Problem 1 (qiuzhen-qualify 2023-fall)

(a) Consider a Hamiltonian $\hat H_0$ with eigenvalues $\{E_i\}$ and corresponding eigenstates $|E_i\rangle$. Write down the time dependent Schrodinger equation satisfied by a general state $|\Psi(t)\rangle$. Consider a small time dependent perturbation $\hat V(t)$, so that the full Hamiltonian is $\hat H = \hat H_0 + \hat V(t)$. By transforming to the "interaction picture", obtain the time-dependent Schrodinger equation satisfied by the state $|\psi_I\rangle$ in interaction picture.

(b) Now consider an initial state $|\Psi(t=-\infty)\rangle = |E_0\rangle$, where $|E_0\rangle$ is the ground state of $H_0$. Compute to leading order in $\hat V(t)$ the state at time $t$. What is the probability amplitude of finding the system in eigenstate of $H_0$ of energy $E_i$ to linear order in $V$ as $t \to \infty$? Suppose that $\hat V(t) = \hat V_\omega \cos \omega t$ for some frequency $\omega$ and Hermitian operator $\hat V_\omega$. What are the necessary conditions that the amplitude of finding the system in the excited state $|E_i\rangle$ non-vanishing?

(c) The Lagrangian density for the scalar Yukawa theory is given by
$$
L = \frac{1}{2} (\partial_\mu \phi)(\partial_\nu \phi)\eta^{\mu\nu} - \frac{m^2}{2}\phi^2 + \bar\psi(i\gamma_\mu \partial^\mu - M)\psi - g\bar\psi\psi\phi - \frac{\lambda}{4!}\phi^4
\tag{1}
$$
where $\eta_{\mu\nu}$ is the Minkowskian metric, $\partial_\mu \equiv \partial/\partial x^\mu$, and $m$ is the mass of the scalar field, while $M$ is the mass of the Dirac field in the Lagrangian above.

i. Find the superficial mass dimension for each term of the Lagrangian supposing that we live in $3+1$ space-time dimensions. Is this theory renormalisable? Write down the renormalised Lagrangian density as a sum of the bare Lagrangian density and appropriate counter-terms. Write down the Feynman rules for the theory. (Use the renormalised mass in the propagators).

ii. Obtain the transformation of the action under Parity ($P$). Is the action invariant under $P$?

iii. The self energy of the spinor field at one-loop is given by the diagram below. Estimate the divergence of the diagram by considering cutting off the loop momentum integral (hint: one has to continue $p^0$ to $i p^0$, so that the 4-momentum becomes a Euclidean vector.) at $\sqrt{p\,p}=\Lambda$ (this $\sqrt{p\,p}$ is the length of the Euclidean 4-momentum vector after the analytic continuation hinted before) and obtain the leading order dependence on $\Lambda$ as $\Lambda \to \infty$. Write down also a counter term appropriate for this diagram.

Problem 2 (qiuzhen-qualify 2023-fall)

(50 points) In this problem you are asked to show that Einstein's general relativity reduces to Newtonian gravity in an appropriate limit.

(a) Consider a particle moving in spacetime with metric $g_{\mu\nu}$. Its mass is small so that its presence does not affect the spacetime metric. The trajectory $x^\mu(s)$ of the particle (where $s$ is a parameter) is such that the proper time
$$
\tau = \int \sqrt{-g_{\mu\nu}\frac{dx^\mu}{ds}\frac{dx^\nu}{ds}}\, ds .
\tag{2}
$$
is minimized. Derive the differential equation satisfied by $x^\mu(\tau)$.

(b) Suppose that gravity is weak so that there is a coordinate system $(x^0,x^1,x^2,x^3)$ such that the metric $g_{\mu\nu}$ is a small pertubation of the Minkowski metric $\eta_{\mu\nu} = \mathrm{diag}(-1,+1,+1,+1)$:
$$
 g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}, \qquad |h_{\mu\nu}| \ll 1 .
\tag{3}
$$
Find the Christoffel symbols $\Gamma^{\mu}_{\nu\rho}$ to the first order in $h_{\mu\nu}$.

(c) Assume that the particle is moving slowly:
$$
\left|\frac{dx^i}{d\tau}\right| \ll \left|\frac{dx^0}{d\tau}\right| , \quad i=1,2,3 .
\tag{4}
$$
Furthermore, assume that $g_{\mu\nu}$ is static. Show that we can identify
$$
 h_{00} = -2\Phi ,
\tag{5}
$$
where $\Phi$ is a potential of a conservative force in the sense of Newtonian mechanics.

(d) Denote matter energy momentum tensor by $T_{\mu\nu}$. Write down the Einstein equation in terms of the Ricci scalar, the Ricci tensor and $T_{\mu\nu}$. Using (3) and the Einstein equation, express $T_{00}$ up to linear order in $h_{00}$.

Problem e (qiuzhen-qualify 2023-fall)

(e) Now we ask what kind of potential $\Phi$ is created by a massive object. Consider an object moving at speed much smaller than the speed of light. Its energy–momentum tensor can be approximated as

$$T_{\mu\nu}=\rho U_\mu U_\nu,$$

where $\rho$ is the rest-frame energy density and $U^\mu$ is the four-velocity. Suppose that $\rho$ is small. Show that $\Phi$ satisfies the Poisson equation

$$\nabla^2\Phi=4\pi G\rho.$$

Therefore, $\Phi$ can be interpreted as the gravitational potential for Newtonian gravity.