Single Particle Expectation of Energy Momentum Tensor

[Diracobama2181]

Homework Statement

Currently trying to explicitly calculate the following for a non interacting spin 0 field

$$\bra{\overrightarrow{P'}}\hat{T}_{\mu v}\ket{\overrightarrow{P}}
$$
where
$$\hat{T}_{\mu v}=\partial^{\mu}\Phi \partial^{v}\Phi-g^{\mu v}L$$

Relevant Equations

For this setup,
$$L=\frac{1}{2}\partial_{\mu}\Phi \partial^{\mu}\Phi-\frac{1}{2}m^2\Phi^2 $$
$$\Phi=\int\frac{d^3k}{2\omega_k (2\pi)^3}(\hat{a}(\overrightarrow{k})e^{-ikx}+\hat{a}^{\dagger}(\overrightarrow{k})e^{ikx}))$$
$$\ket{\overrightarrow{P}}=a^{\dagger}\ket{0}$$
and $$[a(\overrightarrow{k'}),a^{\dagger}(\overrightarrow{k})]=2\omega_k (2\pi)^3 \delta^3(\overrightarrow{k'}-\overrightarrow{k})$$

$$\hat{T}_{\mu v}(x)=e^{i\hat{P}x}\hat{T}_{\mu v}(0)e^{-i\hat{P}x}$$,
so $$\bra{\overrightarrow{P'}}\hat{T}_{\mu v}(x)\ket{\overrightarrow{P}}=e^{iP'x}\bra{\overrightarrow{P'}}\hat{T}_{\mu v}(0)\ket{\overrightarrow{P}}e^{-i\hat{P}x}$$
Now,
$$\partial^{\mu}\Phi=\int\frac{d^3 k_1}{2\omega_{k_1} (2\pi)^3}(-ik_1^{\mu}\hat{a}(\overrightarrow{k_1})e^{-ik_1 x}+i k_1^{\mu}\hat{a}^{\dagger}(\overrightarrow{k_1})e^{ik_1 x}))$$
and $$\partial^{v}\Phi=\int\frac{d^3 k_2}{2\omega_{k_2}(2\pi)^3}(-ik_2^{v}\hat{a}(\overrightarrow{k_2})e^{-ik_2 x}+ik_2^{v}\hat{a}^{\dagger}(\overrightarrow{k_2})e^{ik_2 x}))$$
so
$$\bra{\overrightarrow{P'}}\partial^{\mu}\Phi(0)\partial^{v}\Phi(0)\ket{\overrightarrow{P}}=\bra{0}\int\frac{d^3 k_1 d^3 k_2}{4\omega_{k_1}\omega_{k_2} (2\pi)^6}(-k_1^{\mu}k_2^{v}\hat{a'}\hat{a}_{k_1}\hat{a}_{k_2}\hat{a}^{\dagger}+k_1^{\mu}k_2^{v}\hat{a'}\hat{a}_{k_1}^{\dagger}\hat{a}_{k_2}\hat{a}^{\dagger}
+k_1^{\mu}k_2^{v}\hat{a'}\hat{a}_{k_1}\hat{a}_{k_2}\hat{a}^{\dagger}-k_1^{\mu}k_2^{v}\hat{a'}\hat{a}_{k_1}^{\dagger}\hat{a}_{k_2}^{\dagger}\hat{a}^{\dagger})\ket{0}
=k^{\mu}k^{v'}+k^{\mu'}k^{v}+\int d^3 k_1 d^3 k_2 k_1^{\mu}k_2^{v}\delta^3(\overrightarrow{k_1}-
\overrightarrow{k_2})\delta^3(\overrightarrow{k'}-\overrightarrow{k})$$
Then, using
$$\Phi^2(0)=\int \frac{d^3k_1 d^3k_1}{4\omega_{k_1}omega_{k_2} (2\pi)^3}(\hat{a_{k_1}}\hat{a_{k_2}}+\hat{a_{k_1}}\hat{a_{k_2}}^{\dagger}+\hat{a_{k_1}}^{\dagger}\hat{a_{k_2}}+\hat{a_{k_1}}^{\dagger}\hat{a_{k_2}}^{\dagger})$$
and
$$\bra{\overrightarrow{P'}}\Phi^2(0)\ket{\overrightarrow{P}}=\bra{0}\int\frac{d^3k_1 d^3k_1}{4\omega_{k_1}\omega_{k_2} (2\pi)^6}(4(2\pi)^6\omega_{k_1}\omega_{k_2}\delta^3(\overrightarrow{k_1}-\overrightarrow{k_2})\delta^3(\overrightarrow{k'}-\overrightarrow{k})+4(2\pi)^6\omega_{k_1}\omega_{k_2}\delta^3(\overrightarrow{k_1}-\overrightarrow{k})\delta^3(\overrightarrow{k'}-\overrightarrow{k_2})+4(2\pi)^6\omega_{k_1}\omega_{k_2}\delta^3(\overrightarrow{k_2}-\overrightarrow{k})\delta^3(\overrightarrow{k'}-\overrightarrow{k_1}))\ket{0}=2+\int d^3 k_1 d^3 k_1\delta^3(\overrightarrow{k_2}-\overrightarrow{k_1})\delta^3(\overrightarrow{k'}-\overrightarrow{k}))$$
Do these calculations seem correct so far? And if so, how do I go about renormalizing the equations to get rid of the divergences. Thank you.

 

[Nate810]

A:You're on the right track, but there's still some cleaning up that needs to be done. First you need to write the expression for $\hat{T}_{\mu v}$ in terms of the creation and annihilation operators:$$\hat{T}_{\mu v}(x)=\partial_\mu\Phi(x)\partial_v\Phi(x)-\eta_{\mu v}\Phi^2(x).$$Now you can calculate each term separately. For the first term, we have\begin{align*}\bra{\overrightarrow{P'}}\partial^\mu\Phi(0)\partial^v\Phi(0)\ket{\overrightarrow{P}}&=\int\frac{d^3k_1d^3k_2}{4\omega_{k_1}\omega_{k_2}(2\pi)^6}(-ik_1^\mu k_2^v\bra{0}\hat{a}(\overrightarrow{k_1})\hat{a}(\overrightarrow{k_2})\hat{a}^\dagger(\overrightarrow{P})\hat{a}^\dagger(\overrightarrow{P'})\ket{0}\tag{1}\\&+ik_1^\mu k_2^v\bra{0}\hat{a}^\dagger(\overrightarrow{k_1})\hat{a}(\overrightarrow{k_2})\hat{a}^\dagger(\overrightarrow{P})\hat{a}^\dagger(\overrightarrow{P'})\ket{0}\tag{2}\\&-ik_1^\mu k_2^v\bra{0}\hat{a}(\overrightarrow{k_1})\hat{a}^\dagger(\overrightarrow{k_2})\hat{a}^\dagger(\overrightarrow{P})\hat{a}^\dagger(\overrightarrow{P'})\ket{0}\tag{3}\\&+ik_1^\mu k_2^v\bra{