Why is Symmetry So Important in Physics?

The concept of symmetry plays an incredibly central role in physics. Most people outside of the field have no idea it’s important for physics at all. It’s just not something science journalists bring up very often. Indeed, symmetry seems more like something an interior decorator or a graphic designer would be interested in than someone whose job is to use math to predict the results of precise scientific experiments and to understand the laws of nature!

While those who have had some exposure to the subject (say, through a few introductory college classes) likely have encountered some form of “symmetry arguments” as a common technique for solving physics or math homework problems, most are still unaware of how central a role it really plays. Even if they have heard a professor allude to its great importance once or twice, it’s hard to appreciate why that would be the case without having a more in depth experience of physics. For those who have been involved in active research in physics, especially in theoretical physics, most are very aware of its deep importance. But there is still an air of mystery about symmetry; many of us have found its importance both surprising and intriguing, and have spent a lot of time wondering why nearly everything in physics is connected to symmetry in one way or another.

Having spent a good number of years thinking about this, I think I’ve come up with some pretty interesting explanations for why symmetry is important in physics. There are many ways of explaining it, but most lead back to a similar set of issues involving the difference between reality and the language we use to describe reality. This is one big piece of the reason I’ve chosen the title “Physics as a Foreign Language” for my blog and book.

First, let me help convince you, if you’re not already, that symmetry is important and central in physics. One of the best online sources for news in the particle physics industry is Symmetry Magazine. This is a joint publication between SLAC and Fermilab, the 2 biggest particle accelerators in the US. It’s funded by the Department of Energy. Due to a decision by US politicians in the 1990’s to abandon building a bigger accelerator, in recent years CERN in Europe has now stolen the spotlight from these US accelerators; but the magazine is still widely read even by physicists living and working outside the US. The particle physicists at SLAC and Fermilab could have chosen any name for a magazine describing their industry. It could have been called Particle Magazine, Quantum Magazine, Subatomic Physics Magazine or many other things indicative of particle physics. But in fact it’s called Symmetry Magazine. Why?

You might think the choice was arbitrary, perhaps it’s just a catchy name which happened to get picked. But no, it was picked because symmetry is deeply central to particle physics (and many other areas of physics, which I’ll get to in a bit). To understand why let me start by explaining what a symmetry is.

Probably the simplest example of symmetry most people can understand intuitively from common experience is bilateral symmetry. Most faces of human beings have approximate bilateral symmetry. That means that the left and right sides look like mirror images of each other. If someone has a scar or a pimple on one side and not the other, this breaks the symmetry. (This is one example of why symmetry is often associated with aesthetic beauty. And there may be deeper reasons for that, but that is outside the scope of our present focus.) This symmetry is a type of reflection symmetry. In mathematics, a reflection is defined as a transformation you can do which reflects an image across a particular axis, resulting in a reversed image. (Often in mathematics what I’m calling an “image” is not necessarily an ordinary visual image, it can be something more abstract and general.) In this case the axis of reflection is an imaginary line drawn down the center of the face. If, after you perform this reflection the image in question remains exactly the same, then it is said to have reflection symmetry.

Another slightly more complicated, but also very intuitive, version of symmetry is the symmetry found in many snowflakes and mandalas. This is a form of rotational symmetry. With perfect continuous rotational symmetry, you would be able to rotate an image in a plane around a center point in that plane by any angle and the image should remain the same. However, the only images that have this continuous rotational symmetry are concentric rings of different colors/material —by comparison pretty boring. Snowflakes and mandalas have a more interesting discrete rotational symmetry associated with rotations by a fixed angle, some fraction of the whole 360 degree circle. Every time you rotate by this fixed angle, you get back to the original image. If there are imperfections (as there often are in snowflakes) in some sectors of the circle and not others, then this symmetry is broken but as long as it is still mostly symmetric we could still call it an “approximate symmetry”. Slightly broken symmetries are very important for physics. I’ll explain why later.

With these simple examples in mind, I can now explain the more general mathematical definition of symmetry that’s used in physics. A symmetry is any transformation you can make on something that leaves some property of that something the same.

Usually in physics the thing we’re talking about is either a physical system or an abstract coordinate system that would be used to describe that system from some point of view. The former is called an active transformation while the latter is called a passive transformation. Active transformations are thought of as affecting the physical world itself while passive transformations affect our description of it, our perception of it, 0r our point of view. If you watch a scene in a movie where the camera rotates around the actors that’s a passive transformation. But if the actors happen to be standing on a stage that’s rotating like a merry-go-round and the camera is stationary that’s an active transformation. The two may look indistinguishable to the audience if you can’t see anything in the background in the shot.

This will be an important theme in this blog. In physics, these two cases — one which affects our description of reality and one which affects reality itself — often become indistinguishable. This is what I mean by the subtitle where language meets reality. The distinction between language and reality in modern physics has become very blurry and interconnected — often it’s impossible to say where one begins and the other ends. The mathematical language I’m referring to in this case is the coordinate system — a map constructed out of symbols that is attempting to represent reality but in doing so ends up adding some necessary redundancy. This is the downside of trying to use words and symbols to represent something that is beyond words and symbols — although so far it appears to be the best we can do.

The mathematical language used to talk about symmetry in physics is called group theory. Group theory is an area of mathematics which everyone with at least a Bachelor’s degree in mathematics is usually familiar with, but since it’s not taught in high school (despite the basics of it being no more difficult than high school math) or required of non-math majors, most people outside of math or physics professions have not heard of it.

The examples of symmetries I’ve given above are all examples of transformations that happen in physical space — the ordinary 3 dimensions within which we live our everyday lives. But in particle physics, there are other kinds of transformations that happen in other kinds of spaces called “internal spaces”. And there are also transformations that happen in time (viewed as the 4th dimension in physics, and closely related to and interconnected with the 3 space dimensions).

For all of these transformations, regardless of whether they happen in regular space, in time, or in an internal space, there is a symmetry associated with each type of transformation which can be performed. And one of the biggest mathematical breakthroughs important for physics of the early 20th century, Noether’s Theorem, says that for every symmetry in physics there is a corresponding conservation law. For example, time translation symmetry gives rise to the conservation of energy. Space translation symmetry gives rise to conservation of momentum. Rotational symmetry gives rise to conservation of angular momentum.

Many of the internal symmetries of physics have to do with the forces of nature. For example, there’s a symmetry that in the language of group theory is called U(1), and in language much closer to plain English is called electromagnetism. The conservation law associated with this symmetry is the conservation of electrical charge. If you count up all of the positively charged particles in the universe (like protons) and subtract all of the negatively charged particles (like electrons) you’d better get zero. (Or at least, if you got zero in the past it better stay zero!)

This theorem has become so important now that just about all of what we consider the “laws of physics” are these kinds of conservation laws which come directly from symmetries.

Indeed, the modern way in which particle physicists construct theories always starts with the first step of choosing a set of symmetries. For what’s known as the Standard Model of particle physics the ingredients used are the symmetries known in group theory as SU(3), SU(2), and U(1). These correspond to the forces of nature known as the strong force, the weak force, and electromagnetism. When theoretical particle physicists come up with theories that go beyond the Standard Model, they do so by adding other ingredients in the form of additional symmetries. For example, in graduate school I worked on a grand unified theory proposed by my advisor called the Pentagon Model which was based on 3 copies of the symmetry group SU(5), where the smaller groups mentioned above resided inside one of the three copies.

I also worked quite a bit on a symmetry known as supersymmetry, which is still theoretical but if found would solve/explain several of the most important outstanding unsolved problems in particle physics today. When the LHC (Large Hadron Collider) at CERN was built, the argument for building it was based on the potential for discovering new physics which was motivated by a number of independent arguments. The strongest argument presented to those responsible for funding the construction of the collider was that we would be very likely to find the Higgs boson. The second strongest argument was that we would have a good chance of finding supersymmetry particles. The Higgs boson has subsequently been found, so that argument proved to be valid. We still haven’t found supersymmetry, but there are still many in the particle physics community who think we might soon, and many more who think that even if we don’t it may exist at energies which LHC cannot reach. (Incidentally, the logo for this blog displays the mathematical equations — called anti-commutation relations — which define supersymmetry.)

I have focused on particle physics here, my own area of expertise, and I hope you can see why it’s an important branch to consider since it sits at the foundations of most of the other branches. But what about the other branches of physics? What about cosmology (the study of the large scale structure of the universe) for instance? In cosmology, the focus of study is on a 4th force of nature, gravity. Gravity is usually ignored by particle physicists and set to zero, since it has a negligible effect on particles circulating at nearly the speed of light in giant cyclotrons. But for cosmologists, it’s the most important force at work!

Well as you might expect by now, gravity is also the consequence of a symmetry in physics. It goes by the name of general covariance. General covariance is a symmetry that has to do with the geometry of spacetime. This symmetry says that no matter how you change the coordinates of space and time, as long as the transformations are smooth in a specific mathematical sense, the equations of physics will still take the same form. It’s a very beautiful and deep symmetry, and as such it should come as no surprise that it gives rise to one of the most mysterious and fascinating forces of nature we know about. Gravity is the only force with which physicists are still struggling to construct a complete quantum theory.

Particle physics and cosmology are the two areas of physics generally considered the most “fundamental” (in the same sense that physics is considered more fundamental than chemistry or biology). The other areas of physics deal with emergent phenomena which are thought — at least in principle — to ultimately reduce down to either particle physics or cosmology. But is symmetry also important for those more emergent/practical branches of physics? I don’t think it plays quite as central a role there, but yes it is still very important for these emergent areas.

For example, the structure of crystals is a big part of what solid state physicists study. This structure is represented by various symmetries described again by group theory (in this case, discrete groups like the snowflake/mandala example rather than the continuous groups I mentioned which generate the forces of nature). Symmetries show up in all sorts of other emergent areas of physics, as well as having played an important role in the history of how physics developed.

When I mentioned that the term electromagnetism is “closer to plain English” than the group theory term for it, I included the qualifier closer because in plain English the separate terms electricity and magnetism are usually used instead. If the term electromagnetism is used outside of theoretical physics, even in engineering, it’s often used to mean something a bit different (for example, generating a magnetic field with a coil of electrical wire). The reason theoretical physicists use it to mean “the phenomena of electricity and magnetism” is because of a huge breakthrough by James Clerk Maxwell in the 19th century. Maxwell argued purely based on symmetry that these two seemingly different phenomena were really two sides of the same coin. He put together the various empirical equations describing electricity and magnetism and noticed that they were asymmetric. Then he found that if he patched them up by just adding one extra term, a stunningly beautiful symmetry would be restored and suddenly they all made sense in the same framework. In doing this, he unified two big fields of physics which were previously thought to be different, and as a bonus it also ended up unifying optics with electricity and magnetism, since his insights led to the conclusion that light is an electromagnetic wave. Little did he know at the time, this also laid the groundwork for Einstein’s special theory of relativity. The history of physics is filled with this kind of thing. Most of the great unifications that have happened in physics have been due to the discovery of a new symmetry.

If you’ve made it this far, I hope I’ve convinced you now that symmetry is important in physics. So what is my explanation for why it’s important? Because this post has grown a bit longer than I’d originally intended, I will leave this as a cliffhanger and address that in Part 2 of Why is Symmetry So Important In Physics? To be continued…