Researchers at the Indian Institute of Science in Bengaluru have linked Srinivasa Ramanujan's over-a-century-old formulas for pi to contemporary physics, including turbulent fluids and the universe's expansion. Their work, published in Physical Review Letters, reveals unexpected bridges between Ramanujan's intuitive mathematics and conformal field theories. This discovery highlights how pure math can mirror real-world physical phenomena.
Earlier this month, Aninda Sinha, a professor at the Indian Institute of Science in Bengaluru, and his former doctoral student Faizan Bhat published a paper in Physical Review Letters that connects Srinivasa Ramanujan's esoteric mathematics to the physics of turbulent fluids and the expansion of the universe. The key link is π, the irrational number approximately 3.14159265, central to geometry and computations.
Over a century ago, Ramanujan, then an accountant in Chennai, discovered at least 17 distinct infinite series for 1/π. These formulas converge rapidly, with each additional term dramatically improving accuracy. Some underpin the Chudnovsky algorithm, enabling computations of π to over 200 trillion digits on supercomputers.
Sinha explained, “We were interested in the maths behind Ramanujan’s thinking.” Their investigation began in string theory, a framework positing that fundamental particles arise from vibrations of tiny energy strings. While reviewing string-theoretic calculations, they identified incomplete literature results and derived an infinite number of new π formulas.
Sinha noted that strings, like rubber bands, can be stretched in various ways, embedding π in multiple representations. This led them to recognize parallels between Ramanujan's modular equations, elliptic integrals, and special functions and structures in conformal field theories (CFTs). CFTs describe critical phenomena, such as the point where water at 374°C and 221 atm becomes a superfluid, indistinguishable as liquid or gas.
“At the critical point, you cannot actually say which is liquid and which is vapour,” Sinha said. The Ramanujan equations match correlation functions in logarithmic CFTs, forming a bridge between number theory and physics.
Bhat stated in a press release, “[In] any piece of beautiful mathematics, you almost always find that there is a physical system which actually mirrors the mathematics. Ramanujan’s motivation might have been very mathematical, but without his knowledge, he was also studying black holes, turbulence, percolation, all sorts of things.”
Historical precedents abound: 19th-century Riemannian geometry later underpinned Einstein's general relativity, used in GPS today. Fourier transforms, developed for heat flow analysis, now enable digital compression.
Currently, this connection inspires new inquiries in Sinha's group, appearing in expanding universe models. It also suggests efficient representations for other transcendental numbers rooted in physics, though it does not yet resolve major conjectures in number theory or cosmology.