Problem 1 (yau-contest 2023)

Consider the Lagrangian

$$L(x, y, \dot{x}, \dot{y}) = \frac{1}{2}\frac{\dot{x}^2 + \dot{y}^2 + 2(x\dot{y} - y\dot{x})}{x^2 + y^2} \quad (1)$$

(a) Compute the Hamiltonian $H(x, y, p_x, p_y)$, and show the final form can be written as

$$\frac{1}{2} f(x, y)\big[(p_x - A_x(x, y))^2 + (p_y - A_y(x, y))^2\big] \quad (2)$$

for some $f, A_x, A_y$. Find the vector potential $\vec{A}$ and then compute the corresponding magnetic field away from the origin. (Hint: recall that $\vec{B} = \mathrm{curl}\,\vec{A} = \nabla \times \vec{A}$.)

(b) Prove that the Lagrangian $L(x, y, \dot{x}, \dot{y})$ is invariant under two symmetries: rotations and scale transformations.

(c) Derive the conserved quantities for both symmetries.

(d) Rewrite the Lagrangian in polar coordinates, write down the Euler–Lagrange equations and solve them.

Problem 2 (yau-contest 2023)

A particle of mass $m$ in 2 dimensions is confined by an isotropic harmonic oscillator potential of frequency $\omega$, while subject to a weak and anisotropic perturbation of strength $\alpha \ll 1$. The total Hamiltonian of the particle is

$$H = H_0 + V = \frac{p_x^2}{2m} + \frac{p_y^2}{2m} + \frac{1}{2} m \omega^2 (x^2 + y^2) + \alpha m \omega^2 x y \quad (3)$$

(a) When $\alpha = 0$, what are the energies and degeneracies of the three lowest-lying unperturbed states?

(b) Use perturbation theory to correct the energies of the above three states to the first order in $\alpha$.

(c) Find the exact spectrum of $H$. (Hint: you may want to rotate $x$ and $y$ into a new coordinates)

(d) Check that the perturbative results in part b. are recovered from the exact spectrum.

Problem 3 (yau-contest 2023)

One can express the electric fields $\vec{E}$ and magnetic fields $\vec{B}$ in terms of the scalar and vector potentials, $A^\mu = (\phi, \vec{A})$.

(a) Write down the expression of $\vec{E}$ and $\vec{B}$ in terms of $(\phi, \vec{A})$ and show that the result is unchanged under gauge transformation

$$\phi \to \phi + \frac{\partial}{\partial t} f, \quad \vec{A} \to \vec{A} - \nabla f \quad (4)$$

where $f = f(\vec{x}, t)$ is a scalar function.

(b) Show that two of the 4 Maxwell equations of $\vec{E}$ and $\vec{B}$ are satisfied automatically in terms of $A^\mu = (\phi, \vec{A})$.

(c) Derive the equations for the scalar and vector potentials from the remaining Maxwell equations in Lorentz gauge.

$$\frac{1}{c}\,\partial_t \phi + \nabla \cdot \vec{A} = 0 \quad (5)$$

(d) Recall that the Green’s function of the wave equation

$$\left(\frac{1}{c^2}\partial_t^2 - \nabla^2\right) G(t, \vec{r}) = \delta(t)\delta^3(\vec{r}) \quad (6)$$

is

$$G(t - t_0, \vec{r} - \vec{r}_0) = \frac{\theta(t - t_0)}{4\pi |\vec{r} - \vec{r}_0|}\,\delta\!\left(t - t_0 - \frac{|\vec{r} - \vec{r}_0|}{c}\right) \quad (7).$$

Assume a particle of electric charge $e$ moves with trajectory $\vec{R}(t)$ with $\vec{v}(t) = d\vec{R}(t)/dt$. Use the Green’s function to derive the potential $\phi(\vec{r}, t)$ and $\vec{A}(\vec{r}, t)$ of this particle at $(\vec{r}, t)$. You may assume that $|\vec{R}(t)| \ll |\vec{r}|$, $|\vec{R}(t)| \ll c t$ and $|\vec{v}(t)| \ll c$ and expand your result up to the order $O(1/|\vec{r}|)$ and $O(|\vec{v}(t)|/c)$. This is also called non-relativistic and far-field approximations.

Problem 4 (yau-contest 2023)

Consider a one-dimensional system of free massless bosons with one polarization, and the dispersion relation $E_k = \hbar v |k|$, where $v$ is the particle velocity, $k$ wavevector, $E_k$ energy. The particles are not interacting either among themselves or with external scattering potentials. If the system is in equilibrium at temperature $T$,

(a) Assume that the chemical potential $\mu$ is 0, calculate the heat capacity $C$ per unit length.

(b) Repeat the calculations for massive fermions, for which $E_k = \hbar^2 k^2/(2m)$ and the chemical potential $\mu$ is far above the bottom of the energy spectrum: $\mu \gg k_B T$. (Consider one spin direction for the fermions, and you may make reasonable approximations.)

Problem 5 (yau-contest 2023)

Consider the vacuum Einstein’s equation in four dimensional spacetime with a cosmological constant
$$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+g_{\mu\nu}\Lambda=0.$$

(a) Proof that $R_{\mu\nu}=k\,g_{\mu\nu}$ and find out the value of $k$.

(b) Now start with an ansatz of a metric in the following 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 non zero components of the Ricci tensor $R_{\mu\nu}$ and scalar curvature $R$ of this metric.

(c) Assuming that the above ansatz is a solution of the vacuum Einstein equation with cosmological constant $\Lambda$, solve $f(r)$.

(d) Prove that $\partial_t$ and $\partial_\phi$ are Killing vector fields.

Problem 6 (yau-contest 2023)

Consider the $\phi^3$ model with a real scalar field $\phi(x)$ in $3+1$ dimensional Minkowski spacetime with metric $(-,+,+,+)$. Its Lagrangian density is
$$\mathcal{L}=-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi-\frac{1}{2}m^2\phi^2-\frac{1}{6}g\phi^3,$$
where $g$ is a coupling with dimensions of mass.

(a) Write down the propagator and the interaction vertex for this model in momentum space.

(b) Compute the one-loop self-energy graph using dimensional regularization.

(c) 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 $m_R^2$?

You may find the following formula useful
$$(AB)^{-1}=\int_0^1 dx\,[xA+(1-x)B]^{-2},$$
$$\int d^d k\,\frac{1}{(-k^2-2p\cdot k-M^2+i\epsilon)^s}=(-1)^s i\pi^{d/2}\,\frac{\Gamma(s-d/2)}{\Gamma(s)}\,(-p^2+M^2-i\epsilon)^{d/2-s},$$
where $\Gamma(z)$ is the Gamma function which has a simple pole at the origin.
$$\Gamma(z)=\frac{1}{z}-\gamma+\mathcal{O}(z),$$
where $\gamma$ is the Euler constant.