# Energy Momentum Tensor in Phi^3 Theory

Diracobama2181
Homework Statement:
If I have an energy-momentum tensor
$$\hat{T}^{\mu \nu} (x)=\partial^{\mu}\Phi\partial^{\nu}\Phi-\\g^{\mu \nu}\mathcal{L}$$
,with
$$\mathcal{L}=(\frac{1}{2}(\partial_{\mu}\Phi)^2-\frac{1}{2}m^2\Phi^2-\frac{\lambda \Phi^3}{3!})$$
, how would I go about calculating
$$\bra{ \vec{ p'}} T^{\mu,\nu} \ket{ \vec {p}}$$?
Relevant Equations:
##\ket{\vec{p}}=\hat{a}^{\dagger}(\vec{p})\ket{0}## for a free field with ##[\hat{a}({\vec{k})},\hat{a}^{\dagger}({\vec{k'})}]=2(2\pi)^3\omega_k\delta^3({\vec{k}-\vec{k'}})##
$$\bra{ \vec{ p'}} T_{\mu,\nu} \ket{ \vec {p}}=\bra{\Omega}\hat{a}(\vec{p'})(\partial^{\mu}\Phi\partial^{\nu} \Phi-g^{\mu \nu}\mathcal{L})\hat{a}^{\dagger}(\vec{p})\ket{\Omega}=\bra{\Omega}(\hat{a}(\vec{p'})\partial^{\mu}\Phi\partial^{\nu} \Phi\hat{a}^{\dagger}(\vec{p})\\ -\hat{a}(\vec{p'})g^{\mu \nu}(\frac{1}{2}(\partial_{\mu}\Phi)^2-\frac{1}{2}m^2\Phi^2-\frac{\lambda \Phi^3}{3!}))\hat{a}^{\dagger}(\vec{p})\ket{\Omega}$$
In this instance, would this imply that, for example, $$\bra{\Omega}(\hat{a}(\vec{p'})(\frac{1}{2}(\partial_{\mu}\Phi)^2-\frac{1}{2}m^2\Phi^2-\frac{\lambda \Phi^3}{3!})\hat{a}^{\dagger}(\vec{p}))\ket{\Omega}=lim_{t \to \infty(1-i\epsilon)}\frac{\bra{0}[T(\hat{a}(\vec{p'})(\frac{1}{2}(\partial_{\mu}\Phi)^2-\frac{1}{2}m^2\Phi^2-\frac{\lambda \Phi^3}{3!})\hat{a}^{\dagger}(\vec{p}))e^{-i\int \lambda \frac{\Phi^3}{3!}dt}]\ket{0}}{\bra{0}T e^{-i\int \lambda \frac{\Phi^3}{3!}dt}\ket{0}}$$?
Also, how would I go about calculating terms like $$\bra{\Omega}\hat{a}(\vec{p'})(\partial^{\mu}\Phi\partial^v \Phi )\hat{a}^{\dagger}(\vec{p})\ket{\Omega}$$ using Feynman diagrams? Thanks in advance.
Note: $$\ket{\Omega}$$ is the ground state in the interacting theory.

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