Categories: Physics, Quantum mechanics.

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.

They are somewhat related to fundamental solutions, 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, 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, and \([,]_{\mp}\) is a commutator for bosons, and an anticommutator for fermions. We are in the 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 \(t - t'\):

\[\begin{gathered} G_{\nu \nu'}(t, t') = G_{\nu \nu'}(t - t') \\ G_{\nu \nu'}^R(t, t') = G_{\nu \nu'}^R(t - t') \qquad \quad G_{\nu \nu'}^A(t, t') = G_{\nu \nu'}^A(t - t') \\ G_{\nu \nu'}^>(t, t') = G_{\nu \nu'}^>(t - t') \qquad \quad G_{\nu \nu'}^<(t, t') = G_{\nu \nu'}^<(t - t') \end{gathered}\]

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 or the 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 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:

\[\begin{aligned} \comm*{\hat{H}_0}{\hat{\Psi}(\vb{r})} = \frac{\hbar^2}{2 m} \nabla^2 \hat{\Psi}(\vb{r}) \end{aligned}\]

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.

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.
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