---
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, a **Green's function**
can be any correlation function between two given operators,
although it is usually used to refer to the special case
where the operators are particle creation/annihilation operators
from the [second quantization](/know/concept/second-quantization/).
They are somewhat related to
[fundamental solutions](/know/concept/fundamental-solution/),
which are also called *Green's functions*,
but in general they are not the same,
except in a special case, see below.
## Single-particle functions
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 **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:
$$\begin{aligned}
\boxed{
\begin{aligned}
G_{\nu \nu'}^R(t, t')
&\equiv -\frac{i}{\hbar} \Theta(t - t') \expval{\comm*{\hat{c}_{\nu}(t)}{\hat{c}_{\nu'}^\dagger(t')}_{\mp}}
\\
G_{\nu \nu'}^A(t, t')
&\equiv \frac{i}{\hbar} \Theta(t' - t) \expval{\comm*{\hat{c}_{\nu}(t)}{\hat{c}_{\nu'}^\dagger(t')}_{\mp}}
\end{aligned}
}
\end{aligned}$$
Where $\Theta$ is a [Heaviside function](/know/concept/heaviside-step-function/),
and $[,]_{\mp}$ is a commutator for bosons,
and an anticommutator for fermions.
We are in the [Heisenberg picture](/know/concept/heisenberg-picture/),
hence $\hat{c}_\nu$ and $\hat{c}_{\nu'}^\dagger$ are time-dependent,
but keep in mind that time-dependent Hamiltonians are allowed,
so it might not be trivial.
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 $+$ for fermions.
With this, the causal, retarded and advanced Green's functions
can thus be expressed as follows:
$$\begin{aligned}
G_{\nu \nu'}(t, t')
&= \Theta(t - t') \: G_{\nu \nu'}^>(t, t') + \Theta(t' - t) \: G_{\nu \nu'}^<(t, t')
\\
G_{\nu \nu'}^R(t, t')
&= \Theta(t - t') \big( G_{\nu \nu'}^>(t, t') - G_{\nu \nu'}^<(t, t') \big)
\\
G_{\nu \nu'}^A(t, t')
&= \Theta(t' - t) \big( G_{\nu \nu'}^<(t, t') - G_{\nu \nu'}^>(t, t') \big)
\end{aligned}$$
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:
$$\begin{aligned}
G_{ss'}(\vb{r}, t; \vb{r}', t')
&= -\frac{i}{\hbar} \Theta(t - t') \expval{\mathcal{T}\Big\{ \hat{\Psi}_{s}(\vb{r}, t) \hat{\Psi}_{s'}^\dagger(\vb{r}', t') \Big\}}
\\
&= \sum_{\nu \nu'} \psi_\nu(\vb{r}) \: \psi^*_{\nu'}(\vb{r}') \: G_{\nu \nu'}(t, t')
\end{aligned}$$
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.
If the Hamiltonian is time-independent,
then it can be shown that all the Green's functions
only depend on the time-difference $t - t'$:
$$\begin{gathered}
G_{\nu \nu'}(t, t') = G_{\nu \nu'}(t - t')
\\
G_{\nu \nu'}^R(t, t') = G_{\nu \nu'}^R(t - t')
\qquad \quad
G_{\nu \nu'}^A(t, t') = G_{\nu \nu'}^A(t - t')
\\
G_{\nu \nu'}^>(t, t') = G_{\nu \nu'}^>(t - t')
\qquad \quad
G_{\nu \nu'}^<(t, t') = G_{\nu \nu'}^<(t - t')
\end{gathered}$$
We will prove that the thermal expectation value
$\expval*{\hat{A}(t) \hat{B}(t')}$ only depends on $t - t'$
for arbitrary $\hat{A}$ and $\hat{B}$,
and it trivially follows that the Green's functions do too.
Suppose that the system started in thermodynamic equilibrium.
This could sometimes be in the [canonical ensemble](/know/concept/canonical-ensemble/)
(for two-particle Green's functions, see below),
but usually it will be in the
[grand canonical ensemble](/know/concept/grand-canonical-ensemble/),
since we are adding/removing particles.
In the latter case, we assume that the chemical potential $\mu$
is already included in the Hamiltonian.
In any case, at equilibrium, we know that the
[density operator](/know/concept/density-operator/)
$\hat{\rho}$ is as follows:
$$\begin{aligned}
\hat{\rho} = \frac{1}{Z} \exp\!(- \beta \hat{H})
\end{aligned}$$
Where $Z \equiv \Tr\!(\exp\!(- \beta \hat{H}))$ is the partition function.
In that case, the expected value of the product
of the time-independent operators $\hat{A}$ and $\hat{B}$ is calculated like so:
$$\begin{aligned}
\expval*{\hat{A}(t) \hat{B}(t')}
&= \frac{1}{Z} \Tr\!\big( \hat{\rho} \hat{A}(t) \hat{B}(t') \big)
\\
&= \frac{1}{Z} \Tr\!\Big( e^{-\beta \hat{H}} e^{i t \hat{H} / \hbar} \hat{A} e^{-i t \hat{H} / \hbar}
e^{i t' \hat{H} / \hbar} \hat{B} e^{-i t' \hat{H} / \hbar} \Big)
\end{aligned}$$
Using that the trace $\Tr$ is invariant
under cyclic permutations of its argument,
and that all functions of $\hat{H}$ commute, we find:
$$\begin{aligned}
\expval*{\hat{A}(t) \hat{B}(t')}
= \frac{1}{Z} \Tr\!\Big( e^{-\beta \hat{H}} e^{i (t - t') \hat{H} / \hbar} \hat{A} e^{-i (t - t') \hat{H} / \hbar} \hat{B} \Big)
\end{aligned}$$
As expected, this only depends on the time difference $t - t'$,
because $\hat{H}$ is time-independent by assumption.
Note that thermodynamic equilibrium is crucial:
intuitively, if the system is not in equilibrium,
then it evolves in some transient time-dependent way.
If the Hamiltonian is both time-independent and non-interacting,
then the time-dependence of $\hat{c}_\nu$
can simply be factored out as
$\hat{c}_\nu(t) = \hat{c}_\nu \exp\!(- i \varepsilon_\nu t / \hbar)$.
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/).
$$\begin{aligned}
G_{\nu \nu}^>(t, t')
&= -\frac{i}{\hbar} \expval{\hat{c}_{\nu} \hat{c}_{\nu}^\dagger} \exp\!\big(\!-\! i \varepsilon_\nu (t \!-\! t') / \hbar \big)
\\
&= -\frac{i}{\hbar} (1 - f_\nu) \exp\!\big(\!-\! i \varepsilon_\nu (t \!-\! t') / \hbar \big)
\\
G_{\nu \nu}^<(t, t')
&= \mp \frac{i}{\hbar} \expval{\hat{c}_{\nu}^\dagger \hat{c}_{\nu}} \exp\!\big(\!-\! i \varepsilon_\nu (t \!-\! t') / \hbar \big)
\\
&= \mp \frac{i}{\hbar} f_\nu \exp\!\big(\!-\! i \varepsilon_\nu (t \!-\! t') / \hbar \big)
\end{aligned}$$
## As fundamental solutions
In the absence of interactions,
we know from the derivation of
[equation-of-motion theory](/know/concept/equation-of-motion-theory/)
that the equation of motion of $G^R(\vb{r}, t; \vb{r}', t')$
is as follows (neglecting spin):
$$\begin{aligned}
i \hbar \pdv{G^R}{t}
= \delta(\vb{r} \!-\! \vb{r}') \: \delta(t \!-\! t')
+ \frac{i}{\hbar} \Theta(t \!-\! t') \expval{\comm{\comm*{\hat{H}_0}{\hat{\Psi}(\vb{r}, t)}}{\hat{\Psi}^\dagger(\vb{r}', t')}}
\end{aligned}$$
If $\hat{H}_0$ only contains kinetic energy,
i.e. there is no external potential,
it can be shown that:
$$\begin{aligned}
\comm*{\hat{H}_0}{\hat{\Psi}(\vb{r})}
= \frac{\hbar^2}{2 m} \nabla^2 \hat{\Psi}(\vb{r})
\end{aligned}$$
In the second quantization,
the Hamiltonian $\hat{H}_0$ is written like so:
$$\begin{aligned}
\hat{H}_0
&= - \frac{\hbar^2}{2 m} \sum_{\nu \nu'} \hat{c}_\nu^\dagger \hat{c}_{\nu'} \braket{\psi_\nu}{\nabla^2 \psi_{\nu'}}
\\
&= - \frac{\hbar^2}{2 m} \sum_{\nu \nu'} \hat{c}_\nu^\dagger \hat{c}_{\nu'} \int \psi_\nu^*(\vb{r}') \: \nabla^2 \psi_{\nu'}(\vb{r}') \dd{\vb{r}'}
\\
&= - \frac{\hbar^2}{2 m}
\int \Big( \sum_{\nu} \psi_\nu^*(\vb{r}') \hat{c}_\nu^\dagger \Big) \Big( \nabla^2 \sum_{\nu'} \psi_{\nu'}(\vb{r}') \hat{c}_{\nu'} \Big) \dd{\vb{r}'}
\\
&= - \frac{\hbar^2}{2 m}
\int \hat{\Psi}^\dagger(\vb{r}') \: \nabla^2 \hat{\Psi}(\vb{r}') \dd{\vb{r}'}
\end{aligned}$$
We then insert this into the commutator that we want to prove, yielding:
$$\begin{aligned}
\comm*{\hat{H}_0}{\hat{\Psi}(\vb{r})}
&= - \frac{\hbar^2}{2 m} \int \comm{\hat{\Psi}^\dagger(\vb{r}') \: \nabla^2 \hat{\Psi}(\vb{r}')}{\hat{\Psi}(\vb{r})} \dd{\vb{r}'}
\\
&= - \frac{\hbar^2}{2 m} \int \hat{\Psi}^\dagger(\vb{r}') \comm{\nabla^2 \hat{\Psi}(\vb{r}')}{\hat{\Psi}(\vb{r})}
+ \comm{\hat{\Psi}^\dagger(\vb{r}')}{\hat{\Psi}(\vb{r})} \nabla^2 \hat{\Psi}(\vb{r}') \dd{\vb{r}'}
\\
&= - \frac{\hbar^2}{2 m} \sum_{\nu \nu' \nu''}
\Big( \hat{c}_\nu^\dagger \comm*{\hat{c}_{\nu''}}{\hat{c}_{\nu'}} + \comm*{\hat{c}_\nu^\dagger}{\hat{c}_{\nu'}} \hat{c}_{\nu''} \Big)
\psi_{\nu'}(\vb{r}) \int \psi_\nu^*(\vb{r}') \: \nabla^2 \psi_{\nu''}(\vb{r}') \dd{\vb{r}'}
\end{aligned}$$
When deriving equation-of-motion theory,
we already showed that the following identity
holds for both bosons and fermions:
$$\begin{aligned}
\hat{c}_\nu^\dagger \comm*{\hat{c}_{\nu''}}{\hat{c}_{\nu'}} + \comm*{\hat{c}_\nu^\dagger}{\hat{c}_{\nu'}} \hat{c}_{\nu''}
= - \delta_{\nu \nu'} \hat{c}_{\nu''}
\end{aligned}$$
Such that the commutator can be significantly simplified to:
$$\begin{aligned}
\comm*{\hat{H}_0}{\hat{\Psi}(\vb{r})}
&= \frac{\hbar^2}{2 m} \sum_{\nu \nu'} \hat{c}_{\nu'}
\int \psi_\nu^*(\vb{r}') \: \psi_\nu(\vb{r}) \: \nabla^2 \psi_{\nu'}(\vb{r}') \dd{\vb{r}'}
\end{aligned}$$
We know that the $\psi_\nu$ form a *complete* basis,
which implies (see [Sturm-Liouville theory](/know/concept/sturm-liouville-theory/)):
$$\begin{aligned}
\sum_{\nu} \psi_\nu^*(\vb{r}') \: \psi_\nu(\vb{r})
= \delta(\vb{r} - \vb{r}')
\end{aligned}$$
With this, the commutator can be reduced even further as follows:
$$\begin{aligned}
\comm*{\hat{H}_0}{\hat{\Psi}(\vb{r})}
&= \frac{\hbar^2}{2 m} \sum_{\nu \nu'} \hat{c}_{\nu'}
\int \delta(\vb{r} - \vb{r}') \: \nabla^2 \psi_{\nu'}(\vb{r}') \dd{\vb{r}'}
\\
&= \frac{\hbar^2}{2 m} \sum_{\nu'} \hat{c}_{\nu'} \nabla^2 \psi_{\nu'}(\vb{r})
= \frac{\hbar^2}{2 m} \nabla^2 \hat{\Psi}(\vb{r})
\end{aligned}$$
After substituting this into the equation of motion,
we recognize $G^R(\vb{r}, t; \vb{r}', t')$ itself:
$$\begin{aligned}
i \hbar \pdv{G^R}{t}
&= \delta(\vb{r} \!-\! \vb{r}') \: \delta(t \!-\! t')
+ \frac{i}{\hbar} \Theta(t \!-\! t') \expval{\comm{\frac{\hbar^2}{2 m} \nabla^2 \hat{\Psi}(\vb{r}, t)}{\hat{\Psi}^\dagger(\vb{r}', t')}}
\\
&= \delta(\vb{r} \!-\! \vb{r}') \: \delta(t \!-\! t') - \frac{\hbar^2}{2 m} \nabla_\vb{r}^2
\Big( \!-\! \frac{i}{\hbar} \Theta(t \!-\! t') \expval{\comm{\hat{\Psi}(\vb{r}, t)}{\hat{\Psi}^\dagger(\vb{r}', t')}} \Big)
\\
&= \delta(\vb{r} \!-\! \vb{r}') \: \delta(t \!-\! t')
- \frac{\hbar^2}{2 m} \nabla_\vb{r}^2 G^R(\vb{r}, t; \vb{r}', t')
\end{aligned}$$
Rearranging this leads to the following,
which is the definition of a fundamental solution:
$$\begin{aligned}
\Big( i \hbar \pdv{t} + \frac{\hbar^2}{2 m} \nabla_\vb{r}^2 \Big) G^R(\vb{r}, t; \vb{r}', t')
&= \delta(\vb{r} \!-\! \vb{r}') \: \delta(t \!-\! t')
\end{aligned}$$
Therefore, the retarded Green's function
(and, it turns out, the advanced Green's function too)
is a fundamental solution of the Schrödinger equation
if there is no potential,
i.e. the Hamiltonian only contains kinetic energy.
## Two-particle functions
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 **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):
$$\begin{aligned}
\boxed{
\begin{aligned}
C_{AB}(t, t')
&\equiv -\frac{i}{\hbar} \expval{\mathcal{T}\Big\{\hat{A}(t) \hat{B}(t')\Big\}}
\\
C_{AB}^R(t, t')
&\equiv -\frac{i}{\hbar} \Theta(t - t') \expval{\comm*{\hat{A}(t)}{\hat{B}(t')}_{\mp}}
\\
C_{AB}^A(t, t')
&\equiv \frac{i}{\hbar} \Theta(t' - t) \expval{\comm*{\hat{A}(t)}{\hat{B}(t')}_{\mp}}
\end{aligned}
}
\end{aligned}$$
Where the expectation value $\expval{}$ is taken of thermodynamic equilibrium.
The name *two-particle* comes from the fact that $\hat{A}$ and $\hat{B}$
will often consist of a sum of products
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 the two-particle functions
only depend on the time-difference $t - t'$:
$$\begin{aligned}
G_{\nu \nu'}(t, t') = G_{\nu \nu'}(t \!-\! t')
\qquad
G_{\nu \nu'}^R(t, t') = G_{\nu \nu'}^>(t \!-\! t')
\qquad
G_{\nu \nu'}^A(t, t') = G_{\nu \nu'}^<(t \!-\! t')
\end{aligned}$$
## References
1. H. Bruus, K. Flensberg,
*Many-body quantum theory in condensed matter physics*,
2016, Oxford.