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diff --git a/source/know/concept/lehmann-representation/index.md b/source/know/concept/lehmann-representation/index.md index cfc6838..74bd457 100644 --- a/source/know/concept/lehmann-representation/index.md +++ b/source/know/concept/lehmann-representation/index.md @@ -11,11 +11,11 @@ layout: "concept" In many-body quantum theory, the **Lehmann representation** is an alternative way to write the [Green's functions](/know/concept/greens-functions/), obtained by expanding in the many-particle eigenstates -under the assumption of a time-independent Hamiltonian $\hat{H}$. +under the assumption of a time-independent Hamiltonian $$\hat{H}$$. -First, we write out the greater Green's function $G_{\nu \nu'}^>(t, t')$, -and then expand its expected value $\Expval{}$ (at thermodynamic equilibrium) -into a sum of many-particle basis states $\Ket{n}$: +First, we write out the greater Green's function $$G_{\nu \nu'}^>(t, t')$$, +and then expand its expected value $$\Expval{}$$ (at thermodynamic equilibrium) +into a sum of many-particle basis states $$\Ket{n}$$: $$\begin{aligned} G_{\nu \nu'}^>(t, t') @@ -23,13 +23,13 @@ $$\begin{aligned} &= - \frac{i}{\hbar Z} \sum_{n} \Matrixel{n}{\hat{c}_\nu(t) \hat{c}_{\nu'}^\dagger(t') e^{-\beta \hat{H}}}{n} \end{aligned}$$ -Where $\beta = 1 / (k_B T)$, and $Z$ is the grand partition function +Where $$\beta = 1 / (k_B T)$$, and $$Z$$ is the grand partition function (see [grand canonical ensemble](/know/concept/grand-canonical-ensemble/)); -the operator $e^{\beta \hat{H}}$ gives the weight of each term at equilibrium. -Since $\Ket{n}$ is an eigenstate of $\hat{H}$ with energy $E_n$, -this gives us a factor of $e^{\beta E_n}$. +the operator $$e^{\beta \hat{H}}$$ gives the weight of each term at equilibrium. +Since $$\Ket{n}$$ is an eigenstate of $$\hat{H}$$ with energy $$E_n$$, +this gives us a factor of $$e^{\beta E_n}$$. Furthermore, we are in the [Heisenberg picture](/know/concept/heisenberg-picture/), -so we write out the time-dependence of $\hat{c}_\nu$ and $\hat{c}_{\nu'}^\dagger$: +so we write out the time-dependence of $$\hat{c}_\nu$$ and $$\hat{c}_{\nu'}^\dagger$$: $$\begin{aligned} G_{\nu \nu'}^>(t, t') @@ -40,10 +40,10 @@ $$\begin{aligned} \Matrixel{n}{e^{i \hat{H} (t - t') / \hbar} \hat{c}_\nu e^{- i \hat{H} (t - t') / \hbar} \hat{c}_{\nu'}^\dagger}{n} \end{aligned}$$ -Where we used that the trace $\Tr\!(x) = \sum_{n} \matrixel{n}{x}{n}$ -is invariant under cyclic permutations of $x$. -The $\Ket{n}$ form a basis of eigenstates of $\hat{H}$, -so we insert an identity operator $\sum_{n'} \Ket{n'} \Bra{n'}$: +Where we used that the trace $$\Tr\!(x) = \sum_{n} \matrixel{n}{x}{n}$$ +is invariant under cyclic permutations of $$x$$. +The $$\Ket{n}$$ form a basis of eigenstates of $$\hat{H}$$, +so we insert an identity operator $$\sum_{n'} \Ket{n'} \Bra{n'}$$: $$\begin{aligned} G_{\nu \nu'}^>(t - t') @@ -54,10 +54,10 @@ $$\begin{aligned} \matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n} e^{i (E_n - E_{n'}) (t - t') / \hbar} \end{aligned}$$ -Note that $G_{\nu \nu'}^>$ now only depends on the time difference $t - t'$, -because $\hat{H}$ is time-independent. +Note that $$G_{\nu \nu'}^>$$ now only depends on the time difference $$t - t'$$, +because $$\hat{H}$$ is time-independent. Next, we take the [Fourier transform](/know/concept/fourier-transform/) -$t \to \omega$ (with $t' = 0$): +$$t \to \omega$$ (with $$t' = 0$$): $$\begin{aligned} G_{\nu \nu'}^>(\omega) @@ -66,9 +66,9 @@ $$\begin{aligned} \end{aligned}$$ Here, we recognize the integral -as a [Dirac delta function](/know/concept/dirac-delta-function/) $\delta$, -thereby introducing a factor of $2 \pi$, -and arriving at the Lehmann representation of $G_{\nu \nu'}^>$: +as a [Dirac delta function](/know/concept/dirac-delta-function/) $$\delta$$, +thereby introducing a factor of $$2 \pi$$, +and arriving at the Lehmann representation of $$G_{\nu \nu'}^>$$: $$\begin{aligned} \boxed{ @@ -78,7 +78,7 @@ $$\begin{aligned} } \end{aligned}$$ -We now go through the same process for the lesser Green's function $G_{\nu \nu'}^<(t, t')$: +We now go through the same process for the lesser Green's function $$G_{\nu \nu'}^<(t, t')$$: $$\begin{aligned} G_{\nu \nu'}^<(t - t') @@ -88,7 +88,7 @@ $$\begin{aligned} e^{i (E_{n'} - E_n) (t - t') / \hbar} \end{aligned}$$ -Where $-$ is for bosons, and $+$ for fermions. +Where $$-$$ is for bosons, and $$+$$ for fermions. Fourier transforming yields the following: $$\begin{aligned} @@ -97,8 +97,8 @@ $$\begin{aligned} \: \delta(E_{n'} - E_n + \hbar \omega) \end{aligned}$$ -We swap $n$ and $n'$, leading to the following -Lehmann representation of $G_{\nu \nu'}^<$: +We swap $$n$$ and $$n'$$, leading to the following +Lehmann representation of $$G_{\nu \nu'}^<$$: $$\begin{aligned} \boxed{ @@ -108,8 +108,8 @@ $$\begin{aligned} } \end{aligned}$$ -Due to the delta function $\delta$, -each term is only nonzero for $E_n' = E_n + \hbar \omega$, +Due to the delta function $$\delta$$, +each term is only nonzero for $$E_n' = E_n + \hbar \omega$$, so we write: $$\begin{aligned} @@ -119,7 +119,7 @@ $$\begin{aligned} \end{aligned}$$ Therefore, we arrive at the following useful relation -between $G_{\nu \nu'}^<$ and $G_{\nu \nu'}^>$: +between $$G_{\nu \nu'}^<$$ and $$G_{\nu \nu'}^>$$: $$\begin{aligned} \boxed{ @@ -129,7 +129,7 @@ $$\begin{aligned} \end{aligned}$$ Moving on, let us do the same for -the retarded Green's function $G_{\nu \nu'}^R(t, t')$, given by: +the retarded Green's function $$G_{\nu \nu'}^R(t, t')$$, given by: $$\begin{aligned} G_{\nu \nu'}^R(t \!-\! t') @@ -141,7 +141,7 @@ $$\begin{aligned} \end{aligned}$$ We take the Fourier transform, but to ensure convergence, -we must introduce an infinitesimal positive $\eta \to 0^+$ to the exponent +we must introduce an infinitesimal positive $$\eta \to 0^+$$ to the exponent (and eventually take the limit): $$\begin{aligned} @@ -155,7 +155,7 @@ $$\begin{aligned} \end{aligned}$$ Leading us to the following Lehmann representation -of the retarded Green's function $G_{\nu \nu'}^R$: +of the retarded Green's function $$G_{\nu \nu'}^R$$: $$\begin{aligned} \boxed{ @@ -166,7 +166,7 @@ $$\begin{aligned} } \end{aligned}$$ -Finally, we go through the same steps for the advanced Green's function $G_{\nu \nu'}^A(t, t')$: +Finally, we go through the same steps for the advanced Green's function $$G_{\nu \nu'}^A(t, t')$$: $$\begin{aligned} G_{\nu \nu'}^A(t \!-\! t') @@ -177,7 +177,7 @@ $$\begin{aligned} \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) e^{i (E_n - E_{n'}) (t - t') / \hbar} \end{aligned}$$ -For the Fourier transform, we must again introduce $\eta \to 0^+$ +For the Fourier transform, we must again introduce $$\eta \to 0^+$$ (although note the sign): $$\begin{aligned} @@ -191,7 +191,7 @@ $$\begin{aligned} \end{aligned}$$ Therefore, the Lehmann representation of -the advanced Green's function $G_{\nu \nu'}^A$ is as follows: +the advanced Green's function $$G_{\nu \nu'}^A$$ is as follows: $$\begin{aligned} \boxed{ @@ -211,8 +211,8 @@ $$\begin{aligned} \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) \end{aligned}$$ -Note the subscripts $\nu$ and $\nu'$. -Comparing this to $G_{\nu \nu'}^R$ gives us another useful relation: +Note the subscripts $$\nu$$ and $$\nu'$$. +Comparing this to $$G_{\nu \nu'}^R$$ gives us another useful relation: $$\begin{aligned} \boxed{ |