Path integrals from classical momentum paths

Transcription

1 ah negrals rom classcal momenum ahs John egseh Dearmen o hyscs Unversy o New Orleans New Orleans LA 7048 Absrac The ah negral ormulaon o uanum mechancs consrucs he roagaor by evaluang he acon S or all classcal ahs n coordnae sace. A corresondng momenum ah negral may also be dened hrough ourer ransorms n he endons. Alhough hese momenum ah negrals are esecally smle or several secal cases no one has o my knowledge ever ormally consruced hem rom all classcal ahs n momenum sace. I show ha hs s ossble because here exss anoher classcal mechancs based on an alernae classcal acon R. amlon s Canoncal euaons resul rom a varaonal rncle n boh S and R. S uses xed begnnng and endng saal ons whle R uses xed begnnng and endng momenum ons. Ths alernave acon s classcal mechancs also ncludes a amlon-jacob euaon. I also resen some moran ons concernng he begnnng and endng condons on he acon necessary o aly a Canoncal ransormaon. These roeres exlan he alure o he Canoncal ransormaon n he hase sace ah negral. I ollows ha a ah negral may be consruced rom classcal oson ahs usng S n he coordnae reresenaon or rom classcal momenum ahs usng R n he momenum reresenaon. Several examle calculaons are resened ha llusrae he smlcaons and raccal advanages made ossble by hs broader vew o he ah negral. In arcular he normalzed amlude or a ree arcle s ound whou usng he Schrödnger euaon he nernal sn degree o reedom s smly and naurally derved and he smle harmonc oscllaor s calculaed. Inroducon The sandard ormulaon o uanum mechancs assumes robably amludes and deduces amazngly accurae uanave resuls. A sasyng answer o he ualave ueson o wha s a robably amlude s sll lackng. A very comellng dscusson o hs amlude s gven n eynmann and bbs [] where he ah negral ormulaon s resened. Ths more nuve ormulaon looses much o s luser when conroned wh basc uanes such as sn. A very basc derence beween uanum mechancs and classcal mechancs s he alure o Canoncal ransormaons n he hase sace ah negral. Ths derence has movaed much neres because o he oenal o consderably smly calculaons [] [3] [4]. Analogous uanum ransormaons have been recenly ound [3] [4]. In he ollowng I show how a momenum sace ah negral may be consruced by consderng all ossble classcal momenum ahs. Usng a arallel classcal mechancs dened usng an alernae acon R I nd all o he necessary eaures o dene a momenum ah negral as he sum over all momenum ahs o he uany R h e where h s lanck s consan. In he rocess o analyzng hs classcal mechancs I also show why Canoncal ransormaons do no work n he hase sace ah negral. Ths consderably

2 broadened unverse o ossble ah negrals also resuls n some mmedae smlcaons. I wll demonsrae several secc examles o hese raccal advanages and smlcaons ncludng he calculaon o he coordnae ah negral normalzaon consan or a ree arcle whou relyng on he Schrödnger euaon a drec and smle calculaon o a arcle sn roagaor and he smle harmonc oscllaor. Ths new ah negral wll also show how he momenum reresenaon n uanum mechancs may be vewed as a conseuence o hs arallel classcal mechancs. a b c d gure. Shown schemacally are he ahs or R and S where he sold lnes are he crcal ahs and he dashed lnes are arbrary ahs. gure a and b shows ahs or he acon S[] where he coordnae ah has xed begnnng and endng condons. gure c and d shows he ahs or he acon R[] where he momenum ah has xed begnnng and endng condons. I now survey some ars o he classcal heory ha are relevan o hs work. amlon s rncle ells us ha o all he ossble ahs ha a arcle may ake he one ha exremzes he acon S=L[d/d]d s he ah ha s acually ollowed [5]. The acon S has an exrema when S =0.e. L L L d L S d 0. d d The varaon rom he omal ah s xed a he nal and nal mes.e. = =0 see gure a. The ah ha naure selecs s he one ha makes S=0

3 or arbrary and hs mles he Euler-Lagrange euaons o moon. The acual ah ollowed reures he selecon o wo arbrary consans e.g. he nal condons and d d. The above s easly generalzed o many degrees o reedom. To clary he noaon hs dscusson s resrced o one. One moran bene rom hs classcal heory s he lexbly wh resec o coordnae ransormaons ha resuls. Any coordnaes ha sasy amlon s rncle may be used. Even more lexbly s ganed I use he wo ndeenden varables L where he generalzed momenum mlcly denes. Inroducng he amlonan uncon he Lagrangan s exressed as [5] L. Asserng he ndeendence o he arbrary varaons and whle only usng = =0 as shown n gure a and b I can derve amlon's euaons L L S d 0 d. L In reerence [6] s noed ha because o he denon and are relaed so ha he varaons are no ndeenden. The varaons and beng smly relaed by L derenaon when =0 are deenden and he more general relaon means ha and are also deenden. The rs amlon s euaon s hen a resul o he denon. Ths argumen s vald I ake euaon o be a conseuence o amlon s rncle exressed n erms o and. In he amlonan ormulaon however I generalze o be ndeenden allowng a wder class o ossble ahs and ake euaon o be an axomac saemen. Wh ndeenden and as shown n gure boh amlon s euaons are needed o exremze ; and are conneced only by amlon s euaons [7]. The momenum acon R In he above sandard dervaon he varable s unconsraned a he begnnng and endng ons. I may also derve amlon's euaons usng he begnnng and endng condons = =0 as shown n gure c and d rom he unconal R R d Kd. As beore I allow an ndeenden and and R d d d d 0. As can be seen here R=0 only amlon s euaons are sased. There exss anoher varaonal rncle ha uses xed begnnng and endng condons as shown n 3

4 gure c and d. In he case o xed begnnng and endng condons = =0 R s no saonary.e. R= -. The varaons S and R are boh zero only when boh begnnng and endng condons are sased = =0 and = =0.e. xed nal and nal hase sace ons { } and { }. These ons canno be arbrarly chosen. I he nal on { } s seleced hen he wo rs order amlon s euaons unuely deermne a ah. Ths laer condon can only be sased or secal hase sace ons; he ons on he acual ah a. In ac s well known ha selecng and corresondng o he begnnng and endng condons = =0 s oen no enough o secy an acual ah beween wo ons as haens when a ah asses hrough a conugae on a on where wo or more exremals nersec. The nal momenum s hen needed o selec one arcular exremal rom a amly o exremals. The unconal R s relaed o S hrough aral negraon S R d or euvalenly L K 3. d Euaon 3 may be ormally regarded as a Legendre ransorm rom he varables and o he varables and wh K as he new oenal. I relae K o L by usng L he denon and Lagrange s euaon L.e. L L L L dl d d d d d d. Takng he oal derenal o 3 L gves: dl d d d dk d d d d L or dk d d d. 4 Euaon 4 shows ha K s a uncon o and. Comarng euaon 4 o he K L K K oal derenal o K K shows ha and. The second euaon corresonds o he Euler-Lagrange euaon or R whle he hrd L euaon denes he oson varable n analogy o. amlon s euaons may easly be derved by comarng he derenal d o dk and usng K. A ew smle examles however e.g. a smle harmonc oscllaor shows ha hese euaons o moon smly reea he euaons ha dene he varables and nohng new s ound as should be execed snce one o he euaons ha denes a new varable s he euaon o moon. These euaons also gve absurd resuls n -D consan orce moon. Neverheless hs does show ha K lke L may also be consdered a unconal o only one ndeenden uncon. K s only useul however when s consdered a uncon o boh and. Because S and R may boh be consdered unconals o wo ndeenden uncons I may gan greaer lexbly when ransormng varables. These ransormaons called Canoncal ransormaons can be wren as = =. Such a ransormaon s allowed however only S=0 or R=0 s sll sased n erms o. The resulng varaon 4

5 * S d 0 gves amlon s euaons n and * I reure =0 a he begnnng and endng ons because he negraon by ars mus also be done n hs case. Noe however ha reurng ha eher =0 or /=0 a he begnnng and endng ons. Because he begnnng and endng ons occur a arbrary me he laer condon s only ossble = and no n hs wde class o ransormaons. The Canoncal ransormaons reure boh exac begnnng and endng condons or and. I I urher reure = =0 hen hs wder class o coordnae ransormaons he Canoncal ransormaons are ossble. ah negrals and he momenum reresenaon In non-relavsc uanum heory all ahs are consdered ossble wh eual amlude and a hase ha s rooronal o he acon or he gven ah []. The robably amlude or a classcal ah s calculaed drecly rom a classcal Lagrangan usng he ormula =ex[s/h] where h s a characersc uanum o acon lanck s consan. The usual nal and nal condons are dened wh resec o sace coordnaes.e. and. Summng over all ossble ahs or negrang wh an aroraely dened measure yelds he ah negral ha s dened as he roagaor beween and. In oher words I can use S exressed n sace coordnaes o dene roagaors n coordnae sace as s commonly done. Because hs roagaor roagaes a wave uncon n me beween he begnnng and endng ons o he ah negral mlcly gves he Schrödnger wave euaon. The endons o he ah negral corresond o he sace varables n uanum mechancs. Ths ormulaon however has many dcules such as denng he measure o he negral and descrbng he sn varables o a uanum sysem. roagaors may also be smly dened n erms o he uanum ormalsm whou some o hese roblems as he marx elemens beween deren saes a deren mes [8]. In erms o oson saes s G ; =< >.e. he robably amlude o sar a oson and me and go o. Usng he Troer roduc ormula e A = lm N e A/N N where A s an oeraor and N s an neger I can use he me evoluon oeraor o exress G ; =< >=< e -/h >= lm N < e NT e NV >=lm N < e T e V e T e V e T e V e T e V - > where I have se = Nh and =T+V= /m+v. Ths allows he me nerval o be broken no N nervals or me slces o duraon /N and hese me slces go o zero n he lm. I can u hs no he ah negral orm by alernaely lacng he sace and momenum reresenaons o he deny oeraor =d >< and =d >< beween each ar o oeraors where reers o he h me slce. Usng he negral < >=h -/ ex/h resuls n he roagaor: N N lm N G ; d... ex d dd dn h V. N m Because he argumen n he exonenal s he acon or nnesmal.e. 5

6 N N d V d S h h h m G ; can be d 0 wren n he orm lm N d G ; N dd d d... dn h ex d. 0 d Ths negral can be nerreed as a hase sace ah negral DDe S/h over all ossble ahs n hase sace beween a me =0 and a me where D and D are aroraely dened measures or he negral. Because o he orm o he acon gven above he negrals n can be exlcly erormed resulng n he usual orm o he oson sace ah negral ncludng he correc normalzaon amlude. Unorunaely he classcal Canoncal ransormaons o he above acon only roduce nconssences and no he grea smlcaons hey roduce n classcal mechancs. I have shown above ha a Canoncal ransormaon reures ha boh = =0 and = =0. In uanum mechancs however s mossble o know and exacly a he same me.e. hs volaes he esenberg uncerany rncle and exlans why hey do no work or hase sace ah negrals. I I dene a secc and hen I mus nclude all ossble and n he negrals a and. I now show ha he momenum sace ah negral ollows naurally rom he acon R c.. reerence [8]. I do he same analyss n nal and nal momenum saes.e. G ; =< > o ge he robably amlude o sar a momenum and me and end a. In almos he same ses as above exce or he orderng o he < >< > acors I ge G ; = lm N < e T e V e T e V e T e V e T e V > and nally N N lm N G ; d... ex d dd dn h V. N h m I noe ha he argumen n he exonenal s he acon R or nnesmal.e. N N d V d R h h h m. d 0 Ths negral can be nerreed as a hase sace ah negral DDe R/h over all ossble ahs n hase sace beween a me and a me =0 where D and D are aroraely dened measures or he negral. I can hen ge a ure momenum sace ah negral by erormng he negrals over n a smlar manner o he oson sace ah negral. In hs case a secc oenal uncon s needed o roceed. Momenum ah negrals rom classcal ahs The above dscusson suggess ha a momenum sace ah negral can be consruced rom classcal momenum ahs usng he classcal acon R by consderng all he ossble momenum ahs beween and as s done n coordnae sace n reerence []. By assumng wo comlmenary ah negrals a coordnae ah negral and a momenum ah negral oson and momenum varables are laced on an eual oong. Ths vew also drecly races canoncally conugae varables n 6

7 uanum mechancs o classcal mechancs. I now demonsrae wh several elemenary examles how hs exanded unverse o ossble ah negrals resolves some o he dcules n he ah negral ormulaon o uanum mechancs. Ths vew also roduces a rescron or reang more comlex amlonans where he momenum varables do no searae. Ths s accomlshed by nong ha he classcal ahs n momenum and coordnae sace exs n one-o-one corresondence conneced only by amlon s euaons. Each ossble has a corresondng ha may be ound by solvng noe ha and do no reer o crcal ahs bu o arbrary ahs I use he noaon and n Aendx B or arbrary ahs. In uanum mechancs and canno be known smulaneously.e. a comlee knowledge o he oson a a gven me mles comlee gnorance o he momenum a. A an nal me and oson all momena are ossble and all osons are ossble a he end o he nex me ncremen. Summng over all ossble ahs s hen euvalen o negrang over all ossble osons aer each me ncremen exce a he gven nal oson and me. In Aendx B I show ha oson ahs may secy S. Seccally by usng one o amlon s euaons o elmnae I may wre S where solves. In hs way he acon s reaed as only deendng on one o he Canoncal varables. An elemenary examle comes rom he searable class o amlonans o he orm = /m + V. amlon s euaons yeld and n hs case relacng by m / become ncreasngly accurae as 0. Ths s essenally he sandard rocedure used n calculang many roagaors usng coordnae ahs. Ths same reasonng also ales o he momenum ahs where each s relace by ha s a soluon o. The varable s hen relaced n R and he acon R s calculaed or all ossble ahs beween and. I now llusrae hs rocedure wh several elemenary examles. The ree arcle The rs and smles examle o use he above rescron or he momenum ah negral s he ree arcle. The ree arcle s amlonan s = /m so ha 0 or =consan a every me nerval. Summng over all ossble momenum ahs or negrang over all ossble momena a each me exce a he begnnng and endng ons where s gven consrucs he roagaor. As shown n gure I ake he nal momenum o be a me. A me here are n rncle many ossble values o momenum. The soluon o 0 however only allows one ossbly n hs case and he negral a me s d - -. The uncon resrcs he ossble values o o he ossble values n he negraon and he uncon manans causaly. Exendng hs o all N nervals I ge G ; lm N A d d... N... dn n n n n n n ex. m In he above negral each nerval mus have one and one acor o lm he ossble m 7

8 ahs and manan causaly. Noe ha here s one more han here are. Inegrang hs becomes ; lm ex G A N. N m Ths s he same resul gven n eynmann and bbs [] bu here s deduced by summng over all ossble momenum ahs. The normalzaon acor has been urosely le ambguous o demonsrae ha n hs exended ormulaon he normalzaon acor can be ound whou resorng o he Schrödnger euaon or roagaon over a derenal n me. As shown n reerence [] and [8] he momenum and oson roagaors are relaed by ourer ransormaon over he endons hs s essenally because o he endon erms ha relae R o S. By ourer ransormng he above I can nd he oson roagaor amlude o whn a consan o /A ha s dened hrough many ndeenden measuremens as lanck s consan /A = h. 3 n- n = gure. The momenum ah negral s consruced by consderng all ossble momenum ahs. Ths s euvalen o negrang over all ossble momena a each me exce a he begnnng and endng mes. The harmonc oscllaor The amlonan or a harmonc oscllaor = /m + ½m 0 s symmerc n and and he allowed amlon s euaon yelds m. Subsung or I ge. In hs case relacng by / becomes ncreasngly accurae m m 0 as 0. One can see ha hs roblem s euvalen o he roblem o he uadrac amlonan ah negral n coordnae sace. Many mehods have been develoed o calculae a ah negral o hs orm and he resul s smlar. The neresed reader can consul reerences [] [] [8] and many ohers. Alhough no grea advanage s ganed n dong a momenum ah negral n hs case s nsrucve o see how he above rescron s mlemened. In hs case here s an exernal orce so ha he oal 0 8

9 unceranly n oson a each me mles ha all momena are ossble aer he nal me where he nal momenum s gven. Non-relavsc sn One o he advanages o hs exended ormulaon o he ah negral s ha smle sn varable roblems become more racable. Because he coordnae or angle varables are no observable even n rncle I mmedaely go o he momenum roagaor because he generalzed momenum s observable hrough s assocaon wh a arcle s magnec momen. In one dmenson an oeraonal and conceual smlcaon occurs as shown n he ollowng. The amlonan or a sngle snnng arcle s I or I l. As n he case o he ree arcle l 0 or he angular momenum l s consan durng each me nerval. Ths case s ue deren rom lnear momenum; lnear momenum could be observed by measurng a change n oson durng a change n me. Because can be observed o be + or - he drecon o can be deermned. No analogous echnue s avalable here. Because he conugae varable or angular momenum s no observable even n rncle s no ossble o nd he drecon o l. The value o l s consan bu he drecon or sgn s unknown and has wo ossble drecons along he sn axs l. The roagaor can hen be wren as ; ex G l l l d. In consrucng a ossble ah l s aken o be Ih all ahs consan and eual o he nal l over a me nerval as n he lnear momenum case. gure 3 shows one examle o a momenum sace ah where l can be eher +l or l durng each me nerval. I here are N such nervals hen here are N ahs so ha G l lm N ; l C ex l. In order o have a ne roagaor he normalzaon N consan s C=/ N. I l +l -l 0 = 3 4 n- n = gure 3. The momenum ah negral or he sn o a ree arcle s consruced by consderng all ossble momenum ahs. The momenum l s consan or a ree arcle bu has wo ossble orenaons because he angle varable s no observable. A smlar ah s no ossble or lnear momenum. Durng 9

10 each me nerval here are wo ossble values o l=l. One ossble ah n momenum sace s shown above or he roagaor G+l ;+l. I noe ha hs essenally one dmensonal roblem s embedded n hree saal dmensons wh he axs o he one dmensonal sn angular momenum aken o have arbrary orenaon. Ths roagaor mles a wo-sae snor ha may be roaed o an arbrary drecon. A unary roaon o he above roagaor yelds he roagaor n hree dmensons. Ths resul s also he same as ha ound by calculang a me evoluon oeraor. By usng he momenum sace ah negral a conssen heory resuls whou he need o do he dcul ask o consrucng sn rom unknowable angle varables [] [8]. The above sn one-hal ermon may be used o consruc he roagaor or a sn one boson. A arcle comosed o wo dencal snnng arcles has only wo momenum degrees o reedom l and l n a one-dmensonal sysem. The amlonan s l where now l l l so ha l 0 l =l 0 and l 0 I l =l 0. Because he comose angular momenum l s he observable uany s also consan durng each me nerval bu now has hree ossble values +l 0 -l 0 and 0. The roblem o a comose arcle n hree saal dmensons or he roblem o orbal angular momenum have amlonans ha coule angle and angular momenum varables. These roblems could also be reaed usng hs rescron. These more dcul roblems are no reaed here. Concluson When exremzed he acon R S resuls n amlon s euaons or xed begnnng and endng momena oson. Boh o he acons R and S have a amlon- Jacob euaon see Aendx A. S and R have some smlares and derences ha were ound by ung hem no convex orm and examnng her second varaon see Aendx B and C. Usng hs arallel classcal mechancs dened usng he acon R he momenum ah negral may be consruced rom classcal momenum ahs and he momenum reresenaon n uanum mechancs s shown o have s orgn n classcal mechancs. I show ha hs broader vew o he ah negral grealy mroves hs ormulaon o uanum mechancs by elmnang some o he revous weaknesses. In arcular he amlude or a ree arcle s ound whou usng he Schrödnger euaon and he nernal sn degree o reedom s smly and naurally derved. I also hels n undersandng why Canoncal ransormaons do no work n he hase sace ah negral. I graeully acknowledge he dscussons wh Arun Roy and he hosaly o Yves Garrabos a he ICMCB. I also graeully acknowledge and he nancal suor rom he Unversé de Bordeaux. Aendx A I now show ha he classcal mechancs dened rom he unconal R has a amlon-jacob euaon. In he ollowng I revew resuls or he acon S and resen new resuls or he acon R. The oal derenal o any arbrary uncon can be added o a Canoncally ransormed Lagrangan.e. 0

11 0 * d d d d S because 0. ] [ d d d Euang he above negrands yelds: * or. * I he coecens o he generalzed velocy are zero hen S=0 as can be seen n he above euaon and hs mlcly denes a Canoncal ransormaon. The uncon s sad o generae a Canoncal ransormaon. The above argumen only works or =. I can however Legendre ransorm rom o by denng a new generaor S=-. Aer he usual manulaons I nd ha S=S[] and. S S S In arcular I can ransorm o a coordnae sysem where and are xed ons or consan by choosng an S ha makes * =0. Ths condon when I subsue he exresson above or becomes an euaon or S known as he amlon-jacob euaon 0. S S Ths euaon ncludes he consan ha can be consdered a consan o negraon. In ac he generaor S can be shown o be he acon along he acual ah [9]. So he acon S generaes a Canoncal ransormaon rom he hase sace where he ah may have varous geomercal roeres o he hase sace where he dynamcs s a xed on. Ths xed on s drecly relaed o he negraon consans or nal condons o he moon n he orgnal sysem. Snce he nal condons deermne n rncal he moon n he orgnal hase sace no normaon s los n hs ransormaon bu he sysem has been grealy smled. The above amlon-jacob euaon can be solved or S[]. The acon can also be calculaed s known rom he orgnal denon o S and he nal condons and. I noe ha he begnnng and endng condons needed or a Canoncal ransormaons = = = =0 are he same ones ha also allow boh S and R o be smulaneously saonary. In ac I can do a smlar analyss or R I dene he uncon R by =+R mlyng ha d d d dr d d d

12 dr d d d R R R and * R. Selecng an R ha makes * =0 I ge anoher amlon-jacob euaon or R R R 0. The R uncon ha solves hs euaon can easly be nerreed aer akng he oal me dervave.e. dr R R d or R d Kd. Aendx B I know rom sascal mechancs ha he characer o an exrema s exremely valuable knowledge n undersandng a hyscal sysem. There are however some maor dcules assocaed wh mos varaonal rncles esecally when deermnng he ye o exremum. To gan a broader vew o he relaon beween S and R I now begn examnng he yes o bounds. One ye o amlonan ha resuls n global bounds on he acon s he saddle uncon amlonan [0]. In hs case s a mnmum n and a maxmum n. In he ollowng I revew resuls or he bounds on he acon S when s a saddle uncon and resen new resuls or he acon R. I can es wheher a uncon s convex or concave over a gven doman whle relaxng he connuy reuremens by usng he ollow denon o convexy: A uncon R s convex on a b or 0<< and any and n a b. R s he se o real numbers. A uncon s srcly convex he neualy s src or dsnc and. A uncon s concave - s convex. The above denon exresses he geomery o he 'chord above arc' z= + s he chord n beween and whle z= +- s he arc. I I choose any wo ons { } whn he doman {a b} hen any value o he uncon beween and.e. +- s less hen he chord ha connecs and.e I s easly roven ha s derenable he above denon s euvalen o ' 0 or any { }n {a b} where ' = d /d [0]. or a wce derenable uncon he usual convexy es s recovered.e. s convex n a b 0 or all n a b. The convex uncon denon can be exended o uncons o wo or more ndeenden varables. The uncon y s convex wh resec o +- y y+ - y or 0<< and any { } n {a b}. As n

13 he above he aral dervave exss hen s convex wh resec o y- y - - y0. Ths exanded denon also ncludes he saddle uncon case; a uncon yr s convex n and concave n y hen s a saddle uncon. A derenable saddle uncon has boh y- y - - y0 and y - y -y y y y 0. In reerence [0] hese denons are aled o he unconal S. S s wren n erms o he arbrary momenum and he oson o dsngush hem rom her crcal uncons ha exemze S.e. S d where he crcal curve sases amlon's euaons and. The begnnng and endng condons are - = =0 and - = =0. I can consruc anoher unconal rom S[] by resrcng o be a soluon o he euaon. Subsung he soluon = no S I ge he new unconal J[]=S[] and J[]=S[] s he usual Euler-Lagrange orm o he acon. The derence s d. J J I can u he negrand no he orm o a convex or concave uncon by addng and subracng gvng J J d d where I have used aral negraon and. I s made o sasy he nal and nal condons =0 and =0.e. = and = and s a saddle uncon.e. s convex n and concave n hen he global mnmum rncle J[]J[] holds. These same condons can be used o derve anoher global bound. In hs case I dene he unconal G[] = S[] where s he soluon o. ollowng a smlar argumen add and subrac negrae by ars; use = =- =0 and =0 I ge G G d. d 3

14 I s a saddle uncon hen he global maxmum rncle G[]G[] holds. Combnng hese resuls one can see ha s a saddle uncon and sases he nal and nal condons = and = or =0 and =0 hen J J S G G where s used o dene J and s used o dene G. I now do a smlar analyss or he unconal Kd d R ] [ subec o he begnnng condon =0 and endng condon =0. The unconal d K G ] [ ' s consruced usng he soluon o he euaon.e. =. As n he above case G may be u n he orm. ' ' d d G G Usng he above consran amlon s euaon and he begnnng and endng condons =0 and =0 I ge ' ' d d G G.e. ' ' G G or a saddle uncon. The unconal d K J ] [ ' s dened usng ha s a soluon o. In a smlar manner wh he addon o an negraon by ars I oban ' ' d d J J.e. ' ' J J. ung hese ogeher gves:. ' ' ' ' G G R J J Noe ha hese bounds are oose n he sense ha R s bounded rom above by a unconal n and below by a unconal n whereas S s bounded below and above by smlarly dene unconals o and. In concluson or he saddle uncon amlonan he unconal S R s bounded rom below above by an assocaed unconal n and s bounded rom above below by an assocaed unconal n. 4

15 Aendx C The mahemacal es or a maxmum mnmum ec. o a unconal s analogous o he maxmum and mnmum ess or a uncon n elemenary calculus. I he rs varaon dervave s zero a a ah on hen here s an exremum a ha ah on and he sgn o he second varaon second dervave reveals he ye o exremum mnmum maxmum nlecon on ec. []. In analogy o a uncon I can es or he ye o exremum o S by akng he second varaon.e. he varaon o he unconal ha resuls rom he rs varaon o oban L L L S. d! As n elemenary calculus S=0 and S>0 ndeenden o he choce o hen S s a mnmum or ; or S=0 and S<0 ndeenden o he choce o hen S s a maxmum or. I S=0 and all hgher order varaons are zero hen s ndeermnae or hs mehod. The bounds on R and S ound n Aendx B were deermned by he roeres o he amlonan. These bounds were ound usng resrced momenum or oson varables so ha he boundng unconals were deenden on only one conugae varable. I now rea he and uncons as ndeenden as n he dervaon o amlon s euaons and sudy he ye o exrema o S and R. When s derenable I can exress he second varaon S n erms o conugae varables o ge L L L S. d! The dervaves are evaluaed a he crcal and L. In he amlonan ormulaon he ndeenden uncons and are only conneced by amlon s euaons and I may use o ge T S d V MVd!! where M V. I I consder he roblem MV=V he symmerc marx M n euaon can be dagonalzed yeldng wo real egenvalues and. The corresondng wo egenvecors are V and V. The egenvecors can be used o consruc an orhogonal a marx T =I ha dagonalzes M.e. T V and V T =ab. I nd ha b a T T S! V MV d! V M V d a b M d b a b 0 0 a d b a d b d. 5

16 a and b are arbrary because hey conss o lnear combnaons o he ndeenden varaons and. The second varaon wll only have a consan sgn boh egenvalues have consan sgn over he enre crcal ah as can be roven n he ollowng. One o he wo consans e.g. b can be made zero over he enre crcal ah whle he oher could be made o be zero over only a oron o he ah. Any change o sgn o could make S eher sgn by secal choces o a ha are zero whle s one sgn and no zero when s he oher sgn. A smlar argumen also works or. I us so haens ha he condons >0 and >0 or S o be a mnmum or <0 and <0 or S o be a maxmum are he same or when he marx M s osve negave dene durng = -.e durng. I M s osve negave dene or all nal and nal condons hen S s a global mnmum maxmum. Ths shows ha he ye o global exrema one exss only deends on he orm o. Ths can be llusraed wh he smle and raccal amlonan o he orm /m + V makng >0 and = = =0. S s a global mnmum or all and. The second condon s only ossble when s negave concave downward. Ths s an examle o he saddle uncon case reaed above where I ound uer and lower bounds on S. A concave downward oenal s unsable abou he maxmum o he oenal. The moon s unbounded where a arcle acceleraes away rom hs maxmum.e. he moon aroaches he xed on o and. In hs case S s a global mnmum or a globally unsable and reulsve oenal. The bounded moon or a concave uward oenal sable and aracve oenal s neher a global mnmum nor a global maxmum o he acon. In hs and more comlex cases he oenal may have domans where s osve sable aracve regon and oher domans where s negave an unsable or reulsve doman over gven me nervals. When s locally a saddle uncon and s sucenly small he second condon s sased. I s also ossble ha s eual o zero so ha hs mehod s ndeermnae. These ons are analogous o he crcal ons n hermodynamcs []. In he more general case where s a mos second order corresondng o he uadrac knec energy erm he acon has a mnma when - - >0. The ye o exrema clearly deends on he deals o a arcular amlonan. I can e.g. wre A +B+V gvng a global mnma when AA 4A AV. Ths condon does no deend on he B. I s easy o show ha a Canoncal ransormaon does no necessarly reserve he exrema characer o he acon. The acon corresondng o he amlonan = - s a global mnmum by euaon 4. Usng he generaor = gves a oally ndeermnae acon whereas =/ or a doman no ncludng he orgn gves an acon ha s neher a maxmum nor a mnmum. ollowng he dscusson above I can also show he condon or he ye o exrema o R. The second varaon s: 6

17 K K K R! Usng as above I ge d. R.! d By he same argumen as beore R s mnmum maxmum he marx s osve negave dene durng. The amlonan = /m + V has >0 and = =0 so ha he o dagonal elemens are zero. R s a maxmum The second condon s only ossble when + s negave as occurs n a concave downward oenal or a saddle uncon amlonan. I s revealng ha he neral ar o he amlonan = /m + V allows a mnmum or S and a maxmum or R. Ths s smlar o he resuls o Aendx B where he bounds were ound by elmnang one o he Canoncal varables usng one o amlon s euaon. I may be ossble wh more comlcaed and realsc amlonans o caegorze he hase sace no domans dened by he exrema roeres. These comlcaons are even more evden n hgher dmenson where he dagonal elemens o he nxn n essan marx M have he orm and he o-dagonal elemens are also generalzed. Exendng he above argumen he crcal ah s a mnmum maxmum he egenvalues o M are all osve negave durng. Reerences [] R.. eynmann and A. R. bbs uanum Mechancs and ah Inegrals McGraw- ll New York 965. [] L. S. Schulman Technues and Alcaons o ah Inegraon John Wley and Sons New York 98. [3] Mark S. Swanson hys. Rev. A Mark S. Swanson he-h/ [4] V. erwal hys. Rev. Le [5] L. D. Landau and E. M. Lshz Mechancs ergamon ress Oxord 976. [6] A. Sommereld Mechancs Academc ress New York 95. [7]. Goldsen Classcal mechancs Addson-Wesley Readng MA 959. [8]. Klener ah Inegral n uanum Mechancs Sascs olymer hyscs and nancal Markes World Scenc New Jersey 004. [9] A. L. eer and J. D. Walecka Theorecal Mechancs o arcles and Connua McGraw-ll New York 980. [0] A. M. Arhurs Comlemenary varaonal rncles Clarendon ress ; New York 980. [] C. ox An Inroducon o he Calculus o Varaon Dover New York 987. [] L. D. Landau and E. M. Lshz Sascal hyscs ergamon ress Oxord