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Symmetry runs through physics and mathematics: a square looks the same after a quarter turn, the laws of physics look the same whichever way you face. Harish-Chandra spent his life on the mathematics of such symmetries, after starting out as a physicist. This post covers his path, explains the core idea with a worked example from Class 12 matrices, and ends with what a student can take from his way of working.
Who was Harish-Chandra?
The Harish-Chandra Research Institute, a Department of Atomic Energy institute in Prayagraj named after him, gives his dates: born 11 October 1923 in Kanpur, died 16 October 1983 in Princeton, New Jersey. He studied at the University of Allahabad, where reading Paul Dirac's book on quantum mechanics drew him to theoretical physics, and he took a master's degree in 1943.
According to the MacTutor biography from the University of St Andrews, he then worked on theoretical physics under Homi Bhabha at the Indian Institute of Science in Bangalore and published joint papers with him. Bhabha and K. S. Krishnan recommended him to Dirac, and in 1945 he went to Cambridge to study for a PhD under Dirac's supervision.
How did a physicist become a mathematician?
His 1947 thesis, published as Infinite irreducible representations of the Lorentz group in the Proceedings of the Royal Society, classified a family of representations of the symmetry group of special relativity. He then went to Princeton as Dirac's assistant for 1947 to 1948.
MacTutor records that in Princeton he became uneasy that his thesis arguments were not rigorous. When he raised this with Dirac, Dirac replied that he was interested in what nature does, not in proofs. Harish-Chandra took this as a sign that his own strengths lay in mathematics, and he moved over. He was on the faculty of Columbia University from 1950 to 1963, which MacTutor calls his most productive period, and joined the Institute for Advanced Study in Princeton in 1963.
He later reflected on why an outsider might see new things. The Harish-Chandra Research Institute quotes him: "I believe that there is a certain fundamental conflict between the two, and knowledge, by advocating caution, tends to inhibit the flight of imagination. Therefore, a certain naiveté, unburdened by conventional wisdom, can sometimes be a positive asset." The "two" are knowledge or experience on one side and imagination or intuition on the other.
What did he contribute?
R. P. Langlands, writing his Royal Society biographical memoir, described him as one of those responsible for turning infinite-dimensional group representation theory from "a modest topic on the periphery of mathematics and physics into a major field central to contemporary mathematics."
The American Mathematical Society awarded him the Cole Prize in 1954, citing in particular his 1951 paper On some applications of the universal enveloping algebra of a semisimple Lie algebra, according to MacTutor. He was elected a Fellow of the Royal Society in 1973 and received the Indian National Science Academy's Ramanujan Medal in 1974. Rebecca Herb, a mathematician who built her own career on his work, wrote in the Bulletin of the AMS that he published over fifty papers, more than a thousand pages, on this subject, and that his papers were dense but complete: "There were no mistakes, and the notation was always the same."
A worked example: rotations as matrices
Here is the simplest case, using Class 12 matrices. A rotation by angle θ in the plane is the matrix R(θ) with rows (cos θ, −sin θ) and (sin θ, cos θ).
Take R(30°), with rows (0.866, −0.5) and (0.5, 0.866), and R(60°), with rows (0.5, −0.866) and (0.866, 0.5). Multiply them:
Top left: 0.866 × 0.5 + (−0.5) × 0.866 = 0.
Top right: 0.866 × (−0.866) + (−0.5) × 0.5 = −0.75 − 0.25 = −1.
Bottom left: 0.5 × 0.5 + 0.866 × 0.866 = 0.25 + 0.75 = 1.
Bottom right: 0.5 × (−0.866) + 0.866 × 0.5 = 0.
The product has rows (0, −1) and (1, 0), which is exactly R(90°). A 30° turn followed by a 60° turn is a 90° turn, and matrix multiplication reproduces that. In general R(α)R(β) = R(α + β), which you can prove with the compound-angle formulas for cos(α + β) and sin(α + β).
The rotations also form a group: R(0) is the identity, R(−θ) undoes R(θ), and combining any two rotations gives another rotation. Harish-Chandra worked on far harder groups, such as the Lorentz group of relativity and the semisimple Lie groups, where the matrices can be infinite in size, but the question is the same: how can this group of symmetries act?
A common mistake is to assume order never matters. For plane rotations R(α)R(β) = R(β)R(α), but for most matrices, and for rotations in three dimensions, AB and BA differ. The Fermi Concept Library covers every concept in mathematics, including matrices and trigonometric identities, if you want to revise the tools first.
What can a student take from how he worked?
Follow the question, not the label. He started in physics and moved to mathematics when he saw where his strengths lay.
Care about rigour. His move began with noticing that his own arguments were shaky.
Write so others can follow. Herb's description of his papers, complete and consistent, is a standard worth copying in your own solutions.
Let a fresh view help you. He believed a beginner's naiveté can sometimes be an asset.
Symmetry also links to conservation laws in physics; read our profile of Emmy Noether and symmetry, and of his teacher Homi Bhabha.
Try it: multiply R(45°) by itself and check that you get R(90°).
