Of Symmetry, Other Symmetries and Non-Symmetries

One of the most remarkable concepts in modern physics is symmetry. The structure of our world just seems to be organised in different symmetric structures, which, luckily, can be described using mathematics. This approach has been very successful since the mathematical foundation was laid out by Emmy Noether (1882-1935) in 1918 in Noether’s Theorem.

Noether’s Theorem states that every continuous symmetry has a corresponding conserved quantity.
Many words we will have to go through.

What exactly is a ‘symmetry’?
A symmetry is a transformation that does not change the physical theory; we say it leaves the theory invariant. One example would be translations. We move from on place to another and want that the theory describes the same things in both places. Quite a basic assumption. Symmetry can really be understood in an ordinary sense as taught in school. A mirror transforms a picture into itself, if it is symmetric under reflections.

What does ‘conserved quantity’ mean?
A conserved quantity is constant in time, meaning it stays the same for all time. Here the best example would be energy. It can neither be destroyed or created, only put in different forms like heat, potential-, or kinetic energy.

Noether’s Theorem is a powerful tool to gain deep insights into a theory just by considering what symmetries is has. To pick the above examples again, translational symmetry implies conservation of momentum. Conservation of energy appears through translation symmetry in time. Other conserved quantities include e.g. electric charge, although the corresponding symmetry is mathematically more involved than plain translations.

Over the years Noether’s Theorem was reformulated for Field Theories (fields surround us everywhere; one example is the electromagnetic field, which includes light) and analogues have been proved for quantum systems. The list of conserved quantities goes on…

Until now, we only looked at global symmetries. Those are transformations, which are applied the same at every point in space and time. In contrast to global symmetries, we also have local symmetries. Those are transformations, which are different for every point in space and time. Local symmetries are also called Gauge symmetries for historical reasons, and the overall concept haunts students to this day, as it’s an easily misunderstood concept (I’m no exception to this, so take the following with a grain of salt). Note that gauge symmetries are not physically observable! They are just very useful. Without going too deep into the framework of gauge theory, by going from global to local symmetries, we retain a specific freedom, called gauge freedom. In the Standard Model, the theory of elementary particles and their interaction, for example, this freedom manifests itself as exchange particles of fundamental forces (electromagnetic, weak and strong interaction). Those considerations even give rise to whole particles! They are called gauge bosons. The photon (light particle), as well as gluon (discovered 1978/1979) and W and Z bosons (1983) are such gauge bosons.

Maybe we take a moment to appreciate what miracle it is to expect elementary particles from a purely mathematical standpoint. We proclaim certain symmetries, which sound sane and with some mathematical structure surrounding it, we can predict the interaction particles together with well defined properties, such as mass and charge! Moreover, all of the above named gauge bosons were detected already, and the theory verified.

The procedure of going from global to local symmetries sadly comes with the drawback that the particles are massless, because masses would (mathematically) destroy the symmetry. But the W and Z gauge bosons have masses, so there must be some incompleteness of the theory. Indeed this can be resolved using a nice trick (the trick being worth a Nobel price) called the Higgs mechanism. One introduces a particle(-field) called Higgs field, which gives rise to the Higgs boson. This field generates massive particles upon interaction with some of the gauge bosons (namely the ones of the weak interaction).

And this already introduces the next part. The Higgs mechanism relies on the concept of ‘spontaneous symmetry breaking’ (SSB); a little deceiving, as no symmetry is actually broken. SSB happens when a theory obeys a symmetry, but the state of lowest energy, which is usually the one we are interested in, does not obey the same symmetry.
So after SSB the Higgs field assumes a some value, which is non-zero. Since the Higgs field and the weak interaction bosons (W and Z) interact with each other, there appears a mass in the mathematical description, which does not spoil the gauge symmetry!

Now I would like to end this stream of thought with a philosophical consideration. One might ask, why nature seems to be organised in symmetries? The physicists answer is simply, that we do not know! We only describe nature and predict stuff, based on the known stuff. Why things are like they are, is left as an exercise for the philosophers.
Plato already thought that the elements, out of which the world is composed, correspond to certain geometric structures, the Platonic Solids (polyeder with highest possible symmetry). I like to think that, although being wrong in the details, Plato already understood the world at its core: composed of elementary particles, which correspond to some mathematical or geometric symmetry. Now it is our turn to find further symmetries to understand the universe.