1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
|
---
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.
|
