Rohan Kulkarni

Part 2 of 2.

Part 1 of 2.

In the Källén–Lehmann post we showed non-perturbatively that any interacting scalar theory must have a two-point function with an isolated pole at some physical mass $m$ and residue $Z$, and that — generically — neither $m$ nor $Z$ equals its free-theory…

In free scalar field theory the two-point function $\langle 0|T\phi(x)\phi(y)|0\rangle$ has a clean meaning: it is the amplitude for a particle to propagate from $y$ to $x$.

I sat down with Bhavesh Rajpoot from STARGAZER — Astronomy Outreach Initiative to record something we've both been doing informally for years: answering questions from students about how to navigate the MSc Physics program at Heidelberg.

Prerequisites: Part I — Lie Groups and Lie Algebras, Part II — The Lorentz Group, Part III — Spinors, Fields, and the Representations That Matter.

Prerequisites: Part I — Lie Groups and Lie Algebras, Part II — The Lorentz Group, familiarity with the Pauli matrices, and basic quantum mechanics (spin-1/2 systems).

Prerequisites: Part I — Lie Groups and Lie Algebras, familiarity with index notation for vectors and matrices, and special relativity at the level of knowing what a Lorentz boost is.

Prerequisites: Linear algebra (matrix exponentials, commutators), some exposure to group theory (what a group is, what the identity element is), and basic quantum mechanics (angular momentum).

In Part 1, we saw that the path integral for a $U(1)$ gauge field is ill-defined: the kinetic operator $-\Box,\delta_{\mu\nu} + \partial_\mu\partial_\nu$ has zero modes along gauge directions, and the integral overcounts by an infinite factor — the volume…

When you try to quantize electromagnetism using the path integral, you immediately hit a wall: the kinetic operator has no inverse.

This is the fourth and final post in a series on vector and 2-form dark energy.

This is the third post in a series on vector and 2-form dark energy.

This is the second post in a series on vector and 2-form dark energy.

This is the first post in a series on using higher-rank tensor fields — vectors and 2-forms — as dark energy candidates.

Organizational details M.Sc.

Problem: Does BEC occur in 1D or 2D for free particles with periodic boundary conditions?

This is Part 3 of a 3-part series on IPSP Leipzig.

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Problem: Derive the virial expansion for a classical gas of hard spheres, showing how the excluded volume correction emerges and computing higher-order virial coefficients.

Problem: For the scattering process $e^+ e^- \to \mu^+ \mu^-$, work out the Lorentz-invariant Mandelstam variables $s, t, u$ in the center-of-mass frame.

Organizational details M.Sc.

Most of time, everything is homogeneous and isotropic, until it's not...

Here is a brainstormed document of some main things to remember when applying for a Physics Ph.D. program.

Given a general spin-1/2 state $|\Psi\rangle = \begin{pmatrix}\alpha \ \beta\end{pmatrix}$ (with $|\alpha|^2 + |\beta|^2 = 1$), what is the probability of finding the spin positive along $y$?

Spin-orbit coupling is responsible for the fine structure of hydrogen and plays a central role in atomic physics, relativistic corrections, and condensed matter (topological insulators, Rashba effect).

Neutrino oscillations — the fact that a neutrino created as an electron neutrino can later be detected as a muon neutrino — were confirmed experimentally in 1998 (Super-Kamiokande) and earned the 2015 Nobel Prize in Physics.

The density operator is the most general way to describe a quantum system.

In quantum mechanics, the correlation function $C(t)$ provides deep insights into the evolution of a state and its departure from classical behavior.

These three identities show up constantly in quantum mechanics — in the derivation of coherent states, the Baker–Campbell–Hausdorff formula, time evolution of observables, and Weyl quantization.

This is Part 2 of a 3-part series on IPSP Leipzig.

Time reversal is one of the more subtle discrete symmetries in quantum mechanics.

In undergraduate quantum mechanics you typically compute tunneling probabilities for simple barriers — a rectangular "top hat" or a parabola.

Symmetry is one of the most powerful tools in physics — not just for aesthetics, but for hard computational results.

Clebsch-Gordan coefficients are the bridge between the total angular momentum basis $|j, m_j\rangle$ and the uncoupled basis $|l, m_l; s, m_s\rangle$.

In quantum mechanics, the state of a spin-1/2 particle is often represented in the basis of the $z$-component of spin, $S_z$.

Thirty-one years ago Dick Feynman told me about his ‘sum over histories’ version of quantum mechanics.

After several wonderful undergraduate years at Leipzig, I applied to one of the best physics faculties on the planet and one of the most fairy-tale-like cities - Heidelberg.

Note: This blog post is inspired by my own term paper for MIT 8.06x (Applications of Quantum Mechanics).

:::caution[HEADS UP] I graduated from IPSP in 2019 and worked as a TA until March 2020.

Organizational details M.Sc.

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A directed polymer consists of atoms $i = 0, 1, 2, \dots, N$ at positions $(x_i, y_i) \in \mathbb{Z}^2$ of a square lattice.

Organizational details B.Sc.