Evaluating Time-Ordered Product with Wick's Theorem

[Naz93]

Homework Statement

[/B]
Consider a real free scalar field Φ with mass m. Evaluate the following time-ordered product of field operators using Wick's theorem: ∫d^4x <0| T(Φ(x1)Φ(x2)Φ(x3)Φ(x4)(Φ(x))^4) |0>

(T denotes time ordering)

Homework Equations

Wick's theorem: T((Φ(x1)...Φ(xn)) = : (Φ(x1)...Φ(xn)) + all possible contractions :
( : ... : denotes normal ordered product)

The Attempt at a Solution

I'm really confused here. I've seen examples of using Wick's theorem to evaluate products like T(Φ(x1)Φ(x2)Φ(x3)Φ(x4)) , i.e. field evaluated at fixed values of x only ... but here, there is another term that's integrated over all possible x values. And I have no idea how to deal with that - I've never seen anything like that done in the lectures I've had on this stuff.

I know contractions of the form "contraction(Φ(x1)Φ(x2))" give Feynman propagators.

But that's all I've got...

I'm really bad at this stuff, so if someone can help explain / help me work through this from absolute basics (i.e. assuming I know pretty much nothing), that would be really appreciated!

 

[mark726]

Hello! Let me try to explain this step by step.

First, let's rewrite the time-ordered product in normal ordered form using Wick's theorem:

T(Φ(x1)Φ(x2)Φ(x3)Φ(x4)(Φ(x))^4) = :Φ(x1)Φ(x2)Φ(x3)Φ(x4)(Φ(x))^4: + all possible contractions

Now, let's focus on the normal ordered product. We can expand it using the definition of normal ordering:

:Φ(x1)Φ(x2)Φ(x3)Φ(x4)(Φ(x))^4: = Φ(x1)Φ(x2)Φ(x3)Φ(x4)(Φ(x))^4 - <0|Φ(x1)Φ(x2)Φ(x3)Φ(x4)(Φ(x))^4|0>

The first term in this expansion is easy to evaluate, it's just the product of field operators at different points. But the second term involves a vacuum expectation value, which we need to evaluate using Wick's theorem.

Let's look at the second term in more detail:

<0|Φ(x1)Φ(x2)Φ(x3)Φ(x4)(Φ(x))^4|0>

Using Wick's theorem, we can write this as:

<0|Φ(x1)Φ(x2)Φ(x3)Φ(x4)(Φ(x))^4|0> = <0|Φ(x1)Φ(x2)Φ(x3)Φ(x4)|0> + all possible contractions

Now, let's focus on the first term on the right-hand side. We can rewrite it using the definition of vacuum expectation value:

<0|Φ(x1)Φ(x2)Φ(x3)Φ(x4)|0> = Φ(x1)Φ(x2)Φ(x3)Φ(x4) - <0|T(Φ(x1)Φ(x2)Φ(x3)Φ(x4))|0>

Notice that the second term on the right-hand side is the time-ordered product of field operators, which we can evaluate using Wick's theorem. So, let's rewrite it as:

<