Problem 1 (qiuzhen-qualify 2024-spring)
Section: QFT

Poincare group representations (15 points)

Weyl spinors $\psi_L$ and $\psi_R$ have the following transformation laws under infinitesimal rotation $\vec{\theta}$ and boosts $\vec{\beta}$,

$$\psi_L \mapsto \left(1 - i\,\frac{\vec{\theta}\!\cdot\!\vec{\sigma}}{2} - \frac{\vec{\beta}\!\cdot\!\vec{\sigma}}{2}\right)\psi_L,\qquad
\psi_R \mapsto \left(1 - i\,\frac{\vec{\theta}\!\cdot\!\vec{\sigma}}{2} + \frac{\vec{\beta}\!\cdot\!\vec{\sigma}}{2}\right)\psi_R.$$

The identity $\sigma^T = -\sigma^2\sigma\sigma^2$ allows us to rewrite the $\psi_L$ transformation in the unitarily equivalent form

$$\psi' \mapsto \psi'\left(1 + i\,\frac{\vec{\theta}\!\cdot\!\vec{\sigma}}{2} + \frac{\vec{\beta}\!\cdot\!\vec{\sigma}}{2}\right),$$

where $\psi' = \psi_L^\dagger \sigma^2$. The $\left(\tfrac12,\tfrac12\right)$ representation of the Lorentz group can be represented as a $2\times2$ matrix that has the $\psi_R$ transformation law on the left and, simultaneously, the transposed $\psi_L$ transformation on the right. Parametrize this matrix as

$$
\begin{pmatrix}
V^0 + V^3 & V^1 - i V^2 \\
V^1 + i V^2 & V^0 - V^3
\end{pmatrix}.
$$

Show that the object $V^\mu$ transforms as a 4-vector.

Problem 2 (qiuzhen-qualify 2024-spring)
Section: QFT

$\Pi_{\mu\nu}(q)$ via dimensional regularization (35 points)

At one-loop, the photon propagator in QED receives the correction

$$i\,\Pi_{\mu\nu}(q) = - \int \frac{d^4 p}{(2\pi)^4} \, \mathrm{Tr} \left( i\,\gamma^\nu \, \frac{i}{\not p + \not q - m} \, i\,\gamma^\mu \, \frac{i}{\not p - m} \right),$$

where $\gamma_\mu$ is the gamma matrix and $\not p = p^\mu \gamma_\mu$.

i) 10' Show that this expression can be written as

$$i\,\Pi_{\mu\nu}(q) = -i \int \frac{d^4 p}{(2\pi)^4} \, \frac{N_{\mu\nu}}{D},$$

where

$$N_{\mu\nu} = \operatorname{tr}\,[\gamma_\nu(\not p + \not q + m)\gamma_\mu(\not p - m)]\,, \qquad \frac{1}{D} = \int_0^1 d\alpha \, \frac{1}{D},$$

with

$$D = \big[\,l^2 + \alpha(1-\alpha) q^2 - m^2 + i\epsilon\,\big]^2, \qquad \text{where } l = p + \alpha q.$$

ii) 10' Evaluate $N_{\mu\nu}$ and shift momentum integration from $p$ to $l$ as above.

iii) 10' At the integration step over $l$, perform dimensional regularization by promoting to a $d$-dimensional integral. Derive the expression

$$\Pi_{\mu\nu}(q) = (q_\mu q_\nu - \eta_{\mu\nu} q^2)\,\Pi(q^2),$$

when taking the limit $d \to 4$.

iv) 5' Why does this result ensure gauge invariance?

Problem 3 (qiuzhen-qualify 2024-spring)
Section: GR

20 points

a) Let $\xi^a$ be a timelike Killing field, let $\gamma^a$ be a timelike geodesic. Show that the energy $E = -\gamma^a\xi_a$ is conserved along $\gamma$.

b) A symmetric 2-tensor $K_{ab}$ is a Killing tensor if $\nabla_{(a} K_{bc)} = 0$. Let $\gamma^a$ be a geodesic. Show that $K = K_{ab}\gamma^a\gamma^b$ is conserved along $\gamma^a$ if $K_{ab}$ is a Killing tensor.

Problem 4 (qiuzhen-qualify 2024-spring)
Section: GR

30 points

Analysis of null geodesics in the Schwarzschild spacetime with line element

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

Let $\xi^a = (\partial_t)^a$, $\psi^a = (\partial_\phi)^a$ be the static and axial Killing fields, respectively. The energy $E$ and azimuthal angular momentum $L$ of a null geodesic with 4-velocity $u^a$ are given by

$$E = -\xi^a u_a, \qquad L = \psi^a u_a.$$

a) Write an expression for the energy of an equatorial null geodesic ($\theta = \pi/2$) in terms of its azimuthal angular momentum and radial velocity.

b) Find the location of orbiting null geodesics, and analyze their stability.

c) Consider a light ray emanating at $r = 4M$ with unit energy (with respect to a static observer) which is observed at $r = 8M$. Calculate the observed energy.