Let’s begin by thinking about trying to measure the strength of the electromagnetic interaction experimentally. Here we have an electron sitting somewhere in space. We will define the origin to be the location of the electron.

Note here that this is a regular x-y space-space diagram. We can measure the EM interaction strength by bringing in another electron, separated by some distance r, and seeing how much force we need to apply to keep them together. We can do this as a function of distance.

When we actually perform the measurement1 we find that the number characterizing the interaction strength is about ~1/100, more precisely we find 1/137, and we see that it is essentially independent of distance.

The force gets stronger with distance, but if we correct for the expected inverse square law behavior, we see the same overall interaction strength regardless of how far away we measure it; the interaction strength is constant.

So far this has all been for relatively large distances away from the electron. However, as we go to smaller distances something changes. We know that small distances mean big energies, they are inversely related, so we can talk about distances by specifying the energy. When we get to a distance that corresponds to an energy that is about one over the mass of the electron, something new happens: vacuum fluctuations.

We are trying to perform the measurement at smaller and smaller distances. B/c of QM, to probe the small distance we have to put in more and more energy to satisfy the uncertainty principle. At the distance scale that is 1/m_e (or 1/(2m_e) if we are being more precise) we have enough energy to create an electron-positron pair:

Our act of trying to look at the single electron at the origin ends up creating (or more correctly has a significant probability of creating) an electron-antielectron pair around it. Our first diagram of an electron at the origin sitting in “empty space” is not quite right. It’s fine of course at large distance scales, everyday distances. But at small scales, as we saw before, the vacuum is not empty. At small scales we should really be thinking of the electron as surrounded by electron-antielectron pairs that are constantly coming into and out of existence.

So really the picture we should be drawing at distance scales of ~1/m_e is not the electron sitting here by itself, but we should think of the electron as surrounded by a cloud of electron-antielectron pairs:

Now of course the particles in this cloud are themselves electrically charged, so they will interfere with the attempt to measure the charge of the electron at the origin. At large distances, what we are actually measuring is some combined effect of the “bare” electron sitting at the origin plus this cloud of surrounding particles. It’s really the complicated effective EM interaction of this combined picture that’s giving us the value of 1/137.

As we start going to smaller distances, we begin to penetrate the surrounding electron-anti-electron cloud and we start seeing more of the effect of the bare electron at the origin; the charge we are trying to measure is less screened as we get closer and closer. So when we measure the interaction strength at small distances we find that the observed interaction strength begins changing with distance:

This effect is relatively small, but it has been observed and measured fairly precisely. We see that the interaction strength gets somewhat larger; ie the interaction becomes stronger, at small distances. You can see it should get stronger from the cartoon with the particle cloud. Because the anti-electrons have the opposite charge of the electron and because opposite charges attract, the cloud of electron-antielectron pairs tends to be polarized, with the anti-electrons closer to the bare electron than their electron partners. This closer-in oppositely-charged shell tends to reduce the apparent charge of the electron at the origin. When you penetrate this inner cloud, the interaction with the bare charge increases.

These vacuum fluctuations seem crazy, is empty space really not empty? We saw before that they are a direct consequence of combining QM and relativity and we see here that they have actually observable consequences. They lead to a change in the interaction strength with distance, but they also have other effects that have been even more precisely measured.

The electron has an electric charge and a spin, this means it behaves like a little magnet. The theory can predict how strong of a magnet the electron should appear to be. It turns out this quantity can both be calculated and measured extremely precisely. In order for the theory to be so precise we need to take into account the full effect of the cloud; just as before, the cloud is composed of other little magnets that will affect the precise measurement of the bare magnet-at-the-origin. When we make this measurement we find that it agrees extremely well with the prediction - to better than one part in a trillion. This is one of the record holders in science for level of agreement, crucially relying on the idea that these vacuum fluctuations are real. We will see even more evidence of this throughout the series.

In the next post we will look more closely at the strong interaction and see how it is that it can seem so different from the electromagnetic interaction.

The other main inputs that are needed to specify the theory are the strengths of the interactions of the force carriers. We saw a few posts ago that the fundamental forces are governed by simple three-particle stick figure interactions where two fermions and a boson come together. QFT tells us that there is a dimensionless number associated with each of these diagrams that characterizes the overall strength of the interaction.

For the electromagnetic force this number turns out to be about 1/100 (the more precise value is 1/137, which is a favorite number of physicists). The number associated with the weak interaction happens to be about 1/50; for the strong interaction it’s a bit higher, about 1/10. These numbers are directly related to the probability for the process to occur and set the overall strengths for the interactions.

When we first introduced the four forces we saw that, despite all being critically important for our everyday lives, they appeared to be totally different from one another. Some were described by simple mathematical relationships, others weren’t. Some were “obvious” in the sense that we had known about them since antiquity, others we were only able to detect relatively recently. But here, for the first time, we start seeing that the forces are actually described by the theory in the same basic way. They are all described with the same basic language: force carriers with associated stick figure diagrams. And even the numbers associated with the diagrams are not so different: ~1/100, ~1/50 and ~1/10.

This leaves us with what looks like a contradiction. How can it be that the underlying descriptions of the forces in the Standard Model are so similar, and yet they seem so different to us? In the next post we will start seeing through this contradiction. We will discover that the differences that we observe are really a kind of long-distance illusion that ultimately stems from detailed differences in the inputs to the theory.

We have matter particles that are fermions. These are the particles that make up atoms. We have the quarks which make up the nucleus and we group the neutrino with the electron and refer to them collectively as “leptons”. We saw that QM + relativity requires that fermions be half integer spin particles and that is exactly what is observed: the leptons and quarks are all spin 1/2 particles. This is all the stuff that exists in the world.

This stuff interacts with itself through forces and last time we saw that forces are governed by other, force carrier, particles. The force carriers are bosons. QM + relativity requires that bosons be integer spin particles and again this is exactly what is observed: the force carriers for the EM, weak, and strong interactions are all spin 1 particles. I’ve left the graviton off of the list -- we will see why in a future post -- but it too is believed to be an integer spin particle.

The Standard Model, the quantum field theory that explains our world, describes how the matter particles interact with the force carriers. There are beautiful, but complicated, mathematics that governs the nature of these interactions. We won’t go into the detailed form of interactions as they are not relevant for our story. I’ll just note that the form of the interactions are dictated by principles of symmetry and nearly all of the details are a direct consequence of constraints from combining relativity and QM. Once you impose the QM+relativity constraints, the only real freedom we have is in deciding which particles exist, how many forces there are and what their overall strengths are. Once you have selected options from the tiny menu of allowed possibilities, everything else follows from QM+relativity.

It looks like we are done! We have everything we need to describe the periodic table and the interaction of all everyday objects. However, for reasons that we don’t really understand, we see extra copies of the matter particles:

In addition to our electron and neutrino pair, we have two “extra” lepton pairs. We have met one of these before: the muon. We think the muon, which also has an associated muon-like neutrino, is just like an electron but heavier. Like the electron neutrino, the muon neutrino is electrically neutral and has almost no mass. So the muon-neutrino pair is like a copy of the electron-neutrino pair. There is yet another copy, associated with an even heavier version of the electron called a tauon, which also comes with its own neutral light neutrino. We say that the leptons come in three “generations”. We don’t know why there are three generations, but we are pretty confident there aren’t any more.

It’s the same story for the quarks. There are heavier versions of the “up” and “down” quarks that make up protons and neutrons. These quarks have weird names1: the “charm” quark, the “strange” quark, and the “top” and “bottom” quarks. These are just names, labels that we have given to them to keep them straight. These extra quarks also behave like heavier copies of the “first” generation up and down quarks. Again, we are pretty confident there aren’t more than three quark generations.

The following figure shows the masses of all the particles relative to a proton mass.

The extra particles are heavy, especially the top quark, which is more than 100 times the mass of a proton. We don’t understand these numbers. They are inputs to our theory, unconstrained by symmetries or QM+relativity. There are clearly some patterns here: each generation is heavier than the previous; within a generation the charged leptons are lighter than the quarks; the neutrinos are freakishly lighter than everybody. It’s not clear if these patterns mean anything. Perhaps they are telling us something profound about a theory deeper than the SM. Maybe it’s like when we found the periodic table before atoms were understood. Maybe it’s meaningless. We don’t know.

Despite their large masses, we believe all these particles are fundamental, as fundamental as the electron. Although only the first generation of particles makes up the “normal” matter of the periodic table, we will see later that the heavier particles will play a key role in our story.

Unlike anti-matter and the neutrino, these extra generations of matter particles were forced on us by observation, not theory. We went out and did experiments to test our theory and found them. They were not predicted theoretically and were largely unexpected.

So this is the more complete Standard Model. We have two types of matter particles, leptons and quarks, each with three generations. We can now describe all the matter and forces we know about, plus all these additional particles are forced on us by experiment.

This post has focused mainly on the matter particles. Next time we will look more closely at the force carriers and start to understand the differences among the forces from details of the Standard Model.

In the theory all the forces are described in terms of “local” particle interactions. The particles bump into each other at individual points in spacetime. The “force” that is pushing the electrons apart, which in the diagram are not actually touching each other, is the result of a “messenger” particle -- in this case the photon -- which does touch each of the electrons. The naive action-at-a-distance force between two electrons is the result of two action-at-only-a-point interactions of electrons with photons.

The above diagram shows a contribution to the electromagnetic force. We think of this force classically as being between two charges separated by some distance, but here we see that it actually proceeds via the exchange of a messenger particle or “force carrier” particle, in this case the photon. The electromagnetic force between two electrons is really the result of a local exchange of photons.

It turns out that this is true of all the forces.

For the gravitational interaction, when two objects interact gravitationally what’s really happening is that they are exchanging a particle called a “graviton”. We already saw that the electromagnetic interaction proceeds via the photon. It’s the same for the strong and the weak interaction. The strong interaction comes about by the exchange of particles called “gluons”. The weak interaction is a little more complicated, there are actually two varieties of force carriers; you can exchange a neutral messenger particle called a “Z-boson” or an electrically charged force carrier called the “W-boson”. In the case of W-exchange, to conserve electric charge, the electron has to turn into a neutral electron-like particle, called a neutrino ν.

The neutrino is another example of a particle that was forced on us by the theory before it was discovered experimentally. They are needed to properly describe the weak interaction. Neutrinos are fermions and are a weird form of matter particle with no electric charge and almost no mass, but they are believed to be fundamental. Neutrinos have actually been in the news a lot lately; they are mysterious in many ways and there is a big experimental program in the US to study their properties in detail. We will talk more about neutrinos in a later post.

So in addition to the matter particles that make up atoms (electrons and quarks) ... and now neutrinos ... there are particles associated with each of the four forces. These force carrier particles are also believed to be fundamental. They are all bosons: positive sign particles under exchange. We actually already knew they have to be bosons in order to interact with fermions. One of the restrictions required when combining QM and relativity was a limit to the type of interactions allowed. The only allowed interactions of fermions were of the form:

where two fermions come together to interact with a boson. We indeed see that all the forces conform to this picture.

Large collections of bosons behave classically like waves. This is consistent with our experience of light as a wave and the recent studies of gravitational waves. We don’t observe large collections of the weak and strong force carriers because they only interact over very short ranges, more on this in a future post.

It now looks like we have everything we need; we have all the matter particles needed to make up atoms -- plus an extra one forced on us by the weak interaction -- and we can describe the four fundamental forces in terms of interactions with other particles. Next time we will start putting this all together and discover there is a lot we are still missing.

This theory has been a spectacular success. It is the most accurate theory in all of science. It can describe all matter and interactions down to at least10⁻¹⁸ m (100 x smaller than a proton) where we have been able to directly test the theory experimentally. As we will see in later posts, the theory is not just qualitatively correct -- in that it can account for all the matter and interactions that we observe -- but it is also quantitatively accurate and precise; there are certain properties of the world that it can calculate to twelve decimal places of accuracy which we can also happen to measure as precisely, and we see that they agree. So it predicts basic things like “anti-particles should exist” and very specific things like “this is precisely how magnetic electrons are”.

So we have this great theory, but how do we actually use it? What are its outputs? There are two types of predictions the theory makes. The first is basic properties of particles, like how magnetic electrons are. The second is the result of interactions between particles. If we arrange for particles to interact, the theory will predict the probability for the interaction to occur in various ways. The output is a probability because the theory is quantum mechanical.

Here is an example of a simple interaction between two electrons:

Remember, because of QM we have to describe the electrons by thick wedges, not lines. In this example we shoot two electrons toward each other at time t=0, the electrons come together and interact in some way and then fly off in different directions where they are measured at some later time indicated by the dashed line. QFT allows us to calculate the probability for this interaction to occur.

A trivial thing that is often confusing in QFT is that it’s common to swap the axes in our space-time diagrams. So the above picture is represented as:

Of course this is just a convention. This picture is representing the same physical process as before.

A more important point to remember is that we cannot trace the exact trajectories of the particles; that was the big take away from the uncertainty principle. However the theory actually computes the probabilities by assuming a particular path for the electrons and a particular interaction among them and then adds all the possibilities together. For example, one way the electrons might interact is shown below:

Here the electron on the bottom takes a certain path through spacetime and at a particular point emits a photon γ (more about photons in the next post). This photon, shown as the squiggly line, takes a certain path and then strikes the upper electron, again at a particular place in spacetime. The theory assigns a number to this stick figure. The example shown in the stick figure is only one way the electrons might move and interact. QFT tells us we can then compute the full QM amplitude by summing up the numbers associated with all possible stick figures that are consistent with the input and output measurements. The probability is then the square of the amplitude just as in QM.

From now on in in this series I will just draw the stick figures as shorthand.

These stick figures are called Feynman Diagrams. When you see people draw Feynman diagrams, or when I draw them later in this series, it looks like you know exactly where the electron is and how it’s moving, in contradiction to the uncertainty principle. But we should remember that this is just a shorthand; we don’t know exactly where the electrons are at any given time, we don’t know precisely where the photon is, and we are really summing up all the possible trajectories that the particles may have taken.

This material is advanced. It’s typically taught in a year-long graduate course. However, conceptually it’s fairly simple; we are just adding up the contributions for each possible way the interaction can occur. The content of the theory is really in how you assign numbers to the stick figures. Assigning these numbers is actually pretty straightforward; most of the difficulty is in summing up the numbers. The sums are complicated integrals. These integrals turn out to be formally infinite and there is a whole two-semesters-graduate-course worth of machinery in dealing with the integrals and the infinities. These details, while fascinating in their own right, are not needed to understand the big picture.

In the next post we will start applying this machinery to the matter and forces in our world.

Last post we saw what the world must look like if it is both relativistic and Quantum Mechanical. Now we will start looking at what our actual world looks like.

What is our world made of? All the matter in our world is made of atoms. Atoms in turn are made of electrons “orbiting” nuclei. Electrons are negatively charged and responsible for most of the volume of the atom. Electrons are believed to be fundamental; our best theory of the world treats them as fundamental objects.

The nucleus on the other hand is positively charged and is responsible for most of the mass of the atom. Nuclei are not fundamental, they are made of neutrons and protons. We now know that these are also not fundamental, it turns out that they are made of other particles called “quarks”. We think that the quarks are fundamental and they are also treated in our theory as fundamental objects.

So electrons and quarks are the particles that make up all matter. These matter particles are all fermions -- minus-sign particles under interchange. We saw before that a large collection of fermions behave like classical particles, consistent with how everyday objects we see around us behave.

So this is the stuff in our world: different combinations of electrons and quarks. We also need to talk about how these particles interact with each other. We know of four interactions or forces between the particles.

The first is gravity. Gravity has been known since antiquity. Newton taught us that it has a relatively simple mathematical form: inverse square law, with the strength depending on the masses of the particles. Because the masses cannot be negative, gravity always acts in the same direction; it’s always attractive. While gravity is probably the most obvious force we know, as we will see in a future post, it is totally irrelevant for sub-atomic interactions. So for particle physics, we can really just ignore gravity.

Another force is electromagnetism. This is another big one. It was also known since antiquity and can be described by an inverse square law. However, here we can have both attractive and repulsive interactions depending on the relative charges of the particles. We discussed earlier that this is the interaction that holds the electrons and the nucleus together in atoms.

Electromagnetism and gravity have some superficial similarities, but in detail are really quite different from one another. The other two forces look even less like electromagnetism or gravity.

The “strong force” was only discovered relatively recently in the early 1900s. The strong interaction is not obvious because it is only relevant on short distance scales, at the scale of the size of the nucleus or smaller. Here there really is no simple mathematical distance relationship, eg: the form is much more complicated than an inverse square law. The strong force is responsible for holding together the nucleus; it binds the quarks together into protons and neutrons, and is also responsible for binding the protons and neutrons together into nuclei.

It is called the strong force because it is stronger than electromagnetism. You can see that this has to be by thinking about a nucleus. We are taught that nuclei are made of a bunch of protons (all positively charged particles) and neutrons (all neutral particles) closely packed together. The like-charged protons are all repulsed electromagnetically, so there has to be another force that attracts protons, and it’s got to be stronger than electromagnetism if it’s going to keep them from flying off. So the strong force, while maybe not as well known as EM and gravity, is also important. We will see later that the energy that is stored in the strong force is responsible for almost all the mass of matter.

Finally there is a fourth fundamental interaction known as the “weak force”. The weak force is another late comer. It again looks nothing like the others; there isn’t a simple mathematical form like with EM or gravity, and it looks nothing like the strong force. The weak interaction, while obscure, also plays a critical role in our everyday lives. It’s important for radioactive decay and plays a critical role in heating the sun. It’s referred to as the weak interaction because it is weaker than electromagnetism. This weakness is also important to us; it controls how fast stars -- like our sun -- burn through their fuel. The weakness of the weak force allows stars to live long enough to allow planets, chemistry, and life to develop around them.

So all four forces are crucial to our everyday lives and they all look completely different from one another. Next time we will start to discuss the mathematical framework that describes the matter particles and their interactions. This will be a particular instance of QM+relativity theory (referred to as a Quantum Field Theory) that we talked about in the last few posts. This will allow us to discover that the apparent major differences among the forces is mostly a long-distance illusion.

In the last couple of posts we saw some of the major implications of combining QM and relativity. The combination requires anti-particles, this in turn has major implications for our view of empty space. We built up some intuition as to why anti-particles were needed by thinking about causality. In this post I am just going to state some of the other implications of combining QM and relativity. To highlight these implications we will contrast the situation with QM alone vs QM + relativity.

Before the union, QM could accommodate spin. The theory could describe particles with spin; it wasn’t required, but it was possible. The types of spins were restricted, they could only exist as half integers (in units of h), but you could have any half integer you want and your choice is independent of everything else in the theory, e.g.: how the particles interact or the type of particle -- plus-sign “Bosons” or negative-sign “Fermions” -- when you switched labels.

This picture gets dramatically restricted when you add relativity. In QM + relativity, we are forced to talk about spin; the basic building blocks of the theory are particles with spin. The union also forces a connection between the spin of a particle and its type. The integer particles (0, 1, 2, ...) must be Bosons and the half-integer particles (1/2, 3/2, ...) must be Fermions. With just QM we could have spin-1 Fermions or spin-1/2 Bosons, but when you add relativity these options are forbidden.

It gets worse. QM allows you to describe any half-integer spin you’d like, an infinite number of possibilities. When you add relativity only five spins will dominate at high energies. In this limit, only 5 types of spins are allowed: spin 0, 1/2, 1, 3/2 and 2. We go from an infinite list of possibilities to five.

In terms of possible interactions it’s even more dramatic. In QM, any reasonable form of interaction is possible. We can solve the equations of QM with any sort of complicated potential energy function we can dream up, a continuous infinity of possibilities. In fact, most of an undergraduate QM course is solving the wave equations of the theory for different choices of potential energy shapes.

When we include relativity, almost all choices of potential are inconsistent. It turns out the only types of interaction that are allowed are those in which: 1) charge is conserved, 2) the interactions are local (meaning that they take place at a single point in space time; no more action at a distance) and 3) only one of three possible interactions is allowed. As we will see in a future post, these interactions can be described in terms of these stick-figure like diagrams:

We are only allowed to have three of them; we have an interaction where four Bosons come together, an interaction where only three come together, or an interaction where two Fermions and one Boson come together. That’s it! We go from a continuously infinite set of possibilities... to three.

Again all of this is being forced on us from trying to fit the basic principles of QM and relativity together. You can see why it was so hard to see that it actually is possible. Almost all scenarios that are allowed in QM are forbidden when relativity is added. It’s only when we limit ourselves to these specific and detailed constraints do we see that it can actually work.

These restrictions must hold in any possible world. Any imaginable world that is both quantum mechanical and relativistic must follow these non-trivial, non-obvious prescriptions. In the next posts we will start talking about the world that we find ourselves in. What is interesting is that we can now look at the world we actually live in and check if it conforms to this picture.

We saw last time that QM + Relativity violates causality unless each particle has an anti-particle. The anti-particles have the same mass and spin as the particle but all conserved charges are opposite. It turns out that this has huge implications for the theory. In particular it has a huge impact on nothing... on empty space, or the vacuum.

First, let’s back up and talk about what it takes to study empty space. This seems like a stupid question. It doesn’t take much, just look and see if the space is empty or if there is anything there. And this is largely true, until you start checking smaller and smaller regions where QM kicks in.

In a world without QM, with Newtonian physics, we could imagine straightforward ways of exploring smaller and smaller regions of space. For example, we could build tiny robots to probe their local environment, and they could send signals that report back to us what they find. If we want to probe even smaller spaces, we can make smaller robots, or we can make the small robots build yet smaller robots, etc. There is actually famous science fiction playing with exactly this idea. Classically, there is no limit to how far you can “telescope down” with smaller and smaller robots.

However with QM, this whole program becomes impossible. With QM, as you go to smaller and smaller distances at some point the uncertainty principle kicks in. If we want to really be certain that there is nothing in a small region of size Δx, we need to have some probe with a position uncertainty that is smaller than Δx. But the uncertainty principle requires that probes with small uncertainties in x have large uncertainties in p; they are inversely related, the smaller the Δx the larger the Δp. The large Δps in turn require large energies. So QM says that in order to go to smaller and smaller distances we need to use probes that have more and more energy.

By the way, this is the fundamental reason why the LHC is so big. A natural question about particle physics is why do we need such big machines to study such tiny distance scales? The reason is QM. We need large energies, which means large experiments, to probe small distances.

Once QM kicks in the program of building smaller and smaller robots breaks down. The smaller the robot the more energy you need just to be able to hold it together. Eventually the material it is made of cannot withstand the pressure that comes from holding the necessary energy in the small space.

So when we add QM we need high energies to probe small distances. When we add QM and relativity, we just saw, we need antiparticles and now we have a radically new effect.

We are going to smaller and smaller distances by putting in more and more energy. Eventually the amount of energy that we put in becomes larger than twice the electron mass. At this energy there is nothing in the laws of QM or relativity to prevent the creation of an electron anti-electron pair via E=mc². Without anti-particles this could not happen due to charge conservation. You could put in enough energy to produce an electron (or even a pair of electrons) but this would be forbidden by charge conservation. Because anti-particles must have charges exactly opposite of their particle partners, in order to preserve causality, there is now nothing that prevents us from creating particle -antiparticle pairs and QM predicts a non-zero probability for this to happen.

This completely changes our picture of what the vacuum is. We used to think the vacuum is some boring static thing. If there was really nothing in a region of space that was all there was to it. Now we see it is much more subtle. The attempt to verify that there is nothing has a probability of producing particles. The simple act of “looking” at the vacuum, verifying that there is nothing there, makes the vacuum not empty.

With QM + relativity, there is no sense in which the vacuum is empty or static. The act of looking or probing empty space can create particle-anti-particle pairs. Of course it’s QM, so everything is probabilistic, and what you actually observe changes each time. Sometimes when you look you will create the pairs, other times you won’t. When you look at even shorter distances (meaning higher energies) you can create higher mass particle-anti-particle pairs; e.g.: when the energies are greater than twice the muon mass you will start creating muon-anti-muon pairs. So because of QM + relativity, the vacuum is a dynamic place filled with constantly changing particle-anti-particle pairs whose population changes as you go to smaller and smaller distance scales.

This effect will come up over and over again in our story. It sounds crazy, but we know it actually happens; we will see later that this has explicitly been verified experimentally. We will also see that this effect is central to understanding the big picture of how the world works.

With QM, we shouldn’t be thinking of the bullet as an infinitely thin line. One of the big lessons of QM is that we can’t know the position of the bullet and its momentum arbitrarily accurately. So we need to draw the bullet as a wedge.

The wedge has a thickness that will extend somewhat into the “forbidden” region where causality is a problem. This can be avoided if you obey the upper speed limit AND you draw an infinitely thin line, but as soon as you have some finite thickness, there will always be some area that leaks out. This means that there is always some probability that the quantum mechanical bullet will leak into the forbidden region and appear to have gone faster than the upper limit. For everyday objects this QM probability of leaking into the forbidden region is negligible, but the effect is significant for particles which are quantum mechanical and relativistic.

OK, so let’s shoot a relativistic and QM proton instead of a bullet. Now there will be some probability that the proton hits at point B, in the forbidden region.

In particular, some observers will see B happening before A:

The proton hits B before it was sent from A. The theory has to account for this process, but it looks like it cannot be causal. How can we say “A caused B” if B happened first? This is a problem. Causality seems violated; effects happen before their causes. This is a problem because causality is a basic prerequisite for doing science.

One possible reaction to this is

“Too bad. We tried but it just doesn’t work. QM and relativity are just fundamentally incompatible. We suspected this from the beginning. We probably need something completely new, just like we needed for QM. Maybe we need to throw away the principle of relativity and QM and have another round of revolution in physics.”

For the physicists who just saw the revolution of QM happen this was probably the obvious reaction.

Another reaction is to try to fight through and somehow salvage the situation. We need causality, so if we have any hope of making this work, the observer at B has to be able to interpret what is going on as something going from B to A; for them to say “B causes A”.

Maybe this could work. Whether you say “A causes B” or “B causes A” would be observer dependent, but at least we could retain causality.

However notice that in the original interpretation where A is shooting the proton, A is losing positive charge and B is gaining positive charge. So after the interaction A is more negative than it was before and B is more positive. All observers agree with this; charge is independent of reference frame.

So if we are going to allow for the possibility of an interpretation in which B is shooting something at A, the particle that B shoots must have the opposite charge of the particle that A shoots. In our example, both observers agree that the interaction increases B’s charge and reduces A’s charge. The only way to account for the change in charge and for both observers to view the interaction in a causal way is if there are particles that behave just like protons, but have the opposite charge: “anti-protons”.

We just used protons as an example. You can run the same logic using any other particle as the “bullet” and the argument runs through. We can only hope to preserve causality if every particle has a twin “anti-particle” with the same mass and spin but with opposite charge.

So it seemed as though combining QM and relativity immediately leads to an inconsistency: the violation of causality. However, it turns out we can salvage the theory, but only if all particles have anti-matter partners that carry the opposite charge.

This is a concrete prediction and it was shocking at the time. The number of types of particles that the world was made up of would have to double. Not to mention the fact that no anti-matter particles had ever been seen.

Of course this turned out to be the right idea. Here the conservative approach of keeping the traditional principles of QM and relativity wins out over the radical approach of throwing everything away and starting over. But it comes at a price, the world must have both matter and anti-matter. We will see in the next post this has major consequences for the theory.

We will start by thinking about causality. Causality is a fancy word for something you have an intuitive understanding of: what happens at a certain time can only depend on things that have happened before. In particular, an event cannot depend on something that hasn’t happened yet. If someone dies from a gunshot, the gun has to have been shot first. Causality simply says that you can’t die from a gunshot wound unless the gun was shot first.

This seems obvious, so obvious that we often don’t explicitly talk about causality in physics, but it is taken as given and baked into the underlying assumptions. Causality is really just a basic prerequisite to doing science.

Let’s think about causality in relativity. Here is a space-time diagram of a gun going off at A, the bullet (black line) travels to the right and kills someone at B:

We know that the bullet must travel slower than the upper speed limit, so the bullet’s angle has to be larger than the 45-degree line.

This is what the situation looks like to one observer; they see the person killed at point B. Other observers will disagree on when this happens. They will see B taking place at different times and places. However, all observers agree on the invariant between these events, so they all agree the person dies somewhere along the red line:

Here we see that, while the times for all observers are different, relativity is consistent with causality. The different observers disagree on where and when the person died, but they all agree it happened after the gun was fired at A; all the points on the hyperbola are later in time than A. So all observers can agree that “A caused B”.

Notice that the upper speed limit is really critical here. If you could fire a bullet that goes faster than the speed limit, then things go very wrong. A bullet moving faster than the speed limit would travel below the 45-degree line, killing our victim at B.

This naively seems OK; B again comes later in time than A. But remember, in this quadrant the invariant implies different observers will see event B somewhere on this red hyperbola:

The problem is that now to some observers it will look like B happened before A; the person is killed by the bullet before it is fired. So causality would be violated if things can go faster than the upper speed limit.

So causality is OK as long as things are restricted to travel less than the upper speed limit. In the next post we see the problem that arises when we try to add QM to this picture.

We are going to turn now to combining relativity and QM. We will start discussing this very broadly: what are the implications for any possible world which respects these principles? We will then get more specific: what is the world that we actually find ourselves in? We will see that many details of our world are forced on us by simply making QM and relativity consistent.

So why is it so hard to combine QM and Relativity? There are several reasons, but the biggest and most fundamental is that their basic languages are different. In QM, time is special. QM says if you give me the Amplitude at one time, I will tell you what it is at a later time. Time is singled out and given a fundamental role by the theory. Whereas the whole point of relativity is that time is not so special; in fact different moving observers will disagree on the time at which an event takes place.

So naively these theories look completely incompatible. It turns out that it is possible to combine these ideas, but just barely, and it took the field a long time to understand this. Again, I think partly this was because the character of the solution was so different than what was needed to solve the previous generation of problems.

But it turns out — if you dramatically restrict what the world can look like — you can simultaneously accommodate both ideas. Many things that are possible in QM without relativity, or in relativity without QM, turn out to be impossible when you combine the two. These restrictions are general; they apply to any possible world that obeys both the principles of QM and relativity.

The theory that combines the two is referred to as Quantum Field Theory. This subject is quite advanced. We do not teach QFT to undergraduates; it typically takes a couple of advanced graduate-level courses for students to learn the basics. It’s also worth pointing out that QFT is still at the cutting edge of physics and also of math; we are still learning new things about QFT and using QFT to learn new mathematics.

That said, I believe many of the core concepts and physical intuitions behind QFT can be explained and justified in simple terms. This is what I hope to do in the next few posts. I will end this post by giving a broad overview of the consequences of the union and we will start next time going into the logic behind them.

So what are the restrictions needed for the union of QM and relativity? The first is that the theory demands that there must be anti-particles, anti-matter must exist. This was shocking and totally unexpected at the time. It doubles the variety of particles in the universe. We had thought we understood what types of particles the world was made of, but if we are to combine QM and relativity we need to double the list. This in turn radically changes our view of empty space... more about this later.

The combination also implies that spin must play a key role in the theory. QM and relativity are only consistent if the allowed particle spins are limited to a handful of possibilities (as opposed to an infinite number in QM alone) and if there is a fundamental relation between the spin of a particle and its type: fermion or boson.

Finally there are major constraints on the types of interactions between particles that are allowed. Only a very few specific types of interactions will dominate physical processes.

All of these restrictions are completely general. The next few posts will start putting teeth behind these statements. After that we will turn to what our specific world actually looks like, and we will find that it exactly fits the mold needed for one that is both QM and relativistic.

Quantum mechanics says we can talk about particles moving in space and time but they must respect the uncertainty principle. Particles have well-defined values of various properties: mass, charge, and spin, etc. Mass and charge are more-or-less familiar; spin is an intrinsic angular momentum.

Particles behave as if they were spinning1. Classically, the spin of an object can be any real number you want. In quantum mechanics the allowed magnitudes of the intrinsic spin are discrete; particles can have spin 0, 1/2h, 1h, 3/2h ... In general they can have spin n/2 h for any integer n. This quantization of spin is one of the things that gave quantum mechanics its name.

Another important change with quantum mechanics is how we think about identical particles. Classically, we can have two of the same type of particles, electrons say, interact. A space-time diagram of their interaction is below.

At every point in time we know each of the particles’ position AND their momentum. So at any point along the diagram we know which particle is which; which is electron one and which is electron two. The infinitely thin lines of classical physics allow us to uniquely label, or distinguish, particles that are otherwise identical.

In the quantum world, the infinitely thin lines are not allowed. If we know the positions well at t=0, the interaction between identical electrons is described by the following space-time diagram.

We can’t know both the position and momentum of either particle at any given time. So after the trajectories have begun to overlap, we can’t tell which particle is which. Maybe electron one ended up on the right after the interaction, but maybe it came in and was bent to the left and electron two went to the right. At some later time, after the interaction has occurred, we cannot as a matter of principle tell which electron is which.

In quantum mechanics we cannot follow the trajectories of the individual particles precisely; all we have is the red blobs. This means that when the blobs overlap we cannot assign unique labels to identical particles. The theory must treat identical particles as indistinguishable. This is all a direct result of the uncertainty principle.

This has a big consequence. Identical particles cannot be uniquely labeled, so the physics cannot depend on our labelling. QM encodes the physics in terms of probabilities for events to happen and we saw that the probabilities are given by the square of the amplitude: ψ². The fact that the physics must be independent of our labelling means that the probabilities, the squared amplitudes, for our interaction must be the same if we switch labels. In equations,

ψ(e1,e2)² = ψ(e2,e1)²

The constraint is on the square of the amplitude, this means we have two options for how the amplitude itself behaves when we switch label:

ψ(e1, e2) = ± ψ(e2, e1)

When we switch the order we can either get a plus sign, we get the same amplitude back, or we get a minus sign, which will cancel when we square it. So Quantum Mechanics tells us that there are fundamentally two different types of particles: the negative-sign particles — which we call Fermions — and the plus-sign particles — which we call Bosons. We will see later that the Higgs particle must be a plus-sign particle, which is why we call it the Higgs Boson. Again this all comes back to the fact that with QM you can’t trace the individual particle trajectories because of the uncertainty principle.

These two fundamental types of particles explain the confusion of the wave-vs-particle duality. Before QM there seemed to be two different forms of existence: waves (eg: light) and particles (eg: atoms). QM tells us there are only particles, but they are quantum particles that come in two types. It turns out that big collections of Fermions behave like classical particles, whereas big collections of Bosons behave like classical waves. The classical wave-particle distinction is an emergent phenomenon; it’s a reflection of these two fundamental types of quantum particles.

Our model of the atom -- let’s take Hydrogen for simplicity -- was of a negatively charged electron attracted to a positively charged proton by the electromagnetic interaction. The proton’s mass is about a 1000 times larger than the electron’s, so to a good approximation the proton just sits still while the electron moves. The electromagnetic force in this case is an attractive inverse square law, as it is for gravity, and would thus lead to the electron orbiting the proton. The size of the atom would be set by the size of the orbit.

This all looks like we are in pretty good shape. However, the same EM interaction causing the electron to be attracted to the proton also causes the electron to radiate light as it orbits. The electron must constantly change the direction of its velocity to stay in a circle (or an ellipse). This changing velocity is an acceleration, and EM tells us that accelerating charges radiate light1.

The radiated light will carry away some of the electron’s energy. This in turn causes the electron to move closer to the proton, lowering its potential energy. But now the electron is orbiting in an even tighter circle, which means it’s accelerating more than before, so it will radiate even more, which leads to yet smaller orbits, which ... . This quickly leads to a runaway effect where the electron loses all its energy to radiation after less than a billionth of a second. The electron ends up not moving (no kinetic energy) and sitting directly on top of the proton (no potential energy). Here is a spacetime diagram of what happens classically as an electron falls into the proton.

The planetary model of the atom, where the electron is attracted by the EM force, is unstable with a lifetime of about a picosecond. So a straightforward prediction of the classical theory is that atoms shouldn’t exist.

What to do? The obvious things to try to change are either 1) the model of the atom -- maybe atoms aren’t light electrons orbiting heavy nuclei -- or 2) the EM force holding the atom together -- maybe EM is wrong in some way or maybe we need a new force for atoms? It turns out that both the model of the atom and the EM force are correct and the change needed was much more radical, a change to the way electrons are allowed to move in space.

Here is what our atom looks like with QM:

The general behavior is similar to what is predicted classically. As the electron approaches, it begins to orbit the proton; it very quickly loses energy to radiation and orbits at smaller and smaller radii until eventually it is “sitting right on top” of the proton.

The difference is in the end state. Classically the electron is right on top of the proton (Δx = 0) AND not moving (Δp = 0). This infinitely thin line in spacetime is outlawed by QM. The uncertainty principle says there must be a non-zero thickness to the line, and it’s exactly this thickness that sets the size of the atom.

The uncertainty principle says we can either know the position or the momentum precisely, but not both. Here is an example of an amplitude where we know the position precisely:

QM is probabilistic and the theory tells us the probability of finding the particle at a certain place is given by the square of the amplitude. This means that if a particle has a small uncertainty in where it will be measured, a small Δx, its amplitude will be localized in a small region of x. The position is well-defined when the amplitude, and thus the probability (ψ²), is sharply peaked in one place.

The amplitude of a particle with a well-defined momentum looks totally different:

What characterizes the momentum in QM is the distance between peaks in the amplitude. An amplitude with a well-defined momentum has a constant, regular spacing between peaks.

You can immediately see that the notion of well-defined position is totally at odds with that of having a well-defined momentum. If you are well-defined in position, you only have one place where the amplitude is big, one peak; the concept of what is the difference to the next peak is meaningless. If, on the other hand, you have a constant distance to the next peak, then there are many different positions where the particle can be with large probability.

In a real sense, the amplitude of a particle with well-defined momentum is made up of different amplitudes that have different well-defined positions; it is a bunch of little peaks shifted with respect to one another. It’s also true -- but harder to visualize -- that the amplitude of a particle with well-defined position is made up of amplitudes that have different well-defined momentum. How this works is sketched below.

The big idea is that if you add the amplitudes of a bunch of waves that each have different wavelengths, they can give a combined wave that is peaked at one location. The above shows what happens if you add six different waves. As you add more and more waves with shorter wavelengths, the peak gets larger and sharper. So an amplitude that is localized in one position can be thought of as the sum of many amplitudes that each have different well-defined momenta.

Here we see that the uncertainty principle is not a result of our ignorance or our inability to design a careful position-and-momentum detector, it’s a fundamental feature of how position and momentum are encoded in the amplitude. This in turn follows directly from the fact that according to QM the amplitudes evolve in time as waves.

The above examples are the extreme cases, where either the position or momentum is exactly defined. In practice we often deal with particles that have reasonably well-defined position and reasonably well-defined momentum. The uncertainties on both are small but non-zero, bounded by the new QM constant, h. An amplitude for this particle would look like:

It’s fairly well localized. The probability of finding the particle to the far left or far right is essentially zero; the amplitude is zero far from the center of the figure. However, within the central region there are several peaks and the distance between them is fairly constant.

The big upshot of this was discussed in the last post. If we think of quantum particles moving in space and time we cannot use infinitely thin lines. We have to use thick shapes, where the thickness is set by a new constant of nature (h) that appears in the quantum theory.

We will see next time how this solves our problem with the apparent instability of matter.

Before QM we could describe our world in terms of space-time diagrams. For example, the following shows a particle moving at constant velocity:

The particle’s position is changing uniformly, at a specific well-defined rate with time. Classically, we are able to say that the particle has a certain value of position -- we know where it is along the space axis -- and we know its rate of change or the angle it is making with respect to the space axis.

This angle gives the particle’s velocity or, when multiplied by its mass, its momentum.

In the quantum world it turns out that these two quantities, position and momentum, are a kind of opposites to one another. Instead of drawing lines to represent particles in spacetime, QM tells us we need to draw thick wedges. For example, if we know the position of a particle precisely; ie: we know where it is along the space axis at a certain time, then we cannot know -- at the same time -- what direction it’s moving in. In the figure below, we know precisely where the particle is at t=0 and therefore we cannot know its angle, or equivalently, its momentum. Instead of drawing a line, with QM, we are forced to draw triangles.

Another example of this duality between position and momentum is where we know its angle very precisely; we know its momentum very well. QM tells us that in this case we cannot know position precisely. Again instead of the geometric lines of classical physics, we are forced to talk about a thick wedge. Here we know the angle, but not the starting position.

Infinitely straight lines are not allowed by QM. We can know the position precisely, in which case we need these triangles, or we can know the momentum precisely, in which case we must use these thick slanted rectangles.

The above examples are the edge cases, where we know either the position or momentum arbitrarily precisely. Typically in practice for particles like our muons, we will know its position with some finite uncertainty Δx and its momentum with some finite uncertainty Δp.

With QM we can know both things reasonably well, but we can’t know them both arbitrarily precisely at the same time. The theory puts a numerical limit1 on how well we can know the product Δx Δp; the product is bounded by a new constant of nature h: Δx Δp >= h. The non-zero value of h sets the scale for the thickness of the wedges in our diagrams.

An analogous argument can be made in terms of energy and time. In QM these are also a sort of opposite of one another, and the precision with which they can be simultaneously known is bound by a similar equation: ΔE Δt >= h.

The diagrams above give an intuitive understanding of the implications of the uncertainty principle. In the next post we will see where this position-vs-momentum duality comes from in terms of the QM amplitude.

QM tells us that we shouldn’t talk about the trajectories of the electron within an atom. We shouldn’t talk about where the electron is within the atom at a certain time, what is its path and how is it accelerating, like you would do in Newton’s physics (or in Einstein’s relativity). Instead, there is a new mathematical concept called the “Amplitude”, often written as ψ, which is really the fundamental physical entity. It’s the thing that describes everything there is to know about the electron. Everything we see in nature is encoded in the amplitude.

The theory of Quantum Mechanics tells us how the amplitude changes in time and gives us a prescription for translating from the amplitude to concrete physical questions that we might want to ask. QM tells us that amplitudes evolve in time like waves. It also provides a way to convert these amplitudes into probabilities (more on this below).

You can see already that the character of this revolution is much different than relativity. In relativity, we still had these intuitive physical notions of speed, trajectories, etc. QM is much more abstract, much more formal. There is this completely new abstract entity that we have to track in order to describe reality.

QM is also, in almost all cases, much less predictive than Newtonian physics. Instead of telling us what happens, it tells us what the probability is for certain things to happen. QM tells us there is a fundamental, inherent randomness in nature. In Newton’s physics, if you knew the positions and momenta of all the particles at a given time, you could predict with certainty what is going to happen at all later times. Nature, in this picture, is deterministic. QM throws this out the window.

Here is an example using muons, a particle we talked about already in the context of relativity. We can create two muons that have exactly the same amplitude.

We noted before that muons have a finite lifetime, they decay into other particles after a characteristic time. When the muons decay they turn into an electron, a neutrino, and an anti-neutrino1.

If we watch our two muons, we might find that one of the muons decays after 2.1 microseconds with a particular angle θ between the electron and anti-neutrino, whereas the other decays after 2.3 microseconds, with a different angle among its decay products.

Now remember the amplitude contains everything there is to know about the particles. In terms of the theory these two particles are exactly identical. In QM identical initial states can lead to different outcomes. QM cannot predict what is going to happen in any particular event. In our example, it can’t tell us exactly how long the first muon will live and what its decay angle will be.

What QM can predict is the distributions of outcomes of many identically prepared initial states, ie: the relative probability of seeing a muon live a certain amount of time and decay with a particular angle. The amplitude encodes the probability distribution; QM tells us that the probability is given by the square of the amplitude.

Next time we will discuss another major outcome of QM: the uncertainty principle.

Why was QM so much harder than relativity? In relativity, Einstein took concepts that we were already familiar with—space, time, and energy —and taught us the proper way to think about them. Quantum Mechanics, as we will see, was a much more radical change; it threw away much of what came before and invented new, more abstract, mathematically formal concepts to replace them. QM was a much more dramatic departure from the physics that came before it. Like with relativity, an entire series of posts could be devoted to Quantum Mechanics; we will just focus on the big picture changes that were brought by this revolution and the pieces that we will need later for our story.

A good starting point for our discussion is the picture of atoms before the advent of QM. Atoms were known to consist of negatively charged electrons and a positively charged nucleus. Most of the volume of the atom is taken up by the electrons, whereas most of the mass comes from the nucleus. The electromagnetic force -- which was relatively well-understood at the time -- was thought to be responsible for holding atoms together. There was a solid experimental foundation for this basic picture, but there were many problems with this model.

The first problem was that the electromagnetic interaction predicted atoms would not be stable. If you happen to arrange a negatively charged electron to orbit a positively charged nucleus, it should almost immediately, after ~10⁻¹¹s, spiral in and collapse right on top of the nucleus. So it wasn’t clear why matter was stable at all; how were atoms able to keep their size?

Another thing that was strange was the way that atoms interacted with light. We knew that atoms could absorb and emit light; this was predicted by the theory and observed to happen. However, the theory predicted that atoms should interact with all frequencies or colors of light, whereas it was seen that only discrete values of light interacted with atoms. Moreover, different atoms had completely different patterns of colors they interacted with. Here are some examples:

The top continuous spectrum is what you would expect the light interacting with atoms to look like given the theory. Whereas the spectra below were what was actually observed for a few different elements. These unique patterns were incredibly useful for chemists, who could use them to fingerprint elements present in a given sample, but were totally unexpected from the underlying physical model.

Another closely related worry at the time was an inherent wave-vs-particle duality of nature. There were things in physics that looked like particles, e.g. the atoms. There were other things that behaved like waves, e.g. the light that interacted with the atoms. The relationship between these wasn’t clear. Were there really two different fundamental modes of existence? Or was one somehow more fundamental, more real, than the other?

Before quantum mechanics there were many problems in describing even the most basic things in our everyday lives.

There was a long period of confusion, filled with many false starts and wrong turns. There were several ad-hoc and competing ideas that were able to give partial answers which could explain some of the riddles, but not others. Eventually the ideas settled down into a unified and consistent theory, which we will start describing in the next post.

Now you might think, “OK there is a problem with our description of atoms... the solution should either be a modification of the structure of atoms, i.e.: our picture of what they are made of, or maybe the model is right but we need a modification of the electromagnetic force that holds the atoms together.” These are the two obvious things you would think to change. It turns out that both the model and the electromagnetic interaction were correct. The solution was much more radical: it was to change the way the electrons were allowed to move, and it required a completely new framework for all physical processes.

The following is a cartoon of a particle accelerator — like the Large Hadron Collider — accelerating a proton. We do this by constraining the proton to move in a circle and giving it a little kick every time it goes around. Magnetic fields are used to keep the protons in a circle and electric fields are used to give the kicks. The kick or force applied to the proton leads to a small increase in the proton’s momentum every cycle.

The total change in momentum after going around many times is given by the size of the kick times the number of cycles. Also remember the momentum is given by mass times velocity, so we have:

What’s interesting is that the number of cycles can be arbitrarily large; the proton is stable and we are able to keep it in a circle pretty much as long as we want. The force is a fixed number we also control, the size of each kick. So this means the total change in momentum can be arbitrarily large. However, the velocity cannot be arbitrarily large; this was the whole point of Einstein’s relativity.

As you start increasing the proton’s momentum, the kicks increase the proton’s velocity v. As we keep doing this, at some point the proton bumps up against the speed limit; it’s now moving nearly at the speed of light. Now we seem to have a problem: every cycle increases the proton’s momentum by a fixed amount, but the proton’s speed cannot increase, it can’t go above the limit. Once the proton’s velocity is near the maximum upper limit, the only way the proton’s momentum (mv) can be increased is by increasing m. Our notion of mass must also change to accommodate an upper speed limit; mass increases with speed.1

Thinking more carefully about how the mass must change to make this consistent leads directly to the famous E=mc². Again there is a lot more that we could say about relativity and the implications of a finite upper speed limit, but we have now covered the pieces that we need for our story. Next time we will start discussing the other 20th century revolution in physics: Quantum Mechanics.