EVALUATION OF FUNCTIONAL INTEGRALS BY MEANS OF A SERIES AND THE METHOD OF BOREL TRANSFORM

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1 EVALUATION OF FUNCTIONAL INTEGRALS BY MEANS OF A SERIES AND THE METHOD OF BOREL TRANSFORM Jose Javer Garca Moreta Ph. D research studet at the UPV/EHU (Uversty of Basque coutry) Departmet of Theoretcal Physcs & Hstory of Scece Address: P.O Blbao, Vzcaya (Spa) Phoe: () Fax: () E-mal: josegarc@yahoo.es PACS:.3.Lt,.3.Sa, 3.65.Sq Abstract: I ths paper we gve a evaluato of a fuctoal tegral by meas of a seres fuctoal dervatves, frst of all we propose a dfferetal equato of frst order ad solve t by teratve methods, to obta a seres for the defte tegral oe dmeso, after that we exted ths cocept to fte dmesoal spaces, we troduce the fuctoal dervatve ad propose a fuctoal dfferetal equato for the fuctoal tegral whch we solve by terato obtag a seres that cludes the -th order fuctoal dervatve, ths seres ca be mproved by usg the Euler trasform for alteratg seres, also we provde a method usg the Borel trasform for the dverget seres of correctos to the Semclasscal lmt gve by the seres a(, r, t),so t ca be evaluated usg oly a few a s Keywords: fuctoal tegrals, fuctoal dervatves, sum over paths, Borel trasform Seres expresso for Itegrals: I the paper of hs Ph.D Feyma made a ew formulato of Quatum mechacs, ths formulato he exteded the classcal Hamlto,s prcple to Quatum mechacs S[ ]/ by makg the sum of e over all the possble paths of the partcle, ths formulato ca be appled to Quatum Feld Theory, but all cases to obta the propagator for the theory we must calculate fuctoal tegrals of the form: S [ ] D[ ] e Z () S d xl 4 [ ] (, ) Where S s the acto of the system ad L s the Lagraga,D meas that the tegral must be made over all possble paths, s stll ukow how these tegral ca be calculated, there are maly two methods, a semclasscal evaluato of the fuctoal tegral by expadg the acto fuctoal ear ts classcal soluto S=, ths s vald whe h ( ote that the classcal case h s almost ad the ma cotrbuto to the

2 tegral s gve by the that satsfy S=).The other method s to approach the fuctoal tegral by a tegral R d (wth d bg ad fte) ad evaluate t by Motecarlo tegrato, ths paper we wll obta a seres for the defte fuctoal tegral terms of the fuctoal dervatves, frst we wll take as a example oe dmeso the ext lear dfferetal equato: at f ( t) () ay y e at y( t) e ( due t f ( u) Here the dot meas dervatve respect to t ad a ca be a real or complex costat, the expresso () s the formal soluto to ths equato wth C a costat, O the other had we could solve the dfferetal equato by terato the form: d ay y e dt ( ) ( ) at f ( t ) y( t) ( ) d at f ( t) e (3) a dt Ad as y(t) s but a expoetal factor of e -at the tegral of e f(t) we could use (3) to calculate the tegral of the fucto e f(t), for the fte dmesoal case of our tegrals Quatum Feld Theory we have the fuctoal dfferetal equato: F[ ] F[ ] e J d x L J[ ] 4 [ ] ( (, ) ) Where s the Plack costat dvded by, we ca defe the fuctoal dervatve wth the ad of the Drac delta fucto as a lmt wth the form: F[ ( x y)] F[ ] F lm ( y) Now f we proceed by terato the same way we dd wth our ordary dfferetal equato, we could obta a soluto to our fuctoal dfferetal equato the form of a seres volvg fuctoal dervatves (4), but ths soluto F, cludes the fuctoal tegral () so calculatg the seres we could calculate the fuctoal tegral: 4 ( d x ) S[ ] J [ ]/ [ ] [ ] ( ) ( ) F e D e e (4) F F So from (4), we could calculate the fuctoal tegral, we have troduced the otato to express the fuctoal dervatve of -th order, the expresso (4) s a formal seres

3 for the Fuctoal tegral terms of fuctoal dervatves, ote that both seres (3) ad (4) are alteratg seres, to mprove ther covergece we ca use the Euler trasform of alteratg seres the form: ( ) a() ( ) a()! t dtt e a() ( ) m! a! m! m We wll be able to use Euler trasform f a() > a(+) so the seres wll coverge, where for our seres (4) the geeral term s gve by: a [ ]/ ( ) ( ) ( e J ) Note that from (4), we could obta the expresso for the fuctoal tegral but the costat fuctoal, C[]=costat, but whe we apply the defte tegrato betwee two fuctos ths costat wll dsappear, we have proved that exst a seres that gves us the expresso for our fuctoal tegral (), the covergece crtero for our seres s gve by: a( ) a( ) lm J[ ]/ l ( e ) for tedg to fte, the error term after trucatg the seres the -th term s: E [ ]/ ( ) ( ) ( e J ) (5) The formulato of Quatum mechacs by path tegrals s useful as t provdes a way to calculate the propagator by kowg oly the classcal Lagraga of the theory ad tegratg the acto over all possble paths, to make the seres coverge faster we ca use Euler trasform as (4) s a alteratg seres. Although we have used the complex expoetal represetato of the Fuctoal tegral, usg a Wck rotato t t so the Acto becomes S[ ] H[ ] (Hamltoa desty) so the expoetal s real. We also have the problem of computg hgher order Fuctoal dervatves, a expresso usg Gruwald-Letkov dffertegral operator for teger values of q mght help: q ( q ) m F lm ( ) F[ m ( x y)] q> q q ( m ) q m m

4 Seres expresso for Itegrals volvg Borel Trasform: Aother expresso to evaluate the tegrals would be takg the Statoary Phase prcple so oly the path close to the classcal soluto S[ classcal] cotrbute that case we have the soluto: S [ ] S[ classcal ]/ WKB D[ ] e Z I e a(, r, t) (6) a(,r,t)= Where I WKB s the fte dmesoal aalogue to the kow result for Gaussa Itegrals: / T x Ax D[ x] e I WKB det A as Wth det A det S[ classcal] I the sem-classcal lmt the power seres ca be evaluated takg oly a few terms, however f you sum all the seres, you ca see t s dverget, However f you defe: F( r, t, ) a(, r, t) Borel Trasform of the seres sde (6)! B* de F( r, t, ) Ad the Itegral would take the form: S [ ] S[ ]/ classcal D[ ] e Z Ie de F( r, t, ) B* de F( r, t, ) as B * exst ff: There are o poles for (, ), ad F O( e a ) a> A smlar relatoshp betwee a asymptotc seres ad a fucto also happes for the Expoetal tegral: t x e e ( ) E ( x) dt! The alteratg seres has the Borel trasform: x t x x xt ( ) x dte! Usg the result for the Laplace trasform of /(t+) t x s L e E ( s) you fd, that Usg Borel trasform you ca obta the sum for t

5 the seres eve the case t dverges, geeral the coeffcets a(,r,t) come from the expaso of the acto fuctoal ear ts classcal soluto: S[ classcal] S[ ] S[ classcal]... O( S) ad applyg these correctos to the Fuctoal tegral. However, the Borel sum of a seres ca ot always be obtaed easly, a method to get a approxmato usg s, usg the Euler-Abel trasform of a power seres the form: p k k k p ( ) x [( ) a ] x k p p k k ( x) x f ( x) a( ) x ( ) [( ) a ] x (7) p>> The (7) ca be used to accelerate the covergece of the power seres, to f(x) takg p oly a few dffereces [( ) a ], for a()=p() a Polyomal the Euler-Abel trasform would be exact, the geeral case we must have that : p lm [( ) a] p ( the dffereces get smaller as p creases) Usg the dfferece operator as we dd above for our seres derves ad troducg the Laplace trasform of the fucto: t ( t c) p! s c p p p t st cs dt e E ( cs) e ( ) E ( ) p p p x dt E x x e t The correctos (asymptotc seres wth zero radus of covergece ) takes the value (Borel trasform): (,, ) ( ) [( ) ] ( ) p k k k k cs a r t b k k E s e k k! s c c, s (8) Wth a( ) b( )! a(, r, t), ad a error term E=E(p) E dt [( ) b ] t e O E ( cs) e p p p t p st cs p p t p! s c c, s (Workg wth atural uts, after takg the Borel trasform we would set ) I the geeral case f we have two power seres that coverge to the fuctos f(x) ad g(x),the ther Borel trasform s the same as ther usual sum, f we defe:

6 f ( x) a( ) x a( ) g( x) x! f ( s) dtf( t) e st Usg the propertes of the verse Laplace trasform of s f s (/ ) we fd: ( ) ( ) ( ) g x duuf u J xu F k x F u g x ( )( ( ) ( ) Whch s the zeroth order Hakel trasform of F(u ) at a certa k. I case we could evaluate the Fuctoal tegral by Numercal methods defg the we have for our F(r,t,ξ) the estmate: S[ classcal ] d Zumercal ( ) e e d F( r, t, ) d IWKB ( ) ad (,, ) (,, )! F r t a r t Refereces: [] Abramowtz ad Stegu Hadbook of Mathematcal fuctos New York: Dover (97) [] Demdovch B.P ad Maro I.A Fudametos de Aalss Numerco (Spash) Ed: Parfo Madrd (985), ISBN: X [3] Hardy G.H Dverget seres Oxford Claredo Press (949) [4] Pesk M.E & Schröeder D A Itroducto to Quatum Feld Theory, Addso-Wesley (996). [5] Rud, W. Fuctoal Aalyss, McGraw-Hll Scece, 99