Paul Weiss and the genesis of canonical quantization

19 Pages·0.752 MB·English

by  Rickles D., Blum A.

Eur. Phys.J. H 40, 469–487 (2015) THE EUROPEAN
DOI:10.1140/epjh/e2015-60001-5
PHYSICAL JOURNAL H
Paul Weiss and the genesis of canonical
quantization
Dean Rickles1,a and Alexander Blum2
1 Unit for HPS,University of Sydney,2006 Sydney,Australia
2 Max Planck Institutefor theHistory of Science, 14195 Berlin, Germany
Received 15 January 2015 / Received in finalform 14 August 2015
Published online 6 October2015
(cid:2)c EDP Sciences, Springer-Verlag 2015
Abstract. This paper describes the life and work of a figure who, we
argue, was of primary importance duringtheearly yearsof field quan-
tisation and (albeit more indirectly) quantum gravity. A student of
Dirac and Born, he was interned in Canada during the second world
war as an enemy alien and after his release never seemed to regain a
goodfootholdinphysics,identifyingthereafterasamathematician.He
developed a general method of quantizing (linear and non-linear) field
theories based on the parameters labelling an arbitrary hypersurface.
This method (the ‘parameter formalism’ often attributed to Dirac),
though later discarded, was employed (and viewed at the time as an
extremelyimportanttool)bytheleadingfiguresassociatedwithcanon-
icalquantumgravity:Dirac,PiraniandSchild,Bergmann,DeWitt,and
others. We argue that hedeserves wider recognition for this and other
innovations.
1 Who was Paul Weiss?
Paul Weiss (1911–1991) was a brilliant student of Max Born, and later Paul Dirac,
atCambridgeUniversity –indeed, becoming Dirac’sfirstdoctoralstudentas a result
of Born’s leaving Cambridge to take up the Tait Chair of Natural Philosophy at
Edinburgh. Born wrote of him that “he is one of the most learned students I have
evermetinmyfield”andthat“[h]isknowledgeandthethoroughnessofitisamazing,
in physics and mathematics” (letter to Walter Adams, then general secretary of The
SocietyfortheProtectionofScienceandLearningandlaterdirectoroftheLSE,dated
4thFeb,1937:MS.SPSL 286/2)1.Dirac wrotethatWeiss“hasanunusualabilityfor
a e-mail: [email protected]
1 In this recommendation letter, Born goes on to point out that Weiss is so modest that
it is difficult to find this out without knowing him a long time. He also notes, in the same
letter,thathewillbedeliveringaseriesoflecturesattheInstitutPoincar´e(inApril1937),in
which the“main source... willbethisdissertation ofWeiss”(ibid.).Despitethis,however,
Born goes on to state that “[Weiss] is not a genius” but is rather “a very good scholar”.
BornalsobrieflymentionsWeissinhisautobiography,pointingoutthathehadearlierbeen
a research assistant to Fritz Haber ([6], p. 267). (Note that theSPLP archive is held in the
Bodleian Library in Oxford:Weiss’s files are in MS. S.P.S.L. 286/2 and 441/1).
470 The European Physical Journal H
Fig. 1. Paul Weiss (1911–1991). This is the only photograph we have been able to find of
Weiss, from his obituary [image source: SIAM News 24(3): 2 and 6; note that this is the
original photograph, supplied to SIAM News by Prof. Leon Brown – our thanks to Mike
Cowan for making thisavailable to us].
research” (letter to SPSL dated 29.11.36:MS. SPSL 286/2).High praise from two of
the ‘high priests’ of physics; making it more curious that Weiss’s achievements seem
to have been largely forgotten by history.
Despite this high praise, then, Weiss’s is not a name known to most physicists,
nor to historians of physics2. However, as we shall aim to demonstrate in this paper,
he was responsible for some of the pivotal concepts involved in the canonical quan-
tization of gravity (and field theories in general) and was in fact duly acknowledged
in the founding papers and discussions on this approach. However, it does not seem
that Weiss was directly interested in gravitation in his own research, and was more
interested in the general formal structures of physical theories from a mathematical
pointofview–thoughweshouldnotethatWeissdidreviewseveralpapersongeneral
relativity (and even quantum gravity) for Mathematical Reviews and demonstrated
that he had a good understanding of both areas (see, e.g., [41]). Beyond canonical
quantumgravity,someofWeiss’sinnovativetechniqueshavebecomepartofthestan-
dard toolkit of practising physicists. In this paper we hope to draw attention to this
work and to the man in a bid to reverse the neglect.
Weiss was born in Sagan (Silesia) on April 9th, 1911, of Jewish descent. His
Germancitizenshipnonethelessmade himan‘enemyalien’inthe UK duringthe war
years,resultinginhisinternmentinCanada,alongwithmanyothernotablephysicists
and mathematicians (including Hermann Bondi, Klaus Fuchs, Tommy Gold, Walter
Heitler, and Beniamino Segr`e)3. He had been a pupil of Max Born’s in Go¨ttingen
2 Indeed,theonlyreferencewehaveseentohiminthehistoricalliterature(asopposedto
citations to his work, that is) is in a recent article [28] in which the author provides a brief
account of various scientists (Weiss included) interned during the war, and the work of the
SocietyfortheProtectionofScienceandLearning[SPSL]insecuringtheirrelease(onwhich
more below). There is, however, a very brief appreciation of Weiss in [18].
3 This was despitehis well-documented and well-attested willingness to fight and work in
the service of the British military. Wereturn to Weiss’s internmentbelow.
D. Ricklesand A.Blum: Paul Weiss and thegenesis of canonical quantization 471
(from 1929–1930andthen 1931–1933)– fromSeptember 1930to April 1931he spent
a ‘year out’ as an assistant teacher in the German public school system, at the Freie
Schul- und Werkgemeinschaft Letzlingen (between 1931 and 1933 he also spent some
time studying in Paris and Zurich). Born clearly thought enough of him to have him
do a PhD with him at Cambridge – of course, it is highly likely that Hitler’s rise to
powerhadasmuchtodowithWeiss’sdecisiontoleaveforEnglandsinceitseemsthat
his immediate family (certainly his sister and mother) moved to Oxford at around
the same time (as did Born himself). Weiss arrivedin Cambridge (Downing College)
to work with Born in October 1933 and lived close to Born (at Born’s suggestion,
writing from Bolzanoin 1933:[BORN 1/3/2/5]),with Born’s address 246 Hills Road
and Weiss’s address 23 Hills Road4.
Weiss received his PhD in 1936 with a thesis entitled The Notion of Conjugate
VariablesintheCalculusofVariationsforMultipleIntegralsanditsApplicationtothe
Quantisation of Field Physics (a rather largethesis, containing a thoroughliterature
review, but of which an abbreviated version was published as [39]). He stayed on in
Cambridge after receiving his degree for a further 2 years, lecturing on QED at the
Cavendish Laboratory during the academic year 1937–1938.
Born expressed his puzzlement over where Weiss was getting his funds from to
sustain himself after he had completed his Ph.D. Weiss seems to have been from a
very wealthy, industrialist family. There was a connection between the Weiss and
the Tugendhat families5. Hans Weiss, Paul Weiss’s brother, had been married to
Grete L¨ow-Beer, who later married Fritz Tugendhat. There remained a connection
between the families since the firstmarriageresulted in a daughter,Hanna.Both the
Tugendhat’s and the Lo¨w-Beers were famous Brno industrialist families.
Weiss secured a position at Queens University, Belfast, at 300 pounds per annum
for a periodof two terms (January 1938–September1939).PaulEwaldwas in charge
of the department on Weiss’s arrival, and it seems that Weiss was put in charge of
the department while he was there:
Well,thenicearrangementtheyhadinBelfastwastoleaveMass’sassistant,
S. F. Boys, and a German refugee, Paul Weiss, in charge of the department
after having appointed me, so that I could just be in the department and see
howtheyhandleditandwhatwasgoingonandprepare.Thiswasreallyavery
wise and good idea. Paul Weiss is now here in the United States, I think with
G.E.oranywayinSyracuseorthereabouts.Hewasaverygoodmathematician
and physicist6.
AccordingtoaletterEwaldsenttotheSPSLin1940(17thAugust,[MS.SPSL213]),
Weiss lectured on mathematical mechanics for at least one term (for six hours each
week). He also wrote a large paper on quaternions [40] while he held this position,
presenting this workatthe IrishAcademy.This paper involvesa conversionoftensor
equations of special relativity into quaternion equations. By using angular variables
(propertime andretardeddistance),heis abletoderiveaquaternionicversionofthe
classicalequationsofmotionforaradiatingcharge(doneusinghisbelovedvariational
principle techniques).
It was just a very short time after this post in Belfast expired that Weiss was
interned as an enemy alien, after his return to Cambridge. Weiss’s passport was due
4 Though in a letter to the SPSL, dated 11th October 1939, he states that he arrived in
England at theendof August.Hewas awarded an unconditionalpermit on 26th July1939,
so this is entirely possible. Recall that Hitler was elected chancellor of the coalition at the
end of January, 1933.
5 The same Tugendhat family that commissioned Mies van der Rohe to build the Villa
Tugendhat in Brno, and after which thefamous “Tugendhat” chair was designed.
6 https://www.aip.org/history-programs/niels-bohr-library/oral-histories/4596-2
472 The European Physical Journal H
to expire November 1939,thoughhe receivedunconditionalpermissionto stay in the
UK onJuly 1939.Ina letter dated5th September 1939,Weiss had alreadyexpressed
a desire (to the SPSL) to do national service. He did not apply for naturalisation
there and then since he was waiting to find permanent work. He was interned May
12th 1940.
2 Interment and the society for the protection of science
and learning
LikemanyotherGermanacademics,PaulWeiss’s life duringWorldWarII wasmade
considerably less painful by the work of The Society for the Protection of Science
and Learning.This society began, initially envisagedas a temporary venture, as The
Academic Assistance Council [AAC], in May 1933. In the words of Rutherford, then
president of the AAC, its aim was “to assist scholars and scientists who, on grounds
ofreligion,race,oropinion,wereunableto continuetheirworkintheir owncountry”
([33],p.607).Theassistancecamenotintheformofwelfarebutasakindofclearing
house for finding employment for the displaced scholars so that they might continue
to contribute to the deeper “common cause of scholarship” (ibid.). In 1936 the AAC
was established on a permanent basis, with the change of title to SPSL7 and an
expansiontoincluderesearchfellowshipsforexceptionalrefugeescholars.Thesociety
was funded via subscriptions and donations.
Despite being a ‘category C’ refugee (a genuine refugee, not to be viewed as an
enemy or interned in the event of war), Weiss was interned, with other class Cs. It
is possible that there was public pressure, no doubt exacerbated by hysterical news
items (see [24], p. 79). There were several initial camps in the UK that were used,
including the Isle of Man.
Weiss was interned on May 12th 1940 while he was visiting Cambridge. He was
releasedlaterthesameyear,inDecember,andreturnedtoEnglandsoonafter,taking
upatemporaryposition(thatwouldbe madepermanent)atWestfieldCollegeinthe
University of London, in February 1941. He finally became naturalised as a British
citizen in June 1947.
The very process of internment overseaswas not without its dangers.On the 2nd
ofJuly1940theSSArandoraStar wastorpedoedbytheGermans,whilecarrying700
internees bound for Australia (and while it flew a POW flag that the Allies naively
believed would protect it from destruction). Weiss was aboard the SS Ettrick, the
shipthattookHermannBondi,KlausFuchs,E.WalterKellermann,andMaxPerutz
across8. They arrived on 13th July in Quebec, were they were apparently greeted
as enemies, due to inadequate political education. Tommy Gold recalled that the
conditions on the ship were dire:
Densely packed like sardines there, three layers, on the floor and on the
tables, hammocks up above, and in the whole place, I don’t know, there were
maybe ten toilets for 800 people, and they all had dysentery9.
During his internmentWeiss spenthis time intwo camps:L and N.In campL (Cove
Fields, Quebec,overlookingthe St Lawrence)it seems that the living standardswere
7 Reference [42] offers an excellent treatment tracing this transition, from AAC to SPSL.
Since 1997 it has been known as CARA: The Council for Assisting Refugee Academics –
though in 2014 the acronym was reinterpreted to mean “Council for At-Risk Academics”.
8 Klaus Fuchs’ internment experience is briefly discussed in the first chapter of [26]. For
Kellermann’s detailed account see chapters 7 and 8 of [24].
9 https://www.aip.org/history-programs/niels-bohr-library/oral-histories/4627
D. Ricklesand A.Blum: Paul Weiss and thegenesis of canonical quantization 473
reasonably comfortable (there were “even showers” he writes: 20th October, 1940
[SPSL]). The internees in Camp L also established a kind of university (led by Max
Perutz and begun while they were kept initially at a barracksat Bury St Edmunds).
Concerts were also given. Weiss (ibid.) noted that there were a range of improving
activities: vocational training courses, popular lectures, evening concerts and variety
shows, and an art exhibition.
Hermann Bondi, Walter Heitler, and Klaus Fuchs were also held here and gave
lectures(toclassesofaround20people,accordingtoGold).InalettertoaMrSkemp
written after the war (18th Nov. 1945) Weiss mentions that he taught a fair amount
of mathematics during his internment and helped to organise this camp university.
Bondi (taken in 1940) also found the internment experience to be fairly easy
going10. He claims that his ‘noteless’ teaching style was forged there. On his release
Bondi worked with Hoyle and Gold on radar11. Gold writes of this: “We didn’t have
anybooks,andsoteachingfrommemoryandpuzzleswasthekindofthingwedid”12.
Camp N, at Sherbrooke, seems to have been a far unhappier affair. Fuchs and
Bondi too went to Camp N. It was primarily jews, and others that were on friendly
termswithjews,thatweresenthere.CampNwasadisusedrailwayrepairshed.Weiss
claimed that they spent much of their time making repairs, building the necessary
amenities, and queuing. He also noted bitterly that they had been told to do this
maintenance and building work “to prove our loyalty to the British Commonwealth
andthatbydoingthisweshallpreventbadthingsfromhappeningtoourcoreligionists
in Canada” (20th October, 1940 [SPSL]).
Nothing was done by August 1940 (despite several appeals involving high profile
figures), so Paul Weiss’s sister, Helene, stepped in and tried to gather some new
support to petition the home office. One of those who wrote on Weiss’s behalf was
Paul Ewald, at Queens University Belfast. Ewald pointed to Weiss’s repeated efforts
to register for National Service. He wrote: “[p]ersonally I would put full trust in
Weiss’s character both in private matters and with regard to the British political
outlook” (17th August, 1940 – to SPSL: MS. SPSL). Ewald suggested that Weiss
be classified as ‘category 8’, relating to scientists, research workers and those with
academic distinction who might carry work of national importance – Bondi would
10 Bondidescribeshisinternmentexperiencein hisauto-biographyScience, Churchill, and
Me [5], even writing that he found the initial phase of his internment “a bit of a lark” (p.
28).HeestablishedafriendshipwithhislongtimecollaboratorTommyGoldwhilehousedin
thesecamps.GoldgivesabriefaccountofhisinternmentinaninterviewwithSpencerWeart:
https://www.aip.org/history-programs/niels-bohr-library/oral-histories/4627. Itseemsthat
Cambridge wasthewrong placetobesincelocal police forcesweregivenorderstousetheir
own discretion in the matter of enemy aliens. Bondi, Gold, and Weiss were all picked up in
Cambridge,withGoldandWeissunluckilyjustvisitingatthetime.Asmentioned,E.Walter
Kellerman, a fellow physicist internee, has also described his internment experience, in the
same camps, and the events surrounding it in his auto-biography: A Physicist’s Labour in
Warand Peace: Memoirs1933–1999 [24].HeislessdismissiveoftheexperiencethanBondi
(see, especially, pp.83–84).
11 Gold points out the curious twist here, writing that “the whole thing was pretty ab-
surd, to be in internment at one time and then a few months later to go to the most
secret defence establishment” (https://www.aip.org/history-programs/niels-bohr-library/
oral-histories/4627). Likewise Bondi: “there was a very short time from my being behind
barbedwirebecauseIwasso‘dangerous’,tomybeingbehindbarbedwirebecausetheworkI
didwassosecret”(https://www.aip.org/history-programs/niels-bohr-library/oral-histories/
4519).
12 https://www.aip.org/history-programs/niels-bohr-library/oral-histories/4627
474 The European Physical Journal H
Fig. 2. Paul Weiss (1911–1991). Letter written from within theinternment camp.
be reclassified as such13. Weiss’s old tutor from Downing College, Eric Holmes, also
wrote on Weiss’s behalf (pointing to his teaching potential as much as his research)
as did Ralph Fowler.
There were further testimonials (in addition to earlierones) from Bornand Dirac
again:
– Born: “he is a victim of Nazi persecution and hopes for the victory of Great
Britain” (letter to Miss Simpson of SPSL, dated 5th August 1940).
– Dirac: “Mr Weiss ... is strongly anti-Nazi and thoroughly reliable. His ability
could be of great service in the National Cause, and it would be a great pity not
to make use of it” (13th, August 1940).
The instructions for his release reached his sister on December 18th 1940, thanks to
persistentpressureathighlevels,includingtheSecretaryofState.HeleneWeisswrote
13 The minutes of the House of Lords debates over the status of the internees (and the
categories) can be found on the Hansard webpages: http://hansard.millbanksystems.com/
commons/1940/aug/22/internees-1. Inthisreport SirAclandpointsoutthat evenapromi-
nentanti-Nazi,SebastianHaffner,whohadwrittenafamouspsychologicalstudyofNazidom,
was interned.Also SigmundFreud’sson. Thereport makesit abundantlyclear that thede-
cision to intern the vast majority of people in the Canadian (and Australian) camps was a
terrible mistake.
D. Ricklesand A.Blum: Paul Weiss and thegenesis of canonical quantization 475
tothesecretaryMissEsther(Tess)Simpson14 ofSPSLonthe24thofDecember,from
their mother’s home in Oxfordnoting that she wasvery happy to be able to pass the
news on to her. He was officially released in January 1941. Their mother (Babette
Rosenbacher) died in 1943.
3 Post-internment life
Soon after his release,in February 1941,there was some suggestion (relayed through
G.H. Hardy) that Weiss might take up a post at Liverpool. It happened that he
managed to find a position in applied mathematics at Westfield College, in London
University, for the remainder of the war, again with a salary of 300 pounds – he was
initially going to replace a Mr Jackson who was seconded to the Admiralty (letter
fromMaryStocks,PrincipalofWestfieldCollege,dated21stAugust,1946:MS.SPSL
239). This position lasted from 1941–1950.
Weiss married and had children while at Westfield. His wife was Marliese Oppa´,
and they had two children together, and also a third child they adopted, a refugee
from Belsen, related to his wife. His daughter, Ruth Weiss, worked for Bell labs on
someofthefirstprogramminglanguagesinthe 1960s.Shewasalsoapioneerinusing
programming languages to generate computer graphics of mathematical curves and
surfaces.
Weiss was clearly grateful for this position, and to the SPSL, and quickly vol-
unteered his services during the University vacation period writing to Miss Simpson
(on the 15th July 1941) that he wants to do “real war work”. Miss Simpson pointed
out that getting such work as a refugee really was done through personal contact,
andthathe shouldinform“theBritishmathematicians”thatheis seekingsomesuch
work. It is very interesting to see the political power the mathematicians and scien-
tists hadatthistime,behindthe scenes.Weisswasinfactplacedinacentralregister
of “aliens with special qualifications”, though nothing seems to have come of it.
The SPSL retained an interest in Weiss’s career at least until Weiss discovered
his position at Westfield had been made permanent, in 1946. Weiss also applied for
a naturalisation certificate through the SPSL office (now via a new secretary, Mrs
Ursell) in August 1946.It also seems that Helene and Paul Weiss applied in tandem,
both through the offices of SPSL. By this time Helene Weiss had joined her brother
on the staff of Westfield (some time in 1945)15.
Weiss did not completely disappear from physics after the events of the second
world war were at an end. In 1950 Weiss wrote to Born that he had received an
invitation to the Institute for Advanced Study [IAS] at Princeton16. We see in the
14 Miss Simpson later wrote her memoirs, detailing her experience with the SPSL: [9]
(cf. [25], pp.279-80 for more information).
15 Helene Weiss was a philosopher and classicist of some merit, specializing in ancient
philosophy, Aristotle in particular. She had been a student of Martin Heidegger between
1920 and 1934 and achieved some fame as a Heidegger scholar. Another of Weiss’s sisters,
Gertrud, was married to one of the founders of social and organizational psychology, Kurt
Lewin.
16 ItturnsoutthatthishadbeenatBorn’srecommendation(Letterdated30thMay,1950,
writing from London [Born archive 1/3/2/27]). We note that by 1950 the letters between
Born and Weiss had shifted from their native German to English. A reviewer pointed out
that Weiss would not really have identified has ‘a physicist’ by this point, but rather an
applied mathematician. This strikes us as correct, and was borne out by correspondence
with Weiss’s family who suggested that he felt obliged by his circumstances to choose one
or the other(mathematics versusphysics), though not without some reservations.
476 The European Physical Journal H
IAS records17 that Weiss was indeed a visitor at the IAS during 1950–1951,with his
homeaddressthengivenas635ElmStreet,Syracuse10,NY–perhapscoincidentwith
his IAS visit in 1950, he moved to Syracuse to work for General Electric [GE], until
1957. Weiss was employed as an applied mathematician for GE. One of the projects
he worked on, in 1952, was the application of operations research to the solution of
business problems.While there he alsodesignedandpatented (togetherwithCharles
Johnson)a “formrecognitionsystem” basedonthe identification ofcertain invariant
properties (US Patent No. US2968789)18. He left GE in 1958, briefly taking up a
position at AVCO (Aviation Corporation) until 1960, and finally settling at Wayne
State University, in the mathematics department, until his death in 199119.
Now that we have some sense of Weiss the man, and his struggles, we will turn
to those aspects of his work (all done before the war) that are relevant for quantum
gravity research and general methods of field quantization.
4 Weiss’s role in canonical quantum gravity
Weiss’s work,which turned out to be important to the programof a canonicalquan-
tization ofthe gravitationalfield, wasperformedin the mid- to late 1930s.Canonical
quantization rests on the identification of canonical pairs of variables in the classical
theory, be it a point-mechanical or a field theory. The momentum p canonically con-
jugate to a dynamical variable x (be it coordinate or field strength) can simply be
defined from the LagrangianL of the theory as
∂L
p= , (1)
∂x˙
where x˙ is the time derivative of x. This is how Heisenberg and Pauli, in their foun-
dationalworkonquantumfieldtheory,haddefinedthecanonicalfieldmomenta.Just
like Weiss, it should be noted, they only dealt explicitly with the quantization of
electrodynamics,whilealwaysplacingtheirworkinamoregeneralframeworkoffield
quantization, which could later be taken up in the quantization of gravity.
Weiss proposed starting from an alternate definition of canonical momenta. In
point mechanics, given the usual action integral20
(cid:2)
t1
a a
I = L(x ,x˙ ,t)dt (2)
t0
one can perform a variation in which not just the trajectories x(t) are varied (which
leads to the usual Euler-Lagrange equations of motion), but also the values at the
endpoints, x(t0) and x(t1), and the endpoints t0 and t1 themselves. The variation of
the action then contains additional boundary terms:
(δI)boundary =(H(t1)δt+pa(t1)δxa1)−(H(t0)δt+pa(t0)δxa0). (3)
17 The relevant pages are 414–415 of theCommunity of Scholars booklet.
18 GE was not considered a bad option for a research scientist. Both Bethe and Feynman
did periods there in thesummer of 1946.
19 Paul Weiss’s son, Thomas, recalls emigrating from England in 1952, with his mother,
sister, younger brother, and maternal grandmother in November 1952, on the French liner
Ile de France(private communication).
20 Anticipating the generalization to several coordinates, Weiss chose to denote time by x
(and the canonical coordinates by za). We have opted for the more standard notation used
also by Weiss’s source for this method, Cartan [8], and will switch to Weiss’s notation only
for themulti-dimensional case.
D. Ricklesand A.Blum: Paul Weiss and thegenesis of canonical quantization 477
That is, in the boundary terms, the Hamiltonian shows up as the function multiply-
ing the fixed variation of the endpoints δt, while the canonical momenta pa appear
as the functions multiplying the variation of the endpoint value of the correspond-
ing canonical coordinate, δxa and δxa, respectively. Usually, this relation is used in
1 0
Hamilton-Jacobitheory:Whentheactionisevaluatedalongthephysicalpath,giving
Hamilton’s principal function, only the boundary terms in the variation remain and
describe how the principal function, as a function of coordinates and time, delivers
the momentum and the energy through differentiation. The boundary variation can,
however,alsobeused,asWeissremarked,todefinetheHamiltonianandthecanonical
momenta, which are then defined via the integrationboundaries appearing in the ac-
tionintegral.Thisisofnoconsequenceinpointmechanics,wherethereisnofreedom
in choosingthose integrationboundaries:They are alwayssimply two points in time,
immediately implying an equivalence between this definition andthe standardone of
equation (1).
But things were less straightforwardin field theory. Some work had already been
done in this direction, but it still needed to be put together by Weiss. In his 1935
textbook on the calculus of variations [10], Th´eophile de Donder had presented a
generalization of the boundary variation term to action integrals involving multidi-
mensional integrals, as they appear in field theory, with n independent variables xi
and several dynamical variables za, i.e.,
(cid:2)
a a
I = L(z ,∂z /∂xi,xi)dx1...dxn (4)
V
with the integration going over some n-dimensional volume G. The boundary vari-
ation terms then involved the boundary of this integration volume. Weiss now went
beyond de Donder’s rather formal study of these boundary terms and brought forth
a geometrical reading (inspired, of course, by a reading of the multidimensional ac-
tion integral as describing a relativistic field theory in four-dimensional space-time),
where,atallpoints ontheboundarysurfaceS,onechosesacoordinatesystemwhere
n−1 coordinate axes (coordinates ur, the “parameters”) are parallel to the surface,
while the remaining coordinate axis (coordinate w, the “time”) is perpendicular to
the surface. One could then write the boundary variation terms in the form
(cid:3)
a a n−1
(δI)boundary = (Kδw+λrδur+π δz )d u. (5)
S
Here the δw are local, orthogonal variations of the boundary surface, the δur are
local reparameterizations of the surface, while the δza are the variations of the field
variables on the boundary surface. The question was now how the functions mul-
tiplying these variations could be interpreted in terms of canonical variables and a
Hamiltonian, in the same manner as in point mechanics. Here Weiss could build on
the work of the mathematician Georg Prange, who in his 1915 dissertation [30] had
studied the canonical formulation of a dynamical system along a one-dimensional
boundary line, i.e., the generalization of the one-dimensional point mechanical case
(with a zero-dimensional boundary) to the two-dimensional case. Prange’s work had
been generalized to more than two dimensions by Gustave Juvet [22]. Prange and
Juvethadfoundthat(inordertohaveaninitialvalueproblemthatwasneitherover-
nor underdefined) one had to define the momentum canonically conjugate to a field
variable za as
∂L
a
π = , (6)
∂(∂za/∂w)
whichwaspreciselytheexpressionmultiplyingthefieldvariationinWeiss’sboundary
variation.Similarly,K waspreciselythe multi-dimensionalgeneralizationofPrange’s
478 The European Physical Journal H
two-dimensionalHamiltonian, definednot with time derivatives,but with derivatives
with respect to the coordinate normal to the boundary surface, w. Weiss thus con-
cludedthat,justasinpointmechanics,onecouldinfieldtheoryusethevariationson
the integration boundaries of the action in order to define the Hamiltonian and the
momenta canonically conjugate to the field variables21. For a field theory defined on
a multi-dimensional space-time, one can choose arbitrarily curved hypersurfaces as
integration boundary and can thus, as Weiss outlined, arrive at a generalized notion
of Hamiltonian and canonicalmomenta that does not single out the time coordinate,
but rather defines these quantities with respect to some arbitrary hypersurface.
In his first paper on this project[39] (drawnfrom his PhD thesis under Born and
Dirac), Weiss developed this basic idea of extending the notion of conjugate vari-
ables and then used the conjugate pairs obtained in this manner to set up canonical
commutation relations. The method of constructing a quantum field theory by gen-
eralizing the canonical commutation relations of quantum mechanics to field quan-
tities had been pioneered by Werner Heisenberg and Wolfgang Pauli several years
earlier [20,21]. They had, however, not employed the Hamilton-Jacobi-inspired ma-
chineryofboundaryvariations22 andhadbasedtheirworksimplyonageneralization
of regular Hamiltonian mechanics to field theory. They were thus only able to gen-
eralize the commutation relations of quantum mechanics to a field theory (involving
only first-orderderivatives of the field quantities in the Lagrangian,a restrictionalso
applying to Weiss’s method) by imposing equal-time commutation relations of the
form:
[π(r,t),φ(r(cid:2),t)]=−i(cid:2)δ(r−r(cid:2)), (7)
where φ is the field coordinate,π the correspondingcanonicalfield momentum and r
and r(cid:2) are space (three-)vectors. The Lorentz-invariance (and in fact general covari-
ance)oftheirprocedurewasconsequentlynotmanifestandhadtobeprovenposthoc,
via an explicit calculation of changes produced by infinitesimal Lorentz transforma-
tions23. As Weiss pointed out, “[t]he artificial separation of space and time, inherent
in this quantization method, has the disadvantage that the proof of the relativistic
invariance of the result becomes very complicated” ([39], p. 193)24.
Weiss’s procedure, on the other hand, by referring to an arbitrary 3-surface (em-
bedded in spacetime) secured the Lorentz-invariance from the start. The orthogonal
21 Weiss did not pay much attention to the second summand in the boundary variation
term.Theλr arerelatedtothe(physical,notcanonical) momentumofthesystem,butcan,
given the structure of the boundary variation term, be interpreted as momenta canonically
conjugatetothespacecoordinates.ThiswaslaterdonebyDiracandBergmann,asweshall
discuss.
22 On page 4 of reference [20], Heisenberg and Pauli explicitly mention that they consider
the field variations to vanish at the boundaries. They do not even consider the variation of
the boundaries themselves, however.
23 See paragraph 3 of [20]. Heisenberg and Pauli here refer to an additional condition that
needs to be imposed on the Hamiltonian in order to ensure the Lorentz invariance of the
whole scheme (their Eq. (40)). This condition was, however, soon shown to be superfluous
by Pauli’s assistant Leon Rosenfeld [32].
24 On pages 34–35 in [20], Heisenberg and Pauli discussed why they chose the three-
dimensional over the four-dimensional approach, stressing that it allowed for a clean sepa-
ration between kinematics (equal-time commutation relations) and dynamics. This should
not be read as implying that they had considered Weiss’s approach and discarded it; the
comments are clearly directed against the other alternative method of quantization, using
covariant, four-dimensional commutators, as discussed below. In fact, after Weiss restricted
himself to space-like hypersurfaces in his second paper, he had just as clean a separation
between kinematics and dynamicsas Heisenberg and Pauli did.