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author | Prefetch | 2021-11-18 20:07:12 +0100 |
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committer | Prefetch | 2021-11-18 20:07:12 +0100 |
commit | dc3498fd50121eadbdd3ddac5bf950a16e2b50cb (patch) | |
tree | b48f7f76aaaf4993d36eab276f99a3936b49160e | |
parent | c0d352dd0f66b47ee91fb96eaf320f895fa78790 (diff) |
Expand knowledge base
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-rw-r--r-- | sources/know/concept/feynman-diagram/main.tex | 83 |
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@@ -1,5 +1,4 @@ /.hugo_build.lock /public/* -/sources/**/*.aux -/sources/**/*.log -/sources/**/*.pdf +/sources/**/*.* +!/sources/**/*.tex diff --git a/content/know/concept/dyson-equation/index.pdc b/content/know/concept/dyson-equation/index.pdc index be93379..25abe11 100644 --- a/content/know/concept/dyson-equation/index.pdc +++ b/content/know/concept/dyson-equation/index.pdc @@ -63,8 +63,8 @@ Let us assume that $\hat{H}_0$ is simple, such that $G_0$ and $\hat{G}{}_0^{-1}$ can be found without issues by solving the defining equation above. -Suppose we now perturb this Hamiltonian with -a possibly time-dependent operator $\hat{H}_1(\vb{r}, t)$, +Suppose we now add a more complicated and +possibly time-dependent term $\hat{H}_1(\vb{r}, t)$, in which case the corresponding fundamental solution $G(\vb{r}, \vb{r}', t, t')$ satisfies: diff --git a/content/know/concept/feynman-diagram/conservation.png b/content/know/concept/feynman-diagram/conservation.png Binary files differnew file mode 100644 index 0000000..4068faf --- /dev/null +++ b/content/know/concept/feynman-diagram/conservation.png diff --git a/content/know/concept/feynman-diagram/freegf.png b/content/know/concept/feynman-diagram/freegf.png Binary files differnew file mode 100644 index 0000000..88071ac --- /dev/null +++ b/content/know/concept/feynman-diagram/freegf.png diff --git a/content/know/concept/feynman-diagram/fullgf.png b/content/know/concept/feynman-diagram/fullgf.png Binary files differnew file mode 100644 index 0000000..b66ca9e --- /dev/null +++ b/content/know/concept/feynman-diagram/fullgf.png diff --git a/content/know/concept/feynman-diagram/index.pdc b/content/know/concept/feynman-diagram/index.pdc new file mode 100644 index 0000000..eee6bf3 --- /dev/null +++ b/content/know/concept/feynman-diagram/index.pdc @@ -0,0 +1,337 @@ +--- +title: "Feynman diagram" +firstLetter: "F" +publishDate: 2021-11-15 +categories: +- Physics +- Quantum mechanics + +date: 2021-11-15T21:01:46+01:00 +draft: false +markup: pandoc +--- + +# Feynman diagram + +When doing calculations in the context of condensed matter physics and quantum field theory, +**Feynman diagrams** graphically represent expressions +that would be tedious or error-prone to work with directly. +This article is about condensed matter physics. + +Suppose we have a many-particle Hamiltonian $\hat{H} = \hat{H}_0 + \hat{H}_1$, +consisting of an "easy" term $\hat{H}_0$, +and then a "difficult" term $\hat{H}_1$ +with time-dependent and/or interacting parts. +Let $\ket{\Phi_0}$ be a known eigenstate (or superposition thereof) +of the easily solvable part $\hat{H}_0$, +with respect to which we will take expectation values $\expval{}$. + +Below, we go through the most notable components of Feynman diagrams +and how to translate them into a mathematical expression. + + +## Real space + +The most common component is a **fermion line**, which represents +a [Green's function](/know/concept/greens-functions/) $G^0$ +for the simple Hamiltonian $\hat{H}_0$. +Any type of Green's function is possible in theory (e.g. a retarded), +but usually the *causal* function is used. +Let the subscript $I$ refer to the +[interaction picture](/know/concept/interaction-picture/), +and $\mathcal{T}\{\}$ denote the +[time-ordered product](/know/concept/time-ordered-product/): + +<a href="freegf.png"> +<img src="freegf.png" style="width:60%;display:block;margin:auto;"> +</a> +$$\begin{aligned} + = i \hbar G_{s_2 s_1}^0(\vb{r}_2, t_2; \vb{r}_1, t_1) + = \expval{\mathcal{T} \Big\{ \hat{\Psi}_{s_2 I}(\vb{r}_2, t_2) \hat{\Psi}_{s_1 I}^\dagger(\vb{r}_1, t_1) \Big\}} +\end{aligned}$$ + +The arrow points in the direction of time, or more generally, +from the point of creation $\hat{\Psi}{}^\dagger$ +to the point of annihilation $\hat{\Psi}$. +The dots at the ends are called **vertices**, +which represent points in space and time with a spin. +Vertices can be +**internal** (one Green's function entering AND one leaving) +or **external** (either one Green's function entering OR one leaving). + +Less common is a **heavy fermion line**, representing +a causal Green's function $G$ for the entire Hamiltonian $\hat{H}$, +where the subscript $H$ refers to the [Heisenberg picture](/know/concept/heisenberg-picture/): + +<a href="fullgf.png"> +<img src="fullgf.png" style="width:60%;display:block;margin:auto;"> +</a> +$$\begin{aligned} + = i \hbar G_{s_2 s_1}(\vb{r}_2, t_2; \vb{r}_1, t_1) + = \expval{\mathcal{T} \Big\{ \hat{\Psi}_{s_2 H}(\vb{r}_2, t_2) \hat{\Psi}_{s_1 H}^\dagger(\vb{r}_1, t_1) \Big\}} +\end{aligned}$$ + +Next, an **interaction line** or **boson line** represents +a two-body interaction operator $\hat{W}$ (in $\hat{H}_1$), +which we assume to be instantaneous, i.e. time-independent +(in quantum field theory this is *not* assumed), +hence it starts and ends at the same time, +and no arrow is drawn: + +<a href="interaction.png"> +<img src="interaction.png" style="width:60%;display:block;margin:auto;"> +</a> +$$\begin{aligned} + = \frac{1}{i \hbar} W_{s_2 s_1}(\vb{r}_2, t_2; \vb{r}_1, t_1) + = \frac{1}{i \hbar} W(\vb{r}_2, \vb{r}_1; t_1) \: \delta(t_2 - t_1) +\end{aligned}$$ + +We have chosen to disallow spin flipping, +so $W$ does not depend on $s_1$ or $s_2$. +For reference, this function $W$ +has a time-dependence coming only from the interaction picture, +and is to be used as follows to get the full two-body operator $\hat{W}$: + +$$\begin{aligned} + \hat{W} + = \frac{1}{2} \sum_{s_1 s_2} \iint_{-\infty}^\infty \hat{\Psi}_{s_1}^\dagger(\vb{r}_1) \hat{\Psi}_{s_2}^\dagger(\vb{r}_2) + W(\vb{r}_1, \vb{r}_2) \hat{\Psi}_{s_2}(\vb{r}_2) \hat{\Psi}_{s_1}(\vb{r}_1) \dd{\vb{r}_1} \dd{\vb{r}_2} +\end{aligned}$$ + +One-body (time-dependent) operators $\hat{V}$ in $\hat{H}_1$ +are instead represented by a special vertex: + +<a href="perturbation.png"> +<img src="perturbation.png" style="width:35%;display:block;margin:auto;"> +</a> +$$\begin{aligned} + = \frac{1}{i \hbar} V_I(\vb{r}, t, \sigma) +\end{aligned}$$ + +Other graphical components exist representing +more complicated operators and quantities, +but these deserve their own articles. + +In order for a given Feynman diagram to be valid, +it must satisfy the following criteria: + +a. Each vertex must be connected to one or two fermion lines, + at most one of which leaves, + and at most one of which enters. +b. Each internal vertex contains at most one "event"; + which could be $V$ or $W$. + +Finally, we need some additional rules to convert +diagrams into mathematical expressions: + +1. Disallow spin flipping by multiplying + each internal vertex by $\delta_{s_\mathrm{in} s_\mathrm{out}}$. +2. If both ends of a line are at the same time (always the case for $W$), + an infinitesimal $\eta \to 0^+$ must be added + to the time of all creation operators, + so e.g. $G(t, t) \to G(t, t\!+\!\eta)$. +3. Integrate over spacetime coordinates $(\vb{r}, t)$ + and sum over the spin $s$ of all internal vertices, + but not external ones. +4. Multiply the result by $(-1)^F$, + where $F$ is the number of closed fermion loops. +5. Depending on the context, additional constant factors may be required; + sometimes they are changed on-the-fly during a calculation. + +Note that rules 4 and 5 are convention, +just like the factors $i \hbar$ in $G^0$, $G$, $V$ and $W$; +it simply turns out to be nicer to do it this way +when using Feynman diagrams in the wild. + +The combination of rules 2 and 3 means that spin +belongs to lines rather than vertices, +so that a particle with a given spin propagates +from vertex to vertex without getting flipped. + + +## Fourier space + +If the system is time-independent and spatially uniform, +meaning it has continuous translational symmetry in time and space, +then it is useful to work in [Fourier space](/know/concept/fourier-transform/): + +$$\begin{aligned} + G_{s_2 s_1}^0(\vb{r}_2, t_2; \vb{r}_1, t_1) + &= G_{s_1}^0(\vb{r}_2 - \vb{r}_1, t_2 - t_1) \: \delta_{s_2 s_1} + \\ + &= \frac{\delta_{s_2 s_1}}{(2 \pi)^4} \iint_{-\infty}^\infty G_{s_1}^0(\vb{k}, \omega) \: + e^{i \vb{k} \cdot (\vb{r}_2 - \vb{r}_1)} e^{- i \omega (t_2 - t_1)} \dd{\vb{k}} \dd{\omega} + \\ + W_{s_2 s_1}(\vb{r}_2, t_2; \vb{r}_1, t_1) + &= W(\vb{r}_2 - \vb{r}_1) \: \delta(t_2 - t_1) + \\ + &= \frac{1}{(2 \pi)^4} \iint_{\infty}^\infty W(\vb{k}) \: + e^{i \vb{k} \cdot (\vb{r}_2 - \vb{r}_1)} e^{- i \omega (t_2 - t_1)} \dd{\vb{k}} \dd{\omega} +\end{aligned}$$ + +Where we have used an integral representation of +the [Dirac delta function](/know/concept/dirac-delta-function/). +Note the inconsistent sign of the exponent +in the Fourier transform definitions for space and time. + +Working in Fourier space allows us to simplify calculations. +Consider the following diagram and the resulting expression, +where $\tilde{\vb{r}} = (\vb{r}, t)$, and $\tilde{\vb{k}} = (\vb{k}, \omega)$: + +<a href="conservation.png"> +<img src="conservation.png" style="width:40%;display:block;margin:auto;"> +</a> +$$\begin{aligned} + &= (i \hbar)^3 \sum_{s s'} \!\!\iint \dd{\tilde{\vb{r}}} \dd{\tilde{\vb{r}}'} + G_{s_1's}^0(\tilde{\vb{r}}_1', \tilde{\vb{r}}) G_{s s_1}^0(\tilde{\vb{r}}, \tilde{\vb{r}}_1) \delta_{s_1 s_1'} + W(\tilde{\vb{r}}, \tilde{\vb{r}}') + G_{s_2' s'}^0(\tilde{\vb{r}}_2', \tilde{\vb{r}}') G_{s' s_2}^0(\tilde{\vb{r}}', \tilde{\vb{r}}_2) \delta_{s_2 s_2'} + \\ + &= \frac{-i \hbar^3}{(2 \pi)^{20}} + \sum_{s_1 s_2} \!\!\iint \dd{\tilde{\vb{r}}} \dd{\tilde{\vb{r}}'} + \bigg(\! \int \dd{\tilde{\vb{k}}_2} G_{s_1}^0(\tilde{\vb{k}}_2) e^{i \tilde{\vb{k}}_2 \cdot (\tilde{\vb{r}}_1' - \tilde{\vb{r}})} \!\bigg) + \bigg(\! \int \dd{\tilde{\vb{k}}_1} G_{s_1}^0(\tilde{\vb{k}}_1) e^{i \tilde{\vb{k}}_1 \cdot (\tilde{\vb{r}} - \tilde{\vb{r}}_1)} \!\bigg) + \\ + &\qquad\times \bigg(\! \int \dd{\tilde{\vb{p}}} W(\tilde{\vb{p}}) e^{i \tilde{\vb{p}} \cdot (\tilde{\vb{r}}' - \tilde{\vb{r}})} \!\bigg) + \bigg(\! \int \dd{\tilde{\vb{q}}_2} G_{s_2}^0(\tilde{\vb{q}}_2) e^{i \tilde{\vb{q}}_2 \cdot (\tilde{\vb{r}}_2' - \tilde{\vb{r}}')} \!\bigg) + \bigg(\! \int \dd{\tilde{\vb{q}}_1} G_{s_2}^0(\tilde{\vb{q}}_1) e^{i \tilde{\vb{q}}_1 \cdot (\tilde{\vb{r}}' - \tilde{\vb{r}}_2)} \!\bigg) + \\ + &= \frac{-i \hbar^3}{(2 \pi)^{12}} + \sum_{s_1 s_2} \!\!\iint \dd{\tilde{\vb{k}}_1} \dd{\tilde{\vb{k}}_2} + G_{s_1}^0(\tilde{\vb{k}}_2) G_{s_1}^0(\tilde{\vb{k}}_1) + \iint \dd{\tilde{\vb{q}}_1} \dd{\tilde{\vb{q}}_2} + G_{s_2}^0(\tilde{\vb{q}}_2) G_{s_2}^0(\tilde{\vb{q}}_1) + \\ + &\qquad\times + e^{i \tilde{\vb{k}}_2 \cdot \tilde{\vb{r}}_1' - i \tilde{\vb{k}}_1 \cdot \tilde{\vb{r}}_1 + + i \tilde{\vb{q}}_2 \cdot \tilde{\vb{r}}_2' - i \tilde{\vb{q}}_1 \cdot \tilde{\vb{r}}_2} + \!\!\int \dd{\tilde{\vb{p}}} W(\tilde{\vb{p}}) + \bigg( \frac{1}{(2 \pi)^8} \!\!\iint \dd{\tilde{\vb{r}}} \dd{\tilde{\vb{r}}'} + e^{i (\tilde{\vb{k}}_1 - \tilde{\vb{k}}_2 - \tilde{\vb{p}}) \cdot \tilde{\vb{r}}} + e^{i (\tilde{\vb{q}}_1 - \tilde{\vb{q}}_2 + \tilde{\vb{p}}) \cdot \tilde{\vb{r}}'} \bigg) + \\ + &= \frac{-i \hbar^3}{(2 \pi)^{12}} + \sum_{s_1 s_2} \!\!\iint \dd{\tilde{\vb{k}}_1} \dd{\tilde{\vb{k}}_2} + G_{s_1}^0(\tilde{\vb{k}}_2) G_{s_1}^0(\tilde{\vb{k}}_1) + \iint \dd{\tilde{\vb{q}}_1} \dd{\tilde{\vb{q}}_2} + G_{s_2}^0(\tilde{\vb{q}}_2) G_{s_2}^0(\tilde{\vb{q}}_1) + \\ + &\qquad\times + e^{i \tilde{\vb{k}}_2 \cdot \tilde{\vb{r}}_1' - i \tilde{\vb{k}}_1 \cdot \tilde{\vb{r}}_1 + + i \tilde{\vb{q}}_2 \cdot \tilde{\vb{r}}_2' - i \tilde{\vb{q}}_1 \cdot \tilde{\vb{r}}_2} + \!\!\int \dd{\tilde{\vb{p}}} W(\tilde{\vb{p}}) + \: \delta(\tilde{\vb{k}}_1 \!-\! \tilde{\vb{k}}_2 \!-\! \tilde{\vb{p}}) + \: \delta(\tilde{\vb{q}}_1 \!-\! \tilde{\vb{q}}_2 \!+\! \tilde{\vb{p}}) + \\ + &= \frac{-i \hbar^3}{(2 \pi)^{12}} + \sum_{s_1 s_2} \!\!\int \dd{\tilde{\vb{p}}} W(\tilde{\vb{p}}) + \int \dd{\tilde{\vb{k}}_1} G_{s_1}^0(\tilde{\vb{k}}_1 \!-\! \tilde{\vb{p}}) G_{s_1}^0(\tilde{\vb{k}}_1) + \int \dd{\tilde{\vb{q}}_1} G_{s_2}^0(\tilde{\vb{q}}_1 \!+\! \tilde{\vb{p}}) G_{s_2}^0(\tilde{\vb{q}}_1) + \\ + &\qquad\times + e^{i \tilde{\vb{k}}_1 \cdot (\tilde{\vb{r}}_1' - \tilde{\vb{r}}_1)} + e^{i \tilde{\vb{q}}_1 \cdot (\tilde{\vb{r}}_2' - \tilde{\vb{r}}_2)} + e^{i \tilde{\vb{p}} \cdot (\tilde{\vb{r}}_2' - \tilde{\vb{r}}_1')} +\end{aligned}$$ + +Conveniently, the Dirac delta functions that appear from the integrals +represent conservation of wavevector $\vb{k}$ (momentum $\hbar \vb{k}$) +and angular frequency $\omega$ (energy $\hbar \omega$). + +In Fourier space, it makes more sense +to regard the incoming energies and momenta and spins as given, +and only integrate over the internal quantities. +We thus modify the Feynman diagram rules +such that we end up with the following result: + +$$\begin{aligned} + \equiv \frac{-i \hbar^3}{(2 \pi)^4} + \sum_{s} \!\!\int \dd{\tilde{\vb{p}}} W(\tilde{\vb{p}}) + \: G_{s_1}^0(\tilde{\vb{k}}_1 \!-\! \tilde{\vb{p}}) \: G_{s_1}^0(\tilde{\vb{k}}_1) + \: G_{s_2}^0(\tilde{\vb{q}}_1 \!+\! \tilde{\vb{p}}) \: G_{s_2}^0(\tilde{\vb{q}}_1) +\end{aligned}$$ + +Therefore, we say that fermion lines represent $i \hbar G_{s}^0(\vb{k}, \omega)$, +interaction lines $W(\vb{k}) / i \hbar$, etc., +and the other interpretation rules are modified to the following: + +1. Each line has a momentum $\vb{k}$ and energy $\omega$, + and each fermion line has a spin $s$; + these must all be conserved at each vertex. +2. If both ends of a *fermion* line would be at the same time, + multiply it by $e^{i \omega \eta}$, + where $\eta \to 0^+$ is a positive infinitesimal, + so e.g. $G(\tau, \tau) \to e^{i \omega \eta} G(\tau, \tau)$. +3. Integrate over all internal $(\vb{k}, \omega)$, + and sum over all internal spins $s$. + Let each $(\vb{k}, \omega)$ integral contribute a factor $1 / (2 \pi)^4$. +4. Multiply the end result by $(-1)^F$, where $F$ is the number of closed fermion loops. +5. Depending on the context, additional constant factors may be required; + sometimes they are changed on-the-fly during a calculation. + +Note that if the diagram is linear (i.e. does not contain interactions), +then conservation removes all internal variables, +so no integrals would be needed. + + +## Imaginary time + +Feynman diagrams are also useful when working with +[imaginary time](/know/concept/imaginary-time/). +In that case, the meaning of fermion lines is changed as follows, +involving the [Matsubara Green's function](/know/concept/matsubara-greens-function/): + +$$\begin{aligned} + i \hbar G_{s_2 s_1}^0(\vb{r}_2, t_2; \vb{r}_1, t_1) + \:\: &\longrightarrow \:\: + -\!\hbar G_{s_2 s_1}^0(\vb{r}_2, \tau_2; \vb{r}_1, \tau_1) + = \expval{\mathcal{T} \Big\{ \hat{\Psi}_I(\vb{r}_2, \tau_2) \hat{\Psi}_I^\dagger(\vb{r}_1, \tau_1) \Big\}} + \\ + i \hbar G_{s_2 s_1}(\vb{r}_2, t_2; \vb{r}_1, t_1) + \:\: &\longrightarrow \:\: + -\!\hbar G_{s_2 s_1}(\vb{r}_2, \tau_2; \vb{r}_1, \tau_1) + = \expval{\mathcal{T} \Big\{ \hat{\Psi}_H(\vb{r}_2, \tau_2) \hat{\Psi}_H^\dagger(\vb{r}_1, \tau_1) \Big\}} +\end{aligned}$$ + +Where the time-ordering is with respect to $\tau$. +Interaction lines are modified like so: + +$$\begin{aligned} + \frac{1}{i \hbar} W_{s_2 s_1}(\vb{r}_2, t_2; \vb{r}_1, t_1) + \:\: &\longrightarrow \:\: + -\!\frac{1}{\hbar} W_{s_2 s_1}(\vb{r}_2, \tau_2; \vb{r}_1, \tau_1) + = -\frac{1}{\hbar} W(\vb{r}_2, \vb{r}_1; \tau_1) \delta(\tau_2 \!-\! \tau_1) +\end{aligned}$$ + +One-body $V$-vertices are usually not used, +because they are intended for real-time-dependent operators, +but in theory they would get a factor $-1/\hbar$ too. + +For imaginary time, the Fourier transform is defined differently, +and a distinction must be made between +fermionic Matsubara frequencies $i \omega_n^f$ (for $G$ and $G^0$) +and bosonic Matsubara ones $i \omega_n^b$ (for $W$). +This distinction is compatible with frequency conservation, +since a sum of two fermionic frequencies is always bosonic: + +$$\begin{aligned} + G_{s_2 s_1}^0(\vb{r}_2, \tau_2; \vb{r}_1, \tau_1) + &= \frac{\delta_{s_2 s_1}}{(2 \pi)^3} \int_{-\infty}^\infty \frac{1}{\hbar \beta} \sum_{n = -\infty}^\infty + G_{s_1}^0(\vb{k}, i \omega_n^f) e^{i \vb{k} \cdot (\vb{r}_2 - \vb{r}_1)} e^{- i \omega_n^f (\tau_2 - \tau_1)} \dd{\vb{k}} + \\ + W_{s_2 s_1}(\vb{r}_2, \tau_2; \vb{r}_1, \tau_1) + &= \frac{1}{(2 \pi)^3} \int_{-\infty}^\infty \frac{1}{\hbar \beta} \sum_{n = -\infty}^\infty + W(\vb{k}) e^{i \vb{k} \cdot (\vb{r}_2 - \vb{r}_1)} e^{- i \omega_n^b (\tau_2 - \tau_1)} \dd{\vb{k}} +\end{aligned}$$ + +The interpretation in Fourier space is the same, +except that each internal integral/sum +instead gives a constant $1 / \big(\hbar \beta (2 \pi)^3\big)$, +and same-time fermion lines need a factor of $e^{i \omega_n^f \eta}$. + + + +## References +1. H. Bruus, K. Flensberg, + *Many-body quantum theory in condensed matter physics*, + 2016, Oxford. diff --git a/content/know/concept/feynman-diagram/interaction.png b/content/know/concept/feynman-diagram/interaction.png Binary files differnew file mode 100644 index 0000000..9b4661a --- /dev/null +++ b/content/know/concept/feynman-diagram/interaction.png diff --git a/content/know/concept/feynman-diagram/perturbation.png b/content/know/concept/feynman-diagram/perturbation.png Binary files differnew file mode 100644 index 0000000..fa1da9e --- /dev/null +++ b/content/know/concept/feynman-diagram/perturbation.png 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 @@ -32,16 +32,28 @@ 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: + +$$\begin{aligned} + \boxed{ \begin{aligned} G_{\nu \nu'}^R(t, t') &\equiv -\frac{i}{\hbar} \Theta(t - t') \expval{\comm*{\hat{c}_{\nu}(t)}{\hat{c}_{\nu'}^\dagger(t')}_{\mp}} @@ -75,15 +87,19 @@ $$\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}$$ +\end{gathered}$$ <div class="accordion"> <input type="checkbox" id="proof-time-diff"/> @@ -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}$$ diff --git a/content/know/concept/imaginary-time/index.pdc b/content/know/concept/imaginary-time/index.pdc index 68e4e02..55f163a 100644 --- a/content/know/concept/imaginary-time/index.pdc +++ b/content/know/concept/imaginary-time/index.pdc @@ -25,8 +25,8 @@ $$\begin{aligned} &= \frac{1}{Z} \Tr\!\Big( \exp\!(-\beta \hat{H}_{0,S}(t)) \: \hat{A}_H(t) \: \hat{B}_H(t') \Big) \end{aligned}$$ -Where the Hamiltonian $\hat{H}_{0,S}$ is time-independent. -Suppose a time-dependent $\hat{H}_{1,S}$ is added, +Where the "simple" Hamiltonian $\hat{H}_{0,S}$ is time-independent. +Suppose a (maybe time-dependent) "difficult" $\hat{H}_{1,S}$ is added, so that the total Hamiltonian is $\hat{H}_S = \hat{H}_{0,S} + \hat{H}_{1,S}$. Then it is easier to consider the expectation value in the [interaction picture](/know/concept/interaction-picture/): diff --git a/sources/know/concept/feynman-diagram/main.tex b/sources/know/concept/feynman-diagram/main.tex new file mode 100644 index 0000000..6a7b39e --- /dev/null +++ b/sources/know/concept/feynman-diagram/main.tex @@ -0,0 +1,83 @@ +\documentclass[11pt]{article} +\usepackage[utf8]{inputenc} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{physics} + +\usepackage{feynmp} +\DeclareGraphicsRule{*}{mps}{*}{} + +\begin{document} + +\begin{center} + {\Huge \sc Feynman diagrams} +\end{center} + +\begin{fmffile}{freegf} + \begin{fmfgraph*}(110,60) + \fmfleft{v1} + \fmfright{v2} + \fmflabel{$(\mathbf{r}_1,t_1,s_1)$}{v1} + \fmflabel{$(\mathbf{r}_2,t_2,s_2)$}{v2} + \fmf{fermion}{v1,v2} + \fmfdot{v1} + \fmfdot{v2} + \end{fmfgraph*} +\end{fmffile} + +\begin{fmffile}{fullgf} + \begin{fmfgraph*}(110,60) + \fmfleft{v1} + \fmfright{v2} + \fmflabel{$(\mathbf{r}_1,t_1,s_1)$}{v1} + \fmflabel{$(\mathbf{r}_2,t_2,s_2)$}{v2} + \fmf{heavy}{v1,v2} + \fmfdot{v1} + \fmfdot{v2} + \end{fmfgraph*} +\end{fmffile} + +\begin{fmffile}{perturbation} + \begin{fmfgraph*}(110,60) + \fmfleft{i} + \fmfright{o} + \fmf{vanilla}{i,m} + \fmf{vanilla}{m,o} + \fmfv{decoration.shape=hexagram,label=$(\mathbf{r},t,s)$,label.angle=-90}{m} + \fmflabel{$(\mathbf{r},t,s)$}{m} + \end{fmfgraph*} +\end{fmffile} + +\begin{fmffile}{interaction} + \begin{fmfgraph*}(110,60) + \fmfleft{v1} + \fmfright{v2} + \fmflabel{$(\mathbf{r}_1,t_1,s_1)$}{v1} + \fmflabel{$(\mathbf{r}_2,t_2,s_2)$}{v2} + \fmf{boson}{v1,v2} + \fmfdot{v1} + \fmfdot{v2} + \end{fmfgraph*} +\end{fmffile} + +\begin{fmffile}{conservation} + \begin{fmfgraph*}(110,60) + \fmfleft{i1,o1} + \fmfright{i2,o2} + \fmf{fermion,label=$\tilde{\mathbf{k}}_1$}{i1,v1} + \fmf{fermion,label=$\tilde{\mathbf{k}}_2$}{v1,o1} + \fmf{fermion,label=$\tilde{\mathbf{q}}_1$,label.side=left}{i2,v2} + \fmf{fermion,label=$\tilde{\mathbf{q}}_2$}{v2,o2} + \fmf{boson,label=$\mathbf{p}$}{v1,v2} + \fmfdot{v1} + \fmfdot{v2} + \fmfv{label=$\tilde{\mathbf{r}}_1$,label.angle=180}{i1} + \fmfv{label=$\tilde{\mathbf{r}}_1'$,label.angle=180}{o1} + \fmflabel{$\tilde{\mathbf{r}}$}{v1} + \fmflabel{$\tilde{\mathbf{r}}'$}{v2} + \fmfv{label=$\tilde{\mathbf{r}}_2$,label.angle=0}{i2} + \fmfv{label=$\tilde{\mathbf{r}}_2'$,label.angle=0}{o2} + \end{fmfgraph*} +\end{fmffile} + +\end{document} |