Spontaneous symmetry breaking and 't Hooft and Polyakov monopoles

Question

What is spontaneous symmetry breaking from a classical point of view. Could you give some examples, using classical systems.I am studying about the 't Hooft and Polyakov magnetic monopoles solutions, which says that the SU(2) symmetry is spontaneously broken down to U(1) symmetry. What does this mean exactly? Also, why do we consider a mexican hat potential in the yang-mills-higgs lagrangian, whose EOM give us the solutions to the monopoles, using the Euler-Lagrange equation. I tried reading about this from the internet, but most of the sources have reference to quantum systems ( I havent done QM and QFT), and I haven't been able to understand symmetry breaking from the few classical examples that I found.

Answer 1

You've mentioned a number of pretty intense examples of symmetry breaking, but if I'm reading your question rightly, all you are really looking for is "what does symmetry breaking mean when translated to everyday (classical) physics?"

That's actually a pretty easy question if that really is your intent: Symmetry breaking just means being forced to making a choice.

For example, a pencil balanced on its flat eraser end is perfectly symmetric with respect to every possible orientation on the flat surface on which it rests. But if you tip the pencil over, that perfect symmetry is lost, and the pencil must "choose" a specific orientation into which to fall. Once that fall has taken place, the pencil has lost all of its original beautiful symmetry with respect to the plane, and will not be able to regain it unless you can "heat it up" (energy was lost during the fall) and return it to its original upright position.

Any form of crystallization is an other example. Water is statistically isotropic in three dimensions in its liquid form, but as soon as ice begins to form, the molecules must give up their carefree ways and "choose" some very specific orientation. That too is a symmetry break, and if you think about it, it's not that different from the pencil example.

One of my personal favorites is topological, and involves changing the number of available dimensions of an embedding space.

Imagine molding some clay in into a smooth, symmetric band. The two edges of the band are fully symmetric in 3D in the sense that they can always be rotated to replace each other. Now paint one edge red and the other edge blue. Next, transform the band into a 2D space (reduce its embedding space0 by flattening it onto a table surface, trying as best you can to preserve its internal connectivity in the new version.

You will find that a washer-like form is the best you can do, and that means you must make a choice: Red edge on the inside, or blue edge on the inside? The fully symmetric 3D form of the band thus breaks down into two non-exchangeable forms when the dimensionality of its embedding space is reduced to two.

Notice that while the pencil and ice both have an infinite number of choices when their symmetries are broken, in this case only two choices are available. That kind of twofold symmetry breaking is akin to the one between matter and antimatter. That symmetry can similarly be interpreted as the result of "locking down" the time vector of mass-energy in a 4D space, so that in 3D the local time vector must point in either in the same or the opposite direction as classical time.

No matter how exotic the topic sounds in advanced physics, one way or another the same sort of "make a choice" process. Making the choice lowers the energy of the system, but also destroys the lovely symmetry of the higher energy version.

This in a nutshell is also why particle physics has been firmly devoted for many decades to building larger and larger particle accelerators. The higher energies they provide make it possible to search for those lost symmetries found at higher energies.