Problem 1 (qiuzhen-qualify 2024-fall)
Section: Einstein metric

Consider the vacuum Einstein's equation with a cosmological constant

$$R_{\mu\nu} - \tfrac{1}{2} g_{\mu\nu} R + g_{\mu\nu} \Lambda = 0.$$

Prove that $R_{\mu\nu} = k\, g_{\mu\nu}$ where $k$ is a constant and find the value of $k$.

Problem 2 (qiuzhen-qualify 2024-fall)
Section: Einstein metric

Now start with an ansatz of a metric in the form

$$ds^2 = - f(r)\, dt^2 + \frac{1}{f(r)} dr^2 + r^2 \big(d\theta^2 + \sin^2\!\theta\, d\phi^2\big),$$

where $f(r)$ is a polynomial in $r$.

Compute the Christoffel symbols $\Gamma^{\mu}_{\nu\sigma}$ for this metric.

Problem 3 (qiuzhen-qualify 2024-fall)
Section: Einstein metric

Using the metric ansatz

$$ds^2 = - f(r)\, dt^2 + \frac{1}{f(r)} dr^2 + r^2 \big(d\theta^2 + \sin^2\!\theta\, d\phi^2\big),$$

where $f(r)$ is a polynomial in $r$, compute the Ricci tensor $R_{\mu\nu}$ and the scalar curvature $R$ of this metric.

Problem 4 (qiuzhen-qualify 2024-fall)
Section: Einstein metric

Assuming that the metric ansatz

$$ds^2 = - f(r)\, dt^2 + \frac{1}{f(r)} dr^2 + r^2 \big(d\theta^2 + \sin^2\!\theta\, d\phi^2\big)$$

is a solution of the vacuum Einstein equation with cosmological constant $\Lambda$, solve for $f(r)$.

Problem 5 (qiuzhen-qualify 2024-fall)
Section: Einstein metric

For $\Lambda>0$ and $\Lambda<0$, write down all the Killing vectors of the metric using the solution $f(r)$ you found.

Problem 1 (qiuzhen-qualify 2024-fall)
Section: the $\lambda\phi^3$ model

Consider the following Lagrangian density of a real scalar $\phi(x)$ in $3+1$ dimensional spacetime with flat metric $(-,+,+,+)$,

$$\mathcal{L} = -\tfrac{1}{2}\, \partial_{\mu}\phi\,\partial^{\mu}\phi - \tfrac{1}{2} m^{2} \phi^{2} - \tfrac{1}{6} g\, \phi^{3},$$

where $g$ is a coupling with dimensions of mass.

Draw all the one-particle irreducible Feynman diagrams up to 3 external lines and 2 loops.

Problem 2 (qiuzhen-qualify 2024-fall)
Section: the $\lambda\phi^3$ model

For the theory with Lagrangian

$$\mathcal{L} = -\tfrac{1}{2}\, \partial_{\mu}\phi\,\partial^{\mu}\phi - \tfrac{1}{2} m^{2} \phi^{2} - \tfrac{1}{6} g\, \phi^{3},$$

compute the one-loop self-energy graph using dimensional regularization.

Problem 3 (qiuzhen-qualify 2024-fall)
Section: the $\lambda\phi^3$ model

Introducing $m^{2} = m_{R}^{2} + \delta m^{2}$. What is the value of $\delta m^{2}$ if we want to write the one-loop self-energy graph as a finite function of $m_{R}$?

Problem 4 (qiuzhen-qualify 2024-fall)
Section: the $\lambda\phi^3$ model

Compute the one-loop tadpole graph using dimensional regularization for the theory with

$$\mathcal{L} = -\tfrac{1}{2}\, \partial_{\mu}\phi\,\partial^{\mu}\phi - \tfrac{1}{2} m^{2} \phi^{2} - \tfrac{1}{6} g\, \phi^{3}.$$

Problem 5 (qiuzhen-qualify 2024-fall)
Section: the $\lambda\phi^3$ model

What counter-term should you introduce in the Lagrangian to cancel the one-loop tadpole diagram?