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+---
+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.