What you see above is my rendition of “The Treachery of Images,” a famous painting by René Magritte. Painted in 1928-29, it depicts what is unarguably a pipe with a caption that reads, (translated) “This is not a pipe.” Deciphering the painting is not as difficult as it looks. Obviously, the painting is not a pipe. It is a representation of a pipe. This argument could be extended to say that the word “pipe” is not, in the end, a pipe. One could as easily reference all pipes as chicken nuggets and then the statement, “Ceci n’est pas une pipe,” takes on a whole different level of meaning. What I love about the painting is the fact that it captures the inherent limitations of language and the ways we communicate, in nothing more than one sentence and one simple object. It demonstrates beautifully the divide between “reality” and “representation of reality”. As someone studying physics, I find it interesting to look at this separation from the viewpoint of physics.
During my first few months as a Master’s student, I received a very important piece of advice. Theoretical physics includes (surprise, surprise) a ridiculous amount of abstract mathematics. It was while talking about this mathematics, (I didn’t understand what it meant even though I understood how it worked) that I was told that physicists have to train their brains to think abstractly. You can look at particles and see numbers instead of, well, particles. The representation is mathematical. The reality is physical. Maybe it’s just me. But it took me a while to wrap my head around the idea of something essentially non-physical (or abstract) describing perfectly everything physical (or real). It becomes even more interesting when you think about how starting from the math, you can end up predicting the behavior of a particle (and even entirely new particles!). This interplay between mathematical abstractions and real-world phenomena is nowhere as elegant and as tangible as it is in Representation theory (there’s that word again) and Particle Physics. This, I think, is physics’ own response to The Treachery of Images.
If you take a course in (under-)graduate physics (or mathematics), you’re likely to come across something called Group Theory. A Group is basically a set of things that follow some laws. Very precise, I know. But the reason I said things and not, for instance, matrices is because the elements of a group are abstract. They can be anything you want them to be: numbers, matrices, differently colored MnMs. As long as they follow the stated laws, they form a group. As with almost everything else (and given humankind’s borderline alarming infatuation with categorizing things), you can classify Groups into different kinds. For the purpose of this article, we’d look at the ones tagged as Lie Groups. The superiority (heh.) of Lie Groups was noticed first by Wigner in 1968 when he deduced that properties of elementary particles can be connected to the structure of these Lie Groups. Again, something completely abstract had an almost one-to-one correspondence with something physically observed. The examples of this are many and make for a fascinating –if not mathematically arduous – read. But in the spirit of keeping this post accessible to me from a year ago, and keeping it in line with “Ceci n’est pas une pipe,” (this is, after all, a representation of the representation of reality), I’ll try keeping this simple.
The easiest way to visualize the link between Representation Theory and Particle Physics is to go back to Quantum Mechanics. (Yes, everything started with that darned cat.) For a moment, let us forget about the subtle complexities of quantum mechanical systems and just say this much: We have a quantum mechanical system. Now, this could be a single particle in a box or a clowder of Schrodinger’s cats, we really don’t care. What we do care about is the symmetries of the system. One of such symmetries is rotational invariance. If I flip my box around, the particle (or the disgruntled cats) would continue to be a particle (or now particularly disgruntled cats). So rotation is actually a symmetry transformation. It leaves our system unchanged. Two consecutive rotations are equivalent to a single (larger or smaller) rotation and hence another symmetry transformation. If you read up on the laws that things should follow to form a group, you’d see that these symmetry transformations form a symmetry group. Now, we can represent these transformations mathematically, in the form of matrices and this is where the magic (or the math) happens. Since we know the structure of these matrices and how they act on vectors, we can represent our quantum mechanical system by a vector and work out what would happen if we act on this vector by our transformation (formally called an operator). So you see, we have successfully transformed an actual physical state into a slightly involved and sort-of abstract mathematical equation.
The abstraction only grows once you progress from quantum mechanics to quantum field theory. And so we come back to Lie groups. There are again, many kinds of Lie groups with their individual set of laws and behaviors. I’m only just realizing it would take more than a hastily written article to fully delve into that. So I’ll leave you with this:
Amongst the different kinds of Lie groups, there is the SU(3). (I could as easily call it K-9 or R2D2 here since I’m not going into any rigorous mathematical details). This group, remember, is intrinsically abstract. We can, however, using the power of Sauron mathematics, do a lot of things with this group. We could, for example, draw so-called weight diagrams, an example of which you can see in the figure below.
(Scroll over or click on the images to see further details.)
And now for the physics. Murray Gell-Mann, an American Physicist noticed that if you combine particles which have the same spin together, you get something that resembles these weight diagrams of SU(3). He went on to predict a new particle based on this correspondence, and this was only the beginning. A few years later, he and Zweig (another physicist) proposed the Quark Model. It is interesting to note that the only reason quarks were proposed was because they were a useful mathematical tool to deal with SU(3). Again, something which was completely unrelated to physics in its formulation (properties of SU(3) etc.) laid the foundation of solid physical theories.
Physics was probably the last thing on Magritte’s mind when he painted The Treachery of Images. But the connection (to me at least) is thought-provoking. Because of course, (1, 0, 0) is not a quark. Except that it is. And so the distinction between representation and reality gets blurred.
Until next time.
End Note and Other Interesting Things:
In hindsight, I do feel that ideally, this realization should have occurred to me somewhere in eighth grade when I learned that I can write the rolling of a ball as a mathematical equation and maybe it did. But I think we need to be reminded about such things once in a while, if for nothing else then to step back and rediscover the a-ha! moments that make studying Physics worthwhile.
To know more about The Treachery of Images, click here. This is a breathtaking video essay that first introduced me to the painting.
To know more about Group Theory, pick a book. Or go here. (Disclaimer: This is an introductory Group Theory text, so rigorous mathematics –just like death– is inevitable.)
Lastly, comments and feedback welcome (and much appreciated!).