---
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.