One potato, two potato, three potato, four!
Five potato, six potato, seven potato, more!
Counting is one of the oldest skills we learn and most adults cannot consciously remember being unable to account. We learn to count at a young age, and it is hard to recall what we ever found hard about it. To find out how many of something there is, you just enumerate them. For example, every ten years a census is taken of the population. This is used to give a clear account of how many people live in a particular area at a particular time. In a literal sense, you just count heads.
Counting in physics seems easy as well. If I want to count how many atoms are in the room, it may take longer than the age of the universe to actually do it, but the principle is similar. I just count, and count, and count….and after a long, long time I would have the answer.
This post is about one of the ways counting can actually be rather subtle. In physics this goes under the exotic name of `infrared safety’, but the idea is much more general, and also applies to many areas outside physics (as we shall see).
To describe it, here’s a picture. Bang! This is the (slightly processed) result of a collision of two protons at the Large Hadron Collider (copyright CERN’s ATLAS experiment)
At first sight this picture may appear a bit strange. What is going on? What is happening? The relevant part is the two red bunches of particles. These bunches of particles are called `jets’, and they represents a large grouping of particles all heading in approximately the same direction. In colliders such as the LHC, these `jets’ are one of the common manifestations of the strong force that binds quarks and gluons together. If you want to understand the physics of the Large Hadron Collider, and whether any new physics may be lurking there, it is important to understand these jets.
In practice, given the number of collisions at the LHC, it is clear that computer algorithms are necessary to work with jets and to decide what is and what isn’t a ‘jet’ in any one particle collision. Nature doesn’t, by itself, produce particles with a label on them in Times New Roman stating that this set of particles belongs to jet 1 and this other set belongs to set 2. This requires a set of rules – for example, take an energetic particle and draw a cone around it of a certain angle, and say that, by definition, every particle within this is part of the jet.
However, once you have decided what a jet is, then it may seem quite simple to ask: How many particles are there in a jet? After all, we can surely just count them. We look at the jet, we look at all the particles within it, and we count them up. Simple! But this is where the problem lies. To see the problem, consider one of the most well known particles, the photon, the electromagnetic force carrier. Photons have many different energies. The most energetic are called gamma rays, then moving down to X-rays, ultraviolet light, visible light, infrared, microwaves and photons. The less energetic a photon is, the more sensitive the apparatus needed to detect it. Equipment that detects gamma-ray misses X-rays. X-ray detectors miss infrared light. Infrared detectors miss microwaves. In any jet, however defined, there are photons going down to infinitesimally small energies.
So in the end, this means that the question ‘How many particles are in a jet?’ is, by itself, actually meaningless. In that there is any answer, it is infinite – by going down to infinitely small photon energies, you can make the numbers of particle infinitely large. To make the question sensible – and physically meaningful – we have to instead ask ‘How many particles are there that have at least a minimal energy‘? The imposition of this threshold is essential for the question to make sense in the first place.
This is the simplest example of ‘infrared safety’ – which, roughly, says that results of physical observables shouldn’t be sensitive to the addition (or subtraction) of extremely low energy particles into the system. In a more subtle way, beyond simply counting particles, this requirement of ‘infrared safety’ is also crucial to a good definition of ‘what is a jet’. If your classification of where the jets are in a big smackeroo of a collision at LHC energies could change with the addition of a tiddly little microwave photon, your classification algorithm is a bad one. Indeed, infrared safety is one of the key defining properties of modern jet algorithms, such as the anti-kT algorithm.
I introduced this idea of infrared safety through a counting problem – how many particles are there? However, it has broader implications as the underlying effect holds whenever people try to count without putting a lower threshold in. As such, it is the source of many of the pseudo-statistics that permeate public life. A good example of this is counting hate crimes.
Let us switch now from the Large Hadron Collider to closing time at a large city centre pub on Saturday night. Alfie says something to Bob – who doesn’t like it, and responds by punching Alfie in the face. Rather unpleasant – and a clear case of assault. But, this is also something that has happened for as long as young men have gone our drinking at the weekend, and unless you were personally involved you would not think twice about it.
However, call this a hate crime, and you probably react quite differently. What is needed to turn that drunken right hook into a hate crime? This is a legal question with a legal answer – so going to the Crown Prosecution Service, we find
‘Any criminal offence which is perceived by the victim or any other person, to be motivated by hostility or prejudice, based on a person’s disability or perceived disability; race or perceived race; or religion or perceived religion; or sexual orientation or perceived sexual orientation or transgender identity or perceived transgender identity‘
What does this mean? It means that if anyone, anywhere – present or not, for whatever reason and however unjustified, decides that a crime is a hate crime, then it is a hate crime (I have deliberately set the description so that there is no doubt that there is a crime involved). In the case of no crime, then the police record it as hate incident.
Whether you like it or not (and you shouldn’t), this is the legal definition of a `hate crime’. It is also a terrible definition. Why? It has no threshold. The perception doesn’t have to be correct. It doesn’t have to be justified. It doesn’t even have to be reasonable. Any crime committed in the UK can be converted by you, dear reader, into a hate crime. You simply have to report to the police that you believe this crime was committed due to hostility based on race or gender or religion or sexuality – and the beauty of the definition is that your belief explicitly does not require any evidence, reason or logic to support it.
In addition to having no threshold, this legal definition of `hate crime’ also violates the rules of natural justice – under which if A alleges something against B, evidence has to be adduced to support the accusation, the accused can offer a defence, and a neutral judge, jury or panel decides on the truth of the allegations.
Nonetheless, there are lots of people who want to count `hate crimes’, and the statistic seems to have some political weight, despite the fact it is close to meaningless.
We can now make the connection back to jet algorithims. In both cases, the counting problem arises when there is no threshold of `seriousness’ – whether `seriousness’ involve photon energy, a burden of proof, or the seriousness of an allegation. Sensible counting always requires a threshold. If you try and count something with either no threshold or a threshold set at such a low level that it constantly fluctuates, you get nonsense. You measure something – but something that is entirely un-related to what you are trying to measure. This is the case for the pseudo-statistic of ‘numbers of hate crimes’ – which most likely really measures something like social media shares on how to report hate crimes.
(Of course, there are various way this statistic could be converted into a more meaningful one. For example, it could restrict to cases where there is evidence, meeting the legal burden of proof, that a crime was actually motivated by hate.)
There is another topical area – sexual harassment – which is also prone to a similar counting problem. What fraction of people have been sexually harassed? How does this compare to ten years ago? Fifty years ago? If defined properly, such a question is interesting and tells us something useful about society. However, as a counting question its meaningfulness again depends on the use of a consistent threshold and there are many circulating definitions of ‘harassment’ that do not have a lower threshold.
For example, the UK citizens advice bureau says that
“Harassment is when someone behaves in a way which offends you”
Such a broad and open definition is easy to interpret in a way through which every adult could, if they wish, declare themselves to have experienced sexual harassment.
In contrast, the use of more robust thresholds – for example that of sexual assault, as it requires physical contact as a necessary ingredient – would give something which can be meaningfully counted and does result in a usable statistic.
What is the upshot? Counting is subtle, in physics and in life. When you are counting something large and discrete – like potatoes or murders – it is hard to go wrong. However, when you are counting something on a sliding scale that reaches down to infinitesimals, you need a threshold – and if you don’t have one, the number you produce is unlikely to mean anything.