# Phi 4 Theory Renormalization

• Diracobama2181

#### Diracobama2181

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.

Last edited:
A:The quantity you are interested in is the vacuum expectation value of a quartic interaction. This quantity is zero by time-reversal invariance, see e.g. this paper by J. Schwinger.