Problem 1 (yau-contest 2022)

(a) A symmetry transformation in quantum mechanics is represented by a unitary or anti-unitary operator acting on a Hilbert space. The time reversal transformation $\Theta$ relates the wave function at time $t$ to time $-t$. Prove: $\Theta$ is an anti-unitary operator.

(b) Consider state vector $|\psi\rangle$ for a quantum system. A time reversal transformation is represented by an anti-unitary operator $\Theta$. We now consider position space wavefunction $\psi(x)=\langle x|\psi\rangle$, and $\Theta|x\rangle=|x\rangle$. Prove: the position space wave function for $\Theta|\psi\rangle$ is

$$\psi(x)^*.$$

(c) A one dimensional quantum system is invariant under time reversal transformation, and so its Hamiltonian satisfies $\Theta H=H\Theta$. If an energy eigenstate $|\psi\rangle$ has no degeneracy, Prove: it is possible to take the position space energy eigenfunction to be real:

$$\psi(x)^*=\psi(x).$$

Problem 2 (yau-contest 2022)

Consider following quantum Hamiltonian:

$$H_0=\frac{p_1^2}{2m}+\frac{1}{2}m\omega^2 x_1^2+\frac{p_2^2}{2m}+\frac{1}{2}m\omega^2 x_2^2.$$

This is the Hamiltonian for two decoupled harmonic oscillators.

(a) Calculate the eigenstates and eigenvalues for $H_0$ (an energy eigenstate could be labeled as $|n_1,n_2\rangle$).

(b) Assume the creation and annihilation operators for two harmonic oscillators are $a_i^\dagger,a_i$, $i=1,2$. Define following operators

$$J_+=a_1^\dagger a_2,\quad J_-=a_2^\dagger a_1,\quad J_z=\frac{1}{2}(a_1^\dagger a_1-a_2^\dagger a_2).$$

i. Prove that: $[J_z,J_\pm]=\pm J_\pm$, $[J_+,J_-]=2J_z$.

ii. Consider one eigenvalue $E_n$ of $H_0$ (here $n_1+n_2=n$). Prove that: all eigenstates of $E_n$ form an irreducible representation of $su(2)$ Lie algebra, and compute the spin.

(c) Consider following perturbed Hamiltonian ($\lambda$ is small)

$$H=H_0+\lambda x_1^2 p_2^2.$$

Compute the first order correction to the energy for the energy level $n_1+n_2=2.$

Problem 3 (yau-contest 2022)

A Killing vector field $k^\mu \frac{\partial}{\partial x^\mu}$ satisfies the equation

$$k^\lambda \partial_\lambda g_{\mu\nu}+\partial_\mu k^\lambda g_{\lambda\nu}+\partial_\nu k^\lambda g_{\lambda\mu}=0.$$

(a) Prove: $D_\mu k_\nu+D_\nu k_\mu=0$, here $D_\mu$ is the covariant derivative.

(b) For a moving particle in gravitational background with a Killing vector field, Prove: $k^\mu P_\mu$ is a conserved quantity, Here $P_\mu=m\frac{dx^\nu}{d\tau}g_{\mu\nu}$ is the momentum for the free falling particle with trajectory $x^\nu(\tau)$.

Problem 4 (yau-contest 2022)

Consider following metric

$$ds^2=-(1-\frac{2M}{r})dv^2+dr\,dv+r^2 d\Omega^2.$$

Here $d\Omega^2$ is the standard metric on two sphere. Consider the hypersurface defined by $S=r-2M=0$, and a vector field $l=f(x)(g^{\mu\nu}\partial_\nu S)\,\frac{\partial}{\partial x^\mu}$, here $f(x)$ is a non-zero function. Prove:

(a) $l$ is normal to the surface $S$.

(b) $l^2=0$ on the surface $S$.

(c) $\frac{\partial}{\partial v}$ is a Killing vector field.

Problem 5 (yau-contest 2022)

The energy momentum tensor for a relativistic quantum field theory is denoted as $\theta^{\mu\nu}$, which is symmetric and conserved.

(a) Define new current $s^\mu=x_\nu\theta^{\mu\nu}$ and

$$K^{\lambda\mu}=x^2\theta^{\lambda\mu}-2x^\lambda x_\rho\theta^{\rho\mu}.$$

Compute $\partial_\mu s^\mu$ and $\partial_\mu K^{\lambda\mu}$, and explain the condition on $\theta^{\mu\nu}$ so that these new currents are conserved.

(b) Consider a scalar field $\sigma(x)$ which transforms under a scale transformation as

$$\delta\sigma=x^\nu\partial_\nu\sigma+f^{-1}$$

we have following Lagrangian

$$L=L_s=\frac{\mu_0^2}{2}\,\phi^2 e^{2f\sigma}+\frac{1}{2f^2}\,\partial_\nu e^{f\sigma}\,\partial^\nu e^{f\sigma}.$$

The infinitesimal scale transformation on scalar field $\phi$ is $\delta\phi=(1+x^\lambda\partial_\lambda)\phi$. Here $L_s$ is scale invariant part of the Lagrangian. Prove that: the above Lagrangian is scale invariant.

(c) Explain why a classically scale invariant Lagrangian for a quantum field theory may fail to be scale invariant quantum mechanically.

Problem 6 (yau-contest 2022)

Consider following Lagrangian for $N$ scalar fields $\phi^a$, $a=1,\dots,N$:

$$L=\frac{1}{2}\partial_\mu\phi^a\partial^\mu\phi^a-\frac{1}{2}\mu_0^2\phi^a\phi^a-\frac{1}{8}\lambda_0(\phi^a\phi^a)^2.$$

Here the repeated index implies the summation over the index.

Problem unknown (yau-contest 2022)

(a) Write down the propagator and interaction vertex for this model, and write down four point Feynman diagrams up to one loop level.

(b) Define $g_0 = \lambda_0 N$, and compute the order in $g_0$ and $N$ for all the diagrams listed in last question. If we fix the coupling $g_0$, and let $N$ go to infinity, list the leading order Feynman diagrams in $1/N$.