In my previous article, I mentioned how the newer theories of the Quantum World are showing a shift towards being formulated in a language that is more geometric in nature. One of the reasons that we are seeing such a shift is because Nature filled with symmetries, and geometry is one language that allows us to study and analyze those symmetries in a relatively easy manner. In this article, I will focus on symmetries and why they are important to deal with in the study of physics.
A general strategy to solve problems in physics is to find a quantity that is conserved in the system and then we try to find an equation in terms of that quantity. For classical mechanics, in most cases this conserved quantity is energy. In some cases, we see that linear momentum and angular momentum are conserved. Now, let’s ask ourselves why must energy or linear momentum or angular momentum be conserved? We can make some simple arguments. If any of these quantities are not conserved, the universe will either blow up or collapse in on itself. If energy is not being conserved, then there must be a source or sink through which energy can be added or removed. If linear momentum or angular momentum is not conserved, things will either keep slowing down, and stop interacting with each other, or keep speeding up, and the universe will be filled with stuff moving at velocities close to the speed of light. We can clearly see that if any of these were true, our observations of the universe would be drastically different. The arguments that I have presented here are purely physical in nature.
The arguments that I have made, however, are not formal enough. We need to analyze these statements more. These statements come with an asterisk that we do not talk about. Energy is only conserved if we have a closed system. Linear momentum is conserved only if there is no force acting on the system. Angular Momentum is only conserved when there is no torque. How to do we put all of this under one umbrella? The answer was given by Emmy Noether. We look at symmetries.
This idea is pretty hard to explain and is a consequence of some brilliant mathematics. Well, here goes nothing. Let’s try what Einstein did – thought experiments. Imagine that you are sitting on a carousel in space. Say that you know that there are spots that are equidistant from each other, and all those points lie on a circle that is larger than the circle of the carousel and that both these circles are concentric. So, as you would rotate on the carousel, you would expect to see each of the succeeding points at regular time intervals. Now, if there was torque on the carousel, and you started moving faster and faster, you would not see those points on the outer circle at regular intervals anymore. Therefore, as you are rotating, you can clearly tell that the spacetime around you is not the same, there is something outside pushing you along. In more precise terms, you are not rotationally invariant anymore. Rotational invariance is the symmetry mentioned when we talk about conservation of angular momentum. A very similar argument can be constructed to show the conservation of linear momentum, except you consider yourself moving in a straight line and an infinitely long rod (with marks at equal intervals) that runs perfectly parallel to you (parallel with the underlying space being flat so that the parallel lines do not meet). So, if translational symmetry is preserved, then momentum will be conserved. If a force is acting on the system, we will not observe the points at equal time intervals.
In my previous article, I mentioned that temporal symmetry implies that energy is conserved. That is pretty hard to explain, however, I did already try explaining one of the symmetries. I will now tell you about how this would translate into the language of Mathematics. We look at the equation that tells us about the energy of the system. These equations are the Hamiltonian (a very fancy way of saying just add up stuff till you get the total energy of the system) and the Lagrangian (the difference of the kinetic and potential energies). In these equations, if you introduce an infinitesimal perturbation to ‘t’- the time (effectively, replace everywhere you see ‘t’ with ‘t + at’, where ‘a’ is an infinitesimal), and under that transformation, the final equation will still look the same as before. Naturally, there are some tedious calculations that you would have to do, but you can recover it. In Mathematics, the language we use to analyze most of these symmetries is Group Theory. It studies how sets behave when an additional structure is defined for them.
On the idea of symmetries, I would like to mention the Higgs Mechanism. The Higgs Mechanism is what is responsible for some of the mass that we observe in the universe. This mechanism is formally termed as ‘Symmetry Breaking’. In a way, we observe mass because of the broken symmetry. This idea is kind of hard to wrap your mind around, but here it is. We look at what we call the Higgs Lagrangian. We look at the Higgs field, which is a complex scalar field (basically each point in the field is defined by a complex scalar number). Just by looking at this field, we can calculate a Lagrangian for it. Then, we use some calculus to find the minima of the function (this corresponds to minimizing the action). We see that small fluctuations about this minimum are possible because of the fuzziness that is inherent in the Quantum World. It is this fluctuation that results in mass in our universe. The fields corresponding to different particles interact differently with the Higgs field and based off of how they interact, each of them has a mass.
As a side not, I want to mention other strategy to solving problems in Physics – minimizing some quantity. This is not related to symmetries, but I feel that it is worth mentioning here. There is the principle of least time for light, which tells us that if a ray of light were to travel between two points, the path that it would travel by is naturally the path that would take the least amount of time to traverse. Then there was the principle of least action that I spoke about in my previous article. We define a function called the action, as the integral of the Lagrangian with respect to time. This functional is always a minimum for the path of a particle. This is true classically. In the case of Quantum Mechanics, there is an inherent fuzziness in the path that we have to consider. In Feynman’s Path Integral approach, each path is weighted exponentially by the action of that path, and the path that has the least action is the most probable path.
I would like to end on a personal note. In the last article, I covered how surprisingly effective math is, and in this one, I tried to elaborate on a specific aspect of that, symmetries. I think it would be appropriate to end on where I first thought about and encountered this idea. Quite surprisingly, it was in a Chemistry class in 11th grade. In a chapter about the equilibrium of chemical reactions, we studied Le Chatelier’s Principle. I will paraphrase it here, ‘If a constraint is added to a system in which a chemical reaction is occurring, then the reaction will proceed in a way to minimize the effect of the applied constraint’. It seemed like something that would be pretty obvious, and something that felt like it had to be true. Basically, the take away from this article should be that Nature is incredibly lazy. If there is a problem that you want to solve, either you first minimize something or you find something that is conserved and form an equation out of that.
Quantum Circle 