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At the root of things

Modern theories of fundamental interactions describe strong, electromagnetic and weak interactions as quantum field theories with certain kinds of embedded internal symmetries called `gauge symmetries'. This article introduces quantum field theories and gauge symmetries to the uninitiated.


Things behind the things we see
The reader looking at this page must be establishing some sort of interaction with the marks of ink that define the letters on the page. How is this interaction being established? Well, if it is evening, I assume that there is an electric lamp glowing in the room. Light coming out of that is hitting the page, getting reflected, and entering the reader's eyes.
In short, the interaction is being established through light. Suppose we now ask, why is the lamp glowing? When the lamp was switched on, how did the lamp, sitting a few meters away from the switch, get that piece of information? We know the answer to this question. There is a wire connecting the switch and the lamp, which carried an electric current. So in this case, the connection was established through electricity.
What happens when we turn on an electric fan? There are coils of wires inside a fan. When an electric current flows through it, it generates a magnetic field around the coils. If we put a metallic ring within that magnetic field, the field induces a rotation on the ring. Once you have a rotating something, it is easy to fit a few blades on it so that it can send ripples in the air around it. So here it is the magnetic field which acts as an agent in establishing connections.
In the second half of the 19th century, James Clerk Maxwell taught us that these are not independent phenomena. Light, electricity, magnetism: they all are governed by a common set of laws. So we can summarize the statements made in the previous paragraphs by saying that two things can interact with each other through the electromagnetic field.
In the first half of the 20th century, we cracked the mystery of atoms. An atom has a nucleus in some central position, and electrons going around it. How do the electrons know that there is a nucleus somewhere there? Because the nucleus contains protons and neutrons, of which the protons carry positive electric charges. These charges create an electromagnetic field around them. The electrons hover around through this field. So in this case also, the electromagnetic field acts as the matchmaker.
Sometimes if two different substances are brought close together, they react chemically. What happens in a chemical reaction? In short, molecules break up owing to interactions between the atomic electrons, and the atoms reorganize themselves into new molecules. Thus, here also the interaction is electromagnetic.
While I write, I hold a pen in my hand. How do I do that? There is something going on in the atoms and molecules that constitute the fingers of my hand which allows them to put a pressure on the atoms that constitute the pen. It would be hard for me to describe the details -firstly because the processes are complicated and secondly because I am no expert in physiology. What I can say for certain is that some kind of interaction between atoms is responsible for my holding the pen, and these interactions are electromagnetic. It is the same story behind most of the things we do -speaking, walking, sitting down, chewing our food -you name it! But if my pen slips out of my hand and falls on the floor, that's not due to electromagnetic interactions. Here the earth's gravitation is responsible for the phenomenon. Just as a charged particle or a magnet sets up an electromagnetic field around it, a massive particle sets up a gravitational field around it. Because of the gravitational field that the earth creates, the pen in my hand came to know the presence of the earth near it. So, as soon it slipped out of my hand, it went down and hit the floor.
The sun produces a gravitational field around it, and planets feel it and go orbiting the sun. The stars and galaxies in the sky are roaming around in various ways, all because of gravitation. The moon is encircling the earth. How does the moon know about the earth? Well, because of the earth's gravitational field.
We are talking about arguably the most fundamental question of physics. How does any object know about other objects? How does any object relate to others? How do objects influence other objects? How do objects behave under such influences?
If nothing like this happened, if everything in the universe spent their lives without any interaction with anything else, there would have been nothing to discuss in physics. And if fact, there would have been no one to discuss physics, or anything else, either. Because our body is made out of conglomeration of molecules, and the organization and function of those molecules depend crucially on the interaction between them. Without these interactions, nothing could have formed -no sun, no planet, no plant, no insect, no nothing. Things exist, and events happen, because there are interactions. And every phenomenon that we can see with our naked eyes or feel with our other senses can ultimately be explained with only two kinds of interactions: electromagnetic and gravitational.
2 Things beyond the things we see If we try to understand phenomena that take place at scales which are too small for us to see, we realize that only the aforementioned two kinds of interactions are insufficient. Take, for example, the nucleus of an atom. There are protons and neutrons in a nucleus. Protons are charged particles, so two protons repel each other. The magnitude of this repulsion is much much larger than that of the gravitational attractive force that exists between them. It seems then that any nucleus, except the hydrogen nucleus which has only one proton, should break apart because of the repulsive force which dominates. Why doesn't that happen?
The reason must be that there is some other kind of force that is attractive between the members of a nucleus, and which is much larger than the repulsive electromagnetic force between protons. Any proton or neutron creates a field of this force around it. This is called the field of the strong force. Through its effects, i.e., through strong interactions, the net force between protons and neutrons in a nucleus is attractive. And this is why we can obtain nuclei which are long-lived.
Does it mean that any nucleus lives happily forever, being bound by the strong force? No, that's not the case either. We know about the phenomenon of radioactivity. More correctly, it is a class of phenomena in which certain nuclei spontaneously break up, emitting some particles in the process. There are different types of radioactivity. In one type, a neutron in a nucleus decays to give a proton, an electron and a particle called the e-antineutrino. How does that happen? Strong interactions cannot be responsible for this, because strong interactions have nothing to do with the electron. Could it be possible then that the electromagnetic repulsion that we talked about earlier is somehow dominant in these nuclei? But that cannot be the case either, because electromagnetic interactions cannot change a neutron to a proton, or whatever.
Not strong, not electromagnetic. Gravitation is negligibly small at these scales. So there must be a fourth kind of force. That force must be weaker than strong or electromagnetic forces, which is why all nuclei do not disintegrate. And for this reason, we can call this force the weak force.
It should not be concluded, from what has been said above, that strong force manifests itself only through neutrons and protons, or weak force through radioactive nuclei. There are a host of other phenomena in which these forces play crucial roles.
Many of them include other kinds of particles than the ones we have mentioned so far.
In summary, there are four kinds of forces, or four kinds of interactions. The strongest one is known, not surprisingly, as "strong interactions". Electromagnetic interactions are next in the order of strength, followed by weak interactions. And gravitational interactions are so negligibly feeble that we will ignore them for this article.
3 Give and take: the language of quantum field theory In talking about forces, or interactions, we have used the word "field" quite a few times. That is how forces are felt. Maxwell introduced the electromagnetic field in the 19th century. The gravitational force was known since Newton, although the equations governing the gravitational field came to be known only at the beginning of the 20th century, when Albert Einstein formulated his General theory of Relativity. Indeed, about a decade before that, in 1905, Einstein formulated his Special theory of Relativity, and it was clear from it that no theory of interactions can be complete if only the force law between two objects is specified: one also needs the field in order to carry the information about the force from one object to another which may be sitting a distance away from the first.
It would therefore follow that we should need field theories for strong and weak interactions as well. But before that was found, or even attempted, a new revolution took place in physics. It was the quantum revolution. In the language of quantum theory, light behaves as particles dubbed photons. The idea was introduced in 1900 by Max Planck, in an attempt to explain the manner in which hot objects radiate. Then, in 1905, Einstein used photons to explain how certain substances produce an electric current when they are exposed to light, a phenomenon known as "photoelectricity". The explanation was in terms of collisions between electrons in the substance and photons in the light beam. In the collision, the electron gains energy from the photon and comes out of the substance. Since the electrons are charged particles, an electric current is produced.
If this idea is taken seriously, one should be able to describe other electromagnetic phenomena in terms of this photons as well. In classical field theory, one assumes that energy is transported as waves. That description must have a counterpart in which energy would be transported as particles, or quanta, as they were called in the early twentieth century. Besides, the description should be consistent with the requirements of the special theory of relativity. Such theories are called quantum field theories. The quantum field theory of electromagnetic interactions, A B P Q Figure 1: Two current-carrying wires. The arrows indicate the directions of currents. whose formulation began in the late 1920s and reached its climax in the 1940s, is called quantum electrodynamics, or QED for short.
So now let us ask: how would QED explain that a current-carrying wire exerts a force on another current-carrying wire? Let us look at Fig. 1. We have two wires there, carrying currents in directions shown by arrows. In such a situation, a repulsive force will be felt between the wires. The question is: how? The two wires are not touching each other so that information from one wire can pass on to the other. How does one wire then know about the other wire?
A classical physicist would follow Maxwell's theory to answer this question. He would say that when a current is flowing through the wire AB, it creates an electromagnetic field around the wire. Waves in this field is carrying out the information about this wire. The wire PQ falls within this electromagnetic field, and thus it comes to know about the wire AB. The wire AB learns about the wire PQ in exactly the same manner: through the electromagnetic field.
Fine, but we want to understand the same phenomenon from the viewpoint of quantum theory. We don't want to invoke the idea of waves in the electromagnetic field. We want to talk in terms of photons. How should we go about doing it? Now we will have a different story to tell. Let us start by thinking what it means by saying that an electric current is flowing through a wire. There are electrons in the substance from which the wire has been constructed. Those electrons are flowing. Now, in course of the flow, sometimes some electron in the wire AB is emitting a photon. Maybe this photon travels in the direction where the other wire lies, and hits that wire. The photon would then carry the message from the wire AB to the wire PQ. I don't mean to say that a specific electron in the wire AB shoots out only one photon. There are many electrons flowing through the wire, and each of them is emitting a photon once in a while. Not all of them is emitted in the direction of the wire PQ. They are being emitted all around. So, the space around the wire is teeming with photons at any given instant of time. The collection of these photons is what a classical physicist would call an electromagnetic field.
Among these photons, some travel in the direction of the wire PQ. The wire PQ is catching them, or gobbling them up. It cannot possibly catch all photons that come its way, but is able to catch some of them. Similarly, the wire PQ is emitting a lot of photons, and the other wire is catching some of them. The exchange of photons is going on, establishing the interaction between the two wires.
It is not really necessary to think about something as complicated as two wires, with zillions of electrons flowing through them. We can think of just two electrons. Each of the two will somehow feel the effect of the other. And the reason would be the same: one electron would throw some photons which the other one would catch, and vice versa.
A picture is worth more than a thousand words. Richard Feynman showed how to summarize all such words into some simple diagrams. Look at Fig. 2. An electron was passing along the path AKBLC. Along the way, it threw out a photon when it reached the point B. The other electron, starting from the point P, caught this photon at the point Q.
Such pictures are called Feynman diagrams, and they should not be thought of as a photographic depiction of the real event. In Fig. 2, we have not tried to say that the path of the electron has a sharp bend at the point B. The photon in the middle is obviously not traveling along a wavy line. All lines in the figure are symbolic: the wavy line represents the photon, the straight lines represent electrons. The important message that is given in the figure is this: only one photon has been exchanged between two electrons.
We might ask, why did the electron throw out a photon precisely at the point B? Couldn't it have thrown out a photon when it was at K? Or at L? What's so special about B? Similarly, why did the other electron catch the photon at the point Q? What's special about this point?
Nothing. No specialty at all. In fact, the first electron could have thrown out the electron at any point. The probability was there, all along. The photon could have been emitted at K or L or any other point: it just happened that it was emitted at B. Similarly, it is a matter of chance, or probability, that the other electron caught the photon at the point Q. Since the probability exists, could it not have happened that one photon was emitted at B, and another one at L, and the other electron caught both of them? Well, yes, it could have, and Feynman would have represented it by the diagram of Fig. 3a. The second photon could have been exchanged between two different points as well, as shown in Fig. 3b. Note the word "exchanged". We are no longer saying which electron emitted a photon and which one caught it. In all these pictures, we can also think of the photon being emitted by the second electron and caught by the first electron. Many such exchanges can also take place, as shown in Fig. 3c. While calculating the force between two electrons, all such diagrams will contribute. We will have to add all of them up in order to obtain the full interaction between two electrons.
Planck and Einstein, in the first few years of the twentieth century, showed how the idea of light quanta, or photons, can explain various phenomena involving light, or radiation in general. Take, for example, the case of the photoelectric effect, which happens because of scattering between light and electrons. So light, or electromagnetic field, was the actor in that play. And now we see that the electromagnetic field can do more: it can function like the director of a play, who remains behind the wings, but decides how different actors should interact with one another. In other words, now we can describe all properties of an electromagnetic field in terms of photons. This language, or manner, of description is called quantum field theory. In this language, an electromagnetic field is a collection of many many photons. Or, turning this around, we can say that a photon is the quantum, or the particle, corresponding to the electromagnetic field.
What would have happened if, instead of electrons, we had two different particles? We said that at any instant, there is a probability of the electron emitting a photon. The same would be true for any other charged particle. Suppose we consider two d-quarks, which are some kind of particles whose charge is one-third that of the electron. In the language of quantum field theory, it means that at any instant, the probability of a d-quark's emitting a photon would be one-third the probability of an electron emitting a photon. The probability of catching a photon also suffers from the same factor. Thus, if we think that the solid lines in Fig. 2 correspond to d-quarks rather than electrons, we would have one-third of a chance that the quark will emit a photon at the point B, and a further one-third of a chance that this photon will be absorbed at the point Q. So, the process in its entirety would have a probability of 1 3 × 1 3 , or 1 9 , compared to the same process with electrons. The repulsion between two d-quarks would therefore be one-ninth that of the repulsion between two electrons. This is Coulomb's law: the magnitude of the force between two charged particles is proportional to the product of the charges of the particles. More complicated diagrams, like those appearing in Fig. 3, would yield different ratios, but then the contribution of these diagrams are so small to begin with that they hardly matter.
We seem to be getting back all results of classical electromagnetic theory through this new language, sometimes with some small corrections which went unnoticed in the classical version. Let us now ask the question that would prove extremely important in what follows: how do we obtain the law of conservation of charges in the language of quantum field theory?
The answer is very simple. Let us go back to Fig. 2 one more time. What is happening at the point B? An electron emits a photon. The charge of the electron does not change in the process of course. Thus, no change of charge will take place in the process if photon is considered to have zero charge. No problem with charge conservation if an electron emits a photon. No problem if an electron absorbs a photon. No problem if the charged particle is not electron but something else. No problem if the particle emits a hundred photons and absorbs seventeen.

Symmetry, or the wonderful confusion of being
In the early twentieth century, the renowned mathematician Emmy Noether showed that conservation laws are intimately connected with symmetries. For example, the law of conservation of momentum can be derived from translational symmetry of space, i.e., the hypothesis that no point in space is special, and we can set up the origin of a co-ordinate system anywhere we please, with identical consequences. Energy conservation can be derived from homogeneity of time, which is the hypothesis that any instant of time is equivalent to another.
In the same spirit, we can ask, which is the symmetry that is related to electric charge conservation?
Quantum theory considers particles and waves as complementary descriptions of any object. We mentioned that electromagnetic radiation, which used to be considered as waves in classical physics, was described as a collection of particles called photons in the language of quantum theory. Likewise, quantum theory provided a description of electrons and other particles of matter in terms of their associated waves, called matter waves in general. By waves, we mean some quantity that is varying in space and time. When we have waves on the surface of a pond, it is the water level which varies. For electromagnetic waves, the electric field, or equivalently the magnetic field, varies in space and time. For matter waves, the similar quantity is almost universally denoted by the greek letter ψ (psi). But there is a difference. The height of water level or the electric field are represented by ordinary numbers. The matter wave amplitude ψ, on the other hand, is represented by somewhat more complicated things called complex numbers. For the purpose of our discussion, we can think of a complex number as an arrow on a piece of paper, i.e., an arrow in two dimensions. These are not physical dimensions like length, breadth or height; these are some hypothetical directions. The value of ψ at any point at any instant can be represented by an arrow at that point at that instant. The length of the arrow would represent the magnitude of ψ, and the direction of the arrow would represent the direction of ψ in that hypothetical space embodying complex numbers. The length is called the modulus of the complex number, and the direction its phase.
What does ψ mean? If we consider the ψ of some kind of particle at a point, the square of the length of the associated arrow will give probability of finding the particle there. Good, but not enough. We also need to find out the physical implications of the direction of the arrow.
For that, let us suppose that two streams of electrons are coming from two points A and B, falling on a screen, as seen in Fig. 4. If only the beam from A came to a point, the matter wave amplitude would have been ψ A . If it came only from B, the amplitude would have been ψ B . When both come together, the amplitude would then be the sum of the two, i.e., ψ = ψ A + ψ B .
But we have to remember that these objects that we called ψ or ψ A or ψ B are not ordinary numbers. They are complex numbers or arrows. So, if at a point we have the arrows corresponding to ψ A and ψ B which are equal in length but opposite in direction, the effects of the two would cancel at that point and the sum, i.e., ψ, would be zero at that point. It would mean that at that point, there is no way we can find an electron, no matter how hard we look for it. If we look at Table 1, we find that such are the cases at the points marked 2 and 4. On the other hand, if the arrows corresponding to ψ A and ψ B point in the same direction somewhere, they would reinforce each other, and the probability of finding an electron would be high there. This is what happens at the points 1, 3 and 5 of Table 1. If the directions of the two arrows are neither the same nor the opposite, the sum will be somewhere in-between. In summary, if we have two streams of electrons falling in a region, there will be points where we will see a lot of electrons, and other points where we will see less, and some points where we will see none. Such a phenomenon is called interference, and is expected of any wave. It was known to happen to light waves for a long time. For matter waves, the evidence was obtained in the first half of the twentieth century.
Let us think about the whole thing. Suppose we have performed such an experiment. We found out where the probability of getting electrons was zero. We know that at those places, the arrows for the two streams were in opposite directions. In more formal language, we know that the phases were opposite. True, but do we know the individual phases? The answer is 'no'. We know that in this hypothetical space where phases are like directions, if ψ A pointed along the direction of 10 on the face of a clock, ψ B pointed towards 4. If ψ A pointed towards 6, ψ B towards 12, and so on. We don't know more than that. Similarly, at places where the probability of getting electrons was the highest, we know that the arrows corresponding to ψ A and ψ B pointed to the same direction, i.e., the phases were the same. But we don't know what that phase was.
Imagine that we were all asleep at a time when some genies appear and rotate the directions of these arrows everywhere in the universe. Would it make any difference? If we have two arrows and we rotate both of them by the same amount, the angle between them would not change. If they used to be back-to-back, they would remain so. If they were in the same direction, they would continue to be in the same direction after the genies' work. That would mean that the probabilities of finding Table 1: Arrows represent the phases of electrons coming to different points shown in Fig. 4. The source points for the electrons have been shown as a subscript on ψ. A dot on the rightmost column represents a cancellation between the two contributions, whereas a double-lined arrow represents reinforcement.  Table 1, except that an extra half rotation has been introduced for electrons coming from the point A.
the electron at different points would remain the same even after this change. We would not be able to suspect that the genies have made some change.
If a change of something does not produce any change of something else, that is called a symmetry. We have talked about the translational symmetry of space and time earlier. Here we are encountering a symmetry with the phases. The relative phase between two streams of electrons is important, and we can see its consequences. But if all the phases are changed by the same amount, that does not have any effect on the physical universe.
But wait, there is a caveat! The phases must be changed by the same amount everywhere. If instead we change the phase by different amounts at different places, the result will be appreciable. For example, let us go back to the two streams of electrons shown in Fig. 4. And suppose we change the phase of the stream coming from A by a half turn, doing nothing to the stream at B. In Table 2, we have shown what will happen now when the two streams meet. Previously, the arrows at the points 2 and 4 were back-to-back. Now, they would be pointing in the same direction. In practical terms, it means that now we would get maximum number of electrons at a place where we failed to find any electron earlier.
Obviously, the genies need not do something as dramatic as giving a half turn to the phase at one point in order to be felt. As long as they deviate by any small amount from exact equal changes of phases everywhere, there will be a difference in the interference pattern. Turning things around, we can say that as soon as we see a change of interference pattern, we would know that the phase has changed somewhere: the frivolity of the genies would be exposed.
But these genies do not want to be exposed, so they have arranged a deep conspiracy. The point is that, if an electron had emitted or absorbed a photon at the point A, that also would have changed the phase of the electron. That would have caused changes in the interference pattern if that electron had met another electron subsequently.
It means that, if we see a difference in the interference pattern, we cannot immediately conclude that the arrow of ψ has been rotated somewhere. We should be aware that the difference can just as well be caused by the electron emitting or absorbing a photon on the way.
So the bottom line is the following. Earlier, we said that if the genies changed the phases of everything by the same amount everywhere, we could not have known that. Now, we find that even if the phases are not changed by the same amount everywhere, there is no way for us to know that.
This kind of calculated confusion is called gauge symmetry. The name is bad, and makes no sense, because the word 'gauge', in English, means a measuring instrument, as in 'rain gauge'. But if a meaningless concoction get universal acceptance, we cannot but go along with it and think, "what's in a name!" Certainly a name such as phase symmetry would have been much more appropriate, but who is listening?
Anyway, let's go back to the question that appeared near the beginning of this section. It was a question about the symmetry behind the law of conservation of charge. Well, the answer should be obvious now. The symmetry behind this conservation law is the gauge symmetry that we described in this section.

The same wine in a new bottle
Let us repeat what we just said about gauge symmetry. The genies want to change the phases, or the arrows. If they could do that by the same amount everywhere at the same instant, we could not have possibly noticed their work. But that's easier said than done! Just imagine: they will have to change the phase in Calcutta, in Hyderabad, in Paris, in Abidjan and in Hanoi, all by the same amount, at the same time. They will have to do the same behind the clouds, near the sun, away in the galaxies. Oh, that's too much even for a genie! In a more serious tone, we can say that the tenets of the special theory of relativity does not even allow such an operation.
Of course there is nothing against changing the phase in a small region. The genies can do that. But they are afraid that we will get to know what they are doing. So they have devised a particle called 'photon'. Because there are photons, we cannot really tell whether the phases are being rotated.
Of course, as we learned earlier, talking of photons is talking of electromagnetic interactions. Thus we can say that the electromagnetic interactions are results of gauge symmetry. It is a façade to hide the undercurrents of phase rotations.
We are saying the same thing that we said in the last section, but from a different point of view. This is the way that two physicists, Chen-Ning Yang and Robert Mills, described things in 1954. With this new way of looking at things, they could generalize the idea to other kinds of symmetries and hinted that one should try to explain other interactions with such generalizations.
There are three other kinds of interactions, as we described earlier. Barely about a decade and a half after the Yang-Mills prescription, it was seen that weak interactions can be understood through gauge symmetries. And then, in 1974, it was realized that strong interactions can also be explained the same way. The gauge theory of strong, weak and electromagnetic interactions constitute what is known as the standard model of interactions.
Notice that we have left out gravitation. We will comment on it later. Right now, our aim should be to try to understand the standard model, i.e., to understand how gauge theories helped understand strong and weak interactions. Historically the mystery of weak interactions was cracked earlier, as we just described. But we will take an anachronistic approach and describe strong interactions first, for reasons to become obvious as we proceed.

The pillars of strength
Not everything interacts via strong interactions. Said another way, strong interactions cannot affect all kinds of particles. It can affect protons and neutrons, both of which are constituents of atomic nuclei and are therefore collectively called nucleons. It can affect many other kinds of particles, like pions or delta particles. All these particles are collectively known as hadrons.
The idea took its root in the early 1960s that these hadrons are not fundamental particles. They have another level of substructure, i.e., they are made of something more minute. These minute objects are called quarks. It was conjectured that the proton consists of two up (or u) quarks and two down (or d) quarks. For the neutron, the tally is opposite: two d quarks and one u quarks. The scheme can work if the electric charge of the u and the d quarks is 2 3 and − 1 3 that of the proton, respectively. The idea of quarks brought about great simplification in the task of understanding hadrons. For example, the same u and d quarks could explain the occurrence of many other hadrons, including pions and delta particles that we mentioned a little while ago. It was found that there were four kinds of delta particles: with charges 2,1,0 and -1 in units of the proton charge, represented usually by the symbols ∆ ++ , ∆ + , ∆ 0 and ∆ − . With the charges of u and d quarks mentioned above, it is easily seen that the combinations uuu, uud, udd and ddd would fit the bill exactly for the delta particles. To understand the structure of all hadrons that have been discovered so far, we need four more quarks. That's six quarks in all.
A question that arises is this: how can, say, the combination udd represent both neutron and ∆ 0 ? Or uud, for that matter, which seems to represent both the proton and a delta particle of the same charge. The solution of this apparent mystery lies in the fact that protons and neutrons have spin-1 2 , whereas the spin of the delta particles is 3 2 . Spin, or inherent angular momentum, can take only integral or halfintegral values when measured in a certain unit, which is what we are using here. In this preferred unit, each quark has a spin equal to 1 2 . While adding up the spins of individual quarks, we need to remember that the direction of the spin is important as well as the magnitude. So, if we have two of the quarks pointing in a certain direction but the third in the opposite direction, the sum total of the three spins would be 1 2 , which can be the case with the neutron or the proton. On the other hand, if all quarks have their spins pointed in the same direction, the total spin will be 3 2 , which is what the delta particles have. This explains the difference between the nucleons and the delta particles, but it creates a new problem. In order to understand atomic structure, Wolfgang Pauli proposed a hypothesis called the exclusion principle. It asserts that in an atom, two electrons cannot occupy the same state, i.e., cannot have the same combination of energy, angular momentum, and a few other things. The same principle was applied to nucleons in nuclei, and was successful. So there was an expectation that any particle whose spin is 1 2 would obey this exclusion principle. Now, quarks have spin equal to 1 2 . And what do we see if we look at them? Let us look at the particle ∆ − . It contains three d-quarks. According to the exclusion principle, the three should be in three different states. But we said a little while ago that the spin of the deltas in 3 2 , which can be obtained if all three quarks have spins pointed in the same direction. So, as far as spin is concerned, there is no difference between the quarks in the ∆ particles. There is no difference in their orbital motion either. How does exclusion principle work then in this case?
It cannot, obviously, unless we assume that we have not mentioned everything that is required to specify the state of a quark. If we use all quantities that are required to specify the state of an electron in an atom, then, as we saw, exclusion principle goes down the drain. For quarks in a hadron, let us assume that there is an extra quantity which needs to be specified, and let us call this quantity color.
Quarks can come in three colors: red, blue and green. As I say that, let me warn the reader that I don't mean that some quarks share the same visual characteristic as the setting sun, some the autumn sky, and some the leaves on a tree. That's not what we mean by 'color' here. This 'color' is a new property of matter, and has no connection with the sense in which we use the word in everyday language.
Whatever property it is, it saves the exclusion principle for us. As we said, a ∆ − contains three d-quarks. The three live in the same state as long as one does not think of color. And what about color? Well, one of the quarks is red, one is blue, and the other green. If we include color as we have to, this ensures that each quark is in a different state. Same thing can be said about ∆ ++ . It contains three u-quarks, but each with a different color. Now consider there are genies who are trying to confuse us about this novel property called color. They are changing the colors of everything that is colored. Are we going to know about it? If they change all colors consistently at the same time, we would not know. In a ∆ − particle, if the genies changes the red quark to blue, the blue quark to green, and the green to red, there would still be one red, one Figure 5: Gluon exchange mediates strong interactions. The three solid lines on each side are supposed to represent three quarks in a particle like the proton or the ∆'s. The exchanged line represents a gluon.
blue and one green quark in the ∆ − , and we would not face any problem with the exclusion principle or anything else.
But we discussed earlier that these genies are not that efficient. Rather, they cannot be. They cannot change the colors of all quarks everywhere in the same way. Suppose their activities have been limited to the place where there used to be a red quark in a ∆ − , and they have changed it to blue. Since there was a blue quark to start with, this change would cause a problem with the exclusion principle. And if that happens, we would know what the genies have been trying to do surreptitiously.
But the genies would not allow us that pleasure. So they have invented some new kinds of particles called gluons, which play the same role that the photons play in electromagnetic interactions. A quark emitting or absorbing a gluon can change color. Thus, a red quark can change into blue by emitting a gluon, and another quark might change from blue to red by absorbing the same gluon. Exchange of gluons maintain the color, and this is the way that strong interaction is mediated. A schematic figure is given in Fig. 5.
This, by the way, is a gauge symmetry, though with a difference. In the case of electromagnetism, we commented that the emission or absorption of a photon does not change the charge of a particle. In the present case, we said that the emission of a gluon, for example, can change the color of a quark. The emitted gluon carries this information and dumps it on another quark, which then changes color accordingly. Thus, there can be different kinds of gluons. For example, one kind can be called rb, meaning that if such a gluon is emitted, it can turn a red quark into a blue quark. When it is absorbed, it does the opposite thing of course, i.e., it can turn a blue quark into red. Similarly, there would be br gluons, rḡ gluons, and so on. There will be eight kinds in all.
In the mid-1970s, it was hypothesized that the exchange of these gluons is the mechanism by which strong interaction operates. The gauge theory describing strong interactions in this way came to be known as quantum chromodynamics: since 'khroma' means color in greek. Figure 6: Decay of the muon. The results of the decay are the electron, the muneutrino (ν µ ) and the e-antineutrino.

Saying things in pictures
What can we say about weak interactions? Can it be described by a gauge symmetry as well? We have explained electromagnetic interactions by exchange of photons. Similarly, strong interactions are mediated by exchange of gluons. Which particles play the corresponding role for weak interactions? Photons and gluons have spin, and its value is 1 in the unit in which we are specifying all spins. It was assumed that the mediators of weak interactions should also have spin 1. But, unlike photons, these particles could not be uncharged. The charge of the hypothesized particle, in fact, was equal to the charge of the proton. The particle did not have a full proper name: only the letter W (for 'weak', presumably) was used to denote it. Since its charge is positive like that of the proton, W + is a more explicit name. It was known that, to every particle there must be an antiparticle with opposite charge. Thus, corresponding to the W + , there is also a negatively charged W − .
Let us see how these particles help us understand the decay of the muon. The process has been shown in Fig. 6. The muon has thrown out a W − particle. The charge of the muon, in the unit of proton charge, was −1, same as the charge of the W − . Therefore, after emitting the W − , the muon cannot remain a muon: it must turn into some uncharged particle. This is the mu-neutrino or ν µ . And the W − , after a while, has turned into an electron and another uncharged particle, called e-antineutrino. Thus, one obtains three particles in the decay of a muon.
What happens in the case of β-radioactivity? Now we need to look at Fig. 7. Basically, β-radioactivity means the decay of a neutron into a proton, an electron and an antineutrino. The neutron contains three quarks, one u-quark and two dquarks. If one of these d-quarks gets metamorphosed into a u-quark, we will obtain a particle with two u-quarks and a d-quark, which would be the proton. And how can this metamorphosis take place? Well, through the emission of a W − particle. This W − , as in the example of the muon decay, creates an electron and an antineutrino, and that is how we obtain β-radioactivity. These pictures for weak interactions look very much like the corresponding pictures for electromagnetic or strong interactions. Instead of the photon or the gluon, we have the W , which is the only important difference. But then why are weak interactions weak?
The answer that was forwarded was this: the W particles are very heavy. This is in sharp contrast with what we had for strong or electromagnetic interactions. Gluons are all massless, so is the photon: their energy is all kinetic. Because they are massless, it is easy to emit and absorb them. For the W , since the mass is large, the same processes are very much inhibited.
Let us be a bit more explicit. In Fig. 7, we see a proton and a W particle being produced at a point where a neutron is being annihilated. Suppose the initial neutron was at rest. Its kinetic energy was therefore zero. It would still have some energy just because of its mass, which can be obtained by multiplying the mass by the square of the speed of light in the vacuum. For the neutron, this mass-energy is about 940 MeV. If energy has to be conserved, the energies of the proton and the W should also be 940 MeV then. But this is clearly not possible, since, as we know now, the mass-energy of the W is roughly 81000 MeV. Even if we forget about the energy of the proton and possible kinetic energy of the W , we already have a big mismatch.
In classical physics, this would have spelled impossibility of the event. Not so in quantum theory. Note that the W is not produced as a physical particle in the process: it only appears as an intermediate state. Quantum theory allows for a violation of the law of conservation of energy for intermediate states which are not seen in experiments. Only the probability of such occurrences are small when the mismatch of energy is large. In the case we have been talking about, the mismatch is very large, so the process must be very rare. It is very difficult to emit or absorb a W particle. That is why weak interactions are weak, processes that occur due to weak interactions are very rare.
An analogy might help. Suppose the residents of a locality decided that if anyone makes a surprise visit at someone else's home and finds no one at home, the visitor must leave a card carrying his or her name, so that the residents of that home get to know who visited them while they were away.
In another part of the world and in another civilization, the use of paper is unknown: they could write only on stone tablets. They had the same idea of leaving a 'visiting card', but in their case, they had to carry stone tablets with them whenever they wanted to pay a visit to anyone else.
It will be trivial to guess which community of people has more interaction among its members. Photons and gluons are like paper cards, and W particles are like stone tablets. No wonder that weak interactions are so feeble! But we discussed that photons are required by gauge symmetry of electric charge, gluons are required by gauge symmetry involving color. Can we not mandate the W by some similar gauge symmetry?
There is a problem though. If we set up a gauge symmetry in the manner that Yang and Mills showed us, and then introduce some particles as guardian angels of that symmetry, these new particles ought to be massless like the photon or the gluons. But we just said that the W particles are very massive. Hmm, we have a case at hand!

The naughty boys
Indeed, it is true: any gauge symmetry dictates that the gauge bosons associated with it should be massless. The question is: do we always see things that 'should' happen? There are many things which ought to vanish because of some symmetry, and yet they don't. Take, for example, the case of magnets. If we take a lump of iron and rub it with a magnet along a specified direction, the lump of iron turns into a magnet. Why does that happen? Let us start from the question why it does not happen with an ordinary piece of iron. Each atom inside the lump of iron is a miniscule magnet. But normally, the axes of such magnets are oriented randomly, as shown in the left panel of Fig. 8. If a second piece of iron is brought close to this piece of iron, each magnet will try to attract this piece in the direction of its axis.
Since the axes of the magnets are randomly oriented, so will be the forces, and their effects will cancel. As a result, we will see that the piece of iron will not behave as a magnet. Once we rub the lump with another magnet, the internal magnets all get aligned, as shown on the right panel of Fig. 8. In this case, all atomic magnets will pull a nearby piece of iron in the same direction, and we will conclude that we have a magnet at hand. Now imagine a lilliputian scientist sitting inside this magnet. What will he or she observe? All the atomic magnets around the scientist are pointing in the same direction. So the scientist might think that that direction is special compared to others.
We who know the entire story, would not agree. We think that mother Nature does not prefer any direction over other. The atomic magnets inside the piece of iron are oriented in a particular direction because we rubbed the piece along that direction. We have picked one direction over others by rubbing it in that particular direction. Had we rubbed along some other direction, magnetism would have appeared in that direction.
If we had not chosen any direction and rubbed the piece of iron along it, the atomic magnets would have remained randomly oriented, and the total magnetization, summed over all those randomly directed objects, would have been zero. Turning things around, we can say that if the magnetization were zero, the rotational symmetry in the laws of nature would have been apparent even to the lilliputian scientist sitting inside the piece of iron.
But symmetries are not always so conspicuous. Their faces are sometimes hidden under veils. This is exactly what happens when the piece of iron is magnetized. We could rub the piece along any direction. The piece would have developed magnetization along that direction, no matter what the direction might have been. There is symmetry in this respect, and any direction is equivalent. However, the fact remains that we rub along some chosen direction, and magnetization develops along that direction. As a result, symmetry has become hidden. It has made things difficult for the lilliputian scientist: he or she cannot see that Nature has no preference as far as directions are concerned.
A similar thing is happening when we, the not-so-lilliputians, are trying to think about the W particles which mediate weak interactions. There is a gauge symmetry which says that the mass of the W should be zero, just as rotational symmetry says that magnetization should be zero. But we are sitting in an universe where that symmetry is not plainly apparent. With a hidden symmetry, the W particle can have mass, just like a piece of iron can have magnetization.
The physical ideas behind these things developed through the works of many scientists in the 1950s and the 1960s. In 1967, Steven Weinberg created a gauge theory based on such ideas. A few months later, Abdus Salam also independently hit upon the same idea.
And this initiated a kind of a revolution in physics. We said earlier that the gauge theory of strong interactions was discovered a few years later. Thus, with the announcement of the gauge theory of weak interactions, it was realized that gauge theories are useful in anything other than electromagnetic interactions. Of course, the confidence in this theory did not come overnight. In fact, most people doubted the mathematical viability of the theory of hidden symmetries until, in 1971, Gerhard 't Hooft removed such doubts in a brilliant set of papers.
There was a strikingly new feature in the theory of Weinberg and Salam. They said that the mediators of weak interactions are not only the charged particles that we have called W . There is another particle that acts as the mediator, which came to be known as the Z. They are somewhat heavier than the W 's. Like photons, the Z particles do not carry any electric charge. But, unlike photons, it can be emitted and absorbed even by uncharged particles like neutrinos. Thus, neutrino interactions provide the best testing ground for this Z particle. And indeed that was what happened: some interactions involving neutrinos were observed in 1973 which could not have been mediated by a charged mediator like the W : they could only be the result of Z mediation. A decade later, a huge group of scientists working at CERN (Conseil Européen pour la Recherche Nucléaire, or the European Council for Nuclear Research) in Geneva under the leadership of Carlo Rubbia, detected the W and Z particles directly. In other words, they observed processes in which the W and the Z participated not as mediators, but as particles present in the initial or final state of a physical process.

Untied knots, unexplored horizons
We have described the basics of the standard model of particle interactions. As we saw, the model is based on gauge symmetries. In the case of weak interactions, the symmetry is hidden. For electromagnetic and strong interactions, the symmetry is apparent.
The model has been remarkably successful in describing particle phenomena. Very roughly speaking, we have not seen any particle phenomena which violates the basic tenets of this model. There are numerous situations where the predictions of the model can be calculated with high precision, and there the results of the experiments agree with the predictions of the model.
That does not mean that all problems have been solved. There are quite a few open ends, and vigorous research is going on to settle those issues. Here we list some of them.
First comes the issue of neutrino masses. Originally when the standard model was proposed, the neutrinos were assumed to be massless, because experiments at that time could not establish any mass of the neutrinos. Now we know that neutrinos have mass, although the magnitudes are much smaller compared to the mass of other elementary particles like the electron. It is easy to modify the standard model so that the neutrinos come out to be massive, but it is generally believed that the modification should hold the clue of the unusual lightness of neutrinos. Some interesting ideas have been proposed in this regard, but it is not clear how to test these ideas in foreseeable experiments.
If we take the theory of Weinberg and Salam in its original form, even then we have to admit that some key features of this model have not been established experimentally. The theory predicts a spin-0 particle should be left over in the mechanism that provides masses for the W and the Z particles. This particle, dubbed the "Higgs boson", has not been observed yet.
But that's not all that has defied observation. The theory of strong interactions bases itself entirely on the idea of the existence of quarks. This idea has explained so many experimental observations that it is hard to disbelieve it. And yet, it has to be remembered that it has not passed the acid test for any theory or any idea: no one has observed a quark in an experiment.
The reason for this might be that the quarks cannot be freed: they are perennially in bound states which are hadrons. Such things are not unheard of. One pole of a magnet cannot be freed from the other, the poles always come in pairs. Perhaps something similar happens for quarks. Well, perhaps, but that is a speculation. No one has shown that quantum chromodynamics leads us to this conclusion.
To a large extent, the problem lies in the fact that it is very difficult to calculate the effects of strong interactions when the quarks are far apart. Here the word 'far' must of course be taken in context: even the average distance of quarks in a proton would be considered 'far'. If one wants to free a quark by pulling it apart from a hadron, one has to pull it to even larger distances, where calculations are even more difficult and less reliable. The reason for such state of affairs in the strength of the interaction. For weak and electromagnetic interactions, more complicated diagrams for a process always give a much smaller contribution compared to the simplest ones. For example, consider the interaction between two electrons, mediated by photons. We showed some complicated diagrams in Fig. 3, and a very simple diagram, with only one photon exchange, in Fig. 2. But, unless one is worried about very minute corrections, the simplest diagram is all we need. And, even if one is worried about some minute corrections, one has to calculate only a few complicated diagrams, depending on the degree of minuteness that one is interested in. For strong interactions, such rules of thumb do not exist. Numerical calculations, not dependent on Feynman diagrams, can be performed on computers, but they have to make drastic compromises in the nature of the problem in order to reduce the problem in a calculable form.
We discussed the muon earlier. It is about 200 times heavier than the electron, but in all other respects it resembles the electron. There is another particle called the tau which is even heavier than the muon, but has the same properties that the muon and the electron have. The same structure can be seen among the neutrinos, and among quarks. The particle physicists say that these are particles from three generations. But why are there three generations? We do not know.
There are aesthetic problems as well. The standard model, as it is, contains 19 parameters. These parameters cannot be calculated: they have to be determined through experiments. With them as inputs, we can find the results of other questions. But the number 19 does not make one feel very comfortable. The number grows once one has to accommodate neutrino masses. If you have to give so many inputs to a theory, it leaves you with a creepy feeling that perhaps you are missing some deeper understanding which could have cut down on the number of inputs.
Indeed, one of the persistent dreams of physicists is the idea of unification. This is the underlying belief that we will not need different theories for different interactions: one theory will be able to describe all of them. There have been several suggestions regarding this dream, but no experimental confirmation for any of them.
Of course we do not know for sure whether Nature works on a unified scheme. But we know for sure that something is obviously missing in the standard model. At the very beginning, we said that we will not consider gravitational interactions because it is negligible at the scale of elementary particles. While it is true, it is also true that we do not know how to describe gravitation in the form of a quantum theory. Since the 1980s, string theories have raised the hopes of describing gravitation. It is not clear how and whether the other interactions are contained in such theories.
So there are a lot of things to be done, a lot of ground to cover. We have to walk a long way still. What's more, we do not even know whether there is an end of the road. Reflecting on the history of science, we see that whenever we have cracked a mystery at a certain level, new mysteries at a new level have been exposed in front of us. Perhaps the journey is endless, and that is the beauty of the challenge. * I thank Andrzej K. Wróblewski for pointing out a mistake regarding the time of publication of Noether's work. * For the most part, this article is a free translation of a chapter from my Bengali book "ki diye somosto-kichu gorha" (What is everything made of). I have made conscious deviations only in places where the said chapter referred to earlier chapters in the book, and in the final section where some updating was felt necessary.