# Dirac, Harish-Chandra and the unitary representations of the Lorentz group

Mukunda, N (1993) Dirac, Harish-Chandra and the unitary representations of the Lorentz group. In: Current Science, 65 (12). 936-940.

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## Abstract

Harish-Chandra was born in Kanpur in Northern India on October 11, 1923 and had his early education there. At the University of Allahabad, he did his B.Sc. in 1941 and his M.Sc. in 1943. Harish-Chandra worked as a postgraduate research fellow on problems in theoretical physics, under Homi Bhabha, at the Indian Institute of Science at Bangalore in Southern India. Bhabha had been a student of P. A. M. Dirac in Cambridge, England, in the 1930s. Their joint work was largely inspired by ideas of Dirac \ref[Proc. Roy. Soc. London Ser. A 167 (1938), 148--169; Zbl 23, 427], and they extended some of his results. Around 1945 both H. J. Bhabha and Harish-Chandra's teacher, K. S. Krishnan at Allahabad University, recommended him to Dirac for further research work at Cambridge, in particular, to be guided towards a Ph.D. degree. Dirac suggested that he analyse the irreducible unitary (necessarily infinite-dimensional, except in the trivial case) representations of the homogeneous Lorentz group ${\rm SO}(3,1)$ and of its universal covering group ${\rm SL}_2(\bold C)$, the special linear group of two-by-two matrices with complex entries. This topic was given in order understand previous ad-hoc constructions by Dirac of (reducible) unitary representations of ${\rm SO}(3,1)$. According to Dirac, his ideas were inspired by the analysis of the harmonic oscillator problem in quantum mechanics in the work of V. Fock. In quantum physics unitary representations of the various symmetry groups had begun to play a significant role. Harish-Chandra succeeded in solving the problems posed to him, and was awarded his Ph.D. in 1947. He gave a complete classification of the irreducible unitary representations of ${\rm SL}_2(\bold C)$. The shift from the Lie group to its Lie algebra plays a role, algebraic methods are used, and the notion of a (nowadays called) minimal $K$-type occurs, $K={\rm SU}(2)$ a maximal compact subgroup. Independently, the same results were obtained by I. M. Gel\cprime fand and M. A. Na\u\i mark [resp. V. Bargmann]. In 1947, Dirac went to the Institute for Advanced Study in Princeton, USA, and Harish-Chandra went with him as his assistant. It was at that time that his interests changed over from physics to mathematics, and he began to build his fundamental theory of infinite-dimensional representations of semi-simple and reductive groups. This is closely related to Harish-Chandra's achievements in the theory of harmonic analysis on these groups and their homogeneous spaces. This paper gives a detailed account of the early period in Harish-Chandra's professional career. It focuses on the sources for the problem posed to him by Dirac, discusses previous works, notably by E. Majorana in 1932 and Dirac, and gives a short account of the ingredients of the classification of the irreducible unitary representations of ${\rm SL}_2(\bold C)$.

Item Type: Journal Article Division of Physical & Mathematical Sciences > Centre for Theoretical Studies 31 Aug 2004 27 Aug 2008 10:57 http://eprints.iisc.ernet.in/id/eprint/950