Quantisation of the Multidimensional Rotor
E.Abdalla a
and R. Banerjee b
a
Institutode Fsia-USP, C.P.66318, 05315-970S~aoPaulo,Brazil
b
S.N.BoseNationalCentreforBasiSienes
BlokJD,SetorIII,SaltLakeCity,Calutta 700.091India
Reeivedon9September,2000
We reonsider the problem of quantising a partile on the D-dimensional sphere. Adopting a
Lagrangian method of reduing the degrees of freedom, the quantum Hamiltonian is found to
bethe usual Shrodinger operator withoutany urvatureterm. Theequivalenewith the Dira
Hamiltonianapproahisdemonstrated,eitherintheartesianorintheurvilinearbasis. Wealso
brieyommentonthepathintegralapproah.
The problem of the quantisation of the rotor has
been studied sine deades, but remains a subjet of
intense debate. De Witt[1 ℄ studied the path integral
quantization,andproposedthatatermproportionalto
the urvature should be inluded in the Hamiltonian.
There are, however,problems onnetedto the
deni-tionofthe pathintegralusing urvilinearoordinates,
sinequantisinginurvilinearoordinatesleadstonew
terms,evenforfreepartiles. EdwardsandGulyaev[2℄
arefullyonsideredthatproblem,andshowedthatdue
toproblemsintrinsiallyonnetedtothepathintegral
formulation[3℄, pathintegrationquantisationmaylead
todierentresults.
The Dira formulation[4℄ has also been
undertaken[5℄. In this ase one faes the question of
operatorordering,aquestionwhihanonlybetakled
usingdenitepresriptionsbasedongeneralarguments
ashermitiityand generaloordinate invariane. The
Laplae-Beltramioperator has been obtained without
urvature terms in e.g. [5℄, for the speial ase of a
three-dimensionalrotor.
Besides the early developments[2℄, there are
spe-ial denitions of the path integral on the surfae of
the sphere whih do not give rise to the urvature
term[6℄. Severaldiferentresultshavebeenobtainedin
theliterature[6,7,8,9,10℄.
Inspiteofallthese developments,thestatus ofthe
problem is veryonfusing, and there havebeenmany
papers laiming a rejetion of the Dira formalism[6℄
an intrinsi dierene between path integral
formula-tionandoperatorformalism[7℄,oradvoatingdierent
quantizationshemes[9℄. The importane of the
ques-tionanbeappreiatedfromthefatthatithas
impli-ationsinurvedspaequantization,whendeningthe
Wheeler-de-Wittequation[11℄, andin thequantisation
ofsigmamodelLagrangians.
Theaimofthepresentpaperistoanalysethe
prob-lemdiretlyintermsofreduedoordinatesbysolving
the onstraint, bypassing the ambiguities inherent in
eithertheDiraorpathintegralapproahes. The
las-sial reduedspaeis thereforeobtainedin a
straight-forward manner. We then pass to the quantum
for-mulation by using the Laplae-Beltramionstrution,
leadingtotheusual Shrodinger operatorwithoutany
urvatureterm. Theonnetionofourresultswiththe
Dira approah is then established. Inpartiular, the
roleoftheMaskawa-Nakajima[12℄theoremintheDira
formalismintheartesianbasisiselaborated. We
on-lude mentioningthepathintegralformulation,where
the ounterpart of the above approah is the speial
timesliingoftheintegralasexplainedbyT.D.Lee[13℄.
The Lagrangian for a partile of unit mass
on-strained to move on the surfae of a D-dimensional
sphereofradiusRisgivenby
L= 1
2 _ x
_ x
x
x
R
2
=1D ; (1)
wheretheonstraintisimplementedbyaLagrange
mul-tiplier. ThusweuseDira'sonstraintanalysis. Apart
fromthefatthatDirabraketsareplaguedby
order-ingambiguities,theextrationofthephysialvariables
ofthesystemis notverytransparent[5,6,7,10℄.
Here we shall adopt an alternative anonial
ap-proah developed by one of us[14℄ whih is based on
aLagrangianredutionbysystemmatiallyeliminating
the unphysial variables using the onstraint.
Elimi-natingx
D
weobtain
L
r =
1
2 g
ij _ x
i _ x
j =
1
2
Æ
ij +
x
i x
j
R 2
~ x 2
_ x
i _ x
j
: (2)
The onventional anonial formalismis now
applia-ble,andtheanonialHamiltonianis
H=p
i _ x
i L
r =
1
p
i g
ij
p
j
whihgivesthenalexpressionforthelassialredued
Hamiltonianin termsofthemomentaonjugatetox
i .
In order to perform the quantisation the above
Hamiltonianisreplaedbytheorresponding
Laplae-Beltrami operator,beingdened as
^
H=O
LB = 1 2 g 1=4 ^ i g 1=2 g ij ^ j g 1=4 ; (4)
where ^ isthequantum momentum operator
^ i = i~g 1=4 ~ g 1=4 ; (5)
and g is the determinant of the metri g detg
ij = R 2 R 2 ~x 2
. This transitionfrom lassialto quantum
the-ory is based onhermitiity, general oordinate
invari-aneandpreservationofthesymmetryproperties.
It is now simple to obtain the quantum
Hamilto-nian, ^ H= 1 2 p R 2 ~ x 2~ i Æ ij x i x j R 2 R 2 ~ x 2 1=2 ~ j X L 2 2R 2 : (6) d
wheretheangularmomentumoperatoris givenby
L ij = x i p j x j p i
= i~(x
i j x j i
) ; (7)
L iD = p R 2 ~ x 2 p i = L Di =i~ p R 2 ~ x 2 i : (8)
Theonjugatemomentum p
D
doesnotexist inthe
re-duedvariables. Thusthequantum Hamiltonianisthe
onventional Shrodinger operator withoutany urv
a-tureterm. Thisisourentralresult.
Theanalysiswasarriedoutin theartesianbasis,
butitisinstrutivetorepeatitintheurvilinearbasis,
illuminatingtheonnetionwiththeonventionalDira
approah. Using the standard urvilinearoordinates
theoriginalLagrangianisgivenby
L= 1
2 fr_
2 +r 2 _ ' 2 1 +r
2 _ ' 2 D 1 sin 2 ' 2 1 sin 2 ' D 2
g ( r R ) : (9)
Inonstrast tothe artesiananalysis,the solutionoftheonstraint(a onstantradius)hereistrivialand the
reduedHamiltonianobtainedbyfollowingthepreviousstepsisworkedouttobe
H= 1 2 X g ab a b ; (10)
withtheinversediagonalmetrig
ab g 1 ab givenby g ab =R 2 1;sin 2 ' 1 ;sin 2 ' 1 sin 2 ' 2
;;sin 2 ' 1 sin 2 ' 2 sin 2 ' n 2 : (11)
ThequantumHamiltonian,whihisgivenbytheorrespondingLaplae-Beltramioperator,reduesto
^ H= ~ 2 2R 2 n 1 X i=1 i 1 Y j=1 1 sin 2 ' j 1 (sin' i )
n i 1
'
i ( sin'
i )
n i 1
'
i
(12)
whihorrespondsto(6). ToomparewiththeDiraformalismwestartfromtheonstrainedLagrangian(9). The
set ofprimaryonstraintsisnowgivenby
=
=0 ;
1
=r R0 ;
2 =
r
The unphysial anonial set (;
) assoiated with
theLagrangemultiplier isignored. Thisleavesuswith
apairof seond lassonstraints,
1 and
2
, forming
aanonialset,
f
i ;
j g=
ij
: (14)
Thespeialformoftheonstraintsallowsa
straightfor-wardappliationoftheMaskawa-Nakajimatheorem[12℄
to extrat the physial variables and the Hamiltonian
withouttheneedof anyexpliit omputationof Dira
brakets.Using thistheorem,itissimpleto showthat
theanonialpairsaregivenby'
i ;
'i
. Inotherwords,
the Dira brakets among these variables is equal to
their Poisson brakets. The physial Hamiltonian is
nowobtainedfromtheanonialHamiltonianby
pass-ing on to the onstraint shell. This is found to
oin-idewiththereduedHamiltonian(10). Thequantum
Hamiltonianisthenreobtainedfromtheorresponding
Laplae-Beltramioperator.
TheDiraanalysisof thisproblemin theartesian
basisisquitenontrivial,sinetheonstraintalgebrais
nolonger anonial. The Maskawa-Nakajimatheorem
allows nding theanonial transformation whih
en-ables the extration of the anonial pair of variables
withoutanyfurtherambiguities. Theonstraint
stru-tureisgivenby
1 =x
i x
i R
2
=0
2 =x
i p
i
=0 (15)
TheMaskawaNakajimaanonialtr. isjustthe
trans-formationfromtheartesian tothespherialbasis. In
these redened variables the physial hamiltonian is
givenby,
H
phys =
1
2R 2
2
+
1
sin 2
2
'
: (16)
This ompletes the lassial redution. Sine the
above Hamiltonian is expressed in terms of anonial
pairs, theLaplae-Beltramionstrutiongoesthrough
andweexatlyreproduetheanonialHamiltonian.
The main onlusion of the present work is that
thequantumHamiltonianforthemultidimensional
ro-tor isgivenby thepure Shrodinger operator without
any urvature term. This result was obtained in the
Lagrangianformalism[16℄ bydiretlysolvingthe
on-straintandreduingtheunwanteddegreesoffreedom.
The usual ambiguities of the Dira approah in the
artesianbasiswere avoidedbytaking reoursetothe
powerfulMaskawa-Nakajimatheorem [12℄.
It isessentialto pointoutthat thetransition from
thelassialtothequantumtheorywasdoneby
explot-ing the Laplae-Beltrami onstrution, enforing the
onditionsofhermitiity,generaloordinateinvariane,
andpreservationofthelassialsymmetries. Whilethe
passagefromthelassialtothequantumtheoryis
rid-dled with ambiguities, the Laplae-Beltrami operator
terms are ruled out sinethese areinompatible with
thethirdondition. Inpartiular,theyviolatethe
on-servation of thegenerator ofthe equilong
transforma-tion group [9℄. Yet another way of seeing the
impor-tane of the Laplae-Beltramionstrutionis to
om-pareitwiththepathintegralformalism[13℄. The
spe-i time sliing presriptionused in [13℄ is known to
yield onsistent results for the path integral either in
the artesian or urvilinearbasis. Using the
Laplae-Beltramioperatorintheanonialquantisationof
non-abeliangaugetheories[15℄naturallyledtothe
intera-tiontermsoriginallypostulatedbyShwinger [16℄and
subsequently derived in the path integral framework
[13℄. Thedierent methods of reduingthe unwanted
degrees of freedom in the lagrangianand hamiltonian
approahes,yieldompletelyonsistentresults. Thisis
theimportantndingofthepresentworkwhihserves
todispelatleastsomeoftheonfusionandontroversy
existingin theliterature.
Apossibleapproahofdisussingsystemswith
se-ondlassonstraintsistoembedthemintoarstlass
system. Detailsof this proedure in the present
on-texthavebeengivenin[17℄wheretheinvarianeofthe
quantum theory under reparametrisation of the
on-straint surfae has also been studied. Regarding the
pathintegralformulation,thebasiproblemherestems
from the fat that the denition of the path integral
in urvilinear oordinates is rather triky and subtle.
Anapparentlashbetweentheanonialand(anaive)
pathintegralformulationisalreadyseeninthesimplest
of examples, namely a free non relativisti partile in
twodimensions[18℄.
Thus, a lear omputation as the one performed
by Kleinert[6℄ leadsto theorret result. Wehowever
point outthat adistintion anbemade,sothat itis
possibleto treatthepartileeither onornear the
sur-fae of thesphere. For the former asethe boundary
term disappears[6℄ but in the latter, suh a term
ex-ists. Sine in the present analysis the onstraintsare
alwaysstronglyenfored,weareonnedtotheaseof
thepartileexatlyonthesphere. Areentalulation
from amoremathematial pointofviewalsoonrms
ourresult[19℄.
Aknowledgements: thisworkhasbeenpartially
supported by Conselho Naional de Desenvolvimento
Ciento e Tenologio and Funda~ao de Amparo a
Pesquisa doEstado deS~aoPaulo,Brazil. E.A.wishes
to thankthe S.N.Bose NationalCentre for Basi
Si-enes, Calutta,India,forthehospitality.
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