# Blog

## Natural Units

I’m back! I decided to blog this because it didn’t feel substantial enough to have notes on its own and I couldn’t find a proper place for it elsewhere so here I am.

Today, I wanted to discuss natural units i.e. the system under which $c=1$, $\hbar=1$, etc. It became apparent to me that many believe such a system is the result of convenience or laziness but this could not be further from the truth! There are some deep physical insights associated with such a system which I think never weaves its way into standard curriculum and one which I would like to lay down here. First, we must understand how we work with physical units.

## A Natural Limit Definition

Often, the first exposure one gets to rigorous mathematics is the definition of a limit. Let’s consider what this is for a sequence. We say $\lim_{n\rightarrow \infty} a_n = A$ if $\displaystyle \forall \epsilon \in \mathbb{R}^+\quad\exists N \text{ s.t.}\quad\quad n > N \implies |a_n-A| \leq \epsilon$

This, at first sight, is ugly. It takes a while to even understand what it’s saying, longer to see why it works, and much longer to apply it. It’s intimidating to say the least. I feel, however, there is another version that makes the idea of limits simple and very natural giving a deep insight into what a limit really is.

## Philosophy of Quantum Mechanics

Quantum Mechanics since its inception has been one of the most philosophically controversial concepts in all of physics. But what really is so confusing about quantum mechanics? The answer lies in two fundamental principles: locality and realism.

Locality – locality asserts that all information and matter in the universe is limited by the speed of light. No experiment has ever contradicted this principle thus far. One may bring up the idea of quantum entanglement with EPR pairs but if you refer to my post on Quantum Teleportation, I discuss how, even in this case, no information can actually be sent faster than the speed of light preserving locality. An interesting visualization of this is the following with light cones. ## Quaternions Revisited

It has admittedly been quite a while since my last post over a year ago. I thought I would restart the posts by revisiting one of the first topics I discussed on the website: quaternions. My previous post, upon review, seems to be quite uninformative on what the nature and use of them are which I will attempt to show in this post.

## Formalism

Quaternions are a generalization of complex numbers ( $\mathbb{C}$) or hypercomplex numbers and they are denoted with $\mathbb{H}$. Below I write both in their general form. $\displaystyle \mathbb{C}: a+bi$ $\displaystyle \mathbb{H}: a+bi+cj+dk$
where $\displaystyle ijk=i^2=j^2=k^2=-1$

Now, there are 3 “imaginary” components and they are defined by that relation at the bottom. This is super interesting! What does this even mean though? A real number with some sort of 3-dimensional imaginary component? ## Calculus of Variations Part 2: Lines, Bubbles, and Lagrange

In the first part, we discussed the idea of a functional, what it means, and how to find its extrema using the calculus of variations. However, those equations don’t really capture how amazing and applicable calculus of variations really is so the following will be some examples of this. In fact, the drawn out results from the posts The Shape of a String and The Lagrangian are just two cases of the one equation.

## Differential Forms Part 2: Differential Operators and Stokes Theorem

In the first post, we established a general intuition of how forms work and why they may provide a better geometric intuition of what is actually occurring. It was mentioned that these ideas extend the ideas of vector calculus so it seems natural to see how differential operators like gradient, curl, and divergence arise in the context of differential forms. It all comes out of the analysis of the exterior derivative $\textup{d}$. I will stick to 3 dimensions for now and explain the extension into higher dimensions at the end.

## Cool Things

Math is cool. Here are some cool things in math that I don’t think are really extensive enough for their own post but I still want to share.

1. $\textup{d}$

The following is a great reason why. Continue reading Cool Things

## Calculus of Variations Part 1: Establishing the Basis

Calculus of variations is an extremely useful and amazing tool in physics, math, computer science, and a variety of fields. Similar to how regular calculus is focused around functions and differentials, this field focuses on functionals and variations. A functional $\textup{d}$ takes in a function and spits out a number. The following are examples of functionals. Continue reading Calculus of Variations Part 1: Establishing the Basis

## Orders of ∞

The idea of infinity is easy to look over upon first glance. It can simply be defined as the idea that numbers go on forever and it is easy to end there. However, there are very developed and well-defined notions of infinity suggest that there are different orders and types of infinity which come with various properties. The consideration of all these are vital to our understanding of infinite quantities and especially in set theory. The exploration of these transfinite numbers Continue reading Orders of ∞

## Differential Forms Part 1: Dimensions and Notation

Differential forms is a topic that, in some sense, extends ideas presented in vector calculus with more suggestive notation and geometric intuition into higher dimensions. The distinction may seem small and insignificant especially in the third dimension that we live in but its results and implications are quite elegant and can lead to nice formalization of certain results such as Stokes’ Theorem. Continue reading Differential Forms Part 1: Dimensions and Notation