# Chronological and normal ordering

I've realized I'm little bit confused when I want to treat elements like this $$\left<\phi_0|T\{a_p(t)a_p^+(t')V(t_1)V(t_2)\}|\phi_0\right>$$

with

$$V(t)=\dfrac12 \dfrac{1}{(2\pi \hbar)^9}\int d^3p_1 d^3p_2 d^3q v(q):a_{p_1}^+(t)a_{p_2}^+(t)a_{p_2-q}(t)a_{p_1+q}(t):$$

There is chronological ordering ($T\{\}$) and if I would apply Wick's theorem, there will pop up normal ordering and Wick contractions. Without times it would be easy, but together its mess. How is this done?

Also could you recommend me some book that would clarify how to treat this.

Edit: This [NOTES ON WICK’S THEOREM IN MANY-BODY THEORY, Luca Guido Molinari] (part V) could help here. I guess I'll have to look up on commutation relations for operators with different times (depending only on $H_0$, $H=H_0+V$).

The normal ordening is a way to say: ''we throw away the zero-point energy'' (since it becomes infinity and wa say we only look at energy-differences), or to put it in the words of A. Zee: ''Create before you annihalite''.

The chronological ordening comes in when you calculate the Feynman propagator (also called the Green's function), which is basically the "thing" that you are trying to calculate. If you would look at the Green's function, you know that if $L$ represents your Liouville-equation, for example in the case of the wave-equation:$$L=\frac{\partial^2}{c^2\partial t^2}-\Delta^2$$, the equation for the Green's function $G(r|r')$ hence becomes:$$LG(r|r')=\delta(r-r')$$, which we can use to solve the differential equation represented by $L$ by using a convolution (see the Green's function link).

Now with the Feynman-propagator you do the same thing, and it follows an equivalent equation of the above (only now with some generalized delta-function since we don't werk at equal times)

A book that goes deeper into Wick's theorem is the one of W. Greiner (Field Quantization), which is widely spread on the net. On pages 225-233 Wick's theorem is discussed. After Wick's theorem he starts with QED, this also gives an example of the use of Wick's theorem.

Now I don't know what you hope to find in the answers since the proof is basically an immense ammount of bookkeeping, but basically you make an order-expansion of your product of operators.

Wick-theorem states that:$$T(\hat{A_1}\hat{A_2}\hat{A_3}\hat{A_4}\cdots)=:\hat{A_1}\hat{A_2}\hat{A_3}\hat{A_4}\cdots:+\text{normal ordening with single contractions}+\text{normal ordening with double contractions}+\text{normal ordening with triple contractions}+...$$.

This series goes on untill you run out of stuff to contract (note that you need an even number of operators), so basically op to $n/2$, with $n$ the number of operators.

To use Wick's theorem, one goes order by order. Starting from the zero-th order (which is basically normal-ordening, then going first order (one contraction) and so on. Usually some orders and combinations will simply turn out to be zero (for example when you contract two creation of annihalation operators, or when you contract operators of different fields). You can also exclude terms by using the conservation of momentum and energy (some contractions violate this one).

If you want to compare with Feynman-diagrams, then you need to look at the scattering matrix $S$. This one is developped into a series: $$\hat{S}=\mathbb{I}+\sum\limits_{n=1}^\infty \hat{S}^{(n)}$$, (which is basically the power-series decomposition of your unitary-time evolution operator) for each order, you add a vertex and hence more operators, and thus go to a higher order Feynman-diagram.

Well my problem was that I've forgot how to solve this and when I look at it, the following have come to my mind - I've wanted to use Wick's theorem inside the chronological order, but it's bad idea because T and :: would beat each other in some cases (at least I think and I don't know how to deal with that).

Lets for fun prove Wick's theorem for two chronologically ordered operators.

Wick works thanks to the two things

• $a|\left. \phi_0\right>=0$
• commutation (or anti-commutation) relations for annihilation and creation operators (which are fine for E.T. case, but I don't remember if I've ever needed to care explicitly about non equal time commutation relation in QFT course)

$$T(A_1(t_1)A_2(t_2))=\theta(t_1-t_2)A_1A_2+\theta(t_2-t_1)A_2A_1= \theta(t_1-t_2)(a^+_1+a_1)(a^+_2+a_2)+\theta(t_2-t_1)(a^+_2+a_2)(a^+_1+a_1)= \theta(t_1-t_2)(a^+_1a^+_2+a^+_1a_2+a_1a_2+\underline{a^+_2a_1+(-a^+_2a_1}+a_1a^+_2))+\theta(t_2-t_1)(a^+_2a^+_1+a^+_2a_1+a_2a_1+\underline{a^+_1a_2+(-a^+_1a_2}+a_2a^+_1))$$ where were added underlined zeros to the expansion so now first four term make together $:A_1A_2:$ (or $:A_2A_1:$ which is the same thanks to relations for $\left[a^+,b^+\right]=\left[a,b\right]=0$) $$=\ :A_1A_2: + \theta(t_1-t_2)\left[a_1,a^+_2\right]+\theta(t_2-t_1)\left[a_2,a^+_1\right]$$ last two terms are c-numbers and can be written as ground state elements, for the middle term $$\left[a_1,a^+_2\right]=\left<\phi_0|\left[a_1,a^+_2\right]|\phi_0\right>=\left<\phi_0|a_1a^+_2|\phi_0\right>=\left<\phi_0|a_1a^+_2+\underline{a_1^+a^+_2+a_1a_2+a_1^+a_2}|\phi_0\right>=\left<\phi_0|A_1A_2|\phi_0\right>$$ where were freely added terms that gave zero when acting on the ground state, the last similarly $$\left[a_2,a^+_1\right]=\dots=\left<\phi_0|A_2A_1|\phi_0\right>$$

Together it gives $$=\ :A_1A_2: + \theta(t_1-t_2)\left<\phi_0|A_1A_2|\phi_0\right>+\theta(t_2-t_1)\left<\phi_0|A_2A_1|\phi_0\right>$$ $$=\ :A_1A_2: + \left<\phi_0|T(A_1A_2)|\phi_0\right>$$

Wick's theorem version for time ordered operators is the solution, but I have to look into how it ticks with connection to the time (commutation relation for not equal time)...