Wick's Theorem and Bose Operators

[PeroK]

Homework Statement

Use Wick's Theorem to express the string Bose operators $\hat a_p \hat a_q^{\dagger} \hat a_k^{\dagger}$ in terms of normal ordered fields and contractions.

Relevant Equations

Wick's Theorem:
$$T[ABC] = N[ABC + \text{all possible contractions of ABC}]$$

This is problem 18.3 from QFT for the gifted amateur. I must admit I'm struggling to interpret what this question is asking. Chapter 18 has applied Wick's theorem to calculate vacuum expectation values etc. But, there is nothing to suggest how it applies to a product of operators.

Does the question simply mean to calculate $T[\hat a_p \hat a_q^{\dagger} \hat a_k]$?

Any help interpreting the question would be good. Thanks.

 

[McFisch]

My first impuls was to look up the typos in the book, to see if he just missed a T. What I found instead was this sentence in someone's (personal) solutions:

"In these three problems, L&B don’t say explicitly that we’re dealing with time-ordered products, but I assume we must be as otherwise Wick’s theorem doesn’t apply."

I want to agree with this, I also don't see any other way this exercise would make any sense. I guess, it's not too uncommen though. I recently learned some conformal field theory, where (in radial quantisation) T turns into R (radial ordering), which is also just always "assumed" and not explicitely written out.

[Quote taken from: https://www.physicspages.com/Lancaster%20QFT.html (see exercise 18.3)]

 

[PeroK]

My first impuls was to look up the typos in the book, to see if he just missed a T. What I found instead was this sentence in someone's (personal) solutions:

"In these three problems, L&B don’t say explicitly that we’re dealing with time-ordered products, but I assume we must be as otherwise Wick’s theorem doesn’t apply."

I want to agree with this, I also don't see any other way this exercise would make any sense. I guess, it's not too uncommen though. I recently learned some conformal field theory, where (in radial quantisation) T turns into R (radial ordering), which is also just always "assumed" and not explicitely written out.

[Quote taken from: https://www.physicspages.com/Lancaster%20QFT.html (see exercise 18.3)]

Thanks for this. It becomes clearer in the next chapter. Using the interaction picture to evaluate the scattering matrix, the creation operator is applied as some time $-t$ and the annihilation operators are applied at some time $t$ in the limit as $t \rightarrow \infty$. I assume that is what is implied here.

 

[vanhees71]

How can there be a time-ordering operator, if you look at operators that don't depend on time? The problem seems not to be well-posed. Concerning Wick's theorem, see Sect. 3.8 in

https://itp.uni-frankfurt.de/~hees/publ/lect.pdf