Time-ordered product of two normal-ordered products of fields

Question

Suppose you have a scalar field theory with field operators $\phi(x)=\phi(x)_+ + \phi(x)_- $ that can be decomposed into terms of annihilation and destruction operators. Let $$ D(x-y) = <0|T(\phi(x)\phi(y))|0> $$ be the propagator for said theory. I am trying to prove the relation $$ <0|T(:\phi(x)^n::\phi(y)^m:)|0> = n!\: D(x-y)^n \delta_{n,m}. $$ My fírst attempt at a solution was to plug in the definitions of time ordered and normal ordered products, use the decomposition of $\phi(x)$ and the Multinomial theorem for expressing $\phi(x)^n$ and $\phi(y)^m$. After this had lead me nowhere I looked up Wicks theorem and tried to use it. But I don't know what the contraction of normal ordered product with a normal ordered product is. I know you can use it for expressions like $$<0|T(:\phi(x)^n\phi(y)^m:)|0>,$$ but my problem is obviously different from that. Then I tried to prove the relation by complete induction, which failed because I couldn't express the $(n+1)$ term on the left hand side in terms of the result for the case $n$. My last resort was to work this out for the case $n=m=2$ and then work my way up to arbitrary powers. I looked my problem up in Peskin and Schröder's book and also in Schwabel's Advanced Quantum Mechanics, but found nothing than the definitions and the introductory examples. I closely studied the questions

• Time-ordering vs normal-ordering and the two-point function/propagator

• Why/How is this Wick's theorem?

• https://physics.stackexchange.com/q/183978/

• Quick question regarding Wick's theorem

I've been thinking about this problem for man days now and any help or idea, I can start from, would be greatly appreciated.

Answer 1

Hints:

1. The starting point is the 2-point relation $$T(\phi(x)\phi(y)) ~-~:\phi(x)\phi(y): ~=~ C(x,y)~{\bf 1}, \qquad C(x,y)~\equiv~\langle 0 | T(\phi(x)\phi(y))|0\rangle,\tag{1} $$ cf. this Phys.SE post.

2. The relevant Wick's theorem is a nested Wick's theorem $$ T(:\phi(x)^n::\phi(y)^m:)~=~\exp\left( C(x,y)\frac{\partial}{\partial\phi(x)}\frac{\partial}{\partial\phi(y)}\right): \phi(x)^n \phi(y)^m:$$ $$~=~\sum_{r=0}^{\min(n,m)} \frac{n! }{(n\!-\!r)!} \frac{m!}{(m\!-\!r)}\frac{C(x,y)^r}{r!} : \phi(x)^{n-r} \phi(y)^{m-r}:, \tag{2}$$ cf. my Phys.SE answer here. The main point is that when applying the nested Wick's theorem to the lhs. of eq. (2), one should only include all possible contractions between different normal order symbols, and exclude contractions that are purely within the same normal order symbol.

3. Recall the fact that $$ \langle 0 | :\phi(x_1)\ldots \phi(x_n):|0\rangle ~=~\delta_n^0. \tag{3} $$

4. Combine eqs. (2) and (3) to conclude the sought-for identity $$ \langle 0 |T(:\phi(x)^n::\phi(y)^m:)|0\rangle~=~n!~\delta_n^m ~C(x,y)^n. \tag{4}$$