Interactions

Free photons and electrons are not so interesting in themselves: we only get physics because theyinteract with each other. The interaction is actually very simple: it occurs at a single point in spaceand time, and its strength is given by one number, called the electric charge, �e (this minus sign issimply a historical convention). Thus an electron can emit a photon: in pictures with time runninghorizontally

This diagram represents the amplitude for the process: the probablity that the electron emits aphoton is thus proportional to e

2. Similarly, an electron can absorb a photon:

Interactions cont.

Energy and momentum are conserved at each interaction vertex, ensuring that they are conserved inany process involving interacting photons and electrons. The charge is also conserved (since the

photon carries no electric charge). Moreover the charges of left handed and right handed electronsare equal, and as the photon has spin ±~, emission or absorption of a photon changes a left handedelectron into a right handed one or vice versa. This means that processes involving photons andelectrons are invariant under parity: if we exchange all left handed particles with right handed ones,the probabilities are unchanged. In fact if the left handed and right handed electrons coupled tophotons di↵erently the Dirac equation would no longer work, except in the special case of m = 0.Later in the course, we will see that this creates an interesting dilemma.

The emission and absorption of photons allows electrons to interact with each other:

It can be shown that this produces the usual Coulomb interaction of electromagnetism, withpotential e2/4⇡r at large distances. The interaction can reach across large distances because thephoton is massless: when massive particles are exchanged, the interaction is limited to distances oforder ~/mc, essentially because of the uncertainty principle.

Interactions cont.

Compton scattering, on the other hand, happens when an electron successively absorbs or emits aphoton. There are two ways in which this can happen, and to obtain the probability we have to addtheir amplitudes:

Note that the second diagram can also be written like this:

The electron line between the two interactions is now going backwards in time – it is thus ananti-electron! So in this process the photon first splits into an electron and anti-electron, and theanti-electron then combines with the incoming electron to give the outgoing photon. Because chargeis conserved at each interaction, and the photon has zero charge, this means that the anti-electronmust have charge +e. For this reason it is called the “positron”.

Interactions cont.

So electrons and positrons are particles with the same mass m, but with opposite charge ⌥e.Positrons were first seen in cosmic ray showers by Anderson in 1932: in a magnetic field a chargedparticle moves in a spiral, the direction of which depends on its charge, and Anderson observedparticles which were like electrons in every way, except that they spiralled in the opposite direction.

Creation

In today’s lecture we have introduced some new and radical concepts. Firstly, the concept ofanti-matter – particles of the same mass, but opposite charge. Dirac received the Nobel prize in1933 for predicting positrons: Anderson in 1936 for finding them. Secondly, we found that matterand anti-matter annihilate each other, producing radiation (photons) – and conversely that radiationcan create matter and anti-matter.

These fundamental annihilation and creation processes lie at the heart of our understanding ofphysics at an elementary level. At a particle collider, we take matter and anti-matter, and collidethem – not so much to break them apart, but rather to create radiation, which can then be used tocreate new forms of matter and anti-matter. The higher the energy of the collision, the moremassive the particles that can be created. The great achievement at CERN in July 2012 was tocollide protons with su�cient energy that they could create a Higgs boson: before the Higgs couldbe discovered, it first had to be created.