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Mirror Mirror On The Wall

Nature employs a handful of tricks over and over again to achieve complexity across many physically disparate systems. It is almost as if It runs out of ideas. On the other hand, these few principles are simply astounding - both in their frugality and their conceptual depth. In these posts, I will occasionally focus on these profound and general tricks of the natural laws to underscore the unity and elegance of physics. In the process, this will also give the reader a flavor of how theoretical physicists think: an abstraction from a concrete physical situation, a subsequent chain of logical steps guided by a "thought experiment", and the dramatic conclusion one is finally driven to by consistency. This post's topic is about Gauge Symmetry. Prepare yourself for possibly being a bit perplexed at first, then brought to tears (of enjoyment or pain) once you realize the punchline. When I learned this subject many (many) years ago for the first time, I realized immediately that - if physics can be so deep and cool - I really needed to become a physicist professionally.

In 1932, physicists experimentally discovered than the particle known as an electron has an evil twin: we call it the positron. It has identical properties to the electron in every respect except one: when an electron is thrown in front of a magnet, it spirals in a certain direction; its evil twin, the positron, spirals in the opposite direction (see attached image of actual data! the spiral red and green tracks on the left are those of an electron and a positron). We say that the electron and positron feel a "magnetic force"; and that the electron has some attribute called electric charge, say of value 1, while the positron has a value of -1 for its charge. And by writing some equations, this difference in charge is used to account for - by construction - the different spiraling directions. But what is this seemingly arbitrary thing called "charge"? And what is this mysterious magnetic force that seems to be acting through thin air around a magnet?

The electron and positron are basically the same animal except for this contrived "charge" attribute… So, perhaps they are two faces of the same thing? Let's abstract away a bit. What if we can come up with some notion through which we can talk about both particles within the same language. Let's sip a bit of good italian red wine, and imagine the following. Say the electron and positron animals belong to the same species. Imagine each comes with a birthmark that distinguishes it - in the shape of a tiny imaginary circle with a hand on it…like an analogue pressure gauge. Think of it as a tag. When the gauge's only hand points to 12 o' clock, the animal is the electron; when it points to 6 o'clock, we have the positron. But we have a whole circle worth of a gauge; why can't the hand point in any direction, not just noon or 6 o' clock? A bit more wine. So, let's allow that. This is a common conceptual step often used in physics: embed a concept in an abstract context and then generalize. Often, this leads to nowhere; sometimes it leads to a revolution… Now that the hand on the gauge can point in any direction, how should we think of the other time spots on the gauge? So, let's revise our thought experiment. If the hand on the gauge points in any direction, we say we're tagging an electron; when we mirror the hand vertically about the horizontal 9 to 3 o'clock line, it'll be the positron. For example, if we choose 1 o'clock to tag the electron, then the horizontally mirrored position - that is 5 o'clock - would correspond to the positron; 7 o' clock for the electron would give 11 for the positron. And any initial direction of the gauge's hand can be chosen to correspond to an electron.

Let's summarize: we wanted to combine an electron and a positron into a single species - to understand this electric charge thing that is the only thing distinguishing them. So, we came up with an abstract representation in our imagination: a particle tagged with a gauge. Any direction of the hand on the gauge is equivalent to any other. Pick one direction and call it an electron; flip the hand about the 9-3 o' clock line, and you are now talking about a positron. More wine?

Imagine two electrons a large distance apart. Both have the hands on their gauges point say to 1 o'clock by choice. Now, let's say someone sneaks in and rotates the hand on one of the electrons a tiny bit. The fact that the rotation is a tiny bit - as tiny as we want - implies that we should not change the interpretation of the animal as an electron: only a mirror flip of the hand would do that, and a flip ain't a tiny rotation. The other electron doesn't know about what just happened; at least not yet, because it takes some time for information to travel large distances. This means we should be able to choose to associate any orientation of the hand on the gauge with an electron at different points in space simultaneously… We say "the laws of Physics should be local"; what happens near a star far away cannot instantly affect what happens in Los Angeles. But this then means that we need a mechanism for rotating the hand of the gauge as we move an electron across a path in space! How else can an electron be considered to remain an electron as it is dragged across a path where the gauge hand orientation for an electron is different… Could this mechanism be some mysterious force pervading space? If we were to look at the details of this force that is needed to make sense of this logic, it comes out precisely as what we see around us as the electric and magnetic forces! We just discovered the electromagnetic force…

Let's summarize once more: we lumped the electron and positron together and added an abstract tag in the form of an imaginary gauge. We generalized a bit and allowed for arbitrary rotations of the hand of the gauge at different points in space - because information cannot travel instantly across space. This is called a Gauge Symmetry. We realized that this necessitates a mechanism to rotate the hand of a gauge dynamically. This implied the existence of a force, the electromagnetic force.

Let's say we didn't know about the electron and positron. One Saturday afternoon, a bored and socially challenged theoretical physicist just cooked up this crazy idea of particles tagged with gauges. Through the steps we went through, he or she is lead to the concept of Gauge Symmetry, and concludes that there has to be some force in Nature called electromagnetism. From Gauge Symmetry to a force law… the raison d'être of the force law is the symmetry…

Every one of the four forces of Nature we know of - gravity, electromagnetism, the weak force, and the nuclear force - all originate from slight variations of this narrative. Gauge symmetries are the origins of all the forces of Nature. For example, gravity arises from a gauge symmetry in 3D: a sphere of a gauge with its hand pointing in any direction in the full three dimensional span of space. Sometime back in time, someone said: let there be this or that gauge symmetry, and we had forces that bind things together!

Why gauge symmetry at all? That we do not understand yet - but there is a beautiful mathematical context to this that I will avoid talking about right now. We have rephrased the deep philosophical question "why are there forces in Nature?" into a new deeper one "Why are there certain symmetries in Nature?"; that is progress and a triumph of imagination and logic; but the work is certainly not done.

It usually takes me around 20 minutes to write these posts; this one took one full hour… This was the most challenging thing I've written so far… The subject is very difficult to explain because of the level of abstraction required. I needed to come up with a visual picture; in reality, gauge symmetry is described by rather elaborate mathematical equations. Yet it is so simple at its core; beautiful, and so fundamental to the world we live in. The norm is to avoid talking about this subject outside the circle of professional physicists. I do not know how much I succeeded in relaying the power, beauty, and essence of this principle that is all over physics. I hope however that I at least succeeded in giving you a feel of the crazy thought process involved in discovering new great physics from a theoretical and mathematical perspective; and in convincing you that a glass of good wine can sometimes take you a long way in understanding physics.

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