Those Physicists And Their "Physics Proofs"

There are plenty of cases where a proof written down by a physicist is worse than a proof written down by a mathematician, but this is a particularly bad one. In one of my courses, we got to derive the Dirac matrices, which are instrumental in describing spin 1/2 particles. These four matrices are written as $ \gamma $ with an index. One definition of them says that they should satisfy the anti-commutation relations of the Clifford algebra:

\[  \left \{ \gamma^{\mu}, \gamma^{\nu} \right \} \equiv \gamma^{\mu}\gamma^{\nu} + \gamma^{\nu}\gamma^{\mu} = 2 \eta^{\mu\nu} I  \]

where $ \eta $ is the Minkowski metric from special relativity.

\[  \eta = \left [ \begin{tabular}{cccc} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{tabular} \right ]  \]

How big do our matrices have to be in order to satisfy this? They obviously cannot be 1x1 matrices because these are just numbers that commute. It turns out that they have to be at least 4x4 but all published sources I have seen fail at explaining why. I will go through the physics proof that is often given and then set the record straight by writing a real proof. If it appears nowhere else, let it appear here!

So Now The Meaning Of Life Is 137?

Normally when I see an article about numerology, astrology or homoeopathy, I don't give it the time of day. But this one is interesting because it sounds like the author actually made an honest effort to read up on the science related to the fine structure constant and just got it horribly wrong.

The article is The Mystery of 137 and it lives on a site dedicated to the new age philosopher Ken Wilber. Who would've guessed that a site like that would actually have a correct equation that comes up all the time in quantum electrodynamics?

\[ \alpha = \frac{e^2}{4\pi \epsilon_0 \hbar c} \approx \frac{1}{137.036} \]

Yo Dawg, I Heard You Like Guns...

Awhile ago, my friend showed me Pimp My Gun. This site has a Flash driven app that lets you assemble the weapon of your dreams. It is basically a drawing program that has a library of hundreds of firearm components. There is so much room for customization. I tried it out and came up with the following guns:

So we sent a gun salesman to your mother's house, so you can nag mum to buy a magnum!

Jungle camouflaged AUG

Stupid Math Questions

I never turn down a chance to be a smart-ass. One of the best things higher mathematics can teach you is how to go back and correct almost everyone who claimed to be teaching you math. It's almost impossible to a cover a decent amount of material in a math course without sacrificing correctness. This is true in grade school when you learn tons of stuff that isn't real math and it is true in grad school when writing one proof that is perfectly rigorous takes two weeks. Here are some common questions that need to be rephrased before they make any sense. The links point to where I found the questions but they could've come from anywhere. If they look like they were taken straight out of your high school calculus textbook, they probably were.

Stupid

Original version of question 1

Smart

Better version of question 1

Simple Language Benchmark

As some of you may know, the programming languages I use the most are C and Python. One reason for this is popularity - I want to learn something that will help me edit the programs I use. I also think it's good to know at least one compiled language and one interpreted language. Interpreted languages or "scripting languages" are more convenient in most respects but they take longer to run. I already knew Python would be slower than C but I wanted to see how much slower.
Plot showing how long C and Python codes took to diagonalize various matrices.
To make the above plot, I used C and Python codes to diagonalize an n by n matrix and kept track of their execution times. Once you get past the small matrices, the trend that begins to emerge is that Python is ~30 times slower than C.

Making Circuits

As part of a summer internship, I got to put together several electronic components, and for the first time, use something more permanent than a breadboard. I found my first circuit very frustrating because my solder connections kept coming loose and I was told to make it as small as possible. But seriously... am I so used to learning about algebraic varieties and Feynman diagrams that I have become allergic to learning a real transferable skill?

The need for my first circuit arose because the photodiode that we used to measure the power in various lasers had a proportionality constant that was too small. For every Watt of power, the diode was calibrated to put only $ 0.5 \textup{mV} $ across two pins. We wanted to amplify this to a larger value. The component typically used for these applications that you can buy off the shelf is the operational amplifier.

Applying An Epidemiological Model

As those of you who read my third most recent post will know, I recently became excited about methods for predicting the spread of diseases mathematically. When I learned about compartmental models, I began searching for tips on how they could best be applied to real data. I stumbled upon a solution on Abraham Flaxman's blog, Healthy Algorithms.

In Abraham's post, he presents some code that will estimate the parameters in a dynamical system using Bayesian Inference - the most elegant thing to come out of statistics since the Central Limit Theorem. Also present is an exercise challenging the reader to estimate the parameters of a 1967 smallpox outbreak in Nigeria.

If you want to do this exercise without a spoiler then stop! Otherwise, keep reading and I will tell you how I approached the problem while making some random remarks on the strengths and weaknesses of this particular fitting routine.

It's Getting Hot In Here

In a recent post, we used trigonometry to derive the length of a day on the Earth as a function of the observer's latitude and the time of year. As promised, I want to continue modelling the Earth's orbit to see what it can tell us about temperature. The simplest explanation for the seasonal variation of temperature comes from the concept of solar flux. To see what this means, think about taking a ray of sunlight shining on the Earth and decomposing it into two components - one parallel to the Earth's surface and one perpendicular. Decomposing a ray of sunlight into two components to find the flux. Most of the sunlight going into the ground means this location will be hot, whereas most of the sunlight going along the ground means this location will be cold. The temperature due to direct sunlight is therefore proportional to the cosine of the angle between the ray and the outward normal to the Earth. To actually solve for a temperature, we would have to multiply the intensity of the light by $ \cos \theta $ and integrate this over a region of interest. Our main concern will be solving for $ \cos \theta $, allowing us to express the temperature at one time relative to the temperature at another time without knowing the absolute intensity.

Reaffirming My Own Existence

This site is under construction so don't judge me! Actually judge me all you want because a good site is always under construction. This site will contain many of my ramblings like pointing out chiasmus and if I put a decent amount of work into it, it might just have a small effect on someone's life!

Modelling An Epidemic

Two and a half years ago, when I read the research interests of my statistics prof, I noticed that he had become interested in analyzing epidemiological models. Now, I might finally understand what he was talking about.

SIR model reaching the disease free equilibrium.

If we let S be the population of individuals who are susceptible to the disease, I be the population infected with it and R the population that has recovered, it is not too big a stretch to say that this plot appears to follow the progression of a non-lethal disease. Only a small number of people have the disease at the beginning, but this number grows because the disease is contagious. People who have recovered are immune to further infection meaning that the epidemic eventually dies out.

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