What began in 1914 as elegant, efficient formulas for calculating the irrational constant has resurfaced in cutting-edge research at the Indian Institute of Science (IISc) where physicists have uncovered structures in his work that mirror the mathematics behind today’s high-energy physical theories.
Ramanujan’s work, which predated modern physics by decades is now showing up in logarithmic conformal field theories, a class of mathematical models that describe scale-invariant systems that look the same at different magnifications. Such theories are central to understanding critical phenomena like fluid turbulence, percolation and aspects of black hole physics.
Ramanujan’s legacy: From notebooks to black holes
In 1914, before departing Madras for Cambridge, Ramanujan published a paper containing 17 remarkable infinite series for calculating π. These formulas were notable for their computational efficiency: with only a handful of terms, they produced many accurate digits of pi, far surpassing the techniques of the time.
Over the past century, these formulas have remained important in mathematics and computation, underpinning techniques such as the Chudnovsky algorithm used to compute π to trillions of digits. Yet the question of why such formulas exist boggled mathematicians for decades.
Recent work by a team at IISc, led by professors including Aninda Sinha and researcher Faizan Bhat has shifted focus from computational use to physical meaning. Rather than treating Ramanujan’s formulas as abstract mathematical inventions, the researchers asked whether they might naturally arise from the mathematics of physical systems.
Bridging pure math and physical reality
The IISc team discovered that the same mathematical structures underlying Ramanujan’s π formulas also appear in a broad family of conformal field theories (CFTs), particularly logarithmic CFTs. These theories describe systems that exhibit scale invariance, meaning their behaviour remains consistent regardless of the level of zoom.
Scale invariance is crucial in many physical contexts. For example, water at its critical point where liquid and vapour become indistinguishable displays this symmetry. So do processes such as turbulence in fluids and percolation, and certain theoretical treatments of black holes.
Physicists showed that Ramanujan’s formulae crop up within the equations used in these logarithmic CFTs. By exploiting this shared mathematical backbone, they could compute important quantities in the theories more efficiently, a method that echoes Ramanujan’s own style of deriving precise results from compact expressions.
Implications for physics and mathematics
The study’s findings, published in Physical Review Letters suggest that Ramanujan’s century-old mathematics could be used to accelerate and simplify calculations in modern high-energy physics. This is crucial not only for practical computations but also for understanding the deep connections between number theory and physical law.
While the formulas do not directly solve black hole mysteries like the information paradox, their appearance in the underlying mathematics of the theories means they may offer new tools for exploring complex physical behaviour in extreme environments.
Ramanujan’s legacy thus continues to expand from pure number theory to a “hidden universe” of physical phenomena, where ancient mathematical insights inform tomorrow’s physics.
