Our favorite constant, Pi (π), Description of the relationship between the circumference of a circle and its diameter, has taken on a new meaning.
The new representation is the result of twists in string theory and attempts by two mathematicians to better describe particle collisions.
“Our initial efforts were not aimed at finding a way to look at Pi,” says Aninda Sinha of the Indian Institute of Science (IISc), who co-authored the new work with her colleague Arnab Priya Saha of IISc.
“We were just studying high-energy physics in quantum theory and trying to develop a model with fewer and more precise parameters to understand how particles interact. We were excited when we got a new way to look at pi.”
Because pi is a mathematical constant, its value has not changed, no matter how irrational the number may be. Over time, we have simply gotten more precise information about its exact value, most recently reaching the number 105 trillion.
This new work by Saha and Sinha postulates a new series representation of Pi, which they say provides a simpler way to extract Pi from calculations used to decipher the quantum scattering of high-energy particles hurled around in particle accelerators.
In mathematics, a series specifies the components of a parameter such as pi so that mathematicians can quickly determine the value of pi from its constituent parts. It is like following a recipe, adding each ingredient in the right amount and order to make a tasty dish.
Unless you have the recipe, you don’t know what ingredients a meal consists of or how much to add and when.
Finding the right number and combination of components to represent pi has puzzled researchers since the early 1970s, when they first tried to represent pi this way, “but quickly gave up because it was too complicated,” Sinha explains.
Sinha’s group investigated something completely different: ways to mathematically represent the interactions of subatomic particles with as few and simple factors as possible.
Saha, a postdoctoral fellow in the group, addressed what he called the “optimization problem,” trying to describe these interactions – which create all sorts of strange and hard-to-detect particles – in terms of different combinations of mass, vibrations and the wide range of their irregular motions, to name a few.
A so-called Feynman diagram helped to solve the problem. It represents the mathematical expressions that describe the exchange of energy between two particles that interact and scatter with each other.
This led not only to an efficient model of particle interactions that captured “all essential string properties down to a certain energy,” but also to a new formula for pi that closely resembles the first series representation for pi in recorded history, proposed in the 15th century by the Indian mathematician Sangamagrama Madhava.
The findings are purely theoretical at this point, but could be of practical use.
“One of the most exciting prospects of the new representations in this paper is to use appropriate modifications of them to re-examine experimental data on hadron scattering,” Saha and Sinha write in their published paper.
“Our new representation will also be useful in the context of sky holography,” the pair add, referring to an intriguing but still hypothetical paradigm that attempts to reconcile quantum mechanics with general relativity through holographic projections of spacetime.
For the rest of us, it is a consolation that researchers can now describe in more detail what exactly the famous irrational number is made of.
The study was published in Physical examination letters.