Feynman diagrams

Feynman diagrams enormously helped physicists in visualizing and calculating the probabilities of the processes described by QED.

But they did not cure one important ailment suffered by the theory: When you add the contributions from the infinite number of different histories, you get an infinite result.

(If the successive terms in an infinite sum decrease fast enough, it is possible for the sum to be finite, but that, unfortunately, doesn’t happen here.)

In particular, when the Feynman diagrams are added up, the answer seems to imply that the electron has an infinite mass and charge. This is absurd, because we can measure the mass and charge and they are finite.

To deal with these infinities, a procedure called renormalization was developed. The process of renormalization involves subtracting quantities that are defined to be infinite and negative in such a way that, with careful mathematical accounting, the sum of the negative infinite values and the positive infinite values that arise in the theory almost cancel out, leaving a small remainder, the finite observed values of mass and charge.

These manipulations might sound like the sort of things that get you a flunking grade on a school math exam, and renormalization is indeed, as it sounds, mathematically dubious.

One consequence is that the values obtained by this method for the mass and charge of the electron can be any finite number. That has the advantage that physicists may choose the negative infinities in a way that gives the right answer, but the disadvantage that the mass and charge of the electron therefore cannot be predicted from the theory.

But once we have fixed the mass and charge of the electron in this manner, we can employ QED to make many other very precise predictions, which all agree extremely closely with observation, so renormalization is one of the essential ingredients of QED.

An early triumph of QED, for example, was the correct prediction of the so-called Lamb shift, a small change in the energy of one of the states of the hydrogen atom discovered in 1947.

The success of renormalization in QED encouraged attempts to look for quantum field theories describing the other three forces of nature. But the division of natural forces into four classes is probably artificial and a consequence of our lack of understanding. People have therefore sought a theory of everything that will unify the four classes into a single law that is compatible with quantum theory.

This would be the holy grail of physics.

One indication that unification is the right approach came from the theory of the weak force. The quantum field theory describing the weak force on its own cannot be renormalized; that is, it has infinities that cannot be canceled by subtracting a finite number of quantities such as mass and charge.

However, in 1967 Abdus Salam and Steven Weinberg each independently proposed a theory in which electromagnetism was unified with the weak force, and found that the unification cured the plague of infinities.

The unified force is called the electroweak force. Its theory could be renormalized, and it predicted three new particles called W+, W–, and Z0. Evidence for the Z0 was discovered at CERN in Geneva in 1973.

Salam and Weinberg were awarded the Nobel Prize in 1979, although the W and Z particles were not observed directly until 1983.

The strong force can be renormalized on its own in a theory called QCD, or quantum chromodynamics.

According to QCD, the proton, the neutron, and many other elementary particles of matter are made of quarks, which have a remarkable property that physicists have come to call color (hence the term “chromodynamics,” although quark colors are just helpful labels—there is no connection with visible color).

Quarks come in three so-called colors, red, green, and blue. In addition, each quark has an anti-particle partner, and the colors of those particles are called anti-red, anti-green, and anti-blue. The idea is that only combinations with no net color can exist as free particles.

There are 2 ways to achieve such neutral quark combinations.

A color and its anti-color cancel, so a quark and an anti-quark form a colorless pair, an unstable particle called a meson. Also, when all the three colors (or anti-colors) are mixed, the result has no net color.

Three quarks, one of each color, form stable particles called baryons, of which protons and neutrons are examples (and three anti-quarks form the antiparticles of the baryons). Protons and neutrons are the baryons that make up the nucleus of atoms and are the basis for all normal matter in the universe.

QCD also has a property called asymptotic freedom, which we referred to, without naming it, in Chapter 3. Asymptotic freedom means that the strong forces between quarks are small when the quarks are close together but increase if they are farther apart, rather as though they were joined by rubber bands.

Asymptotic freedom explains why we don’t see isolated quarks in nature and have been unable to produce them in the laboratory. Still, even though we cannot observe individual quarks, we accept the model because it works so well at explaining the behavior of protons, neutrons, and other particles of matter.

After uniting the weak and electromagnetic forces, physicists in the 1970s looked for a way to bring the strong force into that theory. There are a number of so-called grand unified theories or GUTs that unify the strong forces with the weak force and electromagnetism, but they mostly predict that protons, the stuff that we are made of, should decay, on average, after about 1032 years.

That is a very long lifetime, given that the universe is only about 1010 years old. But in quantum physics, when we say the average lifetime of a particle is 1032 years, we don’t mean that most particles live approximately 1032 years, some a bit more and some a bit less. Instead, what we mean is that, each year, the particle has a 1 in 1032 chance of decaying.

As a result, if you watch a tank containing 1032 protons for just a few years, you ought to see some of the protons decay. It is not too hard to build such a tank, since 1032 protons are contained in just a thousand tons of water.

Scientists have performed such experiments.

It turns out that detecting the decays and differentiating them from other events caused by the cosmic rays that continually shower us from space is no easy matter. To minimize the noise, the experiments are carried out deep inside places such as the Kamioka Mining and Smelting Company’s mine 3,281 feet under a mountain in Japan, which is somewhat shielded from cosmic rays.

As a result of observations in 2009, researchers have concluded that if protons decay at all, the proton lifetime is greater than about 1034 years, which is bad news for grand unified theories.

Since earlier observational evidence had also failed to support GUTs, most physicists adopted an ad hoc theory called the standard model, which comprises the unified theory of the electroweak forces and QCD as a theory of the strong forces.

But in the standard model, the electroweak and strong forces act separately and are not truly unified. The standard model is very successful and agrees with all current observational evidence, but it is ultimately unsatisfactory because, apart from not unifying the electroweak and strong forces, it does not include gravity.

It may have proved difficult to meld the strong force with the electromagnetic and weak forces, but those problems are nothing compared with the problem of merging gravity with the other three, or even of creating a stand-alone quantum theory of gravity. The reason a quantum theory of gravity has proven so hard to create has to do with the Heisenberg uncertainty principle, which we discussed in Chapter 4.

It is not obvious, but it turns out that with regard to that principle, the value of a field and its rate of change play the same role as the position and velocity of a particle. That is, the more accurately one is determined, the less accurately the other can be.

An important consequence of that is that there is no such thing as empty space. That is because empty space means that both the value of a field and its rate of change are exactly zero.

(If the field’s rate of change were not zero, the space would not remain empty.) Since the uncertainty principle does not allow the values of both the field and the rate of change to be exact, space is never empty. It can have a state of minimum energy, called the vacuum, but that state is subject to what are called quantum jitters, or vacuum fluctuations—particles and fields quivering in and out of existence.

One can think of the vacuum fluctuations as pairs of particles that appear together at some time, move apart, then come together and annihilate each other. In terms of Feynman diagrams, they correspond to closed loops. These particles are called virtual particles.

Unlike real particles, virtual particles cannot be observed directly with a particle detector. However, their indirect effects, such as small changes in the energy of electron orbits, can be measured, and agree with theoretical predictions to a remarkable degree of accuracy. The problem is that the virtual particles have energy, and because there are an infinite number of virtual pairs, they would have an infinite amount of energy.

According to general relativity, this means that they would curve the universe to an infinitely small size, which obviously does not happen!

This plague of infinities is similar to the problem that occurs in the theories of the strong, weak, and electromagnetic forces, except in those cases renormalization removes the infinities.

But the closed loops in the Feynman diagrams for gravity produce infinities that cannot be absorbed by renormalization because in general relativity there are not enough renormalizable parameters (such as the values of mass and charge) to remove all the quantum infinities from the theory.

We are therefore left with a theory of gravity that predicts that certain quantities, such as the curvature of space-time, are infinite, which is no way to run a habitable universe. That means the only possibility of obtaining a sensible theory would be for all the infinities to somehow cancel, without resorting to renormalization.

In 1976, a possible solution to that problem was found. It is called supergravity. The prefix “super” was not appended because physicists thought it was “super” that this theory of quantum gravity might actually work. Instead, “super” refers to a kind of symmetry the theory possesses, called supersymmetry.