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VALENTINE BARGMANN April 6, 1908-July 20, 1989 BY JOHN R. KLAUDER LET S ASK BARGMANN! With that phrase aciciressec! to me by my Princeton thesis Divisor I was lee! to my first real encounter with Valentine Bargmann. Our ques- tion pertainec! to a mathematical fine point clearing with quantum mechanical HamiTtonians expressed as clifferen- tial operators, en c! we got a prompt, clear, en c! clefinitive answer. Valya the common nickname for Valentine Bargmann was aireacly an establishec! en c! justly renownec! mathematical physicist in the best sense of the term, en c! his acivice was wiclely sought by beginners en c! experts alike. It was, of course, a thorough preparation that brought Valya to his well-cleservec! reputation. Valya Bargmann was born on April 6, 190S, in Berlin, en c! stucliec! at the University of Berlin from 1925 to 1933. As National Socialism began to grow in Germany, he mover! to SwitzerIancI, where he receiver! his Ph.D. in physics at the University of Zurich uncler the guidance of Gregor Wentzel. Soon thereafter he emigratec! to the Uniter! States. It is noteworthy that his passport, which wouIc! have been revoke! in Germany at that time, hac! but two clays left to its valiclity when he was acceptec! for immigration into the Uniter! States. He soon joiner! the Institute for Advance c! 37

38 BIOGRAPHICAL MEMOIRS Stucly in Princeton, en c! in time was acceptec! as an assistant to Albert Einstein. Along with Peter Bergmann, Bargmann analyzed five-di- mensional theories combining gravity en c! electromagne- tism at a classical level. During WorIc! War II, Bargmann worker! on shock wave studies with John von Neumann en c! on the inversion of matrices of large dimension with von Neumann en c! Deane Montgomery. Bargmann taught infor- mally at Princeton beginning in 1941. He receiver! a regu- lar appointment as a lecturer in physics in 1946 and re- mained at Princeton essentially for the rest of his career. Bargmann worker! with Eugene Wigner on relativistic wave equations en c! together they clevelopec! the justly famous Bargmann-Wigner equations for elementary particles of ar- bitrary spin. In 1978 Bargmann en c! Wigner jointly receiver! the first Wigner Mecial, an aware! of the Group Theory en c! Funciamental Physics Foundation. Besicles this honor, Bargmann was electec! to the National Academy of Sciences in 1979 and won the Max Planck Medal of the German Physical Society in ~ 988. Valya was a gentle en c! moclest person en c! he was a taTentec! pianist. At social occasions it was not uncommon for Valya to perform solo or accompany other musicians. His lectures were renownec! for their clarity en c! polish. Among the prizes! series of lectures were those on his ac- knowledged specialties, such as group theory (e.g., the Lorentz group en c! its representations, en c! ray representa- tions of Lie groups), as well as seconc! quantization. In math- ematics, his most influential work was on the irreclucible representations of the Lorentz group. This work has server! as a paradigm for representation theory ever since its ap- pearance. Bargmann also macle important contributions to several aspects of quantum theory. He was a stellar example of the European tradition in mathematical physics in the

VALENTINE BARGMANN 39 spirit of Hermann WeyT, von Neumann, en c! Wigner. A book in Bargmann's honor offers expert comments on a number of topics clear to the heart of Valya.i Bargmann hac! his less serious sicle as well. No better example of that can be given than the story toIc! by Gerarc! G. Emch, who arriver! at Princeton in ~ 964 to begin a postcloctoral year with Valya. Emch also arriver! with a newly minter! "theorem," which he proucIly presented to the mas- ter. No sooner hac! the theorem been lair! out than Bargmann was really with a counterexample. Sorely clisappointecI, Emch retreated for home that clay en c! continues! to stucly the matter. At 3 a.m. Emch's phone range. The caller, Bargmann, heartily laughed when Emch quickly picked up the phone. He then saicI, "I thought you wouIc! still be up. Go to bee! en c! get some sleep. I have fount! an error in my counter- example. We can discuss it tomorrow!" Valya Bargmann publisher! a moclest number of papers by contemporary stanciarcis, but he nevertheless was instru- mental in opening several distinct fielcis of investigation. His paper on establishing a limit on the number of bounc! states to which an attractive quantum mechanical potential may leac! has spawner! a minor industry in the research on such issues. His paper clearing with distinct potentials that exhibit iclentical scattering phase shifts reclirectec! research in inverse scattering theory in which it hac! been previously assumed that the scattering phase shifts would uniquely char- acterize the potential. His stucly of the unitary irreclucible representations of the noncompact group SL (2,R) have prover! not only invaluable in their own right but have servec! as a moclel of how such representations are to be sought for more general noncompact Lie groups. Shortly after com- pleting the work on SL(2,R) he also completed a manu- script on the relater! group SL(2,C). This work, however, was never publisher! because inclepenclent work by Israel

40 BIOGRAPHICAL MEMOIRS Gel'fanc! ant! Mark Naimark covering the same grounc! reacher! the publisher aheac! of Bargmann's planner! sub · ~ mission. To a large extent, Bargmann tenclec! to write either short notes or long, extensive articles. When he felt he really hac! something to say it seems he would become cliciactic, thor- ough, and complete. Thus, his papers on SL(2,R) and the factor representation of groups were both long papers by Bargmann's stanciarcis. However, he saver! his longest en c! most sustainer! stucly until the 1960s, when he clealt with one of the subjects for which he will long be remembered. It is to this set of papers en c! a brief sketch of some of their principal novelty that I wouic! like to turn my attention at this point. I have chosen to outline two mathematical argu- ments, because on the one hanc! they are relatively simple en c! on the other hanc! they are universal en c! profound. From 1961 onward, Valya publisher! several papers cleal- ing with the foundations en c! applications of Hilbert space representations by holomorphic functions now commonly known as Bargmann spaces (or sometimes as Segal-Bargmann spaces in view of an essentially parallel analysis of the main features by Irving Segal). We can outline a few of the prin- cipal icleas in such an analysis by first starting with the fol- lowing backgrounc! material. The basic kinematical opera- tors in canonical quantum mechanics for a single degree of freedom may be taken as the two Hermitian operators Q en c! P. which obey the funciamental Heisenberg commuta- tion relation [Q,P]-QP-PQ = ihI c, ill, where li denotes Planck's constant h/2lrv, and where I de- notes the unit operator. In the Schroclinger approach to quantum mechanics these operators are represented as

VALENTINE BARGMANN 41 Q ~ tic and P ~ -iliO/a~c acting on complex-valued func- tions ~'c) that belong to the Hilbert space L2(R, deck com- posec! of those functions for which I ooV(x) ~ ~(X)4X < an, Given any two such functions ~ en c! if, which are sufficiently smooth en c! vanish at infinity, then the Hermitian character of tic en c! -Lana follow from the properties that | [x¢(x) ] ~ ~(x)dx =| ¢(x) ~ xv(x)dx, _oo _oo and Joo [-ih~(x)/ Ox] ~ v(x)dx _oo = - itch yr(X) Woo + I ¢(X)* (-i~)ayt(x)/ ax do ~ _oo = J ¢(x, ~ [-i~av / ax ]dx. _oo An elementary yet important alternative combination of the basic operators Q en c! P is given by A_(Q+iP)/:, Al_(Q-iP)/:, where AT denotes the Hermitian acljoint of the operator A. The basic commutation relation between Qanc! P then leacis to [A,At]=AAt-AtA=I, which is often chosen as an alternative starting point. In- cleecI, VIaclimir Fock in 1928 recognizec! that this form of the commutation relation may be represented by the ex

42 BIOGRAPHICAL MEMOIRS pressions A ~ O/Oz en c! AT ~ z acting on a space of analytic functions f~zJ clefinec! for a complex variable z. While it is true that this representation of the operators A en c! AT sat- isfies the correct commutation relation, it is unclear how z en c! O/Oz can be consiclerec! acljoint operators to one an- other, especially in light of the fact clemonstratec! above that Ice = tic en c! `-i~a/a~ = `-i~a/a~' hole! in the Schroclinger representation. Bargmann was among the first to show clearly how one can justify the relation zi = O/Oz. To that enc! Bargmann clefinec! a Hilbert space F of holomorphic functions f~zJ restrictec! so that Jam Jam ff )*f(Z)e-~Z~24X4Y<oo _oo _oo where ~z~2 = adz, z = tic + iy, and the domain of integration is over the plane R2. We observe that a general function of tic en c! y can be viewoc! as a (relatecI) general function of z = tic + iy and zig= ~c- iy. A holomorphic function gig) depends on only one such variable, z, or stated otherwise Og~z)/Oz~ _ O along with the complex conjugate relation Og~z)~/Oz _ O. Let f en c! g be two holomorphic functions en c! consider the following integral (with the limits implicit) ,f,ltzg~z)] * faze dx dy = ,f,fg~z) * z * faze dx dy = ~ ~ g(~z) * f(~z)(~-d / Oz e- ~ dx dy - .1 .1 = ,[,[ (d / Oz)Eg~z) f(~z)]dx dy e-~z~2 = ~ ~ gyp) * Elf (z) / ~z]e~ ~ dx dy,

VALENTINE BARGMANN 43 which shows that zi = O/Oz as required. With the introcluc- tion of the given inner product for two holomorphic func- tions, Bargmann put the heuristic notion that z en cl O/Oz were acljoint operators onto a firm mathematical founcia- tion. All separable en c! infinite-climensional Hilbert spaces are isomorphic. The relation between the space L2(R, deck en c! the space F can be given in the following form. To each L2 associate the expansion 00 yr(x) = ~,anhn(x), n=0 Joo an = hn (x)yr(x)dx _00 in terms of the complete, real, orthonormal set of Hermite functions {hn('c)} Oeach element of which is implicitly cle- finec! by the fact that 00 exp(_s2 + 2sx-V2 x2 ) = pi/4 ~, (n! )-1/2 In hn (X). n=0 Moreover, 00 Elan 12=J~INJ('C)I2 d'`<oo. n=0 Clearly, ~ ~ L2 uniquely determines the sequence {an} O. To each such sequence we associate the holomorphic func- tion 00 f (Z) = ~ anon / An!, n=0 which converges absolutely for all z. In a cleciclec! advance, Bargmann was able to put this association to an expanclec! use as follows. The space of temperer! test functions consists of those complex functions u('cJ such that u ~ Chancy or dsu('`~/~'ts

44 BIOGRAPHICAL MEMOIRS ~ O as tic ~ + so for all nonnegative r and s. Each such function is realized by the expansion 00 u('c) = ~bnhn('`)7 n=0 where nrb ~ O as not so for all r (i.e., bn falls to zero faster than any inverse power). Dual to the space of tempered test functions is the space of tempered distributions, a special class of generalized functions. If D('cJ denotes such a gener- alized function, then D admits the formal expansion 00 D(x)= ~ c! h (x), n = 0 where {an} O is a sequence of polynomial growth (i.e., Edna < R + Sn for suitable R and S). Generally, D is only a gener- alized function (e.g., D( ~c) = ~ ( tic - y) when dn = hn(y), etc. ), not an ordinary function, and, although the left-hand side of the relation 00 Ir budn = | u(X) ' D(x)dx n=0 is well defined, the right-hand side does not exist as a tradi tional integral. Bargmann realized, however, that the action of tempered distributions on test functions did indeed possess a genu- ine integral representation in terms of holomorphic func- tions. For that purpose let 00 u(Z) _ Ir bnZn / n=0 denote the image of u('cJ in F. Furthermore, we define 00 D(z)->~ dnZ / ~ n=0 as the image of the generalized function D('c). Since Idnl < R

VALENTINE BARGMANN 45 + En it follows that the series cleaning D(zJ converges every- where en c! thereby clefines a holomorphic function. More- over, it follows that 00 00 | | u(z) ~D(z)e IZl dxdy/ in= 2,bndn . _00 _00 n=0 Thus, the holomorphic function representation enclowoc! with the Bargmann inner product provides an explicit inte- gral representation for the action of an arbitrary temperer! distribution, a feature entirely unavailable in the usual form of generalizes! functions en c! formal integrals. The discussion just concluclec! regarding two topics cleal- ing with holomorphic function spaces illustrates the pen- etrating simplicity of Bargmann's approach to mathemati- cal physics. Would that we had more like him today, I occasionally miss the opportunity to "ask Bar~mann." a THANKS ARE EXTENDED to Gerard G. Emch, Elliott H. Lieb, Barry Simon, and Arthur S. Wightman for their input and/or comments that have found their way into this article. NOTE 1. E. H. Lieb, B. Simon, and A. S. Wightman, eds. Studies in Mathematical Physics, Essays in Honor of Valentine Bargmann. Princeton, N. J .: Princeton University Press, 1976.

46 BIOGRAPHICAL MEMOIRS SELECTED BIBLIOGRAPHY 1934 Uber den Zusammenhang zwischen Semivoktoren and Spinoren und die Reduktion der Diracgleichung fur Semivoktoren. Helv. Phys. Acta 7:57-82. 1936 Zur Theorie des Wasserstoffatoms. Z. Phys. 99:576-82. 1937 Uber die durch Elektronenstrahlen in Kristallen angeregte Lichtemission. Helv. Phys. Acta 10:361-86. 1941 With A. Einstein and P. G. Bergmann. On the five-dimensional rep- resentation of gravitation and electricity. In Theodore con Barman Anniversary Volume, pp. 212-25. Pasadena: California Institute of Technology. 1944 With A. Einstein. Bivector fields. Ann. Math. 45:1-14. 1945 On the glancing reflection of shock waves. Applied Mathematics Panel Report No. 108. 1946 With D. Montgomery and J. von Neumann. Solution of linear sys- tems of high order. Report to the Bureau of Ordinance, U. S. Navy. 1947 Irreducible unitary representations of the Lorentz group. Ann. Math. 48:568-640.

VALENTINE BARGMANN 1948 47 With E. P. Wigner. Group theoretical discussion of relativistic wave equations. Proc. Natl. Acad. Sci. U. S. A. 34:211-23. 1949 Remarks on the determination of a central field of force from the elastic scattering phase shifts. Phys. Rev. 75:301-303. On the connection between phase shifts and scattering potential. Rev. Mod. Phys. 21:488-93. 1952 On the number of bound states in a central field of force. Proc. Natl. A cad. Sci. U. S. A. 38:961-66. 1954 On unitary ray representations of continuous groups. Ann. Math. 59: 1-46. 1959 With L. Michel and V. Telegdi. Precession of the polarization of particles moving in a homogeneous electromagnetic field. Phys. Rev. Lett. 2:435-36. 1960 Relativity. In Theoretical Physics in the Twentieth Century (Paul) Memo- rial Volume), eds., M. Fierz and V. F. Weisskopf, pp. 187-98. New York: Interscience Publishers. With M. Moshinsky. Group theory of harmonic oscillators. I. The collective modes. Nu cl. Phys. 18:697-712. 1961 With M. Moshinsky. Group theory of harmonic oscillators. II. The integrals of motion for the quadrupole-quadrupole interaction. Nu cl. Phys. 23:177-99. On a Hilbert space of analytic functions and an associated integral transform. Part I. Commun. Pure Appl. Math. 14:187-214.

48 BIOGRAPHICAL MEMOIRS 1962 On the representations of the rotation group. Rev. Mod. Phys. 34:829- 45. 1964 Note on Wigner's theorem on symmetry operations. 7. Math. Phys. 5:862-68. 1967 On a Hilbert space of analytic functions and an associated integral transform. Part II. A family of related function spaces application to distribution theory. Commun. Pure Appl. Math. 20:1-101. 1971 With P. Butera, L. Girardello, and J. R. Klauder. On the complete- ness of the coherent states. Rep. Math. Phys. 2:221-28. 1972 Notes on some integral inequalities. Helv. Phys. Acta 45:249-57. 1977 With I. T. Todorov. Spaces of analytic functions on a complex cone as carriers for the symmetric tensor representations of SO(n). f. Math. Phys. 18:1141-48. 1979 trinnerungen eines Assistenten Einsteins. Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, Jahrgang 124, Heft 1, pp.39-44. Zurich: Druck und Verlag Orell Fussli Graphische Betriebe AG.