arXiv:quant-ph/0208026 v1 5 Aug 2002
P ath Integrals and Their Application to Dissipative Quantum Systems Gert-Ludwig Ingold Institut f ur Physik, Universit at Augsburg, D-86135 Augsburg to b e published in \Coheren tE v olution in Noisy En vironmen ts", Lecture Notes in Ph ysics, http://link.spri nger. de/s eries /lnpp c Springer V erlag, Berlin-Heidelb erg-New Y orkGert-LudwigIngold
Institutf urPhysik, UniversitatAugsburg,D-86135Augsburg,Germany
1 Introduction
Thecouplingofasystemtoitsenvironmentisarecurrentsubject inthis
collec-tionoflecturenotes.Theconsequencesofsuchacouplingarethreefold.Firstof
all,energymayirreversiblyb e transferredfromthesystemto theenvironment
therebygivingrisetothephenomenonofdissipation.Inaddition,the uctuating
forceexertedbytheenvironmentonthesystemcauses uctuationsofthesystem
degreeoffreedomwhichmanifestitselfforexampleasBrownianmotion.While
thesetwoe ectso ccurb othforclassicalaswellasquantumsystems,thereexists
athird phenomenon which issp eci c to thequantumworld.As aconsequence
of the entanglement b etween system and environmentaldegrees of freedom a
coherentsup erp ositionofquantumstatesmayb edestroyedinapro cess referred
to as decoherence. This e ect is of majorconcern ifone wantsto implementa
quantumcomputer.Therefore,decoherence isdiscussed indetailinChap.5.
Quantum computation,however, is by no means the only topic where the
couplingto anenvironmentisrelevant.Infact,virtuallynorealsystemcan b e
consideredascompletelyisolatedfromitssurroundings.Therefore,the
phenom-ena listed in the previous paragraph play arole inmany areas of physics and
chemistryandaseriesofmetho dshasb eendevelop edtoaddressthis situation.
Someapproacheslikethemasterequationsdiscussed inChap.2areparticularly
well suited if the coupling to the environmentis weak, a situation desired in
quantumcomputing.Ontheotherhand,inmanysolidstate systems,the
envi-ronmentalcouplingcan b eso strongthat weakcouplingtheories are nolonger
valid.Thisistheregimewherethepathintegralapproachhasproventob every
useful.
Itwouldb eb eyondthescop eofthischaptereventoattempttogivea
com-pleteoverviewoftheuseofpathintegralsinthedescriptionofdissipative
quan-tumsystems.Inparticularforatwo-levelsystemcoupledtoharmonicoscillator
degrees of freedom,theso-calledspin-b oson mo del,quiteanumb erof
approx-imations have b een develop ed which are useful in their resp ective parameter
regimes.Thischapterratherattemptsto give anintro ductionto pathintegrals
forreaders unfamiliarwithbutinterestedinthis metho danditsapplicationto
dissipativequantumsystems.
Inthis spirit,Sect. 2gives anintro ductionto path integrals.Some asp ects
discussed in this section are not necessarily closely related to the problem of
weelab orateonthegeneralideaofthecouplingofasystemtoanenvironment.
Thepathintegralformalismisemployedtoeliminatetheenvironmentaldegrees
of freedom and thus to obtain ane ective description of the systemdegree of
freedom.Theresultsprovidethebasisforadiscussionofthedamp edharmonic
oscillatorinSect.4.Startingfromthepartitionfunctionwewillexamineseveral
asp ects ofthisdissipative quantumsystem.
Readers interested inamorein-depth treatmentof thesubject of quantum
dissipationarereferred to existingtextb o oks.In particular,we recommendthe
b o ok by U. Weiss [1] which provides an extensive presentation of this topic
togetherwithacomprehensivelistofreferences.Chapter4of[2]mayserve asa
moreconciseintro ductioncomplementarytothepresentchapter.Pathintegrals
are discussed in a whole variety of textb o oks with an emphasis either on the
physicalorthemathematicalasp ects.Weonlymentiontheb o okbyH.Kleinert
[3] which gives adetailed discussion ofpath integrals andtheir applicationsin
di erentareas.
2 Path Integrals
2.1 Intro duction
Themostoften usedand taughtapproach to nonrelativisticquantum
mechan-ics is based on the Schrodinger equation which p ossesses strong ties with the
the Hamiltonian formulationof classicalmechanics. Thenonvanishing Poisson
bracketsb etween p ositionandmomentuminclassicalmechanicsleadus to
in-tro ducenoncommutingop eratorsinquantummechanics.TheHamiltonfunction
turnsintotheHamiltonop erator,thecentralobjectintheSchrodingerequation.
One of the mostimp ortanttasks isto nd theeigenfunctions of theHamilton
op erator and the asso ciated eigenvalues. Decomp osition of a state into these
eigenfunctionsthenallowsusto determineitstimeevolution.
As an alternative, there exists a formulation of quantum mechanics based
on the Lagrange formalismof classical mechanics with the action as the
cen-tral concept. This approach, which was develop ed by Feynman in the 1940's
[4,5],avoidstheuseofop eratorsthoughthisdo esnotnecessarilymeanthatthe
solution of quantummechanicalproblems b ecomes simpler. Instead of nding
eigenfunctions of aHamiltonianone now has to evaluate afunctionalintegral
which directly yields the propagator required to determinethe dynamicsof a
quantumsystem.Since therelationb etween Feynman'sformulationand
classi-calmechanicsisveryclose,thepathintegralformalismoftenhastheimp ortant
advantageofprovidingamuchmoreintuitiveapproachaswe willtrytoconvey
tothereader inthefollowingsections.
2.2 Propagator
Inquantummechanics,one often needs todetermine thesolutionj (t)i ofthe
time-dep endentSchrodingerequation
i~ @j i
whereHistheHamiltoniandescribingthesystem.Formally,thesolutionof(1) mayb e writtenas j (t)i=T exp i ~ Z t 0 dt 0 H(t 0 ) j (0)i: (2)
Here,thetimeorderingop eratorT isrequiredb ecausetheop eratorscorresp
ond-ing to the Hamiltonianat di erent times in general due not commute.In the
following,wewillrestrictourselvestotime-indep endentHamiltonianswhere(2)
simpli esto j (t)i=exp i ~ Ht j (0)i: (3)
Astheinsp ectionof(2)and(3)demonstrates,thesolutionofthetime-dep endent
Schrodingerequationcontainstwoparts:theinitialstate j (0)iwhich servesas
aninitialconditionandtheso-calledpropagator,anop eratorwhichcontainsall
informationrequired todeterminethetimeevolutionofthesystem.
Writing(3)inp ositionrepresentation one nds
hxj (t)i= Z dx 0 hxjexp i ~ Ht jx 0 ihx 0 j (0)i (4) or (x;t)= Z dx 0 K(x;t;x 0 ;0) (x 0 ;0) (5)
withthepropagator
K(x;t;x 0 ;0)=hxjexp i ~ Ht jx 0 i: (6)
It is preciselythis propagatorwhich isthe central object ofFeynman's
formu-lationofquantummechanics.Before discussingthepathintegralrepresentation
ofthepropagator,itisthereforeusefulto take alo okatsomeprop ertiesofthe
propagator.
Insteadofp erformingthetimeevolutionofthestatej (0)iintoj (t)iinone
step as was doneinequation(3), one couldenvisagetop erformthis pro cedure
intwosteps by rstpropagating theinitialstate j (0)i up to anintermediate
timet
1
and takingthe newstate j (t
1
)ias initialstate forapropagation over
thetimet t
1
.Thisamountstoreplacing(3) by
j (t)i=exp i ~ H(t t 1 ) exp i ~ Ht 1 j (0)i (7) orequivalently (x;t)= Z dx 0 Z dx 00 K(x;t;x 00 ;t 1 )K(x 00 ;t 1 ;x 0 ;0) (x 0 ;0): (8)
Comparing(5)and (8),we ndthesemigroupprop ertyofthepropagator
K(x;t;x 0 ;0)= Z dx 00 K(x;t;x 00 ;t )K(x 00 ;t ;x 0 ;0): (9)
0 t 1 t x 0 x
Fig.1. According tothesemigroupprop erty(9)thepropagatorK(x;t;x
0
;0)may b e
decomp osedintopropagatorsarrivingatsometimet
1
atanintermediatep ointx
00 and
propagatorscontinuingfromtheretothe nalp ointx
This result is visualized in Fig. 1 where the propagators b etween space-time
p ointsare depicted bystraight lines connecting thecorresp onding two p oints.
Attheintermediatetimet
1
onehastointegrateoverallp ositionsx
00
.Thisinsight
willb eofusewhenwediscussthepathintegralrepresentationofthepropagator
lateron.
Thepropagator containsthe complete informationab outthe eigenenergies
E n
andthecorresp ondingeigenstatesjni.Makinguseofthecompletenessofthe
eigenstates,one ndsfrom(6)
K(x;t;x 0 ;0)= X n exp i ~ E n t n (x) n (x 0 ) : (10)
Here, thestardenotescomplexconjugation.Not onlydo esthepropagator
con-tain the eigenenergies and eigenstates, this informationmay also b e extracted
fromit.Tothisend,weintro ducetheretarded Greenfunction
G r (x;t;x 0 ;0)=K(x;t;x 0 ;0)
Θ
(t) (11)where
Θ
(t)istheHeavisidefunctionwhichequals1forp ositiveargumenttandis zero otherwise. Performing a Fourier transformation,one ends up with the
sp ectralrepresentation G r (x;x 0 ;E)= i ~ Z 1 0 dtexp i ~ Et G r (t) = X n n (x) n (x 0 ) E E n +i" ; (12)
where "is anin nitely smallp ositive quantity.According to (12),thep oles of
the energy-dep endent retarded Green functionindicate the eigenenergies while
thecorresp ondingresiduacanb e factorizedintotheeigenfunctionsatp ositions
2.3 FreeParticle
Animp ortantsteptowardsthepathintegralformulationofquantummechanics
can b e made by considering thepropagator of a free particle of mass m. The
eigenstates ofthecorresp ondingHamiltonian
H=
p 2
2m
(13)
aremomentumeigenstates
p (x)= 1 p 2 ~ exp i ~ px (14)
with a momentumeigenvalue pout of a continuous sp ectrum. Inserting these
eigenstates intotherepresentation(10)ofthepropagator,one ndsbyvirtueof
Z 1 1 dxexp( iax 2 )= r ia = r a exp i 4 (15)
forthepropagatorofthefree particletheresult
K(x f ;t;x i ;0)= 1 2 ~ Z dpexp i ~ p 2 2m t exp i ~ p(x f x i ) = r m 2 i~t exp i ~ m(x f x i ) 2 2t : (16)
Itwasalready noted byDirac[6]that thequantummechanicalpropagator
and the classical prop erties of a free particle are closely related. In order to
demonstratethis, we evaluate theaction ofa particlemovingfrom x
i
to x
f in
timet.Fromtheclassicalpath
x cl (s)=x i +(x f x i ) s t (17)
ob eyingtheb oundaryconditionsx
cl (0)=x i andx cl (t)=x f
,thecorresp onding
classicalactionisfoundas
S cl = m 2 Z t 0 dsx_ 2 cl = m 2 (x f x i ) 2 t : (18)
This result enables us to express the propagator of a free particle entirely in
termsoftheclassicalactionas
K(x f ;t;x i ;0)= 1 2 i~ @ 2 S cl (x f ;t;x i ;0) @x f @x i 1=2 exp i ~ S cl (x f ;t;x i ;0) : (19)
This result isquiteremarkableandone mightsusp ect that it isdueto ap
ecu-liarityofthefreeparticle.However,sincethepropagationinageneralp otential
(intheabsence ofdeltafunctioncontributions)mayb edecomp osedintoaseries
ofshort-timepropagationsofafree particle, theresult (19)mayindeedb e
em-ployedtoconstructarepresentationofthepropagatorwheretheclassicalaction
app earsintheexp onent.Intheprefactor,theactionapp earsintheformshown
2.4 PathIntegralRepresentationofQuantumMechanics
Whileavoidingtogoto o deeplyinto themathematicaldetails,we nevertheless
wanttosketch thederivationofthepathintegralrepresentationofthe
propaga-tor. Themainideaistodecomp ose thetimeevolutionover a nite timet into
N slicesofshorttimeintervalst=t=N wherewewilleventuallytakethelimit
N !1.Denotingtheop eratorofthekineticandp otentialenergybyT andV,
resp ectively, we thus nd
exp i ~ Ht = exp i ~ (T +V)t N : (20)
For simplicity,we willassume that theHamiltonianis time-indep endent even
thoughthefollowingderivationmayb egeneralizedtothetime-dep endentcase.
Wenowwouldliketo decomp osetheshort-timepropagatorin(20)into apart
dep ending onthekineticenergy andanother part containingthep otential
en-ergy. However, since the two op erators do not commute,we have to exercise
somecaution.FromanexpansionoftheBaker-Hausdor formulaone nds
exp i ~ (T +V)t exp i ~ Tt exp i ~ Vt + 1 ~ 2 [T;V](t) 2 (21)
where termsoforder(t)
3
andhigherhaveb een neglected.Sincewe are
inter-ested inthe limitt!0,wemayneglect the contributionof thecommutator
andarrive attheTrotter formula
exp i ~ (T +V)t = lim N!1 [U(t)] N (22)
withtheshorttimeevolutionop erator
U(t)=exp i ~ Tt exp i ~ Vt : (23)
What we have presented here is, of course, at b est amotivationand certainly
do es not constitute a mathematical pro of. We refer readers interested in the
detailsofthepro ofandtheconditionsunderwhichtheTrotterformulaholdsto
theliterature[7].
Inp ositionrepresentation onenowobtainsforthepropagator
K(x f ;t;x i ;0)= lim N!1 Z 1 1 0 @ N 1 Y j=1 dx j 1 A hx f jU(t)jx N 1 i::: hx 1 jU(t)jx i i : (24)
element hx j+1 jU(t)jx j i= x j+1 exp i ~ Tt x j exp i ~ V(x j )t = r m 2 i~t exp i ~ m 2 (x j+1 x j ) 2 t V(x j )t : (25)
Wethusarriveatour nalversionofthepropagator
K(x f ;t;x i ;0)= lim N!1 r m 2 i~t Z 1 1 0 @ N 1 Y j=1 dx j r m 2 i~t 1 A exp 2 4 i ~ N 1 X j=0 m 2 x j+1 x j t 2 V(x j ) ! t 3 5 (26) wherex 0 andx N
shouldb eidenti edwithx
i
andx
f
,resp ectively.The
discretiza-tionofthepropagatorusedinthisexpressionisaconsequence oftheform(21)
of the Baker-Hausdor relation. In lowest order in t, we could have used a
di erentdecomp ositionwhichwouldhaveledtoadi erentdiscretizationofthe
propagator.Foradiscussionofthemathematicalsubtleties werefer thereader
to[8].
Remarkingthat the exp onent in(26) containsa discretized version of the
action S[x]= Z t 0 ds m 2 _ x 2 V(x) ; (27)
wecan writethisresult inshortnotationas
K(x f ;t;x i ;0)= Z D xexp i ~ S[x] : (28)
The action (27) is a functional which takes as argument a function x(s) and
returns anumb er,theactionS[x].Theintegral in(28)thereforeisafunctional
integral where one has to integrate over all functions satisfyingthe b oundary
conditionsx(0)=x
i
andx(t)=x
f
. Sincethese functions represent paths,one
refers tothiskindoffunctionalintegralsalsoaspathintegral.
The three lines shown in Fig. 2 represent the in nity of paths satisfying
theb oundary conditions.Amongthemthe thickerline indicatesasp ecialpath
corresp ondingtoanextremumoftheaction.Accordingtotheprincipalofleast
actionsuchapathisasolutionoftheclassicalequationofmotion.Itshould b e
noted,however,thateventhoughsometimesthereexistsauniqueextremum,in
generalthere mayb emorethanone orevennone.Ademonstrationofthisfact
willb eprovidedinSect.2.7wherewewilldiscussthedrivenharmonicoscillator.
Theother paths depicted inFig. 2may b e interpreted as quantum
uctu-ations around the classicalpath. As we willsee in Sect. 2.8, theamplitude of
these uctuations is typicallyof the order of
p
t
Fig.2. Thethick line representsaclassical path satisfying the b oundary conditions.
The thinner lines are no solutions of the classical equation of motion and may b e
asso ciated withquantum uctuations
Beforeexplicitlyevaluatingapathintegral,wewanttodiscusstwoexamples
whichwillgive ussomeinsightintothedi erence of theapproacheso ered by
theSchrodinger andFeynmanformulationofquantummechanics.
2.5 Particleona Ring
We con ne aparticle of mass m to aring of radius R and denote its angular
degree offreedomby .This systemisdescrib ed bytheHamiltonian
H= ~ 2 2mR 2 @ 2 @ 2 : (29)
Requiring thewave functionto b e continuous and di erentiable,one nds the
stationarystates ` ( )= 1 p 2 exp(i` ) (30)
with`=0;1;2;::: andtheeigenenergies
E ` = ~ 2 ` 2 2mR 2 : (31)
These solutionsof thetime-indep endentSchrodinger equation allowus to
con-struct thepropagator
K( f ;t; i ;0)= 1 2 1 X `= 1 exp i`( f i ) i ~` 2 2mR 2 t : (32)
Wenowwanttoderivethisresultwithinthepathintegralformalism.Tothis
end we willemploythepropagatorof thefree particle. However,animp ortant
di erenceb etweenafreeparticleandaparticleonaringdeservesourattention.
i f i f n=0 n=1
Fig.3. Onaring,theangles
f
and
f
+2 nhavetob eidenti ed.Asaconsequence,
thereexistin nitely manyclassicalpathsconnecting twop ointsonaring,whichmay
b eidenti edby theirwinding numb er n
pathsconnecting
i
and
f
. Allthese pathsare top ologicallydi erent andcan
b echaracterizedbytheirwindingnumb ern.Asanexample,Fig.3showsapath
forn=0andn=1.Due totheirdi erent top ology,these twopaths(and any
twopathscorresp ondingto di erentwindingnumb ers) cannotb e continuously
transformedintoeachother.Thisimpliesthataddinga uctuationtooneofthe
classicalpathswillnever changeitswindingnumb er.
Therefore,we haveto sumoverallwindingnumb ersinorderto accountfor
allp ossible paths.Thepropagatorthusconsists ofasumover free propagators
corresp ondingtodi erentwindingnumb ers
K( f ;t; i ;0)= 1 X n= 1 R r m 2 i~t exp i ~ mR 2 2 ( f i 2 n) 2 t : (33)
Here,thefactorRaccountsforthefact that,incontrasttothefreeparticle,the
co ordinate isgivenbyanangleinstead ofap osition.
Thepropagator(33)is2 -p erio dicin
f
i
andcantherefore b eexpressed
intermsofaFourierseries
K( f ;t; i ;0)= 1 X `= 1 c ` exp[i`( f i )] : (34)
TheFourierco eÆcientsarefoundtoread
c ` = 1 2 exp i ~` 2 2mR 2 t (35)
which proves the equivalence of (33) with our previous result (32). We thus
have obtained the propagatorof a free particle on a ring b oth by solvingthe
Schrodinger equation and by employing pathintegral metho ds.These two
ap-proaches make use of complementary representations. In the rst case,this is
0 x i x f L 1 2 3 4 5
Fig.4. The re ection at the walls of a b ox leads to an in nite numb er of p ossible
trajectoriesconnectingtwop ointsin theb ox
2.6 Particleina Box
Another textb o ok exampleinstandard quantummechanicsis theparticle ina
b oxoflengthLcon nedby in nitelyhigh wallsatx=0andx=L. Fromthe
eigenvalues E j = ~ 2 2 j 2 2mL 2 (36)
withj=1;2;::: andthecorresp ondingeigenfunctions
j (x)= r 2 L sin j x L (37)
thepropagatorisimmediatelyobtainedas
K(x f ;t;x i ;0)= 2 L 1 X j=1 exp i ~ 2 j 2 2mL 2 t sin j x f L sin j x i L : (38)
It to ok some time until this problem was solved within the path integral
approach[9,10].Here,wehavetoconsiderallpathsconnectingthep ointsx
i and
x f
withinap erio doftimet.Duetothere ectingwalls,thereagainexistin nitely
manyclassicalpaths, veofwhicharedepictedinFig.4.However,incontrastto
thecaseofaparticleonaring,these pathsarenolongertop ologicallydistinct.
Asaconsequence,we maydeformaclassicalpathcontinuouslytoobtainoneof
theotherclassicalpaths.
If,forthemoment,wedisregardthedetailsofthere ectionsatthewall,the
motionoftheparticleinab oxisequivalenttothemotionofafree particle.The
fact that paths are foldedback on themselves can b e accounted for by taking
into account replicas of the b ox as shown in Fig. 5. Now, the path do es not
necessarily endat x
(0) f
=x
f
but atone ofthemirrorimagesx
(n) f
where nisan
arbitrary integer. In order to obtainthe propagator,we willhave to sum over
1 2 3 4 5 x ( 2) f x ( 1) f x f x (1) f x (2) f x i
Fig.5. Insteadofaparticlegettingre ectedatthewallsoftheb oxonemaythink of
afreeparticlemovingfromthestarting p ointintheb oxtotheendp ointinoneofthe
replicasoftheb ox
canseethatforano ddnumb er2n 1ofre ections,theendp ointliesat2nL x
f
andthecontributiontothefullpropagatortherefore isgivenby
K (2n 1) (x f ;t;x i ;0)= r m 2 i~t exp i ~ m(2nL x f x i ) 2 2t : (39)
Ontheotherhand,foranevennumb er2nofre ections, theendp ointislo cated
at2nL+x f andwe nd K (2n) (x f ;t;x i ;0)= r m 2 i~t exp i ~ m(2nL+x f x i ) 2 2t : (40)
However,itisnotobviousthatjust summingupthepropagators (39)and(40)
forallnwilldothejob.
Inorderto clarifythis p oint,we startwiththesomewhat simplersituation
ofjust onewallandtakealo okatallpathsrunningb etween x
i
andx
f
intime
t.As canb e seen fromthespace-time diagraminFig.6there arepathswhich
do not cross the walland which therefore contribute to the pathintegral. On
the other hand,there exist also pathswhich cross the wallan even numb erof
times.Sincethese pathssp end sometimeinthe forbiddenregion, they donot
contributetothepathintegral.
Itrequiressomethinkingtoensurethatonlypathsnotcrossing thewallare
takenintoaccount.Ourstrategywillconsistin rstwritingdownapropagator
K free
whichdisregardsthewall.Then,wehave tosubtracto thecontributions
of allthepaths which cross thewall.This can b edone byconstructing apath
withthesameactionastheoriginalpath.Tothisendwe take theoriginalpath
upto thelastcrossing withthewallandthencontinue alongthemirrorimage
oftheoriginalpath.Wethusendupatthemirrorimage x
f
oftheoriginalend
p ointx
f
.Note that apathrunning fromx
i
to x
f
necessarilycrosses thewall
at leastonce.As aconsequence, subtractingthepropagatorb etween these two
p ointseliminates all originalpaths which do not remain inthe region x > 0.
Wetherefore obtainourdesired result,thepropagatorK
wall
inthe presence of
a wall,by subtracting a propagator going to the re ected end p ointfrom the
unconstrainedpropagatortotheoriginalendp oint[9,10,11]
x x i x f 0 x f
Fig.6. Apath crossingthewalliscancelledbyapathrunning tothemirror p oint of
theendp oint
This result b ears muchresemblance withthemetho dof imagecharges in
elec-trostatics.Aftergivingitsomethought,this shouldnotb e to osurprisingsince
thefreeSchrodinger equationandthePoissonequationareformallyequivalent.
Accordingtothemetho dofimagechargesonemayaccountforametallicplate
(i.e. the wall)by putting a negative charge(i.e. the mirrored end p oint)
com-plementingthep ositive charge(i.e.theoriginalend p oint).For thepropagator
thisresults inthedi erence app earingin(41).
Letusnowcomebacktoourin nitelydeepp otentialwellwithtwowalls.This
problemcorresp ondstotheelectrostaticsofachargeb etweentwoparallelmetal
plates. In this case, the metho d of image charges leads to an in nitenumb er
of charges of alternating signs. The original p ositive charge gives rise to two
negativechargeswhichareeachanimagecorresp ondingtooneofthetwometal
plates. Inaddition,however,these imageshave mirrorimagescorresp onding to
theothermetalplateandthispro cess hastob ecarriedonadin nitum.
Expressing the propagator of the particle in the b ox in terms of the free
propagator works in exactly the same way. A path intersecting b oth walls is
subtracted twice, i.e.one timeto ooften.Therefore, one contributionhasto b e
restored which is donebyaddinganother endp oint. Continuingthe pro cedure
oneendsupwithanin nitenumb erofendp oints,someofwhichwehaveshown
inFig. 5.As a consequence, we can attribute a signto each end p ointin this
gure. The general rule which follows from these considerations is that each
re ection atawallleadstofactor 1.Thepropagatortherefore canb e written
as K(x f ;t;x i ;0)= r m 2 i~t 1 X n= 1 " exp i ~ m(2nL+x f x i ) 2 2t exp i m(2nL x f x i ) 2 # : (42)
Thesymmetries K(x f +2L;t;x i ;0)=K(x f ;t;x i ;0) (43) K( x f ;t;x i ;0)= K(x f ;t;x i ;0) (44)
suggest toexpandthepropagatorintotheFourierseries
K(x f ;t;x i ;0)= 1 X j=1 a j (x i ;t)sin j x f L : (45)
ItsFourierco eÆcientsareobtainedfrom(42)as
a j (x i ;t)= 1 L Z L L dx f sin j x f L K(x f ;t;x i ;0) = 2 L sin j x i L exp i ~ E j t (46)
wheretheenergiesE
j
aretheeigenenergies oftheb oxde ned in(36).Inserting
(46)into(45)we thusrecoverourprevious result(38).
2.7 DrivenHarmonic Oscillator
Eventhoughthesituationsdealtwithintheprevioustwosectionshaveb een
con-ceptuallyquiteinteresting, we couldinb othcases avoidtheexplicitcalculation
ofapathintegral.Inthepresentsection,we willintro ducethebasictechniques
needed to evaluate path integrals.As an example, we willconsider the driven
harmonic oscillator which is simple enough to allow for an exact solution. In
addition, the propagator willb e of use inthe discussion of damp ed quantum
systems inlatersections.
Ourstartingp ointistheLagrangian
L= m 2 _ x 2 m 2 ! 2 x 2 +xf(t) (47)
of aharmonicoscillator with massm and frequency ! . Theforce f(t) mayb e
duetoanexternal eld,e.g.anelectric eldcouplingviadip oleinteractiontoa
chargedparticle.Inthecontextofdissipativequantummechanics,theharmonic
oscillator could represent a degree of freedom of the environment under the
in uenceof aforceexerted bythesystem.
Accordingto(28)weobtainthepropagatorK(x
f
;t;x
i
;0)bycalculatingthe
actionforall p ossiblepathsstartingat timezero atx
i
and endingat timet at
x f
.Itisconvenienttodecomp ose thepaths
x(s)=x
cl
(s)+(s) (48)
intotheclassicalpathx
cl
satisfyingtheb oundaryconditionsx
cl (0)=x i ,x cl (t)= x f
anda uctuatingpartvanishingattheb oundaries,i.e.(0)=(t)=0.The
classicalpathhastosatisfytheequationofmotion
mx +m!
2
obtainedfromtheLagrangian(47).
Foranexactlysolvableproblemlikethedrivenharmonicoscillator,wecould
replace x
cl
by any path satisfying x(0) = x
i
, x(t) = x
f
. We leave it as an
exercise to the reader to p erform the following calculation with x
cl
(s) of the
drivenharmonicoscillatorreplacedbyx
i +(x
f x
i
)s=t.However,itisimp ortant
to note that within the semiclassical approximation discussed in Sect. 2.8 an
expansion around the classical path is essential since this path leads to the
dominantcontributiontothepathintegral.
With(48)we obtainfortheaction
S= Z t 0 ds m 2 _ x 2 m 2 ! 2 x 2 +xf(s) = Z t 0 ds m 2 _ x 2 cl m 2 ! 2 x 2 cl +x cl f(s) + Z t 0 ds mx_ cl _ m! 2 x cl +f(s) + Z t 0 ds m 2 _ 2 m 2 ! 2 2 : (50)
Forourcaseofaharmonicp otential,thethirdtermisindep endentoftheb
ound-aryvaluesx
i
andx
f
aswellasoftheexternaldriving.Thesecondtermvanishes
as aconsequence of the expansion aroundthe classicalpath.This can b e seen
bypartialintegrationandbymakinguseofthefactthatx
cl
isasolutionofthe
classicalequationofmotion:
Z t 0 ds mx_ cl _ m! 2 x cl +f(s) = Z t 0 ds mx cl +m! 2 x cl f(s) =0: (51)
We now pro ceed in two steps by rst determining the contribution of the
classicalpathandthenaddressingthe uctuations.Thesolutionoftheclassical
equationofmotionsatisfyingtheb oundaryconditionsreads
x cl (s)=x f sin(! s) sin(! t) +x i sin(! (t s)) sin(! t) (52) + 1 m! Z s 0
dusin(! (s u))f(u)
sin(! s) sin(! t) Z t 0 dusin(! (t u))f(u) :
A p eculiarity of the harmonic oscillator in the absence of driving is the
ap-p earance of conjugate p oints at times T
n
= ( =! )n where n is an arbitrary
integer.Sincethefrequency oftheoscillationsisindep endentoftheamplitude,
thep ositionoftheoscillatoratthesetimesisdeterminedbytheinitialp osition:
x(T 2n+1 ) = x i and x(T 2n ) = x i
. This also illustrates the fact mentioned on
t
Fig.7. Inaharmonicp otentialalltrajectoriesemergingfromthesamestartingp oint
convergeatconjugatep ointsatmultiplesofhalfanoscillation p erio d
Thetaskofevaluatingtheactionoftheclassicalpathmayb e simpli edby
apartialintegration S cl = Z t 0 ds m 2 _ x 2 cl m 2 ! 2 x 2 cl +x cl f(s) = m 2 x cl _ x cl t 0 Z t 0 ds m 2 x cl x cl + m 2 ! 2 x 2 cl x cl f(s) = m 2 (x f _ x cl (t) x i _ x cl (0))+ 1 2 Z t 0 dsx cl (s)f(s) (53)
where wehavemadeuseoftheclassicalequationofmotiontoobtainthethird
line.Fromthesolution(52)oftheclassicalequationofmotionwe get
_ x cl (0)=! x f x i cos(! t) sin(! t) 1 msin (! t) Z t 0 dssin (! (t s))f(s) (54) _ x cl (t)=! x f cos(! t) x i sin(! t) + 1 msin (! t) Z t 0 dssin (! s)f(s): (55)
Insertinginitialand nalvelo cityinto(53)we ndfortheclassicalaction
S cl = m! 2sin(! t) (x 2 i +x 2 f )cos (! t) 2x i x f + x f sin(! t) Z t 0 dssin (! s)f(s)+ x i sin(! t) Z t 0 dssin(! (t s))f(s) 1 m!sin(! t) Z t 0 ds Z s 0
dusin(! u)sin (! (t s))f(s)f(u):
(56)
As asecond step, we have to evaluate the contribution of the uctuations
whichisdeterminedbythethirdtermin(50).Afterpartialintegrationthisterm
b ecomes S (2) = Z t ds m 2 _ 2 m 2 ! 2 2 = Z t ds m 2 d 2 ds 2 +! 2 : (57)
Here, the sup erscript `(2)'indicates that this term corresp onds to the
contri-bution of second orderin . Inviewof theright-handside it isappropriate to
expandthe uctuation
(s)= 1 X n=1 a n n (s) (58) intoeigenfunctionsof d 2 ds 2 +! 2 n = n n (59) with n (0)= n
(t)=0.As eigenfunctionsof aselfadjointop erator, the
n are
completeandmayb echosenorthonormal.Solving(59)yieldstheeigenfunctions
n (s)= r 2 t sin n s t (60)
andcorresp ondingeigenvalues
n = n t 2 +! 2 : (61)
Weemphasizethat (58)is nottheusualFourier seriesonanintervaloflength
t.Such anexpansioncouldb eusedintheform
(s)= r 2 t 1 X n=1 h a n cos (2 n s t ) 1 +b n sin(2 n s t ) i (62)
which ensures that the uctuations vanish at the b oundaries. We invite the
reader toredo thefollowingcalculationwiththeexpansion(62)replacing(58).
Whileattheendthesamepropagatorshouldb efound,itwillb ecomeclearwhy
theexpansionintermsofeigenfunctionssatisfying(59)ispreferable.
Theintegrationover the uctuations now b ecomes an integration over the
expansionco eÆcientsa
n
.Insertingtheexpansion(58)intotheactionone nds
S (2) = m 2 1 X n=1 n a 2 n = m 2 1 X n=1 n t 2 ! 2 a 2 n : (63)
Asthisresult shows,theclassicalactionisonlyanextremumoftheactionbut
not necessarily a minimum although this is the case for short time intervals
t < =! . The existence of conjugate p oints at times T
n
= n =! mentioned
ab ovemanifestsitselfhereas vanishingoftheeigenvalue
n
.Thentheactionis
indep endentofa
n
whichimpliesthat foratimeintervalT
n allpathsx cl +a n n
witharbitraryco eÆcienta
n
aresolutionsoftheclassicalequationofmotion.
Afterexpansion of the uctuations intermsof the eigenfunctions(60),the
propagatortakestheform
K(x f ;t;x i ;0)exp i ~ S cl Z 1 Y da n ! exp i ~ m 2 1 X n a 2 n ! : (64)
Inprinciple,weneedtoknowtheJacobideterminantofthetransformationfrom
thepathintegraltotheintegralovertheFourierco eÆcients.However,sincethis
Jacobi determinant is indep endent of the oscillator frequency ! , we may also
compare with the free particle. Evaluating the Gaussian uctuationintegrals,
we nd for theratio b etween the prefactors of thepropagators K
!
and K
0 of
theharmonicoscillatorand thefree particle,resp ectively,
K ! exp[ (i=~)S cl ;! ] K 0 exp[ (i=~)S cl ;0 ] = r D 0 D : (65)
Here,wehaveintro ducedthe uctuationdeterminantsfortheharmonic
oscilla-tor D=det d 2 ds 2 +! 2 = 1 Y n=1 n (66)
andthefreeparticle
D 0 =det d 2 ds 2 = 1 Y n=1 0 n : (67)
Theeigenvaluesforthefreeparticle
0 n = n t 2 (68)
are obtained from the eigenvalues (61) of the harmonic oscillator simply by
setting thefrequency ! equalto zero. Withtheprefactor ofthe propagatorof
thefree particle
K 0 exp i ~ S cl;0 = r m 2 i~t (69)
and(65),thepropagatoroftheharmonicoscillatorb ecomes
K(x f ;t;x i ;0)= r m 2 i~t r D 0 D exp i ~ S cl : (70)
Forreadersunfamiliarwiththeconceptofdeterminantsofdi erentialop
era-torswementionthatwemayde nematrixelementsofanop eratorbyprojection
ontoabasisasisfamiliarfromstandardquantummechanics.Theop erator
rep-resented inits eigenbasis yields adiagonal matrix withthe eigenvalues onthe
diagonal.Then,asfor nitedimensionalmatrices,thedeterminantisthepro duct
oftheseeigenvalues.
Each of the determinants(66)and (67)by itself diverges. However,we are
interestedintheratiob etween themwhichiswell-de ned[12]
D D 0 = 1 Y 1 ! t n 2 ! = sin(! t) ! t : (71)
Inserting this result into (70)leads to the propagator of the driven harmonic
oscillatorinits nalform
K(x f ;t;x i ;0)= r m! 2 i~sin(! t) exp i ~ S cl = r m! 2 ~jsin(! t)j exp i ~ S cl i 4 +n 2 (72)
withtheclassicalactionde nedin(56).TheMorseindexninthephasefactoris
givenbytheintegerpartof! t= .Thisphaseaccountsforthechangesinsignof
thesinefunction[13].Here,onemightarguethatitisnotobviouswhichsignof
thesquarero otonehastotake.However,thesemigroupprop erty(9)allowsto
constructpropagatorsacrossconjugatep ointsbyjoiningpropagatorsforshorter
timeintervals.Inthisway,thesignmayb e determinedunambiguously[14].
Itisinterestingtonotethatthephasefactorexp( in =2)in(72)impliesthat
K(x f ;2 =! ;x i ;0)= K(x f ;0;x i ;0)= Æ(x f x i
),i.e. thewavefunctionafter
one p erio dof oscillationdi ers fromtheoriginalwave functionbyafactor 1.
Theoscillatorthusreturnstoitsoriginalstateonlyaftertwop erio dsverymuch
likeaspin-1/2 particlewhichpicksupasignunder rotationby2 andreturns
to its original state only after a4 -rotation. This e ect mightb e observed in
thecaseoftheharmonicoscillatorbylettinginterferethewavefunctionsoftwo
oscillatorswithdi erentfrequency [15].
2.8 SemiclassicalApproximation
The systems considered so far have b een sp ecial in the sense that an exact
expression for the propagatorcould b e obtained. This is a consequence of the
fact thatthep otentialwasat mostquadraticintheco ordinate.Unfortunately,
inmostcasesofinterestthep otentialismorecomplicatedandapartfromafew
exceptions an exact evaluationofthe pathintegral turnsoutto b e imp ossible.
Tocop ewithsuch situations,approximationschemeshaveb een devised.Inthe
following,wewillrestrictourselvestothemostimp ortantapproximationwhich
isvalidwhenever thequantum uctuationsaresmallor,equivalently,whenthe
actionsinvolvedarelargecomparedtoPlanck'sconstantsothatthelattermay
b econsidered tob esmall.
Thedecomp ositionofageneralpathintotheclassicalpathand uctuations
around it as employedin(48) in the previous section was merely a matter of
convenience. For the exactly solvable case of a driven harmonic oscillator it
is not really relevant how we express a general path satisfying the b oundary
conditions. Within the semiclassical approximation,however, it is decisive to
expandaroundthepathleadingtothedominantcontribution,i.e. theclassical
path. From a more mathematical p oint of view, we have to evaluate a path
integraloverexp(iS=~)forsmall~.Thiscanb edoneinasystematicwaybythe
x= p
Re exp(i x )
1 1
Fig.8. Instationary phase approximation only asmall region around theextremum
contributes totheintegral. Fortheexampleshownhere,theextremumlies atx=0
At this p oint it may b e useful to give a brief reminder of the metho d of
stationaryphase.Supp ose we wanttoevaluatetheintegral
I( )= Z 1 1 dxg (x)exp i f(x) (73)
inthelimitofverylarge .Insp ectionofFig.8,wheref(x)=x
2
,suggeststhat
thedominantcontributionto theintegralcomes fromaregion,inourexample
of size 1=
p
, around the extremal (or stationary) p oint of the function f(x).
Outside ofthisregion,theintegrandisrapidlyoscillatingandthereforegivesto
leadingorderanegligiblecontribution.Sinceforlarge ,theregiondetermining
theintegral isvery small,wemayexpandthefunctionf(x) lo callyaroundthe
extremumx 0 f(x)f(x 0 )+ 1 2 f 00 (x 0 )(x x 0 ) 2 +::: (74) and replace g (x) by g (x 0
). Neglecting higher order terms, which is allowed if
f 00
(x 0
)isoforderone,weareleftwiththeGaussianintegral
I( )g (x 0 )exp i f(x 0 ) Z 1 1 dxexp i 2 f 00 (x 0 )(x x 0 ) 2 = s 2 jf 00 (x 0 )j g (x 0 )exp h i f(x 0 )+i 4 sgn f 00 (x 0 ) i ; (75) where sgn(f 00 (x 0
))denotes thesignof f
00 (x
0
). Iff(x) p ossesses morethanone
extremum, one has to sum over the contributions of all extrema unless one
extremumcanb eshowntob edominant.
We nowapply the stationary phase approximationto pathintegrals where
1=~playstheroleofthelargeparameter.Sincetheactionisstationary at
clas-sicalpaths,weareobligedtoexpress thegeneralpathas
x(s)=x
cl
(s)+(s); (76)
where x
cl
Withthisdecomp ositiontheactionb ecomes S= Z t 0 ds m 2 _ x 2 V(x) = Z t 0 ds m 2 _ x 2 cl V(x cl ) + Z t 0 ds mx_ cl _ V 0 (x cl ) + Z t 0 ds m 2 _ 2 1 2 V 00 (x cl ) 2 +::: (77)
It isinstructive to comparethis result with theaction(50)for thedriven
har-monicoscillator.Again,the rsttermrepresentstheclassicalaction.Thesecond
term vanishes as was shown explicitly in (51) for the driven oscillator.In the
general case,one can convinceoneselfbypartialintegrationofthekineticpart
and comparisonwith the classical equation of motion that this term vanishes
again.Thisisofcourse aconsequence ofthefactthattheclassicalpath,around
whichweexpand,corresp onds toanextremumoftheaction.Thethirdtermon
the right-hand-side of(77)is theleadingorder terminthe uctuations as was
thecasein(50).Thereishoweveranimp ortantdi erence since foranharmonic
p otentialsthesecond derivative of thep otential V
00
is notconstantand
there-fore thecontribution ofthe uctuations dep ends on theclassicalpath.Finally,
ingeneraltherewillb ehigherordertermsinthe uctuationsasindicatedbythe
dotsin(77).Thesemiclassicalapproximationconsistsinneglectingthesehigher
ordertermssothat afterapartialintegration,we getfortheaction
S sc =S cl 1 2 Z t 0 ds m d 2 ds 2 +V 00 (x cl ) (78)
where theindex`sc'indicates thesemiclassicalapproximation.
Before deriving the propagator insemiclassical approximation,we have to
discusstheregimeofvalidityofthisapproximation.Sincethe rsttermin(78)
givesonlyrisetoaglobalphasefactor,itisthesecondtermwhichdeterminesthe
magnitudeof thequantum uctuations.For thistermto contribute,we should
have
2
=~.1sothat themagnitudeoftypical uctuations isat mostoforder
p
~ .Thetermofthirdorderinthe uctuationsisalreadysmallerthanthesecond
order term by afactor (
p ~ )
3
=~ =
p
~ . If Planck's constant can b e considered
tob e small,we mayindeed neglectthe uctuationcontributionsofhigherthan
secondorderexceptforoneexception:Itmayhapp enthatthesecondorderterm
do es notcontribute,ashasb eenthecase attheconjugatep ointsforthedriven
harmonic oscillator in Sect. 2.7. Then, the leading nonvanishing contribution
b ecomesdominant.Forthefollowingdiscussion, we willnotconsiderthislatter
case.
InanalogytoSect.2.7weobtainforthepropagatorinsemiclassical
approx-imation K(x f ;t;x i ;0)= r m r D 0 exp i S cl (79)
where D=det d 2 ds 2 +V 00 (x cl ) (80) andD 0
isthe uctuationdeterminant(67)ofthefree particle.
Eventhoughitmayseemthatdeterminingtheprefactorisaformidabletask
since the uctuationdeterminantforagivenp otentialhastob eevaluated,this
taskcanb egreatlysimpli ed.Inaddition,thefollowingconsiderationso erthe
b ene tofprovidingaphysicalinterpretationoftheprefactor.Inourevaluation
of the prefactor we followMarinov [16]. The mainidea is to make use of the
semigroupprop erty (9)ofthepropagator
C(x f ;t;x i ;0)exp i ~ S cl (x f ;t;x i ;0) (81) = Z dx 0 C(x f ;t;x 0 ;t 0 )C(x 0 ;t 0 ;x i ;0)exp i ~ S cl (x f ;t;x 0 ;t 0 )+S cl (x 0 ;t 0 ;x i ;0)
where the prefactor C dep ends on the uctuationcontribution. We now have
toevaluatethex
0
-integralwithinthesemiclassicalapproximation.Accordingto
thestationaryphaserequirementdiscussed ab ove,thedominantcontributionto
theintegralcomes fromx
0 =x 0 (x f ;x i ;t;t 0 )satisfying @S cl (x f ;t;x 0 ;t 0 ) @x 0 x 0 =x 0 + @S cl (x 0 ;t 0 ;x i ;0) @x 0 x 0 =x 0 =0: (82)
Accordingtoclassicalmechanicsthesederivativesarerelatedtoinitialand nal
momentumby[17] @S cl @x i xf;tf;ti = p i @S cl @x f xi;tf;ti =p f (83)
sothat (82)can expressed as
p(t 0 ")=p(t 0 +"): (84) The p oint x 0
thus has to b e chosen such that the two partial classical paths
can b e joinedwith acontinuousmomentum.Together they therefore yield the
completeclassicalpathand inparticular
S cl (x f ;t;x i ;0)=S cl (x f ;t;x 0 ;t 0 )+S cl (x 0 ;t 0 ;x i ;0): (85)
This relationensures that thephase factors dep endingon theclassicalactions
onb othsidesof(81)areequal.
Afterhavingidenti edthestationary path,we havetoevaluatetheintegral
overx
0
in(81).WithinsemiclassicalapproximationthisGaussianintegralleads
to C(x f ;t;x i ;0) C(x f ;t;x 0 ;t 0 )C(x 0 ;t 0 ;x i ;0) (86) = 1 2 i~ @ 2 @x 2 S cl (x f ;t;x 0 ;t 0 )+S cl (x 0 ;t 0 ;x i ;0) 1=2 :
Inordertomakeprogress,itisusefultotakethederivativeof(85)withresp ect
tox
f
andx
i
.Keepinginmindthatx
0
dep ends onthese twovariablesone nds
@ 2 S cl (x f ;t;x i ;0) @x f @x i = @ 2 S cl (x f ;t;x 0 ;t 0 ) @x f @x 0 @x 0 @x i + @ 2 S cl (x 0 ;t 0 ;x i ;0) @x i @x 0 @x 0 @x f + @ 2 @x 2 0 S cl (x f ;t;x 0 ;t 0 )+S cl (x 0 ;t 0 ;x i ;t) @x 0 @x i @x 0 @x f : (87)
Similarly,one ndsbytaking derivativesofthestationaryphase condition(82)
@x 0 @x f = @ 2 @x f x 0 S cl (x f ;t;x 0 ;t 0 ) @ 2 @x 2 0 S cl (x f ;t;x 0 ;t 0 )+S cl (x 0 ;t 0 ;x i ;0) (88) and @x 0 @x i = @ 2 @x i x 0 S cl (x 0 ;t 0 ;x i ;0) @ 2 @x 2 0 S cl (x f ;t;x 0 ;t 0 )+S cl (x 0 ;t 0 ;x i ;0) : (89)
These expressions allow to eliminate thepartialderivatives of x
0
with resp ect
tox
i
andx
f
app earingin(87)andone nallyobtains
@ 2 @x 2 0 S cl (x f ;t;x 0 ;t 0 )+S cl (x 0 ;t 0 ;x i ;0) 1 (90) = @ 2 @x i @x f S cl (x f ;t;x i ;0) @ 2 S cl (x f ;t;x 0 ;t 0 ) @x f @x 0 @ 2 S cl (x 0 ;t 0 ;x i ;0) @x i @x 0 :
Inserting this result into (86), the prefactor can b e identi ed as the so-called
VanVleck-Pauli-Morettedeterminant[18,19,20]
C(x f ;t;x i ;0)= 1 2 i~ @ 2 S cl (x f ;t;x i ;0) @x f @x i 1=2 (91)
sothat thepropagatorinsemiclassicalapproximation nallyreads
K(x f ;t;x i ;0) (92) = 1 2 ~ @ 2 S cl (x f ;t;x i ;0) @x f @x i 1=2 exp i ~ S cl (x f ;t;x i ;0) i 4 +n 2
where the Morse index ndenotes the numb er of signchanges of @
2 S cl =@x f @x i
Aswe have alreadymentionedab ove,derivatives oftheactionwith resp ect
top ositionarerelatedtomomenta.Thisallowstogiveaphysicalinterpretation
oftheprefactorofthepropagatorasthechange oftheendp ointofthepathas
afunctionoftheinitialmomentum
@ 2 S cl @x i @x f 1 = @x f @p i : (93)
A zero of this expression, or equivalently a divergence of the prefactor of the
propagator,indicatesaconjugatep ointwheretheendp ointdo esnotdep endon
theinitialmomentum.
Toclosethissection, we wantto comparethesemiclassicalresult (92)with
exact resultsforthefree particleandtheharmonicoscillator.Inourdiscussion
of the free particle inSect. 2.3we already mentionedthat thepropagatorcan
b eexpressedentirelyintermsofclassicalquantities.Indeed,theexpression(19)
forthepropagatorofthefree particleagreeswith(92).
For theharmonicoscillator,we knowfromSect.2.7 thattheprefactor do es
notdep endonap ossiblypresentexternalforce.Wemaytherefore considerthe
action(56)intheabsence ofdriving f(s)=0which thenreads
S cl = m! 2sin(! t) x 2 i +x 2 f cos (! t) 2x i x f : (94)
Takingthederivativewithresp ect to x
i
andx
f
one ndsfortheprefactor
C(x f ;t;x i ;0)= m! 2 i~sin(! t) 1=2 (95)
which is identical with theprefactor inour previous result (72). As exp ected,
forp otentialsat mostquadraticintheco ordinate thesemiclassicalpropagator
agreeswiththeexactexpression.
2.9 Imaginary TimePathIntegral
In the discussion of dissipative systems we will b e dealingwith a system
cou-pled toalargenumb erofenvironmentaldegrees offreedom.Inmostcases, the
environmentwillact like a largeheat bath characterized by atemp erature T.
Thestate of theenvironmentwilltherefore b e givenbyan equilibriumdensity
matrix.Occasionally,we mayalso b einterested intheequilibriumdensity
ma-trixofthesystemitself.Such astate mayb e reachedafter equilibrationdue to
weakcouplingwithaheatbath.
In order to describ e such thermal equilibrium states and the dynamics of
the system on a unique fo oting, it is desirable to express equilibriumdensity
matricesintermsofpathintegrals.Thisisindeedp ossibleasone recognizesby
writingtheequilibriumdensityop erator inp ositionrepresentation
(x;x 0 )= 1 hxjexp( H)jx 0 i (96)
withthepartitionfunction
Z=
Z
dxhxjexp( H)jxi: (97)
Comparingwith thepropagatorinp ositionrepresentation
K(x;t;x 0 ;0)=hxjexp i ~ Ht jx 0 i (98)
one concludes that apart from the partition function the equilibrium density
matrixisequivalenttoapropagatorinimaginarytimet= i~ .
Afterthesubstitution =is theactioninimaginarytime i~ reads
Z i~ 0 ds " m 2 dx ds 2 V(x) # =i Z ~ 0 d " m 2 dx d 2 +V(x) # : (99)
Here and in the following, we use greek letters to indicate imaginary times.
Motivatedbytheright-handsideof(99)wede netheso-calledEuclideanaction
S E [x]= Z ~ 0 d h m 2 _ x 2 +V(x) i : (100)
Eventhoughonemightfearalackofintuitionformotioninimaginarytime,this
resultsshowsthatitcansimplyb ethoughtofasmotionintheinvertedp otential
in real time.Withthe Euclidean action (100) we now obtainas animp ortant
result the path integral expression for the (unnormalized) equilibriumdensity
matrix hxjexp( H)jx 0 i= Z x(~ )=x x(0)=x 0 Dxexp 1 ~ S E [x] : (101)
Thiskindoffunctionalintegralwasdiscussed asearlyas1923byWiener[21]in
thecontextofclassicalBrownianmotion.
Asanexamplewe consider the(undriven)harmonicoscillator.There is
ac-tuallynoneedtoevaluateapathintegralsince we knowalreadyfromSect. 2.7
thepropagator K(x f ;t;x i ;0)= r m! 2 i~sin(! t) exp i m! 2~ (x 2 i +x 2 f )cos (! t) 2x i x f sin(! t) : (102)
Transforming the propagatorinto imaginary timet ! i~ and renaming x
i
andx
f
intox
0
andx,resp ectively,oneobtainstheequilibriumdensitymatrix
(x;x 0 ) (103) = 1 Z r m! 2 ~sinh (~ ! ) exp m! 2~ (x 2 +x 02 )cosh (~ ! ) 2xx 0 sinh(~ ! ) :
Thepartitionfunctionisobtainedbyp erformingthetraceas
Z=
Z
dxhxjexp( H)jxi=
1
whichagreeswiththeexpression Z= 1 X n=0 exp ~! n+ 1 2 (105)
based ontheenergy levelsoftheharmonicoscillator.
Sincethepartitionfunctionoftenservesasastartingp ointforthecalculation
ofthermo dynamicprop erties,itisinstructive totakeacloserathowthis
quan-titymay b e obtainedwithinthe path integral formalism.Ap ossible approach
isthe onewe just havesketched. By meansofanimaginarytimepathintegral
one rstcalculateshxjexp( H)jxiwhichis prop ortionaltotheprobabilityto
ndthesystematp ositionx.Subsequentintegrationoverco ordinatespacethen
yieldsthepartitionfunction.
However,thepartitionfunctionmayalsob edeterminedinonestep.Tothis
end, weexpand aroundthep erio dic trajectory withextremal Euclideanaction
which in our case is given by x( ) = 0. Any deviation will increase b oth the
kinetic and p otentialenergy and thus increase the Euclidean action.All other
trajectories contributing to the partition function are generated by a Fourier
seriesontheimaginarytimeintervalfrom0to~
x( )= 1 p ~ " a 0 + p 2 1 X n=1 a n cos ( n )+b n sin( n ) # (106)
where wehaveintro duced theso-calledMatsubarafrequencies
n = 2 ~ n: (107)
This ansatz should b e compared with (62) for the uctuations where a
0 was
xedb ecausethe uctuationshadtovanishattheb oundaries.Forthepartition
functionthisrequirementisdropp ed sincewehavetointegrateoverallp erio dic
trajectories. Furthermore, we note that indeed with the ansatz (106) only the
p erio dictrajectoriescontribute.Allotherpathscostanin niteamountofaction
dueto thejumpattheb oundaryas we willseeshortly.
Insertingthe Fourier expansion (106) into the Euclideanaction of the
har-monicoscillator S E = Z ~ 0 d m 2 _ x 2 +! 2 x 2 (108) we nd S E = m 2 " ! 2 a 2 0 + 1 X n=1 ( 2 n +! 2 )(a 2 n +b 2 n ) # : (109)
AsinSect.2.7wedonotwanttogointothemathematicaldetailsofintegration
measuresandJacobideterminants.Unfortunately,thefreeparticlecannotserve
as a reference here b ecause itspartition function do es not exist.We therefore
contentourselves withremarkingthatb ecauseof
1 ! 1 Y 1 2 n +! 2 = ~ P 1 n=1 2 n 1 2sinh(~ ! =2) (110)
theresult oftheGaussian integralovertheFourierco eÆcientsyieldsthe
parti-tionfunctionuptoafrequencyindep endentfactor.Thisenablesustodetermine
thepartitionfunctioninmorecomplicatedcasesbypro ceedingasab oveand
us-ingthepartitionfunctionharmonicoscillatoras areference.
Returningtothedensitymatrixoftheharmonicoscillatorwe nallyobtain
byinsertingthepartitionfunction(104)intotheexpression(103)forthedensity
matrix (x;x 0 )= s m! ~ cosh (~ ! ) 1 sinh(~ ! ) (111) exp m! 2~ (x 2 +x 02 )cosh (~ ! ) 2xx 0 sinh (~ ! ) :
Without path integrals, this result would require the evaluationof sums over
Hermitep olynomials.
Theexpression forthe density matrix(111) can b e veri edinthe limitsof
highandzero temp erature.Intheclassicallimitofveryhightemp eratures,the
probabilitydistributioninrealspaceis givenby
P(x)= (x;x)= r m! 2 2 exp m! 2 2 x 2 exp [ V(x)]: (112)
We thus have obtainedtheBoltzmanndistributionwhich dep ends only onthe
p otentialenergy.Thefactthatthekineticenergydo esnotplayarolecaneasily
b eundersto o dintermsofthepathintegralformalism.Excursionsinaveryshort
time~ costto omuchactionandaretherefore stronglysuppressed.
Intheopp osite limitof zero temp eraturethe density matrixfactorizes into
apro ductofgroundstate wavefunctions oftheharmonicoscillator
lim !1 (x;x 0 )= m! ~ 1=4 exp m! 2~ x 2 m! ~ 1=4 exp m! 2~ x 02 (113)
asshould b eexp ected.
3 Dissipative Systems
3.1 Intro duction
Inclassicalmechanicsdissipationcanoftenb eadequatelydescrib ed byincluding
a velo city dep endent damping term into the equation of motion. Such a
phe-nomenologicalapproach isnolonger p ossibleinquantummechanicswhere the
Hamiltonformalismimpliesenergy conservation fortime-indep endent
Hamilto-nians. Then, a b etter understanding of the situation is necessary in order to
arriveat anappropriatephysicalmo del.
Adamp edp endulummayhelpus tounderstand themechanismof
under-molecules in the air surrounding the p endulum's mass. We may consider the
p endulum and the air molecules as one large system which,if assumed to b e
isolatedfromfurtherdegreesoffreedom,ob eysenergyconservation.Theenergy
of the p endulum alone,however, willingeneral not b e conserved. This single
degreeoffreedomisthereforesubject todissipationarisingfromthecouplingto
otherdegrees offreedom.
Thisinsightwillallowusinthefollowingsectiontointro duceamo delfor a
systemcoupledtoan environmentandto demonstrateexplicitlyits dissipative
nature. In particular,we willintro duce thequantities needed foradescription
which fo cuses on the system degree of freedom. We are then ina p osition to
return to the path integral formalismand to demonstrate how it mayb e
em-ployed to study dissipative systems. Starting from the mo del of system and
environment,the latter willb e eliminatedto obtain areduced description for
thesystemalone.Thisleaves uswith ane ective actionwhichformsthebasis
ofthepathintegraldescription ofdissipation.
3.2 EnvironmentasCollectionofHarmonic Oscillators
A suitablemo delfordissipative quantumsystems should b othincorp orate the
idea of a coupling b etween system and environment and b e amenable to an
analytic treatment ofthe environmentalcoupling. These requirementsare met
byamo delwhichnowadaysisoftenreferredtoasCaldeira-Leggettmo del[22,23]
eventhoughithasb eendiscussedintheliteratureundervariousnamesb eforefor
harmonicsystems[24,25,26,27]andanharmonicsystems [28].TheHamiltonian
H=H S +H B +H SB (114)
consistsofthreecontributions.TheHamiltonianofthesystemdegreeoffreedom
H S = p 2 2m +V(q ) (115)
mo dels a particle of mass m moving in a p otential V. Here, we denote the
co ordinate byq tofacilitatethedistinctionfromtheenvironmentalco ordinates
x n
whichwewillintro duceinamoment.Ofcourse,thesystemdegreeoffreedom
do esnothavetob easso ciatedwitharealparticlebutmayb equiteabstract.In
fact,asubstantialpartofthecalculationstob e discussed inthefollowingdo es
notdep endonthedetailedformofthesystemHamiltonian.
TheHamiltonianoftheenvironmentaldegreesoffreedom
H B = N X n=1 p 2 n 2m n + m n 2 ! 2 n x 2 n (116)
describ es acollectionof harmonicoscillators.While theprop ertiesof the
envi-ronmentmayinsomecasesb e chosen onthebasisofamicroscopicmo del,this
linear electric element should b e welldescrib ed by aHamiltonianof the form
(116).Ontheotherhand,theunderlyingmechanismleadingtodissipation,e.g.
inaresistor, mayb emuchmorecomplicatedthanthatimpliedbythemo delof
acollectionofharmonicoscillators.
Thecouplingde nedbytheHamiltonian
H SB = q N X n=1 c n x n +q 2 N X n=1 c 2 n 2m n ! 2 n (117)
isbilinearinthep ositionop eratorsofsystemandenvironment.Therearecases
where the bilinear couplingis realistic, e.g.for an environment consisting of a
linear electric circuit like the resistor just mentionedor fora dip olar coupling
to electromagnetic eld mo des encountered in quantumoptics. Withinamore
general scop e, this Hamiltonian mayb e viewed as linearization of anonlinear
couplinginthelimitofweakcouplingtotheenvironmentaldegrees offreedom.
Aswas rstp ointedoutbyCaldeiraandLeggett,anin nitenumb erofdegrees
of freedomstill allowsforstrongdampingevenifeach environmentaloscillator
couples onlyweaklytothesystem[22,23].
Anenvironmentconsistingofharmonicoscillatorsasin(116)mightb e
criti-cized.Ifthep otentialV(q )isharmonic,onemaypasstonormalco ordinatesand
thus demonstratethat after some timea revival of theinitialstate willo ccur.
ForsuÆcientlymanyenvironmentaloscillators,however,thisso-calledPoincare
recurrence timetendstoin nity[29].Therefore,evenwithalinearenvironment
irreversibilityb ecomesp ossibleat leastforallpracticalpurp oses.
Thereader mayhavenoticedthatinthecouplingHamiltonian(117)aterm
is present which only contains anop erator actingin the systemHilb ert space
butdep ends onthecouplingconstantsc
n
.Thephysicalreason fortheinclusion
of this term liesina p otential renormalizationintro duced by the rst termin
(117).Thisb ecomesclearifweconsider theminimumoftheHamiltonianwith
resp ect tothesystemandenvironmentco ordinates.Fromtherequirement
@H @x n =m n ! 2 n x n c n q ! =0 (118) weobtain x n = c n m n ! 2 n q: (119)
Usingthisresult todeterminetheminimumoftheHamiltonianwithresp ect to
thesystemco ordinatewe nd
@H @q = @V @q N X n=1 c n x n +q N X n=1 c 2 n m n ! 2 n = @V @q : (120)
Thesecondtermin(117)thusensuresthat thisminimumis determinedbythe
barep otentialV(q ).
de-eliminationofthe environmentaldegrees of freedomleads indeed to adamp ed
equationofmotionforthesystemco ordinate.Wep erformtheeliminationwithin
theHeisenb erg picture wheretheevolutionofanop eratorA isdeterminedby
dA dt = i ~ [H ;A]: (121)
FromtheHamiltonian(114)weobtaintheequationsofmotionforthe
environ-mentaldegreesoffreedom
_ p n = m n ! 2 n x n +c n q _ x n = p n m n (122)
andthesystemdegree offreedom
_ p= @V @q + N X n=1 c n x n q N X n=1 c 2 n m n ! 2 n _ x= p m : (123)
Thetrick for solving theenvironmentalequations of motion (122) consists
intreatingthesystemco ordinateq (t)asifitwereagivenfunctionoftime.The
inhomogeneousdi erentialequationthenhasthesolution
x n (t)=x n (0)cos (! n t)+ p n (0) m n ! n sin(! n t)+ c n m n ! n Z t 0 dssin ! n (t s) q (s): (124)
Insertingthisresult into(123) one ndsane ective equationofmotionforthe
systemco ordinate mq Z t 0 ds N X n=1 c 2 n m n ! n sin ! n (t s) q (s)+ @V @q +q N X n=1 c 2 n m n ! 2 n (125) = N X n=1 c n x n (0)cos(! n t)+ p n (0) m n ! n sin(! n t) :
By apartialintegrationof thesecond termonthe left-handsidethis equation
ofmotioncanb e castintoits nalform
mq+m Z t 0 ds (t s)q (s)_ + @V @q =(t) (126)
withthedampingkernel
(t)= 1 m N X c 2 n m n ! 2 n cos (! n t) (127)
andtheop erator-valued uctuatingforce (t)= N X n=1 c n x n (0) c n m n ! 2 n q (0) cos (! n t)+ p n (0) m n ! n sin(! n t) : (128)
The uctuatingforce vanishes if averagedover athermaldensity matrixof
theenvironmentincludingthecouplingtothesystem
h(t)i B+SB = Tr B (t)exp (H B +H SB ) Tr B exp (H B +H SB ) =0: (129)
Forweakcoupling,onemaywanttosplito thetransienttermm (t)q (0)which
isofsecond orderinthecouplingand writethe uctuatingforceas [30]
(t)=(t) m (t)q (0): (130)
Thesode ned force(t) vanishesifaveragedovertheenvironmentalone
h(t)i B = Tr B (t)exp ( H B ) Tr B exp( H B ) =0: (131)
An imp ortantquantity to characterize the uctuating force is the
correla-tionfunctionwhich againcan b e evaluated for with resp ect to H
B
+H
SB or
equivalentlyfor with resp ect to H
B
alone.With (128) and (130) we get the
correlationfunction h(t)(0)i B = X n;l c n c l x n (0)cos (! n t)+ p n (0) m n ! i sin(! n t) x l (0) B : (132)
Inthermalequilibriumthesecondmomentsaregivenby
hx n (0)x l (0)i B =Æ nl ~ 2m n ! n coth ~ ! n 2 (133) hp n (0)x l (0)i B = i~ 2 Æ nl ; (134)
sothat thenoisecorrelationfunction nallyb ecomes
h(t)(0)i B = N X n=1 ~c 2 n 2m n ! n coth ~ ! n 2 cos (! n t) isin(! n t) : (135)
Theimaginarypartapp earinghereisaconsequenceofthefactthattheop erators
(t)and(0)ingeneraldonotcommute.Thecorrelationfunction(135)app ears
It is remarkable that within areduced description for the systemalone all
quantitiescharacterizingtheenvironmentmayb eexpressedintermsofthesp
ec-traldensityofbathoscillators
J(! )= N X n=1 c 2 n 2m n ! n Æ(! ! n ): (136)
As anexample,thedampingkernel mayb e expressed intermsofthis sp ectral
densityas (t)= 1 m N X n=1 c 2 n m n ! 2 n cos (! n t)= 2 m Z 1 0 d! J(! ) ! cos(! t): (137)
For practicalcalculations, itis therefore unnecessary to sp ecify allparameters
m n ;! n and c n
app earing in (116) and (117). It rather suÆces to de ne the
sp ectraldensityJ(! ).
Themostfrequently usedsp ectraldensity
J(! )=m ! (138)
is asso ciated with the so-called Ohmicdamping. This term is sometimes
em-ployedto indicate aprop ortionalityto frequency merely at lowfrequencies
in-stead of over the whole frequency range. In fact,in any realistic situationthe
sp ectral density willnot increase like in (138) for arbitrarily high frequencies.
It is justi ed to use the term \Ohmic damping"even if (138) holds only b
e-lowacertainfrequency providedthisfrequency ismuchhigherthanthetypical
frequencies app earinginthesystemdynamics.
From(137)one ndsthedampingkernelforOhmicdamping
(t)=2 Æ(t); (139)
which renders (126) memory-free. We thus recover the velo city prop ortional
dampingtermfamiliarfromclassicaldamp ed systems.Itshould b e notedthat
thefactoroftwoin(139)disapp earsup onintegrationin(126)since(137)implies
thatthedelta functionissymmetricaroundzero.
At this p oint, we want to brie y elucidate the originof the term \Ohmic
damping".Let us consider the electric circuit shown in Fig. 9 consisting of a
resistance R, acapacitance C and aninductance L. Summingupthe voltages
around the lo op, one obtains as equation of motion for the charge Q on the
capacitor L Q+R _ Q+ Q C =0; (140)
whichshowsthatanOhmicresistorleadsindeedtomemorylessdamping.These
considerationsdemonstratethatevenwithoutknowledgeofthemicroscopic
R
L C
Fig.9. LC oscillatorwithOhmicdamping duetoaresistorR
Thesp ectraldensity(138)forOhmicdampingunfortunatelydivergesathigh
frequencies which,as alreadymentioned,cannot b e the case in practice. Even
in theoretical considerations this feature of strictly Ohmic damping maylead
to divergencies and acuto isneeded for regularization.One p ossibilityis the
Drude cuto ,wherethesp ectraldensity
J(! )=m ! ! 2 D ! 2 +! 2 D (141)
ab ovefrequencies oftheorderof !
D
issuppressed. Thecorresp onding damping
kernelreads (t)= ! D exp( ! D jtj): (142)
Thisleadstomemorye ectsin(126)forshorttimest<!
1 D
.Forthelong-time
b ehaviour, however, only the Ohmic low frequency b ehaviour of the sp ectral
density(141)isrelevant.IfaDrudecuto isintro ducedfortechnicalreasons,the
cuto frequency !
D
shouldb emuchlarger thanallotherfrequencies app earing
intheprobleminordertoavoidspuriouse ects.
Therelation (136) b etween thesp ectral density and the \microscopic"
pa-rameters impliesthat one mayset c
n =m n ! 2 n
withoutloss of generalitysince
the frequencies !
n
and the oscillator strengths c
2 n =2m n ! n
can still b e freely
chosen. This sp ecial choice for the coupling constants has the advantage of a
translationallyinvariantcoupling[31]
H=H S + N X n=1 p 2 n 2m n + m n 2 ! 2 n (x n q ) 2 : (143)
Furthermore,we nowcandeterminethetotalmassofenvironmentaloscillators
N X n=1 m n = 2 Z 1 0 d! J(! ) ! 3 : (144)
If the sp ectral density of bath oscillators at small frequencies takes the form
J(! )!
, thetotalmassofbathoscillatorsisin nitefor 2.Inparticular,
theparticlewillb ehave forlongtimeslikeitwerefreealb eitp ossessinga
renor-malized mass due to the environmental coupling [32]. We emphasizethat the
divergence of the total mass for 2 is due to an infrared divergence and
therefore indep endent ofahigh-frequencycuto .
Itisalsousefultoexpressthep otentialrenormalizationintro ducedin(117)in
termsofthesp ectraldensityofbathoscillators.From(136)itisstraightforward
toobtain q 2 N X n=1 c 2 n 2m n ! 2 n = q 2 Z 1 0 d! J(! ) ! : (145)
Thistermisin niteforstrictlyOhmicdampingbutb ecomes nitewhena
high-frequency cuto isintro duced.
Finally,one ndsforthenoise correlationfunction(132)
K(t)=h(t)(0)i B =~ Z 1 0 d! J(! ) coth ~ ! 2 cos (! t) isin(! t) : (146)
Intheclassicallimit,~!0,thiscorrelationfunctionreducestothereal-valued
expression
K(t)=mk
B
T (t); (147)
where we havemadeuseof (137).For Ohmicdampingthisimpliesdelta
corre-lated,i.e.white,noise.
Inthe quantum case, thenoise correlationfunction is complexand can b e
decomp osed intoitsrealandimaginarypart
K(t)=K
0
(t)+iK
00
(t): (148)
Employingonce more(137),one immediately ndsthat theimaginary part is
relatedtothetimederivative ofthedampingkernel by
K 00 (t)= m~ 2 d dt : (149)
ForOhmicdamping,therealpart reads
K 0 (t)= m (~ ) 2 1 sinh 2 t ~ (150)
whichimpliesthatatzerotemp eraturethenoiseiscorrelatedevenforlongtimes.
Thenoise correlation thenonlydecaysalgebraicallylike1=t
2
muchincontrast
totheclassicalresult (147).
3.3 E ective Action
alone.Thissectionwillb edevotedtoadiscussionofthecorresp ondingpro cedure
withinthepathintegral formalism.
Westarttoillustratethebasicideabyconsideringthetimeevolutionofthe
fulldensitymatrixof systemandenvironment
W(q f ;x nf ;q 0 f ;x 0 nf ;t)= Z dq i dq 0 i dx ni dx 0 ni K(q f ;x nf ;t;q i ;x ni ;0) (151) W(q i ;x ni ;q 0 i ;x 0 ni ;0)K (q 0 f ;x 0 nf ;t;q 0 i ;x 0 ni ;0)
which is induced by the two propagators K. Here, the co ordinates q and x
n
refer againto the systemand bath degrees of freedom,resp ectively. The
envi-ronmentisassumedtob einthermalequilibriumdescrib edbythedensitymatrix
W B
while thesystemmayb e ina nonequilibriumstate .If we neglect initial
correlationsb etween systemand environment,i.e. ifwe switch onthecoupling
after preparationof theinitialstate,theinitialdensity matrixmayb e written
infactorizedform W(q i ;x ni ;q 0 i ;x 0 ni ;0)=(q i ;q 0 i )W B (x ni ;x 0 ni ): (152)
Sinceweareonlyinterestedinthedynamicsofthesystemdegreeoffreedom,
wetrace outtheenvironment.Thenthetimeevolutionmayb eexpressed as
(q f ;q 0 f ;t)= Z dq i dq 0 i J(q f ;q 0 f ;t;q i ;q 0 i ;0)(q i ;q 0 i ) (153)
withthepropagatingfunction
J(q f ;q 0 f ;t;q i ;q 0 i ;0)= Z dx nf dx ni dx 0 ni K(q f ;x nf ;t;q i ;x ni ;0) (154) W B (x ni ;x 0 ni )K (q 0 f ;x nf ;t;q 0 i ;x 0 ni ;0):
Here, thetrace has b een p erformedbysetting x
nf
=x
0 nf
and integrating over
theseco ordinates.Thepropagatorsmayb eexpressedasrealtimepathintegrals
while theequilibriumdensity matrixof thebathis given by apathintegralin
imaginary time. Performing the path integrals and the conventional integrals
app earing in(154) one nds a functionaldep ending on the system path. The
imp ortantp ointisthat thisfunctionalcontainsallinformationab outthe
envi-ronmentrequiredtodeterminethesystemdynamics.
Forfactorizinginitialconditions,thepropagatingfunctionJ hasb een
calcu-latedbyFeynmanand Vernon[33]onthebasisoftheHamiltonian(114).More
general initialconditionstaking into account correlations b etween system and
environmentmayb e consideredaswell[34].
Instead of deriving the propagating function we will demonstrate how to
tracetheenvironmentoutoftheequilibriumdensitymatrixofsystemplus
We start from the imaginary time path integral representation of the full
equilibriumdensitymatrix
W (q ;x n ;q 0 ;x 0 n )= 1 Z Z Dq N Y n=1 Dx n ! exp 1 ~ S E [q ;x n ] (155)
where thepathsrunfromq (0) =q
0 and x n (0)=x 0 n toq (~ ) =qand x n (~ )= x n
. TheEuclideanactioncorresp onding to themo delHamiltonian(114) reads
inimaginarytime S E [q;x n ]=S E S [q ]+S E B [x n ]+S E SB [q;x n ] (156) with S E S [q ]= Z ~ 0 d m 2 _ q 2 +V(q ) (157) S E B [x n ]= Z ~ 0 d N X n=1 m n 2 _ x 2 n +! 2 n x 2 n (158) S E SB [q ;x n ]= Z ~ 0 d q N X n=1 c n x n +q 2 N X n=1 c 2 n 2m n ! 2 n ! : (159)
The reduced density matrix of the system is obtained by tracing over the
environmentaldegreesoffreedom
(q ;q 0 )=Tr B W (q ;x n ;q 0 ;x 0 n ) = 1 Z Z Dq Z N Y n=1 dx n I N Y n=1 Dx n exp 1 ~ S E [q ;x n ] (160)
wherethecircleonthesecondfunctionalintegralsignindicatesthat onehasto
integrate over closed paths x
n
(0) = x
n
(~ ) =x
n
when p erformingthe trace.
Thedep endenceontheenvironmentalcouplingmayb emadeexplicitbywriting
(q ;q 0 )= 1 Z Z Dqexp 1 ~ S E S [q] F[q ] (161)
where the in uence functionalF[q ] describ es the in uence of the environment
onthesystem.Here, the partitionfunctionZ should notb econfused withthe
partitionfunctionZ
ofsystemplusenvironment.Therelationb etweenthetwo
quantitieswillb ediscussed shortly.
Since the bath oscillators are not coupledamong each other, thein uence
functionalmayb edecomp osedintofactorscorresp ondingtotheindividualbath
oscillators F[q]= N Y 1 Z n F n [q] (162)
where Z n = 1 2sinh(~ ! n =2) (163)
isthepartitionfunctionofasinglebathoscillator.Thein uencefunctionalofa
bathoscillatorcanb e expressed as
F n [q]= Z dx n I Dx n exp 1 ~ S E n [q ;x n ] (164)
withtheaction
S E n [q ;x n ]= Z ~ 0 d m n 2 " _ x 2 n +! 2 n x n c n m n ! 2 n q 2 # : (165)
The partitionfunctionZ of the damp edsystemis relatedto the fullpartition
function Z
by the partition function of the environmental oscillators Z
B = Q N n=1 Z n according to Z =Z =Z B
. Inthelimitofvanishingcoupling, c
n
=0,
the in uence functional b ecomes F[q] = 1 so that (161) reduces to the path
integralrepresentationofthedensitymatrixofanisolatedsystemas itshould.
Apartfromthep otentialrenormalizationtermprop ortionaltoq
2
,theaction
(165)describ es adrivenharmonicoscillator.Wemaythereforemake useofour
results fromSect.2.7.Afteranalyticcontinuationt! i~ in(56)andsetting
x i =x f =x n
one ndsfortheclassicalEuclideanaction
S E;cl n [q]=m n ! n cosh (~ ! n ) 1 sinh(~ ! n ) x 2 n c n Z ~ 0 d sinh(! n )+sinh(! n (~ )) sinh(~ ! n ) x n q ( ) c 2 n m n ! n Z ~ 0 d Z 0 d sinh (! n (~ ))sinh(! n ) sinh(~ ! n ) q ( )q ( ) + c 2 n 2m n ! 2 n Z ~ 0 d q 2 ( ): (166)
Inviewoftherequired integrationover x
n
one completesthesquare
S E;cl n [q]=m n ! n cosh (~ ! n ) 1 sinh(~ ! n ) (x n x (0) n ) 2 Z ~ 0 d Z 0 d K n ( )q ( )q ( ) + c 2 n 2m n ! 2 n Z ~ 0 d q 2 ( ) (167) where x (0) n
do es notneedtob e sp eci edsinceit dropsoutafterintegration.
Theintegral kernel app earingin(167)followsfrom(166)as
K n ( )= c 2 n 2m n ! n cosh ! n ~ 2 sinh ~ ! n =K n (~ ) (168)
