# How Does Wick's Theorem Apply to Time-Independent Bose Operators?

• PeroK
In summary, Wick's theorem is used to calculate vacuum expectation values etc. in QFT, but this is not what is asked in the question.
PeroK
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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.

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)]

JD_PM and PeroK
McFisch said:
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.

## 1. What is Wick's Theorem?

Wick's Theorem is a mathematical tool used in quantum field theory to simplify the calculation of expectation values of composite operators. It allows for the separation of normal-ordered and contractions terms in the expansion of an operator.

## 2. How does Wick's Theorem work?

Wick's Theorem works by breaking down an operator into a sum of normal-ordered terms and all possible contractions between creation and annihilation operators. The normal-ordered terms are then evaluated using the usual rules of quantum mechanics, while the contractions are simplified using the commutation and anticommutation relations of the creation and annihilation operators.

## 3. What are Bose operators?

Bose operators are mathematical operators used in quantum field theory to describe particles that follow Bose-Einstein statistics. They are defined as linear combinations of creation and annihilation operators and have specific commutation relations that allow for the simplification of calculations using Wick's Theorem.

## 4. How are Bose operators related to Wick's Theorem?

Bose operators are essential in the application of Wick's Theorem as they allow for the separation of normal-ordered and contraction terms. The commutation relations of Bose operators also play a crucial role in simplifying the contractions and evaluating the expectation values of composite operators.

## 5. What are the applications of Wick's Theorem and Bose operators?

Wick's Theorem and Bose operators are widely used in quantum field theory to calculate expectation values of composite operators and to study the behavior of particles that follow Bose-Einstein statistics. They are also used in other areas of physics, such as condensed matter physics and statistical mechanics, to simplify calculations and understand the properties of systems with many interacting particles.

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