Noether's Theorem

A brilliant result that filled an unprecedented gap in our understanding...

Noether’s theorem is unarguably one of the most important results in physics.

It was formulated by a female German mathematician, Emmy Noether, in 1918. Emmy was a mathematician of the highest quality and even Einstein called her a genius. However, not without a reason…as we will come to see.

The Noether’s theorem, to put it simply, describes that all the qualities that are conserved, are conserved because of a cause. Conservation of energy, momentum, angular momentum…they are all causes…

What are the effects?

Continuous symmetries.

To really understand the beauty of this simple yet elegant idea, we are going to dive into the world of symmetries.

Now, imagine, for a moment, that you are walking inside of a huge flat square garden, that is identical on all four sides. A garden, large enough that you cannot possibly walk to one of its ends. No matter where you look, it’s the same. No matter how much you walk, in any arbitrary direction, it’s the same. That is a continuous symmetry. In this case, it’s translational symmetry or translational invariance.

If we consider our space to be that flat square garden, it will have translational symmetry. Of course, you would have omitted any kind of physical object that can be used as a reference point, from such a universe.

Okay so, translational symmetry is an effect…then what’s the cause? Conservation of Momentum.

Yes! Momentum is not conserved in space (over large distances!), because it is not perfectly flat. On a perfectly flat surface, the gravitational field remains the same, and any object on it will continue to move on it in a perfectly straight line forever…obviously, that is not possible in real life.

There are other important continuous symmetries: rotational and time symmetry for the conservation of angular momentum and energy, respectively.

The second one is particularly interesting. It goes like this: in a closed system, the total energy remains constant over time. In other words, energy can neither be created nor destroyed.

However, it has been proven that our universe does not conform to the time symmetry either. A ton of evidence is there to back this up, i.e., the imbalance between matter and anti-matter, and the universe expansion.

So, turns out, and this was a mind-boggling revelation to me too, energy can be created or lost in a time-variant system! It is possible that this might explain the origins of Dark Energy — the driving force behind the universe’s expansion.

So, this is a wrap. My closing thoughts are, Emmy Noether deserves a ton of respect and admiration. She was a genius, in the words of Einstein, and a creative marvel…to whom we should be thankful for giving us this graceful deduction.