- #1

QFT1995

- 30

- 1

- Homework Statement
- Compute the following

$$\mathcal{T} \langle {0}| \prod_i^Ne^{\imath \beta_i \phi(x_i)} |0 \rangle$$ without contractions at the same point

- Relevant Equations
- Wick's theorem

If I'm computing

$$\mathcal{T} \langle 0 | \prod_i^Ne^{\imath \beta_i \phi(x_i)} | 0\rangle $$

where the contractions at the same spacetime point are ignored, can I simply insert a complete set of states (product now outside of expression) between each exponential to give

$$\mathcal{T} \prod_i^N \langle 0 |e^{\imath \beta_i \phi(x_i)} | 0\rangle$$

and then the only terms not contributing to contractions at the same spacetime point is the term 1 in the exponential which gives

$$\mathcal{T} \prod_i^N\langle 0 |1 | 0\rangle = 1$$

since

$$ \langle 0 | 0\rangle=1$$

or is this wrong?

$$\mathcal{T} \langle 0 | \prod_i^Ne^{\imath \beta_i \phi(x_i)} | 0\rangle $$

where the contractions at the same spacetime point are ignored, can I simply insert a complete set of states (product now outside of expression) between each exponential to give

$$\mathcal{T} \prod_i^N \langle 0 |e^{\imath \beta_i \phi(x_i)} | 0\rangle$$

and then the only terms not contributing to contractions at the same spacetime point is the term 1 in the exponential which gives

$$\mathcal{T} \prod_i^N\langle 0 |1 | 0\rangle = 1$$

since

$$ \langle 0 | 0\rangle=1$$

or is this wrong?