Energy-Momentum Tensor in Phi^3 Theory

[Diracobama2181]

TL;DR Summary

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.

 

[eljose79]

I would first clarify that the equations provided are from quantum field theory, specifically from the quantization of a free scalar field. The first equation is the momentum state of a particle in terms of creation and annihilation operators, and the second equation is the matrix element of the energy-momentum tensor in terms of these operators.

To answer the first question, yes, the expression given for the matrix element can be interpreted as a limit of a time-ordered product of operators. This is known as the LSZ reduction formula, and it is used to extract physical scattering amplitudes from quantum field theory calculations.

To calculate terms like the one given in the second question using Feynman diagrams, one would need to first understand the Feynman rules for the theory being studied. These rules dictate how to assign mathematical expressions to each vertex and propagator in the Feynman diagram. Then, one would need to follow the rules for calculating the amplitude, which involves summing over all possible diagrams and integrating over the loop momenta. The final result would be a mathematical expression that can be compared with the desired matrix element.