ABEL RESUMMATION, REGULARIZATION, RENORMALIZATION AND INFINITE SERIES Jose Jvier Grci Moret Grdute studet of Physics t the UPV/EHU (Uiversity of Bsque coutry) I Solid Stte Physics Addres: Prctictes Ad y Grilb 5 G P.O 644 489 Portuglete Vizcy (Spi) Phoe: () 34 685 77 6 53 E-mil: osegrc@yhoo.es ABSTRACT: We Study the use of Abel summtio pplied to the evlutio of ifiite series d ifiite (diverget) itegrls, we give severl emples of how we c obti regulriztio for the cse of diverget sums d itegrls. Keywords: = Abel sum formul,abel-pl formul, poles, ifiities, reormliztio, regulriztio, multiple itegrls, Csimir effect. Abel summtio for diverget series: Give power series of the form < we defie the Abel resummtio of the series which is coverget o the regio s the limit = = lim A( S), if such limit eist we will sy tht the series summble to the vlue A(s). is Abel As emple let be the series [6] + d ( ) = + 3... = = B + d + + () Ufortutely the series is NOT Abel summble, this is due to the pole t = of the fuctio ( ), however Guo [5] studied this series d gve the followig idetity usig epoetil regultor. ε d Γ ( + ) Z( ) ( ) e = = + ε ε + dε e ε =! ()
Where we hve used iside () the Tylor epsio ivolvig Beroulli s umber = B d the epressio for egtive vlues of the Riem e =! B zet fuctio ζ ( ) =. To evlute the Riem zet iside () for egtive vlues we will eed the s π s Riem s fuctiol equtio defied by ζ ( s) = ( π ) Γ( s)cos ζ ( s), π with Γ( s) Γ( s) = si( π s) They itroduce smll prmeter epsilo d fter clcultios te ε, ufortutely for = - Guo s method gives oly ifiite swer e ε = log ε, this is becuse the followig epressios for the -th Hrmoic umber d for the Lplce trsform of the logrithm H = + + +... + γ + log 3 εt γ + log ε dte log t = (3) ε Where γ =.577.. is the Euler-Mscheroi costt. ε If we te () d igore the pole prt we hve tht f. p e = ζ ( ) for every ecept =-, this is precisely the vlue of the series obtied vi Zet regulriztio, so Abel resummtio d Zet regulriztio re lied d give the sme swer for the diverget series provide we igore the poles To study emple of how the regulriztio d reormliztio of the poles is mde we will study the Csimir Effect ε o Csimir effect: The Csimir effect is physicl force due to the qutiztio of Electromgetic fields, see [7], i the simplest versio of the Csimir effect the vcuum Eergy of the system per uit of Are A is give by / π h π 3 π rdr r 3 E hc c = + = A 4π 6 (4) h Here, π 8 c = 3 m / s is the speed of light i the vcuum. 34 h = =.54 J. s is the reduced Plc s costt d
If we use Zet regulriztio [3] we fid the vlue 3 =, if we isert this vlue iside (4) we get the correct eperimetl vlue of Csimir effect E cπ = h Fc d E cπ so = = h. 3 4 A 7 A d A 4 The physicists pproch to Csimir effect is bit more complicte, for emple they use reormliztio d compute the qutity δ hcπ 3 ε 3 εt E = Ediscrete E = e dtt e 3 6 (5) This differece c be computed with the id of the Euler-Mcluri sum formul B (6) ( )! ( ) 3 ε f ( ) f ( ) d = f () f ( ) = e = Or usig the Abel-Pl sum formul with ε f () f ( it) f ( it) 3 ε f ( ) f ( ) d = + i dt f ( ) = e πt e (7) t Γ ( + ) If we retur to Guo s formul (), d we use the idetity dte ε t = + we ε fid the followig. Z( ) Γ ( + ) ε tε e == ( ε ) + dte t = + =! (8) ε So lthough the Abel regulriztio is ot vlid for the series differece, the (9) ε tε = e dte t = ζ ( ) ε Mes perfect sese d is lwys FINITE, lso for the cse =- we fid tht the Hrmoic series is summble d its sum is equl to Euler-Mscheroi costt = γ fter removig the regultor e ε. So, both methods reormliztio d zet regulriztio gives the sme fiite swer, however Zet regulriztio is esier d fster method d c be geerlized to the cse of more geerl opertors, for emple 3
i, h = i ( g g ) ( ) E = ctrce g () The opertor iside () is the Lplce-Beltrmi opertor d i, g = det g g g,, g,, is determit of mtri, equtio () is the epressio for the vcuum eergy of the Lplci opertor i two dimesios. Abel summtio d diverget itegrls: Abel summtio formul c be eteded to obti fiite results for diverget itegrls too, first we eed the formul m m m m m m d = d + i i + i= i= r m r ( m r + ) d r= B Γ ( m + ) ( r)! Γ( m r + ) () Where is positive iteger, d the ifiite sum iside (9) must be uderstood i the sese of Abel regulriztio, so i e ε i= Also this recurrece () is fiite for positive iteger, due to the poles of the Gmm fuctio t the egtive itegers, i cse is positive d rel umber the recurrece () is ifiite d it must be tructed, i this cse we d c lso use the idetity = m m vlid for Re( m ) > m The cse m=- is ot icluded d must be te seprtely, if we te the fiite prt f. p e ε ε e = γ or if we use the epressio f ( ) = iside the + Euler-Mcluri summtio formul f ( ) + f ( ) B ( ) ( ) f ( ) = f ( ) d + ( f ( ) f ( ) ) ( )! () + = Ad tig ito ccout the followig series epsio for the Digmm fuctio Γ '( ) B Ψ ( ) = = log + r () γ Γ( ) r= Ψ = (3) 4
We get the reormlized result for the itegrl with logrithmic divergece i d the form = log, this mes tht i regulrized/reormlized + reorm sese the 3 itegrls d + d d d re equl to For the cse =, which is the first term iside the recurrece formul () we f ( ) + f ( ) d = + e ε = fid tht this is becuse the vlue ζ ( ) = for the Riem zet fuctio, so if we te the fiite prt of the diverget sum we get the fiite vlue f. p e ε = ζ () o Reormliztio/regulriztio theory from diverget series: Usig Abel summtio d formul () we c give esy method to regulrize diverget itegrls of the form f ( ) d, which y perso could uderstd sice it uses very simple mthemtics, this method of reormliztio regulriztio is bsed o the resummtio of diverget series of power of the positive itegers d lso o reltioship i the form of recurrece equtio betwee the diverget itegrl diverget series couterprt d d its discrete, the method is the followig. Split the itegrl bove ito fiite prt f ( ) d plus diverget prt f ( ) d, this c lwys be mde Epd the itegrd iside f ( ) d ito Luret series of the form with coefficiets give by itegrl over the comple ple f ( z) usig Cuchy s theorem [] = dz + πi z Apply itegrtio o ech term of which is vlid d well defied for m C d = m m+ d the formul m 5
Use the regulriztio for the Hrmoic series logrithmic itegrl the the diverget logrithmic itegrl = γ d of the d = log to regulrize d give fiite meig Use formul () to regulrize the diverget itegrls m=,,,..., with Abel resummtio for every m of this series is ust m e ε m d for every, the reormlized vlue m ε e = ζ ( m) so Abel d Zet regulriztio give both the sme results, ecept for the hrmoic series Aother defiitio of the reormlized ifiite series is mde with the Abel-pl sum formul, use Abel-Pl formul to compute the reormlized vlue of the series reorm ε ε e e d whe the regultor epsilo is te to, this results is logue to zet regulriztio. As emple, let be the diverget itegrl d, with c >, the + c reormlized vlue of this itegrl usig formul () would be d = d c d + c d d = c log c c + 6 reg + c + c + c (4) A more complicte -loop itegrl d dy y c be computed with our + y + reormliztio method bsed o the regulriztio d study of diverget series, i this cse, the itegrl hs sub divergece i the vrible which should be reormlized first, the reormlized vlue of this itegrl is d dy = + y + d + + ( + y + )( + ) dyy ( y ) ydy (5) The itegrl iside (5) d = f ( ) is fiite for every positive, to ( + y + )( + ) simplify the clcultios we c replce (pproimte) this itegrl by qudrture formul with -poits so the sum (qudrture) is esier to wor with, for emple if we use the Lguerre qudrture formul, vlid for [, ) see [] d e y ( y + ) y ( y + ) ω (6) ( + y + )( + ) + + y + ( ) ( ) = 6
Now, ech term iside (6) deped o y so we hve to reormlize the diverget itegrls ( is the umber of poits of the qudrture formul used) e ( y + ) y ω dy 4, this hs qurtic divergece Λ, this c be + + y + ( ) ( ) = see if we itroduce cut-off term i the itegrl, we hve coverted -loop itegrl ito ordiry itegrl by usig Numericl method d pplyig the y Abel resummtio d formul () to our origil itegrl d dy + y + o Uderstdig the Csimir effect reormliztio d why the diverget series = ζ ( ) hs fiite physicl vlue: Let be the boudry vlue problem D f d f = f () = f ( π ) = D f = E f d E = (7) The, if we defie the opertor T = D, the sums re the trces of the powers of the opertor T i terms of the spectrl zet fuctio of the Eergies of the eigvlue problem iside (7) = Trce ( T ) = ζ T, L = π ( ) T s ζ s, L = π = E = ζ ( s) (8) The spectrum of problem (7) is discrete, sice we hve imposed the boudry coditios for the eigefuctios f () = f ( L = π ) =, if we te the limit L the spectrum is o loger discrete d the trces re give by itegrl isted of discrete sum, Trce ( T ) = t dt = ζ T, L L, This itegrl is still diverget but if we te the differece betwee the ( epoetil regultor is ssumed), d we c defie reormlized vlue of the diverget series ε tε ζ T, L = π ζt, L = = e dtt e = ζ ( ) (9) Ad for the cse of the Hrmoic series, the differece is ζ T, L = π ζ T, L = = γ which is gi reormliztio of the diverget Hrmoic series, so i the ed we hve oly fiite vlue. 7
This method is use i the evlutio of the fuctiol determit of opertor with discrete set of eigevlues det( A ) = λ, i geerl the epressio log λ is diverget but we c defie the logrithm of the fuctiol determit s the fiite differece (substrctio of the divergece). log A LogC Z(, ) Z(,) = + Z( s, ) = ( + λ ) s s s () Ad C is fiite costt, this method is used for emple to epd the Gmm fuctio d the sie fuctio ito ifiite product over their zeros. π = + Γ ( + ) si( π ) = π () Refereces [] Abrmowitz, M. d Stegu, I. A. (Eds.). "Riem Zet Fuctio d Other Sums of Reciprocl Powers." 3. i Hdboo of Mthemticl Fuctios. New Yor: Dover, pp. 87-88, 97. [] Berdt. B Rmu's Theory of Diverget Series, Chpter 6, Spriger-Verlg (ed.), (939) [3] Elizlde E. Te Physicl Applictios of Spectrl Zet Fuctios, Lecture Notes i Physics. New Series M35 (Spriger-Verlg, 995) [4] Grci J.J ; A commet o mthemticl methods to del with diverget series d itegrls e-prit vlible t http://www.wbbi.et/sciece/moret.pdf [5] Guo L d Zhg B. Differetil Algebric Birhoff Decompositio Ad th reormliztio of multiple zet vlues Jourl of Number Theory Volume 8, Issue 8, August 8, Pges 38 339 http://d.doi.org/.6/.t.7..5 [6] Hrdy, G. H. (949), Diverget Series, Oford: Clredo Press. [7] Prellberg T Mthemtics of Csimir effect vlible olie t http://www.mths.qmul.c.u/~tp/tls/csimir.pdf [8] Shri A. The geerlized Abel-Pl formul pplictio to Bessel fuctios d the Csimir effect e-prit t CERNhttp://cds.cer.ch/record/48795/files/39.pdf 8
[9] Shirov D. ; Fifty Yers of the Reormliztio Group, I.O.P Mgzies. 9