Evaluating Time-Ordered Product with Wick's Theorem

In summary: T(Φ(x1)Φ(x2)Φ(x3)Φ(x4))|0> = <0|:Φ(x1)Φ(x2)Φ(x3)Φ(x4):|0> + all possible contractionsNow, we're back to a vacuum expectation value, but with one less field operator. We can keep using this process until we have no more field operators left. Then we can evaluate the vacuum expectation values using Wick's theorem and put everything back together to get the final result.In summary, to evaluate the given time-ordered product using Wick's theorem, we need to rewrite it in normal ordered form and then evaluate the vacuum expectation values using
  • #1
Naz93
29
2

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!
 
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  • #2


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:

<
 

Related to Evaluating Time-Ordered Product with Wick's Theorem

1. What is Wick's Theorem and why is it important in evaluating time-ordered products?

Wick's Theorem is a mathematical tool used to simplify the evaluation of time-ordered products in quantum field theory. It allows us to break down complex expressions into simpler ones, making calculations more manageable. This is particularly useful in theoretical physics where time-ordered products are commonly used to describe the dynamics of quantum systems.

2. How does Wick's Theorem work?

Wick's Theorem states that any time-ordered product of operators can be written as a sum of normal-ordered products plus a sum of contractions between pairs of operators. Normal-ordered products have the creation operators on the left and the annihilation operators on the right, while contractions represent the overlapping terms between operators in the time-ordered product. By breaking down the time-ordered product into these simpler terms, the evaluation becomes much easier.

3. What are the benefits of using Wick's Theorem?

Using Wick's Theorem can significantly reduce the complexity of calculations involving time-ordered products. It also allows us to identify the most important contributions to a particular quantity, which can help us gain a better understanding of the system being studied. Additionally, Wick's Theorem provides a systematic approach to evaluating time-ordered products, making the process more efficient.

4. Are there any limitations to using Wick's Theorem?

While Wick's Theorem is a powerful tool, it does have some limitations. It is only applicable to time-ordered products that involve creation and annihilation operators. It also assumes that the operators are in a normal-ordered form to begin with. In some cases, the contractions may also result in infinite terms, which require further mathematical techniques to handle.

5. How is Wick's Theorem used in practice?

In practice, Wick's Theorem is used to evaluate time-ordered products in theoretical calculations. It is commonly used in quantum field theory, particle physics, and condensed matter physics. It is also a fundamental concept in perturbation theory, where it allows for the systematic calculation of higher-order corrections to a given quantity. Additionally, Wick's Theorem has applications in other fields such as quantum chemistry and statistical mechanics.

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