A strange pattern running through the equations of physics may reveal something fundamental about the universe or could be a sign that human brains are biased to ignore more complex explanations of reality – or both.
This insight comes from a physicist’s version of Zipf’s law, an observation by linguists that the most common word in a language appears twice as often as the second most common word, three times as often as the third, and so on. In English, for example, the word “the” tends to make up around 7 per cent of any large text, with the next most frequent word, “of”, occurring around 3.5 per cent of the time. What’s more, it turns out that Zipf’s law appears to hold in other situations, such as income distribution or the population of cities.
Now, at the University of Oxford and his colleagues have found that a similar law applies to the symbols used to construct the laws of physics. They looked at three sources of equations: those used in The Feynman Lectures on Physics; a list of equations named after people found on Wikipedia; and a set of proposed equations describing the inflation of the early universe. By treating each symbol and mathematical operator in the equations as a word and ranking their frequency, they could analyse the equations in a similar way to Zipf’s law.
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“You might expect that this [distribution] would differ quite significantly between the three different sets of equations because they come from different places,” says team member at Sorbonne University in France, but to their surprise, that wasn’t the case. Instead, all three sets seemed to fit the same pattern. That wasn’t true when applying the same analysis to randomly generated mathematical expressions.
Exactly what it means for the equations of reality to follow this pattern is unclear. One possible explanation is that it is saying something about how reality actually works, says at the University of California, San Diego. Each equation by itself is extraordinarily accurate at predicting real facts about the universe, so perhaps the overarching pattern that they follow also contains information about the nature of reality, says Chen. “We have invented language, mathematics and symbols for many purposes, but it turns out that physics, or nature, uses only some of the simplest of those.”
Constantin sees supporting evidence for this in the consistency of the pattern they found, even for symbols that occur rarely. “Operators that appear not so frequently – the exponential and the log and the hyperbolic functions and trigonometric functions – all follow the same law. This is surprising,” he says.
Bartlett, however, thinks the result could simply be a statistical by-product of physicists aiming to express their ideas succinctly, rather than revealing anything fundamental about the universe – an explanation also put forward for Zipf’s law. “You want to communicate as much information as possible with as few symbols or in as little time as possible, and the same is sort of true with equations in physics,” he says. “We create operators that we know are useful.”
Chen says that both of these scenarios could be true at the same time. It could also reflect the way that the human brain works, he says, such as preferring simple explanations that can help predict the world. “The tendency is one that leads us to ignore more complex explanations,” he says.
Bartlett and Constantin hope that, regardless of interpretation, the finding could help guide future machine-learning models to discover new laws of physics. “It should make it more efficient, because it’s not having to search through equations that we would know would not stand a chance of being physically realistic,” says Bartlett.
arXiv
