From 42d409fa774efb8206ae5c701d5cbcc4ae1d9cad Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Tue, 14 Sep 2021 21:20:30 +0200
Subject: Expand knowledge base

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+---
+title: "Time-ordered product"
+firstLetter: "T"
+publishDate: 2021-09-13
+categories:
+- Physics
+- Quantum mechanics
+
+date: 2021-09-13T19:58:33+02:00
+draft: false
+markup: pandoc
+---
+
+# Time-ordered product
+
+In quantum mechanics, especially quantum field theory,
+a **time-ordered product** is a product of
+explicitly time-dependent operators,
+subject to certain ordering constraints.
+
+Let us start with an unusual motivation.
+Suppose that some time-dependent operator $\hat{A}(t)$ is defined like so,
+as a product of $N$ time-dependent sub-operators $\hat{a}_n(t)$:
+
+$$\begin{aligned}
+    \hat{A}(t)
+    \equiv \int_0^{t} \hat{a}_1(t_1) \bigg( \int_0^{t_1} \hat{a}_2(t_2) \bigg( \int_0^{t_2} \hat{a}_3(t_3) \bigg( \cdots \bigg)
+    \dd{t_3} \bigg) \dd{t_2} \bigg) \dd{t_1}
+\end{aligned}$$
+
+Crucially, the upper limits of the inner integrals
+depend on the surrounding variables,
+meaning that these integrals cannot simply be reordered.
+
+An interpretation is that the rightmost $\hat{a}_N(t_N)$ is applied first,
+and then $\hat{a}_{N-1}(t_{N-1})$ secondly with $t_{N-1} > t_N$,
+and so on.
+This suggests there is a form of "time-ordering" here:
+the integrals sweep across all relative timings of $\hat{a}_n$,
+but preserve the ordering.
+Indeed, this could be rewritten as a time-ordered product
+(see the [interaction picture](/know/concept/interaction-picture/) for an example).
+
+A more general and intuitive motivation goes as follows.
+Suppose we have a product of $N$ time-dependent operators $\hat{a}_n(t)$,
+each representing a certain event.
+Clearly, we would want to apply them in chronological order:
+
+$$\begin{aligned}
+    \hat{a}_N(t_N) \: \hat{a}_{N-1}(t_{N-1}) \: \cdots \: \hat{a}_2(t_2) \: \hat{a}_1(t_1)
+    \qquad \mathrm{where} \qquad
+    t_N > t_{N-1} > ... > \: t_2 > t_1
+\end{aligned}$$
+
+But what if the ordering of the arguments $t_N, ..., t_1$
+is not known in advance?
+We thus define the **time-ordering meta-operator** $\mathcal{T}$,
+which reorders the operators based on the $t$-values
+such that they are always in chronological order.
+For example:
+
+$$\begin{aligned}
+    \mathcal{T} \big\{ \hat{a}_1(t_1) \: \hat{a}_2(t_2) \big\}
+    \equiv
+    \begin{cases}
+        \hat{a}_1(t_1) \: \hat{a}_2(t_2) & \mathrm{if} \; t_2 < t_1 \\
+        \hat{a}_2(t_2) \: \hat{a}_1(t_1) & \mathrm{if} \; t_1 < t_2
+    \end{cases}
+\end{aligned}$$
+
+This example suggests a general algorithm for $\mathcal{T}$:
+we need to consider every permutation of the operators $\hat{a}_n(t_n)$,
+and leave only the single one that satisfies our demands.
+
+Mathematically, we do this by summing up all permutations,
+and multiplying each term with a product of
+[Heaviside step functions](/know/concept/heaviside-step-function/) $\Theta$,
+which remove the term if the ordering is wrong:
+
+$$\begin{aligned}
+    \mathcal{T} \big\{ \hat{a}_1 \cdots \hat{a}_N \big\}
+    \equiv \sum_{p \in P_N}^{}
+    \Theta\big(t_{p_1} \!\!-\! t_{p_2}\big) \cdots \Theta\big(t_{p_{N-1}} \!\!-\! t_{p_N}\big)
+    \: \hat{a}_{p_1}(t_{p_1}) \: \cdots \: \hat{a}_{p_N}(t_{p_N})
+\end{aligned}$$
+
+With this, our earlier example for two operators $\hat{a}_1$ and $\hat{a}_2$
+takes the following form:
+
+$$\begin{aligned}
+    \mathcal{T} \big\{ \hat{a}_1(t_1) \: \hat{a}_2(t_2) \big\}
+    = \Theta(t_1 - t_2) \: \hat{a}_1(t_1) \: \hat{a}_2(t_2) + \Theta(t_2 - t_1) \: \hat{a}_2(t_2) \: \hat{a}_1(t_1)
+\end{aligned}$$
+
+However, we are still missing an important detail:
+so far, we have quietly been assuming that the operators are bosonic
+(see [second quantization](/know/concept/second-quantization/)).
+To include fermionic operators,
+we must allow the sign of each term to change,
+based on whether the permutation is even or odd:
+
+$$\begin{aligned}
+    \mathcal{T} \big\{ \hat{a}_1(t_1) \: \hat{a}_2(t_2) \big\}
+    = \Theta(t_1 - t_2) \: \hat{a}_1(t_1) \: \hat{a}_2(t_2) \pm \Theta(t_2 - t_1) \: \hat{a}_2(t_2) \: \hat{a}_1(t_1)
+\end{aligned}$$
+
+Where $\pm$ is $+$ for bosons, and $-$ for fermions in this case.
+The general definition of $\mathcal{T}$ is:
+
+$$\begin{aligned}
+    \boxed{
+        \mathcal{T} \big\{ \hat{a}_1 \cdots \hat{a}_N \big\}
+        \equiv \sum_{p \in P_N}^{} (-1)^p
+        \bigg( \prod_{j = 1}^{N-1} \Theta\big(t_{p_j} \!-\! t_{p_{j+1}}\big) \bigg)
+        \bigg( \prod_{k = 1}^N \hat{a}_{p_k}(t_{p_k}) \bigg)
+    }
+\end{aligned}$$
+
+
+
+## References
+1.  H. Bruus, K. Flensberg,
+    *Many-body quantum theory in condensed matter physics*,
+    2016, Oxford.
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