- #1

Diracobama2181

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- Homework Statement
- I am trying to show that ## \bra{P'}\phi^4 \ket{P}##is divergent, where ##\ket{P}## is a free spin 0 boson.

- Relevant Equations
- ##\ket{P}=a^{\dagger}(k)\ket{0}##

##\phi=\int \frac{d^3k(\hat{a}e^{-ikx}+\hat{a}^{\dagger}e^{ikx})}{2\omega_{k}(2\pi)^3}##

##\omega=\sqrt{k^2+m^2}##

From this, I find

$$\bra{P'} \phi^4 \ket{P} = \int \frac {d^3 k_1 d^3 k_2 d^3 k_3 d^3 k_4} {16 \omega_{k_1}\omega_{k_2}\omega_{k_3}\omega_{k_4} (2\pi)^{12} }\bra{0} a_{P'}(a_{k_1} a_{k_2} a_{k_3}a_{k_4}e^{i(k_1+k_2+k_3+k_4)x}+...)a^{\dagger}_{P}\ket{0}$$ (a total of 16 different terms)

Right now, I am trying to figure out how to calculate terms like

## \bra{0}a_{P'}a_{k_1}a^{\dagger}_{k_2}a_{k_3}a^{\dagger}_{k_4}a^{\dagger}_{P}\ket{0} ##. Any examples would be greatly appreciated.

$$\bra{P'} \phi^4 \ket{P} = \int \frac {d^3 k_1 d^3 k_2 d^3 k_3 d^3 k_4} {16 \omega_{k_1}\omega_{k_2}\omega_{k_3}\omega_{k_4} (2\pi)^{12} }\bra{0} a_{P'}(a_{k_1} a_{k_2} a_{k_3}a_{k_4}e^{i(k_1+k_2+k_3+k_4)x}+...)a^{\dagger}_{P}\ket{0}$$ (a total of 16 different terms)

Right now, I am trying to figure out how to calculate terms like

## \bra{0}a_{P'}a_{k_1}a^{\dagger}_{k_2}a_{k_3}a^{\dagger}_{k_4}a^{\dagger}_{P}\ket{0} ##. Any examples would be greatly appreciated.

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