From dc3498fd50121eadbdd3ddac5bf950a16e2b50cb Mon Sep 17 00:00:00 2001 From: Prefetch Date: Thu, 18 Nov 2021 20:07:12 +0100 Subject: Expand knowledge base --- content/know/concept/greens-functions/index.pdc | 68 +++++++++++++++++-------- 1 file changed, 46 insertions(+), 22 deletions(-) (limited to 'content/know/concept/greens-functions') 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}$$ -- cgit v1.2.3