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Diffstat (limited to 'source/know/concept/wicks-theorem/index.md')
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1 files changed, 24 insertions, 24 deletions
diff --git a/source/know/concept/wicks-theorem/index.md b/source/know/concept/wicks-theorem/index.md index 42a1bec..35be8fa 100644 --- a/source/know/concept/wicks-theorem/index.md +++ b/source/know/concept/wicks-theorem/index.md @@ -38,9 +38,9 @@ The normal product of three or more operators works in the same way, but might not be unique depending, on how many of each type there are. -Next, the **contraction** of the operators $A$ and $B$ +Next, the **contraction** of the operators $$A$$ and $$B$$ is defined as the vacuum matrix element, -i.e. the expectation value of $\Ket{0}$: +i.e. the expectation value of $$\Ket{0}$$: $$\begin{aligned} \Expval{A B}_0 @@ -48,8 +48,8 @@ $$\begin{aligned} \end{aligned}$$ Unsurprisingly, a contraction can only be nonzero if -$A = \hat{c}_\alpha$ is an annihilation and $B = \hat{c}_\alpha^\dagger$ -a creation for the same state $\alpha$. +$$A = \hat{c}_\alpha$$ is an annihilation and $$B = \hat{c}_\alpha^\dagger$$ +a creation for the same state $$\alpha$$. Wick's theorem states: **any product of second quantization operators can be @@ -71,7 +71,7 @@ $$\begin{aligned} \end{aligned}$$ Where the negative signs apply to fermions only. -We take the normal product with 0 contractions removed ($\underline{ABCD}$), +We take the normal product with 0 contractions removed ($$\underline{ABCD}$$), then with 1 contraction removed in every possible way (first two lines), then with 2 contractions removed in every possible way (last line), and so on. @@ -94,9 +94,9 @@ $$\begin{aligned} = - \hat{f}_\beta^\dagger \hat{f}_\alpha + \{\hat{f}_\alpha, \hat{f}_\beta^\dagger\} \end{aligned}$$ -This anticommutator is known to be $\delta_{\alpha\beta}$, +This anticommutator is known to be $$\delta_{\alpha\beta}$$, so we can inconsequentially take -its inner product with the vacuum state $\Ket{0}$: +its inner product with the vacuum state $$\Ket{0}$$: $$\begin{aligned} \hat{f}_\alpha \hat{f}_\beta^\dagger @@ -114,8 +114,8 @@ $$\begin{aligned} = \hat{b}_\beta^\dagger \hat{b}_\alpha + [\hat{b}_\alpha, \hat{b}_\beta^\dagger] \end{aligned}$$ -This commutator is known to be $\delta_{\alpha\beta}$, -so we take the inner product with $\Ket{0}$, like before: +This commutator is known to be $$\delta_{\alpha\beta}$$, +so we take the inner product with $$\Ket{0}$$, like before: $$\begin{aligned} \hat{b}_\alpha \hat{b}_\beta^\dagger @@ -127,9 +127,9 @@ $$\begin{aligned} \end{aligned}$$ Which again agrees with Wick's theorem. -Next, we prove that if it holds for $N$ operators, then it also holds for $N + 1$. +Next, we prove that if it holds for $$N$$ operators, then it also holds for $$N + 1$$. To begin with, consider the following statement about right-multiplying -by an extra $A_{N+1}$, with $s = 1$ for bosons and $s = -1$ for fermions: +by an extra $$A_{N+1}$$, with $$s = 1$$ for bosons and $$s = -1$$ for fermions: $$\begin{aligned} \underline{A_1 ... A_N} A_{N+1} @@ -137,11 +137,11 @@ $$\begin{aligned} + \sum_{n = 1}^N s^{n + N} \Expval{A_n A_{N+1}}_0 \underline{A_1 ... A_{n-1} A_{n+1} ... A_N} \end{aligned}$$ -If $A_{N + 1}$ is an annihilation operator, then this is trivial: +If $$A_{N + 1}$$ is an annihilation operator, then this is trivial: appending it does not break the existing normal order, -and $\Expval{A_n A_{N+1}}_0 = 0$ for all $A_n$. +and $$\Expval{A_n A_{N+1}}_0 = 0$$ for all $$A_n$$. -However, if $A_{N + 1}$ is a creation operator, +However, if $$A_{N + 1}$$ is a creation operator, then to restore the normal order, we move it to the front by swapping, which introduces a bunch of (anti)commutators: @@ -155,12 +155,12 @@ $$\begin{aligned} + \sum_{n} s^{n + N} \Expval{A_n A_{N+1}}_0 \underline{A_1 ... A_{n-1} A_{n+1} ... A_N} \end{aligned}$$ -Where $\{[]\}$ is the anticommutator or commutator, +Where $$\{[]\}$$ is the anticommutator or commutator, respectively for fermions or bosons. -If we take Wick's theorem for $N$ operators $A_1 ... A_N$, -and right-multiply it by $A_{N + 1}$, -then each term will contain a product of the form $\underline{A_{v} ... A_{w}} A_{N+1}$. +If we take Wick's theorem for $$N$$ operators $$A_1 ... A_N$$, +and right-multiply it by $$A_{N + 1}$$, +then each term will contain a product of the form $$\underline{A_{v} ... A_{w}} A_{N+1}$$. Using the relation that we just proved, each such product can be rewritten as follows: @@ -171,15 +171,15 @@ $$\begin{aligned} \end{aligned}$$ Inserting this back into Wick's theorem, -we get new terms with contractions of $A_{N+1}$. +we get new terms with contractions of $$A_{N+1}$$. After a lot of rearranging, -the result turns out to just be Wick's theorem for $N\!+\!1$ operators. +the result turns out to just be Wick's theorem for $$N\!+\!1$$ operators. Therefore, -if Wick's theorem holds for $N$ operators, -it also holds for $N\!+\!1$. +if Wick's theorem holds for $$N$$ operators, +it also holds for $$N\!+\!1$$. -We showed that Wick's theorem holds for $N = 2$, -so, by induction, it holds for all $N \ge 2$. +We showed that Wick's theorem holds for $$N = 2$$, +so, by induction, it holds for all $$N \ge 2$$. |