# Section 3: Fields Are Fundamental

Figure 7: When an electron and its antiparticle collide, they annihilate and new particles are created.

Source: © David Kaplan. More info

At a particle collider, it is possible for an electron and an antielectron to collide at a very high energy. The particles annihilate each other, and then two new particles, a muon and an antimuon, come out of the collision. There are two remarkable things about such an event, which has occurred literally a million times at the LEP collider that ran throughout the 1990s at CERN. First, the muon is 200 times heavier than the electron. We see in a dramatic way that mass is not conserved—that the kinetic energy of the electrons can be converted into mass for the muon. E = mc2, again. Mass is not a fundamental quantity.

The second remarkable thing is that particles like electrons and muons can appear and disappear, and thus they are, in some sense, not fundamental. In fact, all particles seem to have this property. Then what is fundamental? In response to this question, physicists define something called a field. A field fills all of space, and the field can, in a sense, vibrate in a way that is analogous to ripples on a lake. The places a field vibrates are places that contain energy, and those little pockets of energy are what we call (and have the properties of) particles.

As an analogy, imagine a lake. A pebble is dropped in the lake, and a wave from the splash travels away from the point of impact. That wave contains energy. We can describe that package of localized energy living in the wave as a particle. One can throw a few pebbles in the lake at the same time and create multiple waves (or particles). What is fundamental then is not the particle (wave), it is the lake itself (field). In addition, the wave (or particle) would have different properties if the lake were made of water or of, say, molasses. Different fields allow for the creation of different kinds of particles.

Figure 8: Ripples in lake from a rock.

Source: © Adam Kleppner. More info

To describe a familiar particle such as the electron in a quantum field theory, physicists consider the possible ways the electron field can be excited. Physicists say that an electron is the one particle state of the electron field—a state well defined before the electron is ever created. The quantum field description of particles has one important implication: Every electron has exactly the same internal properties—the charge, spin, and mass for each electron exactly matches that for every other one. In addition, the symmetries inherent in relativity require that every particle has an antiparticle with opposite spin, electric charge, and other charges. Some uncharged particles, such as photons, act as their own antiparticles.

## A crucial distinction

In general, the fact that all particles—matter or force carriers—are excitations of fields is the great unifying concept of quantum field theory. The excitations all evolve in time like waves, and they interact at points in spacetime like particles. However, the theory contains one crucial distinction between matter and force carriers. This relates to the internal spin of the particles.

By definition, all matter particles, such as electrons, protons, and neutrons, as well as quarks, come with a half-unit of spin. It turns out in quantum mechanics that a particle's spin is related to its angular momentum, which, like energy and linear momentum, is a conserved quantity. While a particle's linear momentum depends on its mass and velocity, its angular momentum depends on its mass and the speed at which it rotates about its axis. Angular momentum is quantized—it can take on values only in multiples of Planck's constant, = 1.05 x 10-34 Joule-seconds. So the smallest amount by which an object's angular momentum can change is . This value is so small that we don't notice it in normal life. However, it tightly restricts the physical states allowed for the tiny angular moment in atoms. Just to relate these amounts to our everyday experience, a typical spinning top can have an angular momentum of 1,000,000,000,000,000,000,000,000,000,000 times . If you change the angular momentum of the top by multiples of , you may as well be changing it continuously. This is why we don't see the quantum-mechanical nature of spin in everyday life.

Now, force carriers all have integer units of internal spin; no fractions are allowed. When these particles are emitted or absorbed, their spin can be exchanged with the rotational motion of the particles, thus conserving angular momentum. Particles with half-integer spin cannot be absorbed, because the smallest unit of rotational angular momentum is one times . Physicists call particles with half-integer spin fermions. Those with integer (including zero) spins they name bosons.

## Not your grandmother's ether theory

Figure 9: Experimental results remain the same whether they are performed at rest or at a constant velocity.

Source: © David Kaplan. More info

An important quantum state in the theory is the "zero-particle state," or vacuum. The fact that spacetime is filled with quantum fields makes the vacuum much more than inactive empty space. As we shall see later in this unit, the vacuum state of fields can change the mass and properties of particles. It also contributes to the energy of spacetime itself. But the vacuum of spacetime appears to be relativistic; in other words, it is best described by the theory of relativity. For example, in Figure 9, a scientist performing an experiment out in empty space will receive the same result as a scientist carrying out the same experiment while moving at a constant velocity relative to the first. So we should not compare the fields that fill space too closely with a material or gas. Moving through air at a constant velocity can affect the experiment because of air resistance. The fields, however, have no preferred "at rest" frame. Thus, moving relative to someone else does not give a scientist or an experiment a distinctive experience. This is what distinguishes quantum field theory from the so-called "ether" theories of light of a century ago.

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