--- title: "Green's functions" sort_title: "Green's functions" date: 2021-11-03 categories: - Physics - Quantum mechanics layout: "concept" --- 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 as follows, where $$\mathcal{T}$$ is the [time-ordered product](/know/concept/time-ordered-product/), $$\nu$$ and $$\nu'$$ are single-particle states, and $$\hat{c}_\nu$$ annihilates a particle from $$\nu$$, etc.: $$\begin{aligned} \boxed{ G_{\nu \nu'}(t, t') \equiv -\frac{i}{\hbar} \Expval{\mathcal{T} \Big\{ \hat{c}_{\nu}(t) \: \hat{c}_{\nu'}^\dagger(t') \Big\}} } \end{aligned}$$ The expectation value $$\Expval{}$$ is with respect to thermodynamic equilibrium. This is sometimes in the [canonical ensemble](/know/concept/canonical-ensemble/) (for some two-particle Green's functions, see below), but usually 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 $$\hat{H}$$. Explicitly, for a complete set of many-particle states $$\Ket{\Psi_n}$$, we have: $$\begin{aligned} G_{\nu \nu'}(t, t') &= -\frac{i}{\hbar Z} \Tr\!\Big( \mathcal{T} \Big\{ \hat{c}_{\nu}(t) \: \hat{c}_{\nu'}^\dagger(t')\Big\} \: e^{- \beta \hat{H}} \Big) \\ &= -\frac{i}{\hbar Z} \sum_{n} \Matrixel{\Psi_n}{\mathcal{T} \Big\{ \hat{c}_{\nu}(t) \: \hat{c}_{\nu'}^\dagger(t')\Big\} \: e^{- \beta \hat{H}}}{\Psi_n} \end{aligned}$$ 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. Depending on the context, we could either be in the [Heisenberg picture](/know/concept/heisenberg-picture/) or in the [interaction picture](/know/concept/interaction-picture/), hence $$\hat{c}_\nu$$ and $$\hat{c}_{\nu'}^\dagger$$ are time-dependent. 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 $$t - t'$$: $$\begin{gathered} G_{\nu \nu'}(t, t') = G_{\nu \nu'}(t - t') \\ G_{\nu \nu'}^R(t, t') = G_{\nu \nu'}^R(t - t') \qquad \quad G_{\nu \nu'}^A(t, t') = G_{\nu \nu'}^A(t - t') \\ G_{\nu \nu'}^>(t, t') = G_{\nu \nu'}^>(t - t') \qquad \quad G_{\nu \nu'}^<(t, t') = G_{\nu \nu'}^<(t - t') \end{gathered}$$