- #1

Diracobama2181

- 75

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- Homework Statement
- Given the energy momentum tensor

##\hat{T}^{\mu \upsilon}## and a Lagrangian with potential ##V=\frac{1}{2}m\phi^2+\lambda\phi^4##,

the quantity ##\bra{\overrightarrow{P'}}T^{\mu v}\ket{\overrightarrow{P}}## should diverge.

How would I go about the 1 loop re-normalization procedure on this term?

- Relevant Equations
- ##\hat{T}^{\mu \upsilon}=\partial^{\mu}\partial^{\upsilon}-g^{\mu \upsilon}L##

##L=\frac{1}{2}\partial_{\mu}\partial^{\mu}-\frac{1}{2}m\phi^2-\lambda\phi^4##

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

Let ##\phi_{+}=\frac{d^3k}{2\omega_k (2\pi)^3}\int(\hat{a}^{\dagger}(\overrightarrow{k})e^{ikx})##

and ##\phi_{-}=\frac{d^3k}{2\omega_k (2\pi)^3}\int(\hat{a}(\overrightarrow{k})e^{-ikx})##.

Then ##\phi^4=\phi_{1}\phi_{2}\phi_{3}\phi_{4}=(\phi_{1+}+\phi_{1-})(\phi_{2+}+\phi_{2-} )(\phi_{3+}+\phi_{3-}) (\phi_{4+}+\phi_{4-})##

Looking only at the term ##\phi_{1-}\phi_{2-}\phi_{3+}\phi_{4+}##, we find

##\bra{\overrightarrow{P'}}\phi_{1-}\phi_{2-}\phi_{3+}\phi_{4+}\ket{\overrightarrow{P}}## diverges. Unsure where to go from here. Any resources would be helpful. Thank you.

and ##\phi_{-}=\frac{d^3k}{2\omega_k (2\pi)^3}\int(\hat{a}(\overrightarrow{k})e^{-ikx})##.

Then ##\phi^4=\phi_{1}\phi_{2}\phi_{3}\phi_{4}=(\phi_{1+}+\phi_{1-})(\phi_{2+}+\phi_{2-} )(\phi_{3+}+\phi_{3-}) (\phi_{4+}+\phi_{4-})##

Looking only at the term ##\phi_{1-}\phi_{2-}\phi_{3+}\phi_{4+}##, we find

##\bra{\overrightarrow{P'}}\phi_{1-}\phi_{2-}\phi_{3+}\phi_{4+}\ket{\overrightarrow{P}}## diverges. Unsure where to go from here. Any resources would be helpful. Thank you.

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