nu(t) \hat{c}_{\nu'}^\dagger(t') e^{-\beta \hat{H}}}{n}
+\end{aligned}$$
+
+Where $\beta = 1 / (k_B T)$, and $Z$ is the partition function
+(see [canonical ensemble](/know/concept/canonical-ensemble/));
+the operator $e^{\beta \hat{H}}$ gives the weight of each term at thermal equilibrium.
+Since $\ket{n}$ is an eigenstate of $\hat{H}$ with energy $E_n$,
+this gives us a factor of $e^{\beta E_n}$.
+Furthermore, we are in the [Heisenberg picture](/know/concept/heisenberg-picture/),
+so we write out the time-dependence of $\hat{c}_\nu$ and $\hat{c}_{\nu'}^\dagger$:
+
+$$\begin{aligned}
+    G_{\nu \nu'}^>(t, t')
+    &= - \frac{i}{\hbar Z} \sum_{n} e^{-\beta E_n} \matrixel**{n}{e^{i \hat{H} t / \hbar} \hat{c}_\nu e^{- i \hat{H} t / \hbar}
+    e^{i \hat{H} t' / \hbar} \hat{c}_{\nu'}^\dagger e^{- i \hat{H} t' / \hbar}}{n}
+    \\
+    &= - \frac{i}{\hbar Z} \sum_{n} e^{-\beta E_n}
+    \matrixel**{n}{e^{i \hat{H} (t - t') / \hbar} \hat{c}_\nu e^{- i \hat{H} (t - t') / \hbar} \hat{c}_{\nu'}^\dagger}{n}
+\end{aligned}$$
+
+Where we used that the trace $\Tr\!(x) = \sum_{n} \matrixel{n}{x}{n}$
+is invariant under cyclic permutations of $x$.
+The $\ket{n}$ form a basis of eigenstates of $\hat{H}$,
+so we insert an identity operator $\sum_{n'} \ket{n'} \bra{n'}$:
+
+$$\begin{aligned}
+    G_{\nu \nu'}^>(t - t')
+    &= - \frac{i}{\hbar Z} \sum_{n n'} e^{- \beta E_n}
+    \matrixel**{n}{e^{i \hat{H} (t - t') / \hbar} \hat{c}_\nu e^{- i \hat{H} (t - t') / \hbar}}{n'} \matrixel**{n'}{\hat{c}_{\nu'}^\dagger}{n}
+    \\
+    &= - \frac{i}{\hbar Z} \sum_{n n'} e^{-\beta E_n}
+    \matrixel*{n}{\hat{c}_\nu}{n'} \matrixel*{n'}{\hat{c}_{\nu'}^\dagger}{n} e^{i (E_n - E_{n'}) (t - t') / \hbar}
+\end{aligned}$$
+
+Note that $G_{\nu \nu'}^>$ now only depends on the time difference $t - t'$,
+because $\hat{H}$ is time-independent.
+Next, we take the [Fourier transform](/know/concept/fourier-transform/)
+$t \to \omega$ (with $t' = 0$):
+
+$$\begin{aligned}
+    G_{\nu \nu'}^>(\omega)
+    &= - \frac{i}{\hbar Z} \sum_{n n'} e^{-\beta E_n} \matrixel*{n}{\hat{c}_\nu}{n'} \matrixel*{n'}{\hat{c}_{\nu'}^\dagger}{n}
+    \int_{-\infty}^\infty e^{i (E_n - E_{n'}) t / \hbar} \: e^{i \omega t} \dd{t}
+\end{aligned}$$
+
+Here, we recognize the integral
+as a [Dirac delta function](/know/concept/dirac-delta-function/) $\delta$,
+thereby introducing a factor of $2 \pi$,
+and arriving at the Lehmann representation of $G_{\nu \nu'}^>$:
+
+$$\begin{aligned}
+    \boxed{
+        G_{\nu \nu'}^>(\omega)
+        = - \frac{2 \pi i}{Z} \sum_{n n'} e^{-\beta E_n} \matrixel*{n}{\hat{c}_\nu}{n'} \matrixel*{n'}{\hat{c}_{\nu'}^\dagger}{n}
+        \: \delta(E_n - E_{n'} + \hbar \omega)
+    }
+\end{aligned}$$
+
+We now go through the same process for the lesser Green's function $G_{\nu \nu'}^<(t, t')$:
+
+$$\begin{aligned}
+    G_{\nu \nu'}^<(t - t')
+    &= \mp \frac{i}{\hbar Z} \sum_{n} \matrixel*{n}{\hat{c}_{\nu'}^\dagger(t') \hat{c}_\nu(t) e^{-\beta \hat{H}}}{n}
+    \\
+    &= \mp \frac{i}{\hbar Z} e^{-\beta E_n} \sum_{n n'} \matrixel*{n}{\hat{c}_{\nu'}^\dagger}{n'} \matrixel*{n'}{\hat{c}_\nu}{n}
+   e^{i (E_{n'} - E_n) (t - t') / \hbar}
+\end{aligned}$$
+
+Where $-$ is for bosons, and $+$ for fermions.
+Fourier transforming yields the following:
+
+$$\begin{aligned}
+    G_{\nu \nu'}^<(\omega)
+    &= \mp \frac{2 \pi i}{\hbar Z} \sum_{n n'} e^{-\beta E_n} \matrixel*{n}{\hat{c}_{\nu'}^\dagger}{n'} \matrixel*{n'}{\hat{c}_\nu}{n}
+    \: \delta(E_{n'} - E_n + \hbar \omega)
+\end{aligned}$$
+
+We swap $n$ and $n'$, leading to the following
+Lehmann representation of $G_{\nu \nu'}^<$:
+
+$$\begin{aligned}
+    \boxed{
+        G_{\nu \nu'}^<(\omega)
+        = \mp \frac{2 \pi i}{Z} \sum_{n n'} e^{-\beta E_{n'}} \matrixel*{n}{\hat{c}_\nu}{n'} \matrixel*{n'}{\hat{c}_{\nu'}^\dagger}{n}
+        \: \delta(E_n - E_{n'} + \hbar \omega)
+    }
+\end{aligned}$$
+
+Due to the delta function $\delta$,
+each term is only nonzero for $E_n' = E_n + \hbar \omega$,
+so we write:
+
+$$\begin{aligned}
+    G_{\nu \nu'}^<(\omega)
+    = \mp \frac{2 \pi i}{\hbar Z} \sum_{n n'} e^{-\beta (E_n + \hbar \omega)}
+    \matrixel*{n}{\hat{c}_\nu}{n'} \matrixel*{n'}{\hat{c}_{\nu'}^\dagger}{n} \: \delta(E_n - E_{n'} + \hbar \omega)
+\end{aligned}$$
+
+Therefore, we arrive at the following useful relation
+between $G_{\nu \nu'}^<$ and $G_{\nu \nu'}^>$:
+
+$$\begin{aligned}
+    \boxed{
+        G_{\nu \nu'}^<(\omega)
+        = \pm e^{-\beta \hbar \omega} G_{\nu \nu'}^>(\omega)
+    }
+\end{aligned}$$
+
+Moving on, let us do the same for
+the retarded Green's function $G_{\nu \nu'}^R(t, t')$, given by:
+
+$$\begin{aligned}
+    G_{\nu \nu'}^R(t \!-\! t')
+    &= \Theta(t \!-\! t') \Big( G_{\nu \nu'}^>(t - t') - G_{\nu \nu'}^<(t - t') \Big)
+    \\
+    &= - \frac{i}{\hbar Z} \Theta(t \!-\! t') \sum_{n n'}
+    \matrixel*{n}{\hat{c}_\nu}{n'} \matrixel*{n'}{\hat{c}_{\nu'}^\dagger}{n}
+     \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) e^{i (E_n - E_{n'}) (t - t') / \hbar}
+\end{aligned}$$
+
+We take the Fourier transform, but to ensure convergence,
+we must introduce an infinitesimal positive $\eta \to 0^+$ to the exponent
+(and eventually take the limit):
+
+$$\begin{aligned}
+    G_{\nu \nu'}^R(\omega)
+    &= - \frac{i}{\hbar Z} \sum_{n n'} \Big( ... \Big) \int_{-\infty}^\infty \Theta(t) e^{i (E_n - E_{n'}) t / \hbar} e^{i (\omega + i \eta) t} \dd{t}
+    \\
+    &= - \frac{i}{\hbar Z} \sum_{n n'} \Big( ... \Big) \int_0^\infty e^{i (E_n - E_{n'}) t / \hbar} e^{i (\omega + i \eta) t} \dd{t}
+    \\
+    &= - \frac{i}{\hbar Z} \sum_{n n'} \Big( ... \Big)
+    \bigg[ \frac{\hbar e^{i (\hbar \omega + E_n - E_{n'}) t / \hbar} e^{- \eta t}}{i (\hbar \omega + E_n - E_{n'}) - \hbar \eta} \bigg]_0^\infty
+\end{aligned}$$
+
+Leading us to the following Lehmann representation
+of the retarded Green's function $G_{\nu \nu'}^R$:
+
+$$\begin{aligned}
+    \boxed{
+        G_{\nu \nu'}^R(\omega)
+        = \frac{1}{Z} \sum_{n n'}
+        \frac{\matrixel*{n}{\hat{c}_\nu}{n'} \matrixel*{n'}{\hat{c}_{\nu'}^\dagger}{n}}{\hbar (\omega + i \eta) + E_n - E_{n'}}
+        \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big)
+    }
+\end{aligned}$$
+
+Finally, we go through the same steps for the advanced Green's function $G_{\nu \nu'}^A(t, t')$:
+
+$$\begin{aligned}
+    G_{\nu \nu'}^A(t \!-\! t')
+    &= \Theta(t' \!-\! t) \Big( G_{\nu \nu'}^<(t - t') - G_{\nu \nu'}^>(t - t') \Big)
+    \\
+    &= \frac{i}{\hbar Z} \Theta(t' \!-\! t) \sum_{n n'}
+    \matrixel*{n}{\hat{c}_\nu}{n'} \matrixel*{n'}{\hat{c}_{\nu'}^\dagger}{n}
+     \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) e^{i (E_n - E_{n'}) (t - t') / \hbar}
+\end{aligned}$$
+
+For the Fourier transform, we must again introduce $\eta \to 0^+$
+(although note the sign):
+
+$$\begin{aligned}
+    G_{\nu \nu'}^A(\omega)
+    &= \frac{i}{\hbar Z} \sum_{n n'} \Big( ... \Big) \int_{-\infty}^\infty \Theta(-t) e^{i (E_n - E_{n'}) t / \hbar} e^{i (\omega - i \eta) t} \dd{t}
+    \\
+    &= \frac{i}{\hbar Z} \sum_{n n'} \Big( ... \Big) \int_{-\infty}^0 e^{i (E_n - E_{n'}) t / \hbar} e^{i (\omega - i \eta) t} \dd{t}
+    \\
+    &= \frac{i}{\hbar Z} \sum_{n n'} \Big( ... \Big)
+    \bigg[ \frac{\hbar e^{i (\hbar \omega + E_n - E_{n'}) t / \hbar} e^{\eta t}}{i (\hbar \omega + E_n - E_{n'}) + \hbar \eta} \bigg]_{-\infty}^0
+\end{aligned}$$
+
+Therefore, the Lehmann representation of
+the advanced Green's function $G_{\nu \nu'}^A$ is as follows:
+
+$$\begin{aligned}
+    \boxed{
+        G_{\nu \nu'}^A(\omega)
+        = \frac{1}{Z} \sum_{n n'}
+        \frac{\matrixel*{n}{\hat{c}_\nu}{n'} \matrixel*{n'}{\hat{c}_{\nu'}^\dagger}{n}}{\hbar (\omega - i \eta) + E_n - E_{n'}}
+        \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big)
+    }
+\end{aligned}$$
+
+As a final note, let us take the complex conjugate of this expression:
+
+$$\begin{aligned}
+    \big( G_{\nu \nu'}^A(\omega) \big)^*
+    = \frac{1}{Z} \sum_{n n'}
+    \frac{\matrixel*{n}{\hat{c}_{\nu'}}{n'} \matrixel*{n'}{\hat{c}_\nu^\dagger}{n}}{\hbar (\omega + i \eta) + E_n - E_{n'}}
+    \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big)
+\end{aligned}$$
+
+Note the subscripts $\nu$ and $\nu'$.
+Comparing this to $G_{\nu \nu'}^R$ gives us another useful relation:
+
+$$\begin{aligned}
+    \boxed{
+        G^R_{\nu \nu'}(\omega)
+        = \big( G^A_{\nu' \nu}(\omega) \big)^*
+    }
+\end{aligned}$$
+
+
+
+## References
+1.  H. Bruus, K. Flensberg,
+    *Many-body quantum theory in condensed matter physics*,
+    2016, Oxford.
diff --git a/content/know/concept/propagator/index.pdc b/content/know/concept/propagator/index.pdc
index 517376c..bcc1de7 100644
--- a/content/know/concept/propagator/index.pdc
+++ b/content/know/concept/propagator/index.pdc
@@ -61,7 +61,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Sometimes the name "propagator" is also used to refer to
-the so-called *fundamental solution* $G$
+the [fundamental solution](/know/concept/fundamental-solution/) $G$
 of the time-dependent Schrödinger equation,
 which is related to $K$ by:
 
@@ -73,9 +73,3 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Where $\Theta(t)$ is the [Heaviside step function](/know/concept/heaviside-step-function/).
-The definition of $G$ is that it satisfies the following equation,
-where $\delta$ is the [Dirac delta function](/know/concept/dirac-delta-function/):
-
-$$\begin{aligned}
-    \Big( i \hbar \pdv{t_f} - \hat{H} \Big) G = \delta(x_f - x_i) \: \delta(t_f - t_i)
-\end{aligned}$$