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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

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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

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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

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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

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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

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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

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'`

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

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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.

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46
BIOGRAPHICAL MEMOIRS
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VALENTINE BARGMANN
1948
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