My Story

With me starting my last year as an undergraduate and the prospect of applying to graduate school looming in my very near future, I got pretty introspective about my journey and relationship with physics, and I thought it would be worth sharing some of my thoughts on this, so here goes nothing!

I have had this dream of doing physics since I was five. In many ways, this was really not my dream, and this is quite natural for anybody who figures out what they want to do when they are this young. It is a dream or a burning passion of whoever had the most influence on you as a child. This was my maternal grandfather’s undying enthusiasm for understanding nature that induced this dream onto me. He used to take me up onto the roof of our home and sky-gaze and tell me stories about the stars that form the most beautiful tapestry in the heavens. But in all honesty, looking back, this prospect of loving something this much that early on is quite scary – I had an answer to a question I didn’t even know that I could ask. The study of physics was just so beautiful that it has stuck with me since. As I grew up, and in the latter part of my teenage years, when I could understand what it meant to love something, that I was fortunate enough that my answer had not yet changed. It is physics through and through.

My relationship with physics was quite interesting, and the relationship has been a bit rocky at best. I was fascinated by the stars, and the universe out there. It was quite easy to explore those as an amateur. There are so many well-made and fascinating videos out there on YouTube and of course, countless Wikipedia pages filled with so much information! There were these particles out there that made up everything, the quarks, leptons, and bosons and I was enthralled by them all. I remember having clippings of all the discoveries reported in the newspapers. The discovery of the Higgs in 2012 took up pages of that scrapbook and made my imagination run wild. Unfortunately, soon after, I was stuck. As a child that young, the world only cares about interests that are tangible, and that was a major hurdle for me. I was never really much of a tinkerer. I preferred solid rules and playing about amongst them. I loved puzzles and riddles, but these are things few care about. This aspect – that there was really no one I could share this enthusiasm with – was a hurdle. I started questioning whether I really loved this subject or not – because I never really did anything tangible to show for it.

It was quite a few years before I landed back on physics, and that was when I understood that there is something called theoretical physics, and that was exactly what I had been so fascinated by. The physics that you study in middle and high school is really nothing interesting. If anything, it was absolutely mind-numbingly boring. I really hated a lot of it and did most of it quite mechanically – because I had to. It also did not help that until 10th grade, I could get amazing grades while putting in bare minimum effort, but once we got to 11th grade, that approach did not work. It was towards the end of 10th grade that things got interesting. I discovered MITOCW and found exactly what I wanted – some amazing resources to study theoretical physics! 11th grade was a tale of two extremes. I performed terribly in school, but when I put in the same amount of effort, I was able to study Quantum Mechanics from MITOCW at a really great pace. Somehow I was doing great and terrible at the same time, and that was the most confusing part. I was just much better at the quantum-y stuff. No matter how much effort I put in, I just did significantly better at anything that has to do with the microscopic world, and this is a trend that continues into college to this day – the courses in which I performed my best are the three courses on Quantum Mechanics, the two courses on Particle Physics and the one course on Quantum Field Theory. In fact, of the three research projects that I have done since coming to Illinois Tech, I most thoroughly enjoyed the one on computing scattering amplitudes in Particle Physics.

I think the other thing that quite struck me in my undergraduate degree is the kind of courses that pushed me, and I had to put in actual work for. Turns out, the undergraduate physics courses were mostly a breeze – purely because they were very superficial. These courses gave me no deep understanding of the subject, they had the shut up and calculate approach. These courses gave me the skills to do calculations, but they rarely taught me what the calculations meant, and more importantly, they didn’t help my understand how to ask questions that are relevant. This isn’t really a flaw of the system I guess, rather it is how undergraduate courses are designed. The graduate physics courses and the undergraduate mathematics courses are where I was pushed a lot more. I had to think about the calculations and problems that I did, and quite often each problem was designed to emphasize a different aspect of the theory that is important to understand. I found those a lot more to my taste and liking. I guess in that way I was very fortunate to have had the advisors that I have had – who never really questioned my passion and always encouraged me to push myself (especially when some of the academic choices that I made were quite objectively stupid).

Anyway, I digress. Coming back to some of the other thoughts that I had. There is something fundamentally unreasonable about doing research, and wanting to stay in academia. There is definitely not enough money, grants are hard to come by and the compensation is definitely not proportional to the effort you put in. It is purely love for the subject, and the puzzles that come with it, that dominates over all of these factors. This line of reasoning is truly unreasonable. When you talk to anyone in theoretical physics about this, their response is quite often just a resigned sigh about the entire lack of funding, but they all still do it anyway! I find this oddly ironic honestly – some of the smartest and most logical people that I have known all do what they do because of a reason that is fundamentally irrational and unreasonable. I guess this is quite symbolic of how you can never quite leave your emotional side in decision-making, no matter how hard you try. If there is something that you really want to do, because of a deep-rooted desire, nothing can really stop you except yourself.

I guess I want to tie it up and end by talking about what set this article off in the first place. This is the fourth year of my undergraduate degree, and I think I have come quite some way. I have learned so much about physics and mathematics, and more importantly, I have learned how to learn. I think that is one of the most important skills that I have picked up in my time here. Now with the looming graduate admissions, this is the end of the beginning. I am going to be taking my last first step as a baby physicist, on my way to being a fully-fledged physicist. There is so much to come, and I am genuinely excited for all that my future has in store for me, and I think this is a good place to end this bit of rambling. Here is to all that is to come!

On Mass

When one starts their study of physics, mass seems like one of those quantities that doesn’t really have an origin; it seems like one of those things that just exists that we must take for granted. Questioning the origins of mass is not something that seems to be something worth asking. The first place where the idea of mass is questioned is in the context of relativity, where the mass squared is an invariant property of the system. By this, I mean that if we have a system, then the mass that we measure for it in any reference frame is the same. The common notion that moving objects gain mass is flawed. The mass that we think is “added on” is just a relic of the fact that the object is in motion. The other place where the property of mass matters is when we talk about the speed of light. Calling it the speed of light is really a misnomer. A more realistic name for it should be the speed of masslessness. Any massless object will always move at the speed of light in that medium. Anything that is massive (in physics, when we say massive, we mean that the object has the property of some positive mass, as opposed to its colloquial use to refer to something of large mass) is doomed to move at speeds that are slower than that. This is about the complete discussion of mass in a classical setting. However, the idea of mass turns on its head when we go down to the really small scales. Things get super wacky and weird, and this is what I want to discuss over the rest of this article.

I want to start off with something that is relatively less nutty – “dynamical mass.” Dynamical mass is something that we see for particles that are not fundamental, that is, particles that are bound states of fundamental particles. Consider, for instance, any of the nucleons – the protons and neutrons. The mass of the nucleon is much larger than the sum of the masses of the quarks that make it up. This extra mass is purely a quantum phenomenon and arises from dynamical properties (hence the name) of the bound quarks. There are many processes that can cause this, for instance, we can have some of the mass arising from the spin-spin interactions of the quarks or we can have the mass arising from electrodynamical processes that arise from the charge interactions of the bound quarks. In order to get a better picture of this, and understand the structure of nucleons, what physicists do is that they throw things at increasingly higher energies at the nucleon. In order to get a better understanding of what we observe, lets do a quick and simple thought experiment. Consider that you are holding a clump of sand that seems to stick together. If you lightly throw sand at it, the clump of sand will look as though it is one chunk and has nothing that makes it up. Now, throw some small pebbles at it really lightly. The clump that you are holding will lightly break apart. So, it will look like it is made up of smaller clumps of sand. However, if you throw a rock really hard at the clump of sand, it will completely disintegrate into sand and then you will understand that it is made up of a bunch of sand that was held together. The idea here is that things look different based on the energies that you probe them at. So, if we lightly throw things at a nucleon, it will look like a solid ball, which is what we thought they were when they were initially discovered. However, once our understanding of the quantum world grew, and we got better particle accelerators, we realized that a nucleon is a bound state of three quarks. Recently, with imporvements to accelerator techonologies, we are able to probe the nucleons at much higher energies. At these scales, the quark looks like a sea of gluons and quarks that are forcibly held together because of asymptotic confinement (for more on this, take a look at the article that I wrote “On Quarks”). You might look at this and claim victory; we have all the extra mass that we need because there are a ton of things in there! And yes, this is indeed the answer. The stuff that we see inside is solely because of the quarks interacting amongst each other while being tightly bound together.

Take a second to digest that, because the next thing that we are going to take a look it is how fundamental particles get their mass and that seems significantly more nuts to me. In order to understand how mass arises at the most fundamental level, you need to recall one of the more basic things you learn in physics – the idea of a degree of freedom (dof). A dof is basically the minimum number of independent quantities that you need to completely specify a system and its dynamics. If you know all the dof’s for an object, you can completely specify its dynamics. In order to understand mass, first, we need to take a look at the massless objects that we know. Consider a photon, the simplest massless object that we all know and love. As we noted earlier, a photon, as a virtue of being massless moves at the speed of light, and anything that is massive cannot do so. So, we have one degree of freedom of the system being constrained here, what we have is effectively a trade-off! With this in mind, we can now look at where one extra dof is getting constrained in our fundamental particles.

Let’s start with fermions. For simplicity, let’s look at spin 1/2 particles. These particles have a property called helicity. In technical jargon, helicity is the projection of the spin along the momentum 3-vector. In simple terms, what we are doing is that we set the axis of the particle to be along its direction of motion, and then look at how the spin of the particle is oriented. The orientaion of the spin can be either clockwise or anti-clockwise (physicists call it right-handed or left-handed, and this becomes the handedness of the particle). Consider a massless fermion. Say we measure some helicity for it. Now, if we flip the direction of motion, the note that its helicity does not change, as it has no mass, so its momentum is determined purely by its wave-number, and the wave-number stays the same on flipping the direction of motion. However, for a massive fermion, if we flip the direction of motion, it’s helicity will flip! This happens because the momentum, which is defined as the mass multiplied by the velocity itself flips as the velocty flips. This tells us something interesting, for a massless fermion, its handedness is a defining property, but that is not so for a massive fermion! Here is our “extra” dof that seems to be missing is expressed as the mass. A massive fermion must have a right-handed and a left-handed component defining it, but only one of them is a measured value. Somehow, one of the helicity dof’s is always missing, and it is that dof that is expressed as mass. This leaves us with an interesting conclusion – mass can only arise if left and right-handed helicities mix! And that is indeed what we have just described, and is also what we observe. For higher spin fermions, the argument is similar, but the mathematics gets more fiddly as we need to keep track of more dof’s in that computation.

Next, we need to deal with bosons. We know of only one spin 0 boson – that is the Higgs Boson. The reason this has a mass is an artifact of the conditions necessary for something called spontaneous symmetry breaking (SSB). SSB is the reason why the W and Z bosons (spin 1 bosons) have a mass. We will start by looking and SSB and then briefly mention how it relates to a Higgs mass. So, SSB, as the name suggests, is when some expected symmetry of the system ceases to exist. The potential term that is used to describe these spin 1 Bosons looks like a Mexican Hat, with a circular trough at the bottom. Now, let us take a look at somewhere really high up on the Mexican Hat. Then from our vantage point, our position is completely specified by three dof’s, the three “coordinates.” Now, if we were at the minimum of the potential (this is the circle at the bottom of the Mexican Hat), then we are constrained to move only in the circle. Here, we only need one number to describe our position as the circle is at constant distance away from the origin and at a constant height. So, we have lost two dof! But wait, this is bad, we had two W bosons and one Z boson, that’s three of them! The idea here is that the two W bosons are particle and anti-particle of each other. So, we need only one of them to have a mass. The other one is automatically guaranteed a mass. So, the process of SSB results in spin 1 bosons getting a mass. Now, I made no mention of where this potential comes from. In order to introduce this potential in our theory, we need to introduce the Mexican Hat potential, and the physical manifestation of such a potential turns out to be a massive spin 0 boson, and it is this boson that we call the Higgs. There is an entire discussion that is warranted here on the nature of the Higgs Boson and such, but I will hold off on that now as that would be a massive digression from the general aim of this article.

These are just some of the mass mechanisms. There are more wacky ways in which things can have mass. I do encourage you all to look these things up and try to get a hang of them. There is another interesting way we can get massive particles, from something called the Kaluza-Klein Compactification, but that deserves an article of its own.

On the Quantum Zeno Effect

Zeno’s Paradox, as the same suggests, comes from Zeno of Elea, in Ancient Greece. He came up with a series of paradoxes to support Parmenides’ doctrine, which pushed for the fact that there was no motion, despite what our common senses and observation tell us. Now, that seems absurd in the face of it – motion is a thing we see on a day-to-day basis – it must exist. There are a bunch of versions of this, but for the purpose of this article, I will stick to one that seems to be the best argument for a motionless reality. Say you want to go a meter to your right. But, before you manage to get there, you first have to cover half the distance. But before you get to half, you have to get a quarter of the way there, and so on. There seems to be no so-called first step that you can take! We can always keep dividing the distance at any point in time by 2, and it will always yield a non-zero value. The argument goes that since you cannot conceivably take the first step, you cannot move, and therefore motion cannot be a thing.

I think one of the most fascinating arguments is to look at this as a geometric series. Since you are halving at every step, and a half is lesser than one, this is a convergent geometric series with a common ratio of half and an initial value of 1. Since this is converging, we can argue that the time taken to cover each step must also go decrease, as a consequence of which we can take the first step and execute the motion. The other proposed solution is that space and time must be discrete, that is there are some smallest distances and some smallest chunk of time that is meaningful to talk about. Anything smaller than that is meaningless. If we look at some smallest distance, then the individual can cover that distance in the smallest time interval, and so, motion can still be a thing.

While all this is really fun to think about, I want, instead, to shift your focus to Quantum Mechanics, where we do see a manifestation of Zeno’s paradox, and this effect is termed as the Quantum Zeno Effect. In order to talk about this meaningfully, we first need to take a quick look at the wavefunction and what it tells us. The Born interpretation tells us that if we integrate the square of the wavefunction over a region of space, then we get the probability of finding the particle in that state. In order to do this, the wavefunction must be expressed as a function of space. There are mathematical tools that allow switching to a different basis, that is, we can transform the wavefunction in order to express it in terms of the momentum or the energy. In that case, taking the square of the wavefunction will tell us the expected value of the momentum or the energy respectively.

Consider a wavefunction and express it in terms of the energy. The wavefunction itself contains information about the probability of finding the wavefunction in each energy state. That is, the wavefunction tells us all the information about the energy states that the particle could potentially be occupying, and the probability with which it is likely to be in those states. If we make a measurement and see the particle to be in one state, then the wavefunction modifies to take that specific value. This is termed as wavefunction collapse. After the measurement is made, the wavefunction relaxes and expands out again to encode information about all the energy states again. Basically, the wavefunction contains information about all energy states and then measuring it makes the wavefunction collapse to one specific state, and after the observation, the wavefunction evolves to encode information about all the states again.

Now, consider this interesting case. Measure the wavefunction at some time. Then, after a very short time period, make that exact same observation again. We expect the measurement to yield that same state again! Indeed, more often than not, we do observe that same state. This is because the likelihood of us measuring that same state again is inversely dependent on how fast we make the second observation. This should make sense, as the longer we wait between taking the measurement, the more time we give the wavefunction to go back to its original state, which had information about all possible energy states that the system can take. This is the Quantum Zeno Effect and this has very interesting implications for something called meta-stable states.

So what are meta-stable states? They are states that seem to be a very stable state, but in reality, they are not the stable state as there can be states that have lower energies than it. In a complex Quantum Mechanical system like, say, an atom or cooled system of atoms, some transitions between states are forbidden because of the nature of the wavefunction of the system itself. This means that we can have two states, say A and B, and state A is at higher energy than state B, but due to the nature of the system, there cannot be a transition from state A to B, so A will seem to be a stable state. Then, A is your meta-stable state. Now in reality, if an electron is in state A, what it ends up doing is that it transitions to some other state from which it can transition down to state B. This happens because all transitions are governed by probabilities, but transitions to the most stable state, B, will have the highest probabilities. So the meta-stable state A has some non-zero half-life.

We now know that if an electron is in a meta-stable state in a system, we would expect it to eventually drop down to the most stable state. Consider the Quantum Zeno Effect here! Start by sending an electron to the meta-stable state, and then make a measurement of the energy. We get the energy of that meta-stable state, as expected. Here is the cool part – measure the state again, within a short time span from when you make the first measurement. We expect to measure that same state again! If we keep repeating the measurement in very short time intervals, we would expect to always get the electron in the meta-stable state. What this means is that by repeated measurements, we can make the meta-stable state exist indefinitely! What we have effectively done is take a state that would be short-lived in general, and just by making the same measurement on it repeatedly, we can drag it out to exist for as long as we want.

This is a great example of how all of Quantum Mechanics seems to be some sort of leap of faith and barrelling headfirst into the realm of counter-intuitive thoughts, but the thing is, our intuition is shaped by our perception of the world, which exists in some classical limit of the quantum theory. The most important thing to take away from this is that one must not dismiss some idea just because it seems counter-intuitive. Instead, take a look at it and see if it is consistent with the theory that is being developed. The better the foundations of the theory are, the more accurate the predictions it will make.

On The Debate On Einstein And Newton

Recently, I read an article on NBC news which had a very catchy title – ‘Einstein proved Newton was wrong about gravity. Now scientists are coming for Einstein’. The title itself set off alarm bells in my head, it’s not exactly accurate. The article itself was more unbiased and told a good tale, but I just wanted to write about the debate and how theories work in physics.

So the thing is, Newton was neither wrong nor was he completely right. Newton proposed that the force of gravity varied with the inverse of the square of the distance between two bodies that have mass, and directly proportional to the product of the masses of the bodies. The force of gravity was defined as a consequence of mass and the further away the two bodies were, the lesser would be the strength of gravitational attraction between the two bodies.

The following experiments substantiated this claim. It helped Kepler calculate the orbits of the planets. It worked when a torsion balance was used to test it. The thing is, this was a perfectly fine theory for certain energy scales. At very low energy scales, Newton’s Laws are a perfectly good description of gravity. Now, there is one glaring issue with Newton’s theory. It does not talk about how gravity works but just tells us how to calculate the force of attraction and figure out how the matter will move under the influence of gravity.

Now, when we observed the orbit of Mercury, we realized Newton’s theory was not the complete truth. There was an unexplained precession in the orbit of Mercury that the inverse square law was not able to predict. According to Newton, the orbit should have been stable, with no precession. This issue, however, was resolved by Einstein’s Theory of General Relativity. His theory has a very accurate description of the precession and gave meaning to gravity. Gravity, he said, influenced the very fabric on which our entire cosmos was built on. An important point to be noted here is that one of the first tests of General Relativity was that whether we could recover Newtonian gravity at large distances, and Einstein’s theory passed this test with flying colors.

Now, moving on, physicists have been having a go at Einstein for decades. Einstein’s theories are also not a complete description of reality. It describes gravity at certain energy scales. It fails inside a Black Hole, and especially when we start asking questions about the singularity and the nature of spacetime itself within the Black Hole.

Many potential theories want to try and figure out what gravity does at those scales. These are theories like Quantum Gravity, Loop Quantum Gravity, String Theory, Shape Dynamics and many more. If any of these theories are going to be an accurate and absolute theory that describes gravity universally, it has to recover Einstein’s theory at the appropriate energy scales.

The thing about modern theories is that it is becoming more and more evident that we cannot figure out a so-called Theory of Everything from scratch. What we are doing now is coming with theories of forces that are valid at some energy scales. The future step may be to try and piece all these together into one equation that talks about everything but that might not be possible. We will probably have bits and pieces of theories, each of which will work at specific energy scales. Right now, we are trying to test Einstien’s theories at extreme energy scales and hoping that shows some cracks which will give us a good starting point to see where we can go. But then again, given how Einstein’s theory worked every time we tried to crack it, there is not much to look forward to. The best we can do is wait and hope.

On Symmetries

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.

On The Unreasonable Effectiveness of Mathematics

This article is going to be quite different from any other that I have written. This one is going to be more philosophical in nature, rather than explaining some aspect or idea in physics. I want to talk about something that has been termed the ‘Unreasonable Effectiveness of Mathematics’. At the face of it, there is really no reason that math should work in modeling our reality. I mean, it is just a set of rules that allow us to calculate so many things, and the weirdest thing is that every branch of mathematics can be traced down to some basic axioms. These axioms were a result of human thought. Axioms, in mathematics, are the most fundamental statements that have to be true, they are often touted as ‘self-evident truths’.

The development of mathematics is quite varied and spread out. It started out, like so many other things, with the Greeks. Pythagoras, one of the greatest minds that the classical world had to offer, studied shapes and was one of the first to study geometry. He was followed by Euclid, whose book ‘Elements’ was the first great mathematical text. It was a seminal work of art, it shaped mathematical thought for centuries, and where it lacked, it provided an impetus for development. Before I get to that, I’d like to discuss the geometry that they studied. These classical geometers studied what are dubbed perfect shapes. Spheres, cubes and cuboids and other shapes that are ‘perfect’, with smooth boundaries and no random bumps.

Euclid’s only shortcoming was his famous fifth postulate. Euclid developed his study of geometry based off of five postulates. The first four postulates were proved over the centuries. However, the fifth postulate stumped everyone who attempted to prove it. This was Euclid’s only, if at all, short-coming. However, it was this exact short-coming that provided the impulse that was needed for further development. With no one able to prove the fifth postulate, people began to question its necessity. I am paraphrasing the fifth postulate here- “Parallel lines intersect at infinity”. At first glance, it seems to make sense. Parallel lines are, by definition, lines that do not intersect. However, consider lines of longitude. On the surface of the earth, lines of longitude are parallel. However, they do intersect at the poles. Turns out, that parallel lines do not intersect only when the underlying surface on which we define the lines are flat. If the underlying surface is not flat, there is no reason for lines that are parallel to not intersect. This might sound familiar to anyone who has read my other articles, this idea came up a lot in the articles in which I spoke about gravity, and that is not a coincidence. Einstein’s theory of gravity is essentially a theory of geometry, where masses change the curvature (geometry) of the underlying manifold, spacetime, and in turn, that curvature of spacetime dictates how matter moves.

The point I am trying to make is very evident here. The study of geometry evolved from the study of shapes and distances and perfect objects. Out of the blue, just by modifying our definition of parallel lines, we unearthed a lot of mathematical beauty, and the new mathematics that we found seemed to work perfectly to describe gravity! Why is the large scale structure of our universe so effectively modeled by geometry? For that matter, our newer theories of the quantum realm are also seeing a shift towards formulation in a more geometric form.

There is also a very fundamental relationship between symmetries and conservation laws that are encapsulated in Noether’s Theorem. A small side note, Emmy Noether was an incredible physicist and mathematician and her results are beautiful. I would highly recommend reading up about her. She showed that for every continuous symmetry, there is a corresponding conservation law. If there is a temporal symmetry, then energy will be conserved in that system. If the system has translation symmetry, then momentum is conserved. The proof here uses the Lagrangian, which is the function of the energy of a system. It is defined as the difference between the kinetic and potential energies. There is the ‘Principle of Least Action’ that tells us that this function, called the Lagrangian has to be at a minimum for any physical system (this is a very hand-wavy statement, but bear with me). Again, that has no obvious reason to be true! It simply is! And it is very surprising why a difference between the kinetic and potential energies should be minimized rather than their sum (the total energy). The thing that surprises me the most is that the Lagrangian approach to solving problems is what we use for our modern theories of the quantum world. And what is even more surprising, is that we just write down a bunch of terms (naturally, the terms that we write down have to obey some basic rules) for the Lagrangian! We literally just guess terms and put them in and then go about this entire process of minimizing the Action and see what we get. Essentially, it is a lot of educated guesswork.

Another thing that I should mention is Euler’s constant – ‘e’. This ‘e’ and the natural logarithm appear everywhere! The exponential appears in the equation for the rate of cooling or heating, it appears in the equations that govern the rate of radioactive decay, it appears in quantum mechanics, (the ground state wave function of the non-relativistic Hydrogen atom is a simple decaying exponential) and so many other places! In fact, the bell-shaped Gaussian curve is described by the square of an exponential function, and it appears as the wave function of a free packet. For that matter, the Gaussian distribution models the variation of velocities of the molecules of a gas! What is even more surprising is that ‘e’ was not found as a constant in some equation of physics, but, it was the result of a limit of an infinite series. There is absolutely no reason why the limit of some series should result in a constant whose exponential function models so many different things.

I guess the point I am trying to make is that Mathematics, which is just an expression of the abstractions of the human mind, has absolutely no business modeling Nature, our world and so many things around us as well as it does. The very language of Physics is now Mathematics. Physics is the study of Mathematics when applied to model the real world. Our current mathematics can even describe Chaotic systems and their behavior! These are questions that we do not deal with on a day to day basis. After all, this is just a fun thing to think about. Maybe this question can be answered once we have a much deeper understanding of our physical theories. Perhaps at the end of all this, we might see a connection between the abstractions that we have formalized and the workings of Nature.

On Wormholes

Wormholes are one of the most common sci-fi topics that you can find in physics. Frequently featured in Star Wars and Star Trek, it is a mind-blowing idea. Wormholes connect two distant points in space-time. That is, it allows us to traverse large distances seemingly instantaneously. Though technically, time travel should be possible, the use of wormholes for doing so is highly debated. Wormholes are not something that just popped out of someone’s head for sci-fi books, they actually could exist.

The governing equation behind Relativity is the Einstein Field Equations, as I have mentioned so many times before. When solved in a certain way, they give us the structure of space-time for certain conditions or matter distributions. These equations have predicted the existence of Black Holes, Gravitational Waves and the weird orbit of Mercury. Under certain conditions, wormholes come out as a solution to these equations. The problem is, we would need negative energy to open them, and substances with negative energy densities to stabilize them. Stabilize them, in this case, means to keep them open long enough for someone or something to traverse it. A wormhole that can be traversed is called a Traversable Wormhole. You can see how much thought went into that name.

For a wormhole to exist, we need at least one extra spatial dimension to exist. This can be illustrated using two examples. The first one, also tells us why the name wormhole was given. So consider a worm, say a maggot, on the surface of the apple. It has a flat, two dimensional surface of the apple over which it can move. Say it wants to get to the other side. It can go all the way across the apple, or it can eat through the apple, and go to the other end. If it eats through the apple, it has made use of the higher third dimension of space to travel.

The other example was used in the movie, Interstellar. Consider a sheet of paper, my favorite thing to simulate a two-dimensional space-time with. The only way to travel is restricted to the surface of the sheet. Mark the point you are at, and the point you want to travel to. Now, if you bend the sheet over, into the third dimension, then you could use the third dimension to travel to the other point faster as a shorter distance has to be covered.

That’s about all that you can say about a wormhole, there is really nothing much left to explain. The idea of wormholes first came up in a paper suggested by Albert Einstein and Nathan Rosen (which is why Wormholes are sometimes called Einstein-Rosen bridges). That paper is referred to as ER.

There is a very interesting idea called ER = EPR. ER refers to the paper mentioned above. EPR is a paper written by Albert Einstein, Boris Podolsky, and Nathan Rosen. In that paper, they question the completeness of Quantum Mechanics as a viable description of reality. As a consequence of that paper, the phenomenon of Quantum Entanglement was found. The reason why entanglement was contradictory was that it allowed information to be transmitted faster than the speed of light. This contradicted General Relativity. However, if entanglements occur with wormholes, then this is not a problem. Information can travel via a wormhole practically instantaneously to a distant point. This would allow entanglement to be allowed in General Relativity too.

Another place where wormholes have become relevant recently are in the study of black holes. Turns out, in certain cases, it is possible to have the inside of black holes be connected to the inside of another black hole via a wormhole. This is a very theoretical idea and detecting this is more or less impossible. And of course, there is the minor inconvenience of having to fall into a Black Hole in order to access it which is not really a good idea.

On Electrons

This article is sort of a continuation of the article on quarks. That one focused on quarks and the theory of its interactions, Quantum Chromodynamics. This article will focus on electrons and the theory of electromagnetism on a small scale and the theory of Quantum Electrodynamics (QED). Feynman, one of the founders of this theory, called it the ‘Jewel of Physics’ for its incredibly accurate predictions. Such accuracy has never been seen before. It predicted some observations accurately to thirteen decimal places!

The electron is a fundamental particle with a unit negative charge and an incredibly small mass. It is a lepton with a spin of magnitude half. The idea of spin is pretty hard to grasp, it is not like the usual classical idea of a spinning object, that cannot happen as calculations show that an electron has to spin at a speed faster than the speed of light. Spin, essentially means that a particle has a non-zero value of angular momentum just by the virtue of existence. Electrons are fermions, which implies that electrons are subject to the Pauli Exclusion Principle; no two electrons in the universe can have the same magnitude of energy. Like gluons in QCD, the carriers of electromagnetic force are virtual photons. Virtual particles are those that do not exist, that is, while we can’t physically show the existence of the particle, we can feel its effects.

First, a note on fermions and bosons. As I mentioned, the Pauli Exclusion Principle applies only to fermions, or particles with half-integral spin, and not to bosons, with integral spin. This idea is purely mathematical, so there’s really not much to explain. The effective wavefunction of fermions is anti-symmetric while, for bosons, it is symmetric. This basically means that if you have a wavefunction of two bosons, and you interchange the bosons, then the wavefunction still remains the same. However, in case of a wavefunction for two fermions, if you switch them, then the wavefunction picks up a negative sign.

Electrons (and other fermions) exhibit an interesting phenomenon, the formation of Cooper pairs. At very low temperatures, two electrons come together and behave as one bound pair. The energy of formation of a cooper pair is very low, so thermal excitations can easily break up the pairs. The electrons in the pair don’t have to be in contact, they behave as one pair even over long distances, like up to hundreds of nanometers. An electron in a metal normally behaves as a free particle. The electrons mutually repel each other and attract the positive ions that make up a metallic lattice. This attraction distorts an ionic lattice, moving some of the ions slightly toward the electron. This increases the positive charge density of the lattice in the vicinity of the electron. This positive charge can attract other electrons. Over long distances, the attraction between electrons due to the displaced ions can overcome the inter-electronic repulsion and can thus pair up. This is at best a semi-classical explanation of the effect. The actual quantum mechanical explanation is much more rigorous and requires a lot of mathematics so I won’t be going into that.

An interesting phenomenon in this theory is the idea of vacuum polarization. The measured charge of an electron is lesser than its actual charge. Because of the charge of the electron, it has its own electric field. Now, the existence of an electric field means that there is more energy in that region than there would be without the electron. This leads to the spontaneous formation of particle-antiparticle pairs around the electron for a short duration of time. However, since this process happens near the electron as long as it is present, it leads to the electron being perpetually surrounded by a covering of particles. The existence of this layer of particles damps out the charge on the electron, and thus the charge that we observe is lesser than the actual charge on the electron.

Before I discuss the positron, I want to mention one last very interesting phenomenon that is only shown by electrons. This is called spin-charge separation. Basically, under certain conditions, the spin and charge on an electron split up. That is, one electron exhibits only charge, and the other electron exhibits only the spin. No electron can exhibit both these properties. An electron, by itself, has charge and spin. Another way looking at this is that the electron itself is bound to the state of two particles, the spinon (carries the spin) and the chargon or holon (carries the charge). An electron in a bound system also has a third particle, the orbiton (carries the orbital angular momentum). In our case, we consider an electron as a bound state of a spinon and chargon. Under certain conditions these particles can become deconfined, that is, they break free and behave as individual particles. These particles, the chargon, orbiton and the spinon are called quasi-particles.

The antiparticle of an electron is the positron, with a unit positive charge. All other properties remain the same. The existence of antiparticles leads to an interesting effect termed as ‘Zitterbewegung’, which is German for ‘trembling motion’. When the relativistic effects are taken into consideration and the wavefunction of an electron is formulated, then, it is observed that the wave-packet solution of the electron interacts with the positron wave-packets, which occupy the negative energy states and this gives rise to rapid oscillations. These oscillations occur with the speed of light. On a side note, this effect is observed not only with electrons but with the relativistic hydrogen atom and has been observed in Bose-Einstein-Condensates (BECs).

QED is one of the only theories that we understand to a great extent, yet there is a lot to be done, in the larger scheme of things. This is just the starting point in the quest to find a Grand Unified Theory.

Stephen Hawking and the Big Bang

On the 14th of March, 2018, one of the greatest cosmologists of our time, Stephen Hawking, passed away. It was a great loss, not only for physics but for science as a whole. There is no better way to honour his memory than to follow in his footsteps, taking the complex concepts of science and making them accessible to the general public. In this article, I will be elaborating upon some of his most influential works concerning the universe. I will focus mostly on his contributions to understanding the universe, and not on his seminal work on black holes as I have covered most of that in my two articles on black holes.

The first of his revolutionary ideas was the notion of the Big Bang. He did this in collaboration with Roger Penrose. Penrose had put forth his singularity theorem, which proposed that at the center of every black hole is a point-sized singularity. Hawking thought that this might be true for the entire universe. He worked out what the universe should have looked like in the past, and concluded that considering the accelerated universe now, the universe must have, at some point of time in the past, been a point like a singularity. This was the idea of the beginning of the universe, the Big Bang. In 1970, the two of them jointly proved that if the universe were to be any of the proposed Friedmann models (the most widely accepted models that describe the universe), then the universe must have started off with the Big Bang.

After this, he shifted his focus to the study of black holes. He came with the concept of Black Hole Thermodynamics and Hawking Radiation, which laid the foundation for the famous Information Loss Paradox of Black Holes.

Following this, Andrei Linde and Alan Guth proposed the inflationary model of the Universe, as a consequence of the Big Bang. The Inflationary model was important as it explained the fluctuations necessary to create the complex matter that we see now. Inflation basically says the universe initially expanded rapidly, but then, the rate of expansion slowed down.

All of this is pretty well known and there are a lot of resources one can find to read up on these ideas. Now, here is one of his ideas that really fascinated me, on which I did not find many articles. The “Hawking-Hartle No Boundary Proposal” of the universe. This idea basically says that the universe has no boundary. In the sense that if you go far enough and long enough in the same direction, you will end up where you started. This is sort of like an ant walking on the surface of an apple. Or, sort of like asking the question, How do I go north?, when you are at the North Pole. This is called a closed universe.

In the paper, they derived what they called the wavefunction of the universe. It seems absurd at first sight, a wavefunction of a particle is generally a function of either its position or momentum. How do you describe either of those two for the entire universe? You don’t. They calculated an integral of all the possible histories. This has an interesting consequence. The wavefunction now has the added information about all the possible different universes that could have come about after the Big Bang! It elegantly includes the idea of a multiverse. The wavefunction looks like any other quantum mechanical wavefunction, it can have places where the value of the wavefunction is zero. These correspond to places where there are breaks between universes. Basically, this corresponds to a boundary between two universes.

An idea like this has it’s fair share of opposition. One argument against this is that observation tells us that the universe is expanding, which means that the universe must be open. We can overcome this problem by choosing a suitable interpretation of the wavefunction. If we think of the wavefunction as a generator of space-time, then any bit of the wavefunction that corresponds to a universe can be thought of as an expanding universe. That is, we take it as the wavefunction generating spacetime in that region. This is, of course, glossing over a lot of technicalities and subtleties, but for an article of this type, that can be accepted.

Stephen Hawking, in a way, was the catalyst for my interest and love for the subject. The first scientific books that I read were the series of books about George, written by the man himself and his daughter Lucy Hawking. Soon after, I read the Grand Design and the Brief History of Time. He is certainly one of the giant figures whose ideas single-handedly drove an entire field in academia. And if we can see further than anyone before us, it is because we stand upon the shoulders of such giants. As he once said, “Science is not only a disciple of reason but, also, one of romance and passion. For not only does God play dice, but…… he sometimes throws them where they cannot be seen.”

On Quarks

The Standard Model of Particle Physics is the most elegant model of nature that we have currently. According to it, everything in the universe and every change in the universe is governed by 17 (confirmed) fundamental particles. Five of which are classified as bosons, and are of two types, there are 4 gauge bosons that are responsible for mediating the four fundamental forces and the one scalar boson, the Higgs Boson that is, in a way, responsible for matter having mass. According to the standard model, we feel a force when the particle mediating that force is exchanged between particles. If two electrons interact via the electromagnetic force, they exchange photons, as photons are the mediator of the electromagnetic force. The W and Z bosons are responsible for the Weak Nuclear Force, that underlies radioactivity and other phenomena, and the gluons, that mediate the Strong Nuclear Force. Gravitons have been predicted to exist (a particle that mediates the gravitational force) but not yet been confirmed.

The other 12 particles have mass and may or may not be charged. The interactions between the particles are very well represented by the famous Feynman diagrams. Feynman diagrams tell us in how many ways particles can interact by showing the incoming particles, the subsequent exchange of bosons and the outgoing particles. Based on some rules, we can tell how likely each of the interactions will be. Technically, there are infinite possible ways an interaction can happen, but some have incredibly low probabilities and can be ignored.

In this article, I will focus only on quarks, a set of 6 of the mass particles and their interaction via the gluons (responsible for the Strong Nuclear Force). The six kinds of quarks are the up, down, top, bottom, strange and charm quarks. They are termed as the flavors of quarks. Very, very creative, I know. The theory that explains this interaction is called Quantum Chromodynamics (QCD). The name is very misleading. There is nothing related to colours at this scale but due to the incredible creativity of physicists (Feynman called them as ‘idiot physicists’), a property that is called ‘colour charge,’ was introduced to explain the interactions between quarks.

Colour charge is a property that is exhibited only by quarks and gluons. It is named so because the interactions are quite analogous to how the three primary colors – red, green and blue interact. Each flavor of quark has three types and a different colour charge. Of course, their antiparticles also show colour. They are called anti-red, anti-blue, anti-green. A combination of all these three colours or the quark of a colour and the corresponding anti-color form a ‘colourless’ quark. This was needed to show how quarks could form hadrons (composite particles made up of three quarks, like protons and neutrons) and mesons (particles composed of quark-antiquark. But they do not annihilate each other as they are not a colour charge and its corresponding anti-colour charge), without violating the Pauli Exclusion Principle (which states that two or more particles with half-integral spins cannot occupy the same quantum state in a quantum system).

Like the idea of electric and magnetic field lines, there are field lines between quarks of different colours. Just like how in classical electrodynamics, one can draw field lines, we can also do so for QCD. The field lines exist between quarks of different colours. These field lines correspond to interactions that hold together quarks to form hadrons and mesons, via gluons. Quarks of different colours exchange gluons and experience the force due to colour charge. Colour charge is felt because of the exchange of ‘coloured’ gluons between the quarks.

Now, to the two most interesting and mind-blowing phenomena that occur in QCD – quark confinement and asymptotic freedom. To study the internal structure of composite particles, we bombard them with high energy particles to split them and study the resultant stream of released particles to understand what it consists of. When protons were bombarded with high energy quarks, we noticed something very weird. The binding energy of quarks inside protons was small. It makes sense to think that we could then observe free quarks right? Wrong. We have not yet isolated any sample of a free quark.

The quarks inside of a proton behaved like free particles. We know this directly from experiments. As we bombarded protons with high energy electron beams, we could clearly observe energy and momentum being imparted to the quarks. The way the quarks moved inside a proton was a lot like you would expect any free particle to move when they undergo similar collisions. The closer you push two quarks together, the weaker they are bound and the more like free particles they will behave. This is something very counter-intuitive. You would expect that when you push stuff closer, each individual particles will behave lesser and lesser like free particles. Instead, the closer you push quarks, the more like free particles they behave. This phenomenon is known as asymptotic freedom.

We can extend asymptotic freedom to get to quark confinement. In asymptotic freedom, we observe that the potential energy of a system of closely bound quarks is less, and as the quarks draw further apart, the potential energy of the system increases. The potential energy of the system is directly dependant on the distance of separation. The further two quarks are apart, the closer they appear to be bound. A very interesting thing happens at this point.

Before I go on, I need to state two things. One is Einstein’s mass-energy equivalence. Mass and energy are one and the same and they can be converted to each other. The other fact is that mass is a very stable expression of energy. Energy wants to exist in the form of mass rather than in the form of kinetic or potential energy. We observe kinetic and potential energy because their magnitudes are not enough to create any appreciable amount of mass at the classical scale. However, in the quantum limit, the energy that corresponds to a quark is not much.

Going back, as we pull two quarks further and further apart, the potential energy of the system increases. One can visualize this like a tube(called the flux tube) connecting two quarks. We will reach a point at which the potential energy between the quarks is sufficient enough to form a quark-antiquark pair. At this point, the tube snaps into two, resulting in the formation of two sets of tubes. One end of each of the flux tubes contains the original two quarks that were being pulled apart and in between the two, because of the high potential energy, a quark and an antiquark pair are formed. Now, each of the flux tube pairs contains one of the original quarks and one of the created particles.

This explains everything about why we cannot observe free quarks and why we observe only bound states of quarks. Every particle likes to be in its lowest possible energy state. So, quarks prefer to be bound, as they are more free that way. The only way to observe free quarks is to pull the quarks apart, but, we cannot do so as if we try to pull a pair of quarks apart the flux tubes split. Clearly, asymptotic freedom and quark confinement explain everything weird about quarks. If you have a more technical knowledge or are looking to understand these phenomena to a greater extent, without a great deal of math, I would highly suggest Franck Wilczek’s Nobel Prize lecture titled “Asymptotic Freedom: From Paradox to Paradigm”. It is a truly wonderful read.

On a slightly unrelated, yet cool note, I have stated that it is gluons that mediates the Strong Nuclear Force. That is not completely true. To be more precise, it is ‘glueballs’, that are responsible for the force. Glueballs are a result of gluons interacting with itself.