Why Is Symmetry So Important In Physics? Part 2

In Part 1, I went over several examples of where and how symmetry is important in physics. This is good for appreciating the significance of the question being asked here. But I didn’t quite make it to the why part —I haven’t yet given an answer.

I should preface this part with a disclaimer. Part 1 summarizes a lot of facts and direct observations of mine that most working physicists — or at least most working particle physicists — are pretty familiar with and I think pretty uncontroversial. Part 2 (my explanation for why symmetry is important in physics) will rely more heavily on my own ideas about how symmetry fits in to the bigger picture of what physics is and how it progresses. And since every physicist’s experience of physics is a bit different, these ideas are bound to be viewed as a bit more controversial. I will be treading on softer ground here, but I’ll do my best not to stray too far into speculative territory.

In my first year of graduate school, our department asked if any of us wanted to volunteer to write an essay on why we decided to go into physics. I wrote one (which I should probably post here at some point, if can still find it) which focused on two key features of how physics is conducted which I found to be a particularly powerful combination: empiricism and reductionism. I thought my essay was pretty good, at least for a first year. I’d like to say it won some kind of an award, but alas — it’s probably just sitting in a stack of other similar essays that never gets read, in some university administrator’s office.

The empiricism part is what distinguishes a branch of the natural sciences like physics from formal sciences such as mathematics or computer science. In a nutshell, it’s not just about the classification or definition of abstract patterns — you actually get to predict the results of real life experiments and test theories against those results. Observation is key! Theories are not proposed in a vacuum, but based on empirical observations. They can live or die by new observations which they may or may not be consistent with. Puzzles arise from our desire to explain those observations in a consistent way (or our inability for present theories to do so). Physics attempts to solve those puzzles. This it has in common with the rest of the natural sciences, as well as the social sciences.

But more relevant for our present topic is the second part, reductionism. Now, in some areas of academia — such as biology, systems theory, or certain parts of philosophy — the term reductionism tends to be used more often as an insult than as a compliment. But in physics, you’re way more likely to find people who wear it as a badge of honor. This isn’t to say that all biologists or philosophers view reductionism in a negative light, or that they routinely call other people — especially physicists — reductionists as an insult. Nor am I implying that all physicists consider themselves reductionists (Nobel Prize winning condensed matter physicist Robert Laughlin wrote a whole book on why he’s not a reductionist.) But it’s noticeably more common to be used in a negative way in these other disciplines than it is in physics. And there are good reasons for that.

There’s a joke about physicists that goes something like this:

Q: How does a physicist milk a cow?

A: Well first let us consider a spherical cow…

This is how some people view physicists. Supposedly, they make a lot of sweeping simplifying assumptions that don’t seem well-motivated and fail to capture a lot of the complexity of the world. Another stereotype is that based on these oversimplified assumptions, physicists come to grandiose conclusions and then arrogantly proclaim to the rest of the world that they’ve figured out or explained a lot more than they really have.

I have to be honest here: there’s a grain of truth in these stereotypes. ;-) It’s not uncommon for this kind of mistake to happen in physics. And when it does, the word reductionism is often trotted out as a way of explaining the supposedly false paradigm that physicists adhere to which is responsible for the commission of these egregious errors. If only physicists could just wake up and smell the glory of holism (the opposite of reductionism)! Perhaps they would realize the error of their ways and then start using their brilliant minds more effectively, kicking them into gear and making faster more efficient progress toward truly understanding the natural world. Or perhaps they could do this by switching to another field which is less dominated by a reductionist paradigm?

I haven’t defined what is meant by either of these terms, reductionism or holism. Rather I’ve just discussed the negative connotations that the former has for some people. I’d like to avoid giving a simple one-line definition of it, because really there are many interconnected things meant by the term in different contexts, and a long history of philosophical debate surrounding it, including many reinterpretations of what it really means. Perhaps I’ll get to that in another post. For now, I’ll just say that the basic idea of reductionism as an approach to gaining knowledge about the world has to do with breaking things into constituent parts and analyzing how the parts of something interact with each other to form the whole. A strong form of reductionism would be that if you understand how the parts work, then you’ve successfully understood everything there is to know about the whole. Holism on the other hand has something to do with the idea that the whole is more than the sum of its parts and hence cannot be understood by such techniques.

I think there is something to be said for the limits of reductionism. There are some cases, especially in biology, where the system under study is just so complex that staring at the parts just doesn’t get you very far toward understanding the function of the whole. The neural structure of the brain may be the ultimate example there. I’m under no illusion that reductionist methods are going to give us a complete understanding of it any time soon. In this sense at least, I sympathize with the holists.

Nevertheless, when it comes to debates on the issue, I almost always find myself more on the side of reductionism, both in practice and in spirit. I very much believe that the full power and beauty of reductionism has been drastically underestimated by most people.

One of the reasons I believe reductionism is underestimated is that many people assume — usually without ever thinking it through — that holism represents a viable alternative technique. As I said there is some truth to the concept of holism in that you can’t always get very far very fast with reductionism. There are limits. However, I see these as not so much limits on reductionism as an effective approach, but as general limits on the possibility for acquiring new knowledge by any means. There is no real alternative to reductionism; it’s simply the best and most powerful tool we have for answering any question which begins with the word “why”. Holism interpreted as a technique is a myth — no such technique exists.

When I think of the way in which these two opposing ideas relate, I think of the 1998 Darren Aronofsky film Pi. I have no idea whether this was Aronofsky’s intended interpretation, but my own interpretation was that throughout most of the film you see a person obsessed with finding mathematical patterns in the world, searching for a theory of everything that will explain and unify all of life into a single mathematical framework. BIG SPOILER ALERT: He ends up driving himself mad and drilling out a part of his own brain. (Ouch, bummer.) After that part of his brain (presumably the part responsible for identifying patterns) has been drilled out, he sits down on a park bench and just appreciates the beauty of the world in a more holistic way, without attempting to interpret or understand it. In other words, he gives up. I was deeply disappointed by this ending, and yet I also think it was thought provoking. I have no problem with those who don’t seek explanations for things, who aren’t curious about why questions and don’t feel the need to interpret or understand anything. If you’re happy with not understanding things, that’s perfectly okay. (Although you probably won’t find this blog very interesting, as it’s going to be aimed at providing a deeper understanding for many things.) There’s even a phrase for that, which is very suggestive of the happiness the protagonist finds at the end of Pi: ignorance is bliss! But that doesn’t stop me from being one of those who does get curious from time to time, and who does seek to understand things at more than the most superficial level. A blessing or a curse, I was born that way and I suspect I will always be that way. Holism may offer bliss, but what reductionism offers is genuine understanding. Also, it might come as a shock to hear this, but many of us reductionists are perfectly capable of sitting down on a park bench and enjoying a sunset too. In fact, I can personally attest that some of us take great pleasure in doing so. Having a deeper understanding of the mysteries of the universe doesn’t detract from that experience in any way, as far as I can tell. Yes, you really can have your cake and eat it too, in this case. :-) I see it as tragic that popular culture, as reflected by Aronofsky’s work and many other similar cultural narratives, doesn’t seem to grok this. :-(

And now let’s return to the question at hand: why is symmetry so important in physics? Symmetry is important because the goal of physics is try to explain various physical phenomena. And the most powerful way of explaining anything — reductionism — is to reduce them down to simpler constituents and then show how these simpler ingredients can be combined to give rise to the original thing you’re trying to explain.

This isn’t just something that happens in physics, I think it happens in any field where explanation — wanting to know the reason for something, as opposed to simply labeling or categorizing what it is — is part of the goal. For example, chemists have explained chemical reactions with the Periodic Table of Elements. Biologists have explained how different physical traits are inherited with DNA. But these fields have a different balance between categorization and explanation than physics does, with more of the former and less of the latter. If you’ve ever taken a class on biology, you’ll notice that it’s very different from taking a class on physics. When it comes time to study for the test, there are a lot of things to memorize. When teaching physics, I found that I constantly had to beg my students not to memorize anything (which they were very much in the habit of from taking non-physics classes) but instead try to understand the concepts. In my experience, biology students usually have a particularly hard time kicking this habit when trying to learn physics. In physics if you understand the concepts then there is very little need for memorization, because you can derive most things (at least at the undergraduate level) from scratch just thinking through it logically.

There are many levels of explanation which form what’s sometimes called a reductionist hierarchy, each level explaining and underpinning the next. Physics just happens to sit at the root of the hierarchy in terms of the natural sciences (one can effectively argue that areas like mathematics or philosophy are further down at the root, but these are usually considered outside of the empirical sciences and are more closely connected to linguistics).

At each level in the hierarchy, a more complex system is reduced to a set of simpler components, the whole being explained in terms of the parts. (Another way of saying this is that the whole emerges from the parts. I take issue with anyone who thinks that emergence is at odds with reductionism, as I see them as two inseparable sides of the same coin). But what do we mean here by “simpler”? That’s tricky to give a one line answer to so I’ll avoid doing so. In physics, one way to quantify this is in terms of the number of degrees of freedom a system has. The phrase degrees of freedom is used to count the number of independent interacting parts of a system. (Another important measure is how many independent input parameters it takes to define a theory, but for now let’s focus on the degrees of freedom.)

Picture this hierarchy as a tree of knowledge, where everything grows out of the root (in plain English I should probably say trunk, although in computer science and in some other contexts the word root is used to mean the part of the tree I mean here). Often, things closer to the root tend to have a smaller number of independent degrees of freedom, ie they are simpler. But how on Earth can something simpler contain the same amount of information as something more complex?? Glad you asked: that’s where symmetry comes in!

When you want to transfer a lot of files over the internet, but you’re worried that it will take too long or not fit on the computer you want to transfer them to, what do you do? You compress them into a smaller file (usually with an extension like .zip, .gz, or for images .jpg) by using one of the standard compression algorithms available on the net. And now we’ve reached a big punchline: symmetry is the compression algorithm for reductionism — it’s how you explain something more complex in terms of something simpler.

More specifically, when there’s a system you want to understand, the way to do that is to analyze it looking for patterns, until you discover some hidden symmetry that wasn’t obvious at first. Symmetries represent a principle that organizes a system, usually connecting different seemingly independent degrees of freedom with each other in a rigorous way. I say seemingly because the message that symmetry has for you is that some of the degrees of freedom you thought were independent are not. There is a correlation between them, which means that the initial counting process involved overcounting. Overcounting is when you count 2 things as different but it turns out they were actually the same — you counted the same thing twice! (Or 3, 4, or 5 times, etc.)

For a more concrete example of overcounting, let’s return to the snowflake mentioned in part 1. Imagine you have a digitized image of a snowflake and it takes up a certain number of bits of information (representing the different pixels in the snowflake, each either on or off, black or white). But then you notice the rotational symmetry: every time you rotate the image by a fixed angle, say 72 degrees, you get back to the same image. 72 degrees is exactly 1/5th of 360 degrees, the whole circle. Then you have a very natural compression algorithm for how to compress the image into a file of a much smaller size. The algorithm is: just store the pixels for 1/5th of the entire circle, and when you want to reproduce the original image just use the same image copied and rotated by 72 degrees, 5 times: the original image will emerge. Your initial counting of the information contained in the snowflake involved overcounting by a factor of 5: you counted the each piece of information exactly 5 times. But unless you notice the symmetry there, you may never have realized the different sectors of the snowflake were connected, so you never knew that the true amount of information contained in it was less than it appeared.

In the case of the snowflake, it’s fairly obvious what the symmetry is so you may not think it’s all that impressive to be able to recognize that and reduce the amount of information contained in it by a factor of 5. But in physics there have been some pretty non-obvious symmetries discovered. I mentioned Maxwell’s discovery of what’s now called the U(1) symmetry of electromagnetism. Without that, the separate realms of electricity and magnetism seemed a lot more complicated and would have taken a lot more words (and equations) to describe. But there was a hidden symmetry there which he was the first to see. It’s a type of symmetry called a gauge symmetry; a similar kind of symmetry is behind each of the other 3 fundamental forces of nature.

I mentioned the distinction between internal symmetries and spacetime symmetries in part 1. Well there’s a different dichotomy one can draw between gauge symmetries (sometimes called local symmetries) and global symmetries. Electromagnetism is due to an internal gauge symmetry. While gravity is due to a spacetime gauge symmetry.

I also mentioned the distinction between passive and active symmetry transformations. Global symmetries have more to do with active symmetry transformations. They represent a case where the actual degrees of freedom in a system have been overcounted. Just like the snowflake example. The actual number of degrees of freedom in the system turns out to be smaller than it would have been without the symmetry. But gauge symmetries are a bit more subtle and mysterious. They include global symmetry transformations, which involve a similar reduction in the number of degrees of freedom; however they also include weirder transformations which are better thought of as passive transformations. They also represent a kind of overcounting or redundancy. But it’s a redundancy in how the description of a system relates to the system itself.

With gauge symmetries, it turns out that there are many different equivalent descriptions for a system and yet usually there is no known preferred one. They each make use of their own coordinate system, their own way of talking about things, their own language. A very simple case of this would be like using polar coordinates (angles and distance from some central location) to describe where someone is in a city as opposed to using rectangular coordinates (how many blocks North and how many blocks East from some starting location). But the more interesting gauge transformations in physics tend to be much more convoluted and can result in two or more very different pictures of what’s going on physically. It becomes difficult to separate out what is part of the description and what is part of the system itself.

Ok, I can’t resist giving you one spooky example of this. There are entities in Quantum Chromodynamics (the mathematical theory of the strong nuclear force, known as QCD for short) called ghosts. Ghosts behave almost like regular particles but they have very bizarre ethereal properties, and when you try to compute the likelihood of them being in a small region of space, you get negative probabilities for the answer. It ends up not mattering because the ghosts aren’t physically detectable. But they are a part of the mathematical framework typically used in order to understand how collisions in particle accelerators work, and depending on what gauge you’re using they may show up frequently or not be there at all. (By gauge, I mean a particular coordinate system which is related to other coordinate systems via the above mentioned gauge transformations.) Ghosts are one type of virtual particle. Most virtual particles (for example, quarks or gluons) are a lot more similar to ordinary particles in that there is always a positive probability for them to be in a particular region of space. But because it’s not quite clear whether virtual particles are just a part of the mathematical framework we use to talk about physics or a part of the physical world itself, this can lead to some pretty lively philosophical debates. ;-)

Let’s return to the spherical cow joke. What happened there and how is it related to the rest of this post? Astute readers may have noticed a striking thing that’s different about a sphere and a cow: the sphere has a lot more symmetry! If you rotate a sphere around by any angle around any axis which passes through its center, you get back the same thing. You can even reflect the whole thing across any plane passing through its center and still get back the same thing. That’s a huge amount of symmetry — in group theory it’s called O(3) — perfect continuous 3 dimensional rotational symmetry plus a bonus reflection symmetry. If you rotate a cow around by any angle, you always get back something different. You won’t have much better luck reflecting it across most planes, unless you happen to chose the vertical plane which passes right through the middle of the cow. Aside from this approximate bilateral symmetry, a cow is highly asymmetric.

The spherical cow is an example of reductionism run amok. The physicist has attempted to reduce the cow to a sphere because the sphere is much simpler. But of course a cow is not at all similar to a sphere, so the reduction doesn’t work. Or at least, only in very limited situations would it work (if you just want to know where in a farmer’s field the cow is, it doesn’t matter what its shape is so you are free to use the simplest shape possible to represent it, a sphere or a point.) For most of the questions one could ask about a cow, such as how to milk it, reducing it to a sphere eliminates all the parts of a cow which are crucial to answering the question.

The process of understanding the world and making sense of it has to do with identifying which parts are just random and which parts follow some kind of pattern. There are a lot of cases in physics where it turns out the world is almost but not quite symmetric. If our world had too much symmetry, it would be too simple to support intelligent life. If it had too little symmetry it would be completely random and there would be no way to understand it at all. As it happens, we’ve got the goldilocks world that has just the right amount of balance between order and chaos, symmetry and asymmetry. (Incidentally, I am using chaos here in the plain English sense, where it’s essentially a synonym of randomness; in physics-speak, it’s not at all equivalent to randomness and represents more the kind of balance I’m talking about between order and randomness.) I see this universal balancing act as one of many examples of the anthropic principle at work — but that’s a subject I must leave for another time.

Because so much of our world is close to being highly symmetric, that means it is comprehensible and there are many interesting patterns which can be identified, reduced, and simplified. I know it sounds like I’m overstating the case here, but basically: this is the reason why knowledge is possible. Physics is about trying to find the simplest description that involves the least number of independent parameters. But in the quest to find that, shortcuts are inevitably taken. These shortcuts are called approximations. Approximations are crucial because they involve paying attention to what’s important while throwing away things that aren’t important: filtering signal from noise. But when something important is accidentally thrown away which should have been included in the model, that’s an example of what’s known as greedy reductionism. That’s the spherical cow. It’s a common mistake to make while on the reductionist path, but what many outsiders miss is that the benefits of this approach far outweigh the occasional missteps. There’s no reason to throw the baby out with the bathwater. In the end, physicists have made amazing progress in simplifying the world down to just a few important ingredients and a small set of natural laws encoded in symmetries. For the most part, these were not things invented by the physicists, rather they were discovered through the process of actively looking for hidden symmetries and trying to figure out what’s important for understanding and what isn’t. (Although the line between invention and discovery is every bit as fuzzy as the line between language and reality; in fact, it’s the same fuzzy line!) For all its faults, they could not have done any of this without the secret sauce known as reductionism.

I’m sure if someone else with a physics background tried to explain why symmetry is important in physics, they would likely use different words than I have, and they might nitpick about this or that in the way I’ve said it. But I think nearly all have wrestled with this issue at one point or another, and I suspect many have come to similar conclusions about why it’s important.