Symmetry: the concept underlying all modern physics

Symmetry is most often known as describing objects which remain the same when reflected around some axis. This can refer to geometric shapes (e.g. a hexagon with 6-fold symmetry), photographs, buildings and so on. This type of symmetry could be described as discrete symmetry.

In physics however, it is continuous symmetries that are the most common. Just like rotating a hexagon by multiples of 60 degrees will leave it looking the same, so will rotating a circle by any angle – hence the word continuous. This concept generalises to all sorts of transformations, not just rotation: consider totally empty space. If a rocket is placed there, the laws of physics governing astronauts inside will not depend on where exactly the rocket is – after all, empty space looks the same no matter where you are within it. This means there’s a continuous translational symmetry of the system and moving the rocket to a different location will not change the outcome of experiments inside.

Now, what’s so useful about symmetries is that they actually lead to some seemingly fundamental properties of physics! In 1918, a German mathematician Emmy Noether proved that every continuous symmetry leads directly to a conserved physical quantity. In the previous example, the homogeneity of empty space leads to the conservation of momentum – meaning if the rocket was set in motion, it would remain in motion at a constant speed and in a straight line. Noether derived a mathematical way of finding the conserved quantity from the transformation, meaning if one finds a new symmetry they can calculate what’s conserved. And so: rotational symmetry of space gives us conservation of angular momentum, time translational symmetry gives us the conservation of energy and the U(1) gauge symmetry gives us the conservation of electric charge.

When trying to discover the fundamental laws of nature, often the only thing physicists can safely assume is some symmetry of the thing they are studying. They might know that some properties does not change when a specific transformation is applied: perhaps some part of the system does not change in time or looks the same from all sides. The work of Emmy Noether enables this assumption to be turned into a mathematically cohesive theory, ready to be tested in experiments. Moreover, the derived theories often predict other results which could not be obtained in other ways.

It is truly remarkable that what seems like a fairly simple statement has led to the creation of many new areas of physics and significantly deepened our understanding of the universe.

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