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diff --git a/source/know/concept/feynman-diagram/index.md b/source/know/concept/feynman-diagram/index.md new file mode 100644 index 0000000..15410f7 --- /dev/null +++ b/source/know/concept/feynman-diagram/index.md @@ -0,0 +1,332 @@ +--- +title: "Feynman diagram" +date: 2021-11-18 +categories: +- Physics +- Quantum mechanics +layout: "concept" +--- + +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%"> +</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%"> +</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%"> +</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%"> +</a> +$$\begin{aligned} + = \frac{1}{i \hbar} V_s(\vb{r}, t) +\end{aligned}$$ + +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%"> +</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. +We have: + +$$\begin{aligned} + G_{s_2 s_1}^0(\vb{r}_2, \tau_2; \vb{r}_1, \tau_1) + &= \frac{\delta_{s_2 s_1}}{(2 \pi)^3} \int_{-\infty}^\infty \frac{1}{\hbar \beta} \sum_{n = -\infty}^\infty + G_{s_1}^0(\vb{k}, i \omega_n^f) e^{i \vb{k} \cdot (\vb{r}_2 - \vb{r}_1)} e^{- i \omega_n^f (\tau_2 - \tau_1)} \dd{\vb{k}} + \\ + W_{s_2 s_1}(\vb{r}_2, \tau_2; \vb{r}_1, \tau_1) + &= \frac{1}{(2 \pi)^3} \int_{-\infty}^\infty \frac{1}{\hbar \beta} \sum_{n = -\infty}^\infty + W(\vb{k}) e^{i \vb{k} \cdot (\vb{r}_2 - \vb{r}_1)} e^{- i \omega_n^b (\tau_2 - \tau_1)} \dd{\vb{k}} +\end{aligned}$$ + +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. |