“She had studied the universe all her life, but had overlooked its clearest message: For small creatures such as we the vastness is bearable only through love.” ― Carl Sagan, "Contact"

# Epsilon Step at a Time

Here are a collection of my articles and lecture notes [none available yet!], some of which were written a long time ago. I am sure there are mistakes lurking in all of them, so beware.

ARTICLES

[1] Where is the Commutation Relation Hiding in the Path Integral Formulation?

The commutation relation between the momentum operator P and the position operator Q is the cornerstone of quantum mechanics. The fact that QP-PQ is nonzero (and imaginary!) is crucial -- quantum mechanics wouldn't work without it.

The Feynman's approach to formulate quantum mechanics by using path integral is more intuitive, but it hides the commutation relation. In this article we explain how to recover the commutation relation from the path integral. We comment on the mathematical difficulties to rigorously formulate Feynman path integral. We also make a foray into stochastic calculus to understand the Euclidean path integral, in which the commutation relation is related to the famous Itô's lemma.

"calculus", by tenshi-no-pocky

[2] What is the Conserved Charge that Corresponds to Lorentz Boost?

Noether's theorem tells us that to each differentiable symmetry of the action of a physical system, there is a corresponding conservation law. The conserved quantity is often called a "charge". For example, translational invariance gives rise to momentum conservation, while time translation invariance gives rise to energy conservation. Both momentum and energy are "charges" in this sense. In special relativity, Lorentz transformations are coordinate transformations between two coordinate frames that move at constant velocity relative to each other. What then, is the conserved charge that corresponds to Lorentz transformation? In particular, what is the conserved charge corresponding to a Lorentz boost (a Lorentz transformation which doesn't involve rotation)? We discuss this issue from two different perspectives: the viewpoint of a field theorist (in the language of Noether's theorem) and that of a relativist (in the language of Killing vectors).

[3] From Dirac Equation to the Quest for Majorana Fermions

The Dirac equation, born out of the mind of Paul Dirac as he was staring at a burning flame, is one of the greatest achievements in theoretical physics, as it predicted the existence of antimatter. Hidden in the equation, which was purportedly remarked by Dirac to be more intelligent than himself, is the possibility that there could exist in Nature, as yet undiscovered fermions -- now called Majorana fermions -- which are antiparticles of themselves. A young promising Italian physicist Ettore Majorana, having peered into the this secret of Nature in 1937, vanished without a trace just a year after during a trip from Palermo to Naples onboard a ship...

This article explains the beautiful idea behind Majorana’s discovery and the current ongoing quest to find Majorana fermions (could neutrino or dark matter be such a particle?). Even if Nature does not have elementary Majorana fermions, solid state physics provides an exciting arena in looking for the next-best-thing: quasiparticles formed from the collective movement of electrons in various materials. We touch upon the possible applications of Majorana fermions to quantum computing. This is essentially the story about Clifford algebra and its representations! Braid theory also makes an appearance.

[4] Cosmology: A Playground of Geometry

This is a rather long note meant as an introduction to cosmology (56 pages of dense text!). With a whole chapter on differential geometry, this note is aimed at more mathematically inclined readers. We cover the cosmological principle and the standard FLRW cosmologies with some mathematical details, as well as more physical topics such as cosmography (i.e., the different notions of "distance" in cosmology). We also discuss the possible topologies of the Universe.

Remark: This is a rather old note with typos and minor mistakes, but unfortunately I lost the TeX file, so I won't be correcting it anytime soon, sorry...