# QFT Wicks theorem contraction -- different fields terms of propagation

• binbagsss
In summary: Since the fields are different, they commute, and the commutator (i.e., contraction) is zero. In summary, when trying to express ##T(\phi(x1)\Phi(x2)\phi(x3)\Phi(x4)\Phi(x5)\Phi(x6))## in terms of Feynman propagators, we must first consider the contraction terms. However, since the fields are different, their commutator is zero, resulting in a zero contraction term. Therefore, the final answer cannot be expressed in terms of the Feynman propagators ##G_F^{\phi}(x-y)## and ##G_F^{\Phi}(x-y)##.
binbagsss

## Homework Statement

I am trying to express ##T(\phi(x1)\Phi(x2)\phi(x3)\Phi(x4)\Phi(x5)\Phi(x6))## in terms of the Feynman propagators ##G_F^{\phi}(x-y)## and ##G_F^{\Phi}(x-y)##

where ##G_F^{\phi}(x-y) =\int \frac{d^{4}k}{(2\pi)^{4}}e^{ik(x-y)} \frac{ih}{-k.k - m^2 -i\epsilon} ##
and ##G_F^{\Phi}(x-y) =\int \frac{d^{4}k}{(2\pi)^{4}}e^{ik(x-y)} \frac{ih}{-k.k - M^2 -i\epsilon} ##

## Homework Equations

[/B]
##<0|T(\phi(x1)\Phi(x2)\phi(x3)\phi(x4)\phi(x5)\phi(x6)) |0> = : non-fully contracted terms : + fully contracted terms = fully contracted terms ##

where ##T## is the time ordering operator

##<0| ## being the vacuum state,

since non-fully contracted terms, i.e. where every field is not involved in a contraction will vanish.

## The Attempt at a Solution

My question is how/ can you contract over two different fields?

I am fine with contractions over two of the same field, but can't find any notes/examples on what to do in the case of two different fields - here ##\phi(x)## and ##\Phi(x)##

In particular, if you can't, the question ask to express the final answer in terms of ##G_F^{\phi}(x-y)## and ##G_F^{\Phi}(x-y)##, and I can't see how you could write the contraction over two different fields in terms of this.

I know ##G^{\phi} (x1-x3) = contraction of the fields \phi(x1) , \phi(x3)##

and similarly that ##G^{\Phi} (x4-x5) = contraction of the fields \Phi(x4) , \Phi(x5)##

but no idea how you would write e.g contraction over the fields ## \phi(x1), \Phi(x5)## in terms of ##G_F^{\phi}(x-y)## and ##G_F^{\Phi}(x-y)##?

Can you contract over two different fields?

Help really appreciated,
(Or a point to some notes on this)

Thanks

binbagsss said:
Can you contract over two different fields?
No.

binbagsss
Orodruin said:
No.

many thanks.

a possible explanation or a point towards one?

I can't type much (on my phone). Go back to how contractions were constructed, i.e., what does a contraction entail?

binbagsss
Orodruin said:
I can't type much (on my phone). Go back to how contractions were constructed, i.e., what does a contraction entail?

oh right, no worries at all, thanks for your help,
so i looked at a derivation of time ordering of a field by splitting a field ##\phi## into its annihilation and creation operator components, and then get normal ordered + commutator terms, the commutator terms being the contraction...oh so commutators of different fields is zero- is this why?

binbagsss said:
oh so commutators of different fields is zero- is this why?
Right.

## 1. What is QFT (Quantum Field Theory)?

Quantum Field Theory (QFT) is a theoretical framework in physics that describes the behavior of particles at a subatomic level. It combines the principles of quantum mechanics and special relativity to explain the interactions between particles and fields.

## 2. What is Wicks theorem in QFT?

Wicks theorem is a mathematical tool used in QFT to simplify calculations involving multiple fields and their interactions. It allows for the breaking down of complicated equations into smaller, more manageable terms.

## 3. How does Wicks theorem work?

Wicks theorem uses the concept of contractions, which represent the interactions between fields. It involves summing over all possible contractions in an equation and then applying a special rule to simplify the resulting terms.

## 4. What is the significance of contractions in Wicks theorem?

Contractions are important in Wicks theorem as they represent the interactions between fields. They are used to determine the strength of these interactions and can help in calculating the probability of certain particle interactions.

## 5. How does Wicks theorem relate to the propagation of fields in QFT?

In QFT, fields propagate or spread out in space and time. Wicks theorem helps in calculating the probability amplitude for this propagation by breaking it down into simpler terms involving contractions. This allows for more efficient and accurate calculations in the study of quantum field theory.

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