How Does Time-Reversal Invariance Affect Phi 4 Theory Renormalization?

[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.

 

[member 553083]

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.