Having written down the equation in the last article, describing the standard model of particle physics (so capturing all the known interactions between all the known fundamental fields), we would now like to make use of it to calculate something about the particles arising in the field. For example, say we want to figure out what happens when an electron ($e$) and a positron (the antimatter partner of the electron, written as $\overline{e}$) collide. Several things could happen, but let us focus on the case where an electron and a positron emerge after the collision.

QFT provides us with the tools to compute exactly how often this process happens. We start with the equation I gave (in the last article) for the photon field interacting with the electron field

and apply the rules of quantum mechanics to it. On the face of it, the maths required is simply horrifying. But the brilliant Richard Feynman found a visual way of organising it, the so-called Feynman diagrams (earning him a Nobel Prize in 1965).

### Physicists can doodle too

Feynman diagrams depicting some of the ways an electron and positron collision can happen

Photons are drawn as wiggly lines, while electrons and positrons are drawn as a straight line with an arrow showing which way negative electric charge is flowing. The incoming and outgoing particles have an extra arrow next to the line, showing the direction of motion.

In principle, Feynman diagrams don't depict what is actually happening in the collision; they are just a useful visual shorthand for mathematical expressions. But it is tempting, and useful for your intuition, to interpret them as movies of the collision, with time on the horizontal axis and space on the vertical axis.

For example, the first diagram looks like an electron and a positron colliding and turning into a photon, which then turns back into an electron and a positron.

The second diagram looks like the two particles exchanging a photon and then continuing on their merry ways.The third diagram is like the first one, except that the intermediate photon spends some time as an electron-positron pair.

Feynman diagrams are useful in calculations
because, although in principle you have
to draw all possible diagrams (which can be
arbitrarily complicated) and add them all up,
in practice the simplest diagrams tend to be
the most important. (You can read more about Feynman diagrams in *Quantum pictures*.)

### The QFT rabbit hole

Quantum field theory contains a huge number
of utterly fascinating details that would
take many pages to explain properly. One of
the great things about QFT is that it is a very
rich subject – starting from its basic principles,
you can reach many surprising conclusions. I
am just going to give you a quick taste of two
of the most important ones: *symmetry* and
*zooming*.

Rotationally symmetric rabbits

In physics, a *symmetry transformation* is a
change that has no observable effect on the
world. For example, if somebody moved the
whole Universe a few metres to the left, or
rotated it by some amount, this would be
completely impossible to detect. Symmetries
are a very rich aspect of QFT. For example,
the mathematical definition of a *charge* – like
electric charge, or the less well-known *hypercharge*
and *isospin* – is just a matter of how a
field changes in a given symmetry transformation.

The Standard Model has quite a lot of
symmetries – a lot of different charges – and
the way they interplay to form the physics
we observe is an intriguing subject where the
Higgs field ends up "breaking" the original
symmetries. This is the *Higgs mechanism*, the
discovery that earned Peter Higgs and François Englert the 2013 Nobel
Prize in physics. (You can read more in *Secret symmetry and the Higgs boson*.)

You might also be surprised to hear that *zooming* takes on a highly important role
in QFT (the technical term for this is *renormalisation*). When we zoom in and out, the theory
changes appearance, so two superficially
different theories might in fact be the same
theory, just at different levels of zoom. Mathematically,
the Standard Model looks rather
zoomed out, so we expect that it is only a
zoomed out version of some unknown, more
fundamental theory. Most theories get less
complicated when you zoom out. The unexpected
discovery that the theory of quarks
and gluons actually gets more complicated
when you zoom out was awarded the Nobel
Prize in 2004. (You can find out more in *Strong but free* and *Going with the flow*.)

*Find out about the last piece of the puzzle in the next article.*

### About the author

Elias Riedel Gårding grew up in Stockholm and chose physics instead of programming for his undergraduate degree because his secondary school physics class was frankly not very good, and he wanted to see what he was missing. He has always been interested in the most basic laws of nature – those of fundamental physics – but it wasn't until his master's degree in theoretical physics that he got to study them properly. He thinks quantum field theory, the basic paradigm of particle physics, deserves to be more widely known, hence this article series.