About the Author /Profile/
Author: Juan Maldacena, Argentine theoretical physicist and professor at the Institute for Advanced Study in Princeton. Translators: Guo Yuanhong and Zhang Siyuan, 2021 and 2019 doctoral students of the Institute of Theoretical Physics of the Chinese Academy of Sciences, respectively, under the supervision of Yang Gang, research directions are quantum field theory and string theory.
Editor's Note
The translators, with Maldasina's consent, translated one of his popular science articles, "The symmetry and simplicity of the laws of physics and the Higgs boson" (arXiv:1410.6753). Due to the length of the article, we will push it three times, this time the first part.
Abstract In the 1950s and 1970s, the idea that "the interaction forces of nature derive from the principle of symmetry" gradually developed, and physicists predicted the existence of the Higgs boson, which was subsequently discovered in 2012. In this popular science, we will explain these symmetries in nature through an economic analogy. In addition, there is another very important element - the Higgs mechanism. This mechanism would make conclusions about symmetry more rigorous, while also revealing many properties of elementary particles.
1. Fairy tales
Our current understanding of particle physics is like the story of Beauty and the Beast. "Beauty" is like a force in nature: such as electromagnetic force, weak force, strong force, gravitational force. They are all based on the principle of symmetry, which we call "gauge symmetry." But we also need the Beast, the Higgs Field. Many mysteries in particle physics are contained in the Higgs field (some would argue that it is "ugly"). However, the description of nature must include both of these things at the same time, because we are the "love crystallization" of "beauty" and "beast".
Figure 1: Particle physics is like Beauty and the Beast. The four basic forces are beauty, and the Higgs field is the beast.
First, we'll introduce the two main actors through some analogies: Beauty (the principle of canonical symmetry) and Beast (the Higgs mechanism). In many popular science articles on particle physics in the past, readers are often led a long way: starting from everyday experience of the world, passing through molecules, atoms, nuclei, etc., following the order in which they have been discovered in history, and gradually delving into the microscopic units that make up matter. And this popular science will try something different, and we will directly descend into a modern point of view - it is a wonderland with very simple rules. Specifically, this popular science article will revolve around the role of normative symmetry and the surprising simplicity of particle physics laws. We know that the models that describe elementary particles are the "Standard Model," and the reason these laws are surprisingly simple is because they explain most universal phenomena, they can also describe the universe from the first millisecond of the universe to the present day, and they can describe almost all known matter except dark matter. These descriptions of the natural world are the result of painstaking experimental work by physicists, and will not be reviewed in this popular science, and interested readers can read other popular science works in particle physics.
2. Symmetry
2.1 Ordinary symmetry
Figure 2: Graphics with various symmetries. (a) The square is symmetrical when rotated 90 degrees. (b) The rectangle is symmetrical only when rotated 180 degrees. (c) Squares are also symmetrical under the reflection transformation along the dotted line. (d) This figure is asymmetrical under any reflection transformation. However, it is symmetrical when rotated 90 degrees. (e) This circle can be symmetrical about rotating at any angle. (f) The symmetry of the ring is the same as that of the circle. (g) This figure has no symmetry.
Let's start with the concept of symmetry. Symmetry in physics is the same as what we call symmetry in everyday life, it is a transformation that keeps an object intact. For example, a square can be rotated 90 degrees, the same as it looked before rotation. This is not the case if we have a rectangle, as shown in Figure 2, where the rectangle needs to be rotated 180 degrees. Of course, there can also be some graphics without symmetry, such as (g) in Figure 2. The circle is very symmetrical, and it remains unchanged after rotating any angle around the center. In addition, two different shapes can have the same symmetry. For example, circles and rings are different shapes, but they have exactly the same symmetry when rotated around the center. So, there may be a situation where we only know what the symmetry of the object is, but we don't know what the object is. But sometimes, simply knowing the symmetry of an object is enough to predict the properties of an object. For example, if a rigid object that is symmetrical under rotation has the symmetry of a circle, then whether the rigid object is a solid cylinder or a hollow cylinder (or some other shape), we can predict that it can roll smoothly on the table. Of course, in other respects, hollow cylinders and solid cylinders may behave differently. For example, one can float in water and the other may not.
And the symmetry in physics that we are going to discuss is a generalization of these symmetries that we are more familiar with, and it is they that determine the interaction forces in nature.
2.2 Electromagnetics Review
Figure 3: (a) An electric field exerts a force on a charged particle in the direction of the electric field. (b) A magnetic field exerts a force on a charged particle in motion that is perpendicular to the magnetic field and the direction of motion. (c) Charged particles eventually make a circular motion around the magnetic field.
Before we begin the discussion, let's review some of the knowledge about electromagnetism. We know that there are electric and magnetic fields, which can be thought of as small arrows at every point in space-time, and that these fields act on charged particles, and we feel their presence. The electric field pushes the charged particles in the direction of the electric field, while the magnetic field acts only on the moving charge. In the presence of a magnetic field, the moving charge feels a force perpendicular to its speed direction (Lorentz force). So, if you're a charged particle, when you move forward in the direction of a vertical magnetic field, you'll feel a force pushing you to the side. At this point, if there is no other force, then you will eventually move along a circular orbit. As a result, charged particles make circular motions in a magnetic field. The whole process is shown in Figure 3.
These electric and magnetic fields have their own dynamics, such as generating electromagnetic waves. Electromagnetic waves are phenomena in which electric and magnetic fields oscillate each other and can propagate in a vacuum, such as radio waves, light, X-rays, gamma rays, etc. I hope you are familiar with these. Don't panic! You don't need to know the detailed equations to understand the following. All you need to know is: (1) there are electric and magnetic fields, which can exist in a vacuum; (2) these fields act on charged particles and affect their motion; (3) electric fields push charged particles in the direction of the electric field; and (4) charged particles make circular motions around the magnetic field.
2.3 Canonical symmetry
Electromagnetism can be seen as a "gauge theory," another narrative of electromagnetism that is useful when we want to generalize this theory to other interaction forces and to describe the quantum mechanical version of electromagnetism. To graphically explain normative symmetry, an economic analogy that has been idealized and simplified can be introduced. Remember, our goal is not to explain the real economic world, but only to explain the real physical world. The good news is that the model is much simpler than the real economic world, which is why physics is simpler than economics!
Figure 4: Each circle is a country with its own currency, they are connected by blue bridges, and there is a bank on each bridge, and each bank has a separate exchange rate, some examples are given in the figure. When you cross the bridge, you should exchange all your money for a new currency.
Next, let's introduce this economic model. Suppose there are countries, each with its own currency. The first hypothesis is that these countries are arranged on a grid of rules for a flat world, as shown in Figure 4, where each country is connected to its neighbors by a bridge. And there is a bank on each bridge, and you need to exchange foreign exchange at the bank, that is, to exchange the currency you originally carried into the currency of the country you are about to enter. And these banks are independent of each other, and there is no central authority to coordinate exchange rates between countries, so each bank can set the exchange rate in any way independently. In addition, banks do not charge commissions. As an example, suppose the currency of the departure country is the US dollar, the currency of the arriving country is the euro, and the bank on the bridge between the two countries sets the exchange rate at 1.5 US dollars = 1 euro, as shown in Figure 4. If you have $15, the bank will exchange it for 10 euros when you cross the border. If you decide to come back, your €10 will be exchanged for $15. So if you go to a neighboring country and come back right away, you can only exchange it for as much money as you did when you set off. The second assumption is that you can only travel from one country to its neighbors, not directly from one country to another. To sum up, you can only walk from one country to another, cross various bridges, and constantly exchange foreign exchange. The final assumption is that the only thing you can bring with you when you move between countries is money, and you can't carry gold, silver, or anything else. Later, this assumption will be relaxed when considering the weak force and the Higgs mechanism, but for now we will only consider the situation where we can only carry money. Let's revisit these assumptions again: (1) arrange countries on a grid; (2) each country has its own currency; (3) these countries are connected by bridges; (4) there is a bank on each bridge to exchange currency; (5) banks can choose any exchange rate they like; (6) banks have no commissions; and (7) they can only carry money when moving between countries. It's fairly simple, all you can do is exchange foreign exchange between countries every time you cross the border. So, where is the symmetry? Let's explain what canonical symmetry is. Imagine a country that has accumulated too many zeros in the face value of its currency, and then they hope to get rid of those zeros. This is quite common in real-world countries with high inflation. So, one day, the local government decided to change the currency unit, for example, now everyone needs to use "Oster" instead of "peso" (two currency units used in Argentina). The government will declare that 1000 pesos is now worth 1 oster, or that 1000 pesos equals 1 oster, and then all prices and exchange rates will change accordingly. For example, as shown in Figure 5, if a banana is worth 5,000 pesos, it is now worth 5 Oster; if your salary is 1 million pesos, it will now be 1,000 Ooster; assuming its neighbor is the United States and the exchange rate is 3,000 pesos = 1 dollar, then now it will be 3 Ooster = 1 dollar. We call this change in the unit of money "symmetry", because after this change nothing really happens, no one becomes richer or poorer, and this change does not bring new opportunities for profit, it is purely for convenience. You can see this "gauge symmetry" on the Argentine banknotes in Figure 6, which we call "gauge" symmetry because it is the symmetry of the units we use to measure, or "normalize" various quantities.
Figure 5: Each country can change its currency unit. Countries that use pesos change their currency to Oster, making 1000 pesos = 1 oster. Then all prices and exchange rates change accordingly: for example, the price of domestic bananas and the exchange rate of the currency against its neighbors will change accordingly.
Figure 6: The symmetry of norms at work: the change from Argentine peso to Auster in the real world.
Moreover, this symmetry is "local", which means that each country can decide to make this change within its own country, without being influenced by its neighbors. Some countries may be more willing to do so than others. In the real world, Argentina has eliminated 13 zeros through various actions of this "gauge symmetry" since the 1960s, so today's 1 peso is equal to the peso of the 1960s. Then we introduce speculators: speculators are people who run between different countries, trying to profit from foreign exchange, so he travels along the route with the highest economic returns. Recall that, based on previous assumptions, he had to travel around and not sit at his desk and make transactions on his computer. If you're a speculator, do you think you can make money in this world? Think about it, how would you do it? At first glance, you don't seem to be making any money. If you come straight back from a country to its neighbors, the money you end up exchanging is the same because the exchange rate remains the same. However, if you pass through more countries and come back, you are likely to make money! For example, let's consider three countries, such as the United States, Europe, and Argentina, and their corresponding currencies, the dollar, the euro, and the peso. Now imagine that these three countries are connected by bridges and the exchange rate is as follows: $1.5 = 1 euro, 1 dollar = 10 pesos, and 1 euro = 10 pesos. In this case, can you make money? Think about it before you keep reading, it's worth the effort! In fact, you can make money in the following way: from Argentina, the principal is 10 pesos, then go to Europe, get 1 euro, go to the United States, get 1.5 US dollars, and finally return to Argentina and get 15 pesos. If it is 10 pesos when departing from Argentina and 15 pesos when returning to Argentina, then the increase in value of the circuit is 1.5 times, that is, the yield is 50%. If you leave with X pesos, you will end up with 1.5X pesos, a factor independent of the currency unit. For example, if the Argentine government changed the monetary unit from peso to Oster, the proceeds of this route would still be 1.5 times the same. You might think that it's because banks stupidly set "wrong" exchange rates that speculators are able to exploit these loopholes. However, this is exactly the rule that we have already articulated. Banks set the exchange rate they want, and then some choices lead to speculative opportunities, while others do not. The mind of the speculators is simple, they only care about making money, they will choose the route that makes the most money. In the above case, the speculators will first travel from Argentina to Europe, then to the United States, and finally back to Argentina, and so on, as in the green route in Figure 7.
Figure 7: Three countries use their respective currencies, and the corresponding exchange rate is given on the blue bridge. Move in the direction indicated by the green line to make a profit. And the yield coefficient of the cycle is 1.5, and the yield is 50%.
Now, in physics, these countries resemble dots or small regions in space. The set of exchange rates is a distribution of the "magnetic potential" throughout space. A situation like Figure 7, which makes money, is called a magnetic field. We can relate the amount of earnings to the magnetic field, and speculators can think of them as electrons or other charged particles, and in the presence of magnetic fields, they make money by circling. The "total return along the loop" can then be equated with the "magnetic flux in the area enclosed by the loop". Now imagine that you are a speculator with debt instead of money. In this case, you will pass through these countries in reverse order! Then your debt will be reduced in the same proportion. In the example given in Figure 7, by moving in the opposite direction, your debt will be reduced by one-third per lap of the cycle. It can be considered that the case of debt corresponds to the positron in physics, which is a particle similar to the electron but has the opposite charge, and in the magnetic field the direction of the positron's circular motion is opposite to the electron. In physics, this story about countries and exchange rates takes place at very, very small spatial distances, much smaller than what can be measured today. When we look at any physical system, even if it's a vacuum, we're looking at those countries from a great distance, so they look like a continuum, as shown in Figure 8. When an electron moves in a vacuum, it moves continuously from one point in space-time to the next. On a very microscopic scale, it will constantly move between different countries, changing the currency it carries and becoming "richer" in the process. In physics, we don't know if the space-time points are as discrete as the previously described states. However, calculations in gauge theory usually assume that these countries are discrete, and then treat them as a continuum when all countries are very close to each other, which is the "limit of continuity".
Figure 8: Grids of countries of different sizes. From left to right, the country is shrinking and the number of grids is increasing. When the country is small enough, the entire mesh can be treated as a continuum.
Electromagnetism is based on a similar normative symmetry. In fact, at every point in space-time, there is a symmetry of circular rotation. It is simply conceivable that each point in space-time has an extra circle, which is an extra dimension, as shown in Figure 9(a). The "state" at each point in space-time can choose a way to independently use this circle to define an angle. More precisely, if a "country" chooses a starting point on its own circle, the position of the other points on the circle can be determined by the angle between the center of the circle and the connection to the starting point, just like choosing a monetary unit in an economic model. In physics, we don't know if the circle is real or if it really has an extra dimension. What we do know is that the symmetry required physically is similar to the symmetry resulting from the extra dimension. In physics, we like to make as few assumptions as possible, and the extra dimension is not a necessary assumption, only symmetry. Moreover, the only relevant quantity is the magnetic potential and the corresponding magnetic field, which tells us how a particle's position on an extra circle changes as it moves from a point in space-time to its adjacent point.
Figure 9: (a) The symmetry of electromagnetic interactions, equivalent to having a circular structure at each point in space-time. Where the black lines intersect, that is, each space-time point, the corresponding circle can be seen as an additional dimension. (b) The symmetry of electromagnetic interactions, equivalent to the structure of a sphere at every point in space-time. We don't know if a circle or sphere really exists as an extra dimension, but canonical symmetry is equivalent to the existence of such an extra dimension. We only need to consider their symmetry, as well as the "exchange rate" associated with the extra dimensions
In electromagnetism, electric and magnetic fields follow Maxwell's system of equations. Corresponding to the economic analogy, this is equivalent to the existence of a requirement for exchange rates, which can be intuitively understood through economic models. Imagine if there were some countries and corresponding exchange rate distributions, and some speculators who carried money around to exchange foreign exchange. Assuming one of these particular bridges is considered, there will be many speculators crossing the bridge from both directions, and these speculators will need to exchange foreign currency at the banks on the bridge. However, if there are more speculators in one direction than speculators in the other, then banks may exhaust one of these currencies. For example, consider that banks are located on a bridge connecting Argentina to the United States, and if there are more speculators who want to exchange dollars than speculators who want to exchange pesos, then banks will run out of dollars. If this happens in the real world, banks will adjust their exchange rates to reduce the number of speculators who want to exchange dollars. In fact, if we assume that the number of speculators along a particular circuit is proportional to the gains they receive in that circuit, then we find that the condition that banks do not exhaust any one currency, or that the net flow of capital on each bridge is zero, is equivalent to Maxwell's equations.
2.4 Weak interaction forces
Let's start with the discussion of weak interaction forces (hereinafter referred to as weak forces). Weak forces play an important role in radioactive decay. For example, a typical decay process is that a free neutron outside the nucleus of an atom takes about 15 minutes to decay into a proton, an electron, and a (anti-electron) neutrino. Compared to other reaction processes that occur on microscopic time scales, this is really a very slow decay. Although the weak interaction is not very relevant to our daily lives, it has played an important role in the early evolutionary history of the universe, although this force is not very strong. In addition, weak forces also play a large role in the transmutation of stellar chemical elements. In fact, all the chemical elements around us (except hydrogen and helium) are "cooked" in stars! Weak forces play a crucial role in this process. Closer to our lives, it can also be said that the weak force is moving the earth's crust plates, mountains and oceans! More precisely, part of the source of geothermal heat is the decay of elements caused by weak forces, and it is geothermal heat that moves the landing ground and creates mountains! Weak forces can also be understood in terms of gauge theory. Here there is a rotational symmetry of a ball at each point in space-time, which we later refer to as a weak ball, see Figure 9(b). We don't really know if the weak sphere really exists, we only know that in the weak interaction, when moving from one point in space to another, we need to specify three quantities, which can be used as three "exchange rates" (later we call them "weak exchange rates"). And this time we are not carrying money, but an object that can be oriented with a weak ball. When we carry an object that is oriented in a weak sphere, from a country to a neighboring country, we need to redirect the thing we are carrying according to the "weak exchange rate". It is worth noting that it takes three quantities to reorient with a weak ball, and it is necessary to determine an axis of rotation (two quantities) and an angle of rotation around the axis of rotation (a third quantity), similar to Euler angle. At this point, these three quantities will correspond to three magnetic fields, and there will no longer be only one magnetic field like the electromagnetic force. Then there will be a series of equations that limit the behavior of these magnetic fields and the corresponding electric fields, which is similar to the electromagnetic force. In 1954, Yang and Robert L. Mills first proposed these equations, but wolfgang E. Pauli fiercely opposed them. Pauli's objection is that the Young-Mills theory predicts the existence of new massless particles, but they are not discovered in nature. Such an ugly fact thus stifles a theory that is so beautiful.
2.5 Why are massless particles?
To understand the reasons for Pauli's objections, we must first review some of the nature of the waves. Waves in a system can have different wavelengths, and wavelength refers to the distance between two adjacent peaks. In a physical system, people are often concerned about how much energy it takes to excite waves of different wavelengths. For a given intensity of the wave, the energy required to excite the wave will depend on its wavelength, as the wavelength becomes longer and longer, its energy consumption will decrease more and more, and when the wavelength is very, very long, the energy required to excite is convenient for the wave to correspond to the mass of the particle directly related. To explain this, the famous mass-energy equation and de Broglie relationship are used, but unfortunately I have not been able to find a very concise way to explain this, and you can believe this conclusion at this point and then read on. In our economic analogy, energy has not yet been discussed, and it can be assumed that energy consumption will increase as the profits of speculators increase. This assumption is intuitive, because the more speculators can make, the harder it will be for banks! Therefore, the thinner the profit generated by the distribution of exchange rates given by a wave, the lower the energy consumed to stimulate this wave. In Figure 10, we give the exchange rate distribution corresponding to the long-wave and short-wave scenarios, respectively. It is important to note that the money earned by speculators per circumnavigation (the green line in Figure 10) comes from the difference in adjacent exchange rates, regardless of the level of the exchange rate itself. In this case, the longer the wavelength, the thinner the profit. As the wavelength increases, the profit will decrease to unprofitable (that is, the energy required for excitation decreases to 0), which means that the corresponding particles of this wave are massless, such as photons are massless for electromagnetic fields. This process of argument holds true not only for electromagnetic interactions, but also for weak interactions. At least for the version of weakness we are currently discussing, this argument is correct...
Figure 10: Examples of long-wave and short-wave long exchange rates. Wave refers to the phenomenon that when moving from left to right, the number in the middle rises or falls. Each segment is a bank located between two countries, and the country is at the intersection of the segment, and the number indicates the exchange rate at which the bridge is crossed in the direction of the arrow. In addition, the total amplitude of the waves is the same, while the exchange rate is between 1 and 1.3. For the case of longer wavelengths, the gain obtained along the smallest loop indicated by the green line is smaller.