# Feynman diagrams

##### 8 minutes • 1569 words

## Table of contents

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.

This leaves 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 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 4 classes is probably artificial and a consequence of our lack of understanding.

People have therefore sought a theory of everything that will unify the 4 classes into a single law that is compatible with quantum theory.

This would be the holy grail of physics.

### The ElectroWeak Theory: W, Z Bosons

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. 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 3 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.

### Quantum Chromodynamics

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 call color, hence the term “chromodynamics”. This has no connection with visible color.

Quarks come in three colors, red, green, and blue.

Each quark has an anti-particle partner with colors 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*.

## Superphysics Note

When all the 3 colors (or anti-colors) are mixed, the result has no net color.

Three quarks, one of each color, form stable particles called baryons.

Examples of baryons are protons and neutrons (and three anti-quarks form the antiparticles of the baryons).

QCD also has a property called asymptotic freedom mentioned in Chapter 3.

It 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
- we have been unable to produce them in the laboratory.*

## Superphysics Note

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.

They mostly predict that protons should decay, on average, after about 1,032 years.

That is a very long lifetime, given that the universe is only about 1,010 years old.

In quantum physics, when we say the average lifetime of a particle is 1,032 years, we don’t mean that most particles live approximately 1,032 years, some a bit more and some a bit less.

Instead, what we mean is that, each year, the particle has a 1 in 1,032 chance of decaying.

As a result, if you watch a tank containing 1,032 protons for just a few years, you should see some of the protons decay.

It is not too hard to build such a tank, since 1,032 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 not easy.

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 1,034 years, which is bad news for grand unified theories.

Earlier observational evidence had also failed to support GUTs.

This is why 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:

- it does not unify the electroweak and strong forces
- it does not include gravity.

It may be 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.*

## Superphysics Note

That principle makes the value of a field and its rate of change play the same role as the position and velocity of a particle.

- 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.

### Virtual Particles

Vacuum fluctuations are like 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, they 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. There are an infinite number of virtual pairs, and so 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!