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Oct162010

On Beauty, History, and Her Story

 

In the early 1600's, two little known European scholars were on the verge of changing the course of human history forever - not through war or politics, but through scholarship. 

Tycho Brahe, a rich Danish nobleman, was obsessed with observing the night sky. He built the most advanced telescope of his time and started recording the positions of heavenly bodies meticulously. He collected remarkably detailed tables of numbers describing the positions of planets; but they were just that - numbers with no physical meaning. He struggled with making sense of his data - as well as with a chronic weakness for alcohol, loose women, and partying… Brahe was the first experimental physicist of history - in the modern sense of this term.  

Concurrently, a poor German scholar, Johannes Kepler, was incessantly trying to understand the rules by which the heavenly objects moved in the sky. But Kepler lacked data, numbers to look through, patterns to discover. He had already become famous for his mathematical skills, but he was dirt poor - with a mother in jail accused of witchcraft… Kepler was the first theoretical physicist of history...

Eventually fate brought Brahe and Kepler together. And after one night of heavy drinking, Brahe dropped dead and Kepler basically stole his data… he pondered over the long tables of numbers - positions of planets wandering the night sky. And from these numbers, Kepler's genius unraveled complex repeating patterns… he formulated his discoveries through three simple laws. And Physics had just been given birth to. As is typical of theoretical physicists, after this work Kepler got obsessed with some ill-conceived mathematical ideas; he eventually died as a war refugee… About a hundred years later, Isaac Newton was to finally see the big picture in an amazing work known as the Principia. Newton wrote: "If I have seen further it is by standing on the shoulders of giants" - referring to Brahe and Kepler.

Since then, Physics has been about observing Nature, measuring it, pondering over the measurements looking for patterns. And patterns are about symmetry. Think of a perfect sunflower, with a set of identical petals. If you rotate it around its stem by one petal, it does not change; it looks the same. We say the sunflower has a rotational symmetry. Since the beginning, physicists understood that symmetry was important for understanding the laws of Nature. After all, it's all about finding repeating patterns in measurements since Brahe's and Kepler's time. And when there's pattern, there's a symmetry: the elegant pattern of circling petals in the sunflower is a reflection of its rotational symmetry. But it was only in the 1900's when we finally understood the depth and importance of symmetry in Nature.

In 1918, a mathematician and physicist by the name of Emmy Noether was to change the way we think about Physics forever. While struggling to overcome discrimination against women, Noether focused on the role of symmetry in the natural laws. She managed to publish a seminal paper - while working from home: no institution would give her a job despite her fame simply because she wore a skirt… In this work, Noether stated and proved a remarkable theorem now named after her: every symmetry in Nature is associated with a quantity that remains constant in time. It is difficult to overstate the depth and beauty of this statement. Every observable that physicists measure and analyze arises from the fact that, in certain situations, the corresponding quantity can remain constant - and hence may be interesting. Noether was saying that Physics amounts to cataloguing the symmetries of Nature, and was providing a concrete prescription on how to proceed.

A table-top Physics experiment is performed at 2pm and leads to some measurements and results. It is then repeated at any later time, say 3pm; and it is found that the results have not changed. This implies that the laws of Physics governing the experiment are unchanged under a time shift or "time translation". There is then a symmetry at work in this setup - much like the case of rotating a sunflower without changing how it looked. Noether's theorem states that there must a quantity in this experiment that does not change in time, that remains constant. And the theorem identifies this quantity: we call it energy… Energy is constant by virtue of time translational invariance!  The reason we talk about the concept of energy at all is simply traced back to a symmetry. Even when energy is not conserved because of a lack of the required symmetry in a situation, we learn to still measure it to explore the new physics responsible for its non-conversation.

If an experiment is performed in my office (not that that'll ever happen); and then repeated in an office nearby with no changes in the results, we say that the laws of Physics at work are unchanged under "space translation". This is then again a symmetry. According to Noether's theorem, there is a constant quantity that we can track and study: we call it momentum… Momentum conservation is a result of space translational invariance!  Even when momentum is not conserved because of a lack of symmetry in a certain situation, we learn to still look at it to explore the new physics responsible for the non-conversation phenomenon: force!

Furthermore, every force of Nature discovered to date is now known to arise from a symmetry principle… The associated Noether constant is the charge of the force: for example, the electric charge for the electric force, mass for gravity. Hence, you quickly realize that Noether's theorem is very fundamental to the way we think in Physics and beyond. Noether's simple statement literally reorganizes the way we view Physics and the world; and provides a systematic tool for identifying patterns in Nature. Very few things in life get this beautiful and elegant.  

A simple reference article: Rubens de Melo Marinho Jr (2006). Noether's theorem in classical mechanics revisited http://arxiv.org/abs/physics/0608264

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