ZETA REGULARIZATION METHOD APPLIED TO THE CALCULATION OF DIVERGENT INTEGRALS

Transcription

1 ZETA REGULARIZATION METOD APPLIED TO TE CALCULATION OF DIVERGENT INTEGRALS s Jose Jvier Grci Moret Grdute studet of Physics t the UPV/EU (Uiversity of Bsque coutry) I Solid Stte Physics Addres: Prctictes Ad y Grilb 6 G P.O Portuglete Vizcy (Spi) Phoe: () E-mil: osegrc@yhoo.es MSC: 97M5, 4G99, 4G, 8T55 ABSTRACT: We study geerliztio of the zet regulriztio method pplied to the s cse of the regulriztio of diverget itegrls for positive s, usig the Euler Mcluri summtio formul, we mge to epress diverget itegrl i term of lier combitio of diverget series, these series c be regulrized usig the Riem Zet fuctio ζ ( s) s >, i the cse of the pole t s= we use property Γ ' of the Fuctiol determit to obti the regulriztio = ( ), with ( ) Γ the id of the Luret series i oe d severl vribles we c eted zet regulriztio to the cses of itegrls f, we believe this method c be of iterest i the regulriztio of the diverget UV itegrls i Qutum Field theory sice our method would ot hve the problems of the Alytic regulriztio or dimesiol regulriztio Keywords: = Riem Zet fuctio, Fuctiol determit, Zet regulriztio, diverget series. ZETA REGULARIZATION FOR DIVERGENT INTEGRALS: Sometimes i mthemtics d physics, we must evlute diverget series of the form k, of course this series is diverget ules Re (k) >, however cses like k= or k=3 pper i severl clcultios of strig theory d Csimir effect, for the cse of 3 Csimir effect [3] the result = ppers to give the correct result for the

2 Fc cπ Csimir force = h here A is the re d d the seprtio betwee the 4 A 4 pltes, c d h re the speed of ligth d the Plck s costt. The ide behid the Zet regulriztio method is to tke for grted tht for every s the idetity s = ζ ( s), follows lthough this formul is vlid ust for Re (s) >, to eted the defiitio of the Riem Zet fuctio to egtive rel umbers, oe eed to use the fuctiol equtio for the Riem fuctio s π s ζ ( s) = ( π ) Γ( s)cos ζ ( s) π Γ( s) Γ( s) = () si( π s) This gives the epressios t s=, the rmoic series fiite vlue =, i = d i = due to the pole i is NOT zet regulrizble, lthough it c be give = γ = , this vlue c be ustified by usig the theory of Zet-regulrized ifiite products (determits), s we shll see lter i the pper o Zet regulriztio for diverget itegrls: m s Let be f = with Re(m-s) < -, the the Euler-Mcluri summtio formul (see [] d ppedi A formul (A.) )for this fuctio reds f f B f f f f ( k)! k = k = (k ) (k ) () m s = ζ ( s m) i m s m s m s m s i= r m r s ( m r s) r= B Γ( m s ) ( r)! Γ( m r s) N (3) ere i formul () ll the series d itegrls re coverget, formul () is usully k k worthless, sice it is trivil to prove tht = for Re(k) >,d the Riem k zet fuctio ζ m s =, so othig ew c be obtied from () i= m s i The ide is to use the Fuctiol equtio () for the Riem d Zet fuctio to eted the defiitio of equtio () to the whole comple ple ecept s=, i cse (m-s) is positive there will be o pole t =, so we c put = d tke the limit s

3 m B m!( m r ) ( r)!( m r )! m m r m r = ζ ( m) (4) r= Formul (4) is the Alytic cotiutio of formul (3) with = d c be used to obti fiite defiitio for otherwise diverget itegrls,ppretly this recurrece equtio hs ifiite umber of terms but the Gmm fuctio hs pole t = d t beig some egtive iteger, lso if the fuctio is of the form f = α with α > we c us the Euler-Mcluri formul directly, sice the sum f ( ) f d its derivtives re fiite. some emples of formul (4) I = I = ζ () = = ζ ( ) = I I I B I = ζ ( ) I = (5) 3 B 3 I3 = ( I ζ ( ) ) I ζ ( 3) B3I = So our method c provide fiite regulriztio to diverget itegrls, with the Aid of the zet regulriztio lgorithm. Also our formule (3) (4) d (5) re cosistet with the usul summtio properties, i fct if Λ m is fiite for fiite Λ d we use the property of the Riem d urwitz Zet fuctio [ ] to get the sum of the k-th powers of o the itervl [, Λ ] Λ m i = ζ ( m) ζ ( m, Λ), i= ζ ( s, Λ ) = ( Λ) s defied for Re(s) > (of course for positive s s Λ the secod term goes to ) m B m!( m r ) ( r)!( m r )! Λ Λ Λ m m r m r = ζ ( m) ζ ( m, Λ) (6) r= B For iteger m (, ) m ζ m = we fid the Beroulli Polyomils, the powers m Λ m m Λ of Λ would ccel the itegrl =, so i the ed i formul (6) we would m get the usul defiitio of Zet regulriztio B () m ( m) = for iteger m. Of m ζ 3

4 course oe could rgue tht simpler regulriztio of the diverget itegrls should s s be I( s) = ( ) = d I( ) = ( ) = log s, this is ust droppig out the term proportiol to log s or iside the itegrl to mke it fiite, however if wi plugged this result ito the Euler-Mcluri summtio formule (3) (4) or (6) the terms ivolvig would ccel d we would filly fid tht ζ ( m) = for every m which clerly is gist the defiitio of zet regulriztio of series, for the cse of the logrihmic divergece, obtied from differetitio with respect to the eterl prmeter this is result of tkig the fiite prt of the itegrl,which ppretly m s k works. For the cse of the itegrls log, we c simply differetite k- times with respect to regultor s iside (3) i order to obti fiite vlues i terms of ζ ( s) d ζ '( s) for egtive vlues of s uless m=- (for other egtive vlues of m we c mke chge of vrible q = ), this is treted i the et sectio o Zet-regulrized determits d the rmoic series: λ = Give opertor A with ifiite set of ozero Eigevlues { } Zet fuctio d Zet-regulrized determit, Voros [] we c defie s { } Tr A = ζ A( s) = λ s dζ A det( A) ep () = λ = ds (7) The proof of the secod formul iside (7) is pretty esy, the derivtive of the log λ Geerlized zet fuctio will be ζ A '( s) = s ow let s=, use the property of λ the logrithm log(. b) = log logb d tke the epoetil o both sides. For the cse of the Eigevlues of simple Qutum rmoic oscilltor i oe dimesio [] λ =, the Zet fuctio is ust the urwitz Zet fuctio, so we c defie zet-regulrized ifiite product i the form dζ ep (, ) = ds dζ (, ) log log π ds = Γ (8) I cse we put = we fid the zet-regulrized product of ll the turl umbers ( ) = π,see [5] if we tke the derivtive with respect to, we would fid the sme regulrized Vlue Rmu did [] precisely Γ ' = ( ) > ( ) Γ rmoic series pper due to logrithmic divergece of the itegrl ( ), if we put m= - iside formul (3), usig regultor s, s we hve the Euler Mcluri summtio formul 4

5 (9) r B r = s s r s ( ) ( ) r= ( r)! u ( ) = Sice s > the itegrl d the series iside (9) will be coverget, ow we c itegrte s over iside (9) d use the defiitio of the logrithm lim log =, to s s Γ ' regulrize the itegrl ( )s s s i terms of the fuctio plus Γ some fiite correctios due to the Euler-Mcluri summtio formul. A fster method is ust simple differetite with respect to iside the itegrl di =, ow this itegrl is coverget for every d equl to d, itegrtio over gi gives the vlue log c plus costt c tht will ot deped o the vlue of iside the itegrl i questio, the proof tht c is uique o mtter wht is comes from the fct tht the differece log b =. b dζ (,) For the cse =, the derivtive of the urwitz Zet is = log ( π ) so if ds we pproimte the diverget itegrl by series, the we c get the regulrized result =. Appretly it seems tht usig two differet regulriztios we get some differet results, the ide is tht if we use the Stirilig symptotic formul B z log Γ ( z) = z log z z log ( π ) r () r= r ( r ) If we tke the derivtive with respect to z iside (), is ow more ppret tht for µ the logrithmic derivtive log here c = log µ is costt obtied from differetitio with respect to to regulrize the diverget itegrl, this costt c must be relted to some physicl costt or i cse the qutity hs dimesio of Eergy the µ must hve lso dimesios of eergy so the logrithm is dimesioless, this costt c would be the oly free dustble prmeter tht would pper iside our clcultios to regulrize itegrls. If is egtive there is etr term due to the vlue log( ) = πi, for more comple k log ( ) logrithmic itegrl oe c use the defiitio log k µ with k the sme eergy scle c = log µ 5

6 So if we tke formul (3) d the idetity iside Zeidler [], pge 55 ζ ( s, ) s C \ { } s ( ) = reg Γ ' s = Γ the we c regulrize y diverget itegrl ( ) m to get fiite result by lytic cotiutio. This result is ustified by tkig the limit s iside the formul for the Digmm ( s) s Ψ ( z) = ( ) Γ ( s ) ζ ( s, z) fuctio Also formlly is possible to get the result = γ, if we itroduce the followig regultor R( s, ) = cosh( slog ), s, iside the regulrized rmoic series R( s, ) ζ ( s) ζ ( s) so we get the limit lim = γ, which is precisely the s Euler-mscheroi costt. o Regulriztio of diverget itegrls f : I geerl, the diverget itegrls tht pper i Qutum Field Theory [] re 4 ivrit uder rottios, for emple 4 d p p m or d p ( p q) m p, if we use 4-dimesiol polr coordites we c reduce these itegrls to the cse d / π d drf ( r) r Γ d / the the UV divergeces pper whe r, here d=4 is the dimesio of the spcetime, depedig o the vlue of d we c hve severl types of divergeces Λ d m drf ( r) r Λ blog Λ, if b = for m = the UV divergeces re qudrtic if m = the divergeces re lier, i cse = d b = the divergeces re 4 of logrithmic type, for emple d p p m hs oly logrithmic divergece i dimesio 4, for lower vlue of the dimesio (d=3 ) this itegrl eists. To study the rte of divergece, we c epd the fuctio ito Luret series vlid k for z, f = c ( ) k is fiite umber d mes tht the fuctio f ( ) hs power lw divergece for big, the the ide to compute diverget itegrl would be this, we dd d substrct Polyomil plus term proportiol to to split the itegrl ito fiite prt d other diverget itegrl, i both cses 6

7 we must lso itroduce regultor ( ) s for turl umber so we mke the itegrls coverget for some Re(s) > b k k s f b ( ) b ( ) b s s ( ) = ( ) () Also we c use the chge of vrible ( ), so the ew limits of itegrtio would be (, ), sice is turl umber, the the followig idetity m s m s m s ( ) = = ζ ( m s) holds for every positive d m i the sese of zet regulrized series. Now, we hve to regulrize the itegrls iside () s = I, this c be mde usig formul (3) to get oly FINITE results for these itegrls I, see (5), i geerl for every this diverget itegrl i the limit s will be of the form ciζ ( i), where i rus from to -, for the logrithmic diverget itegrl, we c regulrize it by usig the idetity = γ plus the Euler-Mcluri summtio formul so we get oly fiite (regulrized) results, we re usig zet regulriztio plus formul (3) to get oly fiite results, this is the mi key of our method. Of course iside () i our substrctio we c iclude o-iteger powers of sice the recursio formul (3) is still vlid for them. The umber of terms k is chose so the first itegrl is FINITE, this first itegrl c be computed by Numericl or ect methods d yields to fiite vlue, the rest of the itegrls re ust the logrithmic d power-lw divergeces, they c be regulrized with the id of formule (3) (4) (5) (7) (9) to get fiite vlue ivolvig lier combitio of ζ ( m) m=,,,...,k d other vlue proportiol to µ log, this pper becuse i reormliztio/regulriztio we must obti oly fiite vlues, so = log, sice the diverget term log is ersed by ddig couterterm to our Lgrgi defiig our qutum theory so, we hve tht the (regulrized) itegrls fter reormliztio must hve vlue equl to, if we isert these itegrls iside the Euler-Mcluri formul the Γ ' we get fiite vlue for the rmoic series = ( ), which is fiite d ( ) Γ it is the oly vlue tht rmoic series c dmit fter regulriztio reormliztio, so eve the Riem d urwitz fuctio hve pole t s= we c get fiite vlue for the rmoic series. i 7

8 As empleto uderstd our method better, we c lyze this simple diverget itegrlwith > s ζ () = Γ B Γ r ' r ζ r r= ( r)! u = s () The first itegrl i () is coverget d hve ect vlue of log, i order Γ ' to regulrize the logrithmic itegrl we hve used the result = ( ) plus ( ) Γ the Euler-Mcluri summtio formul. The mthemticl ustifictio of this is the followig, give diverget itegrl f we itroduce regultor F( s) = f s so the itegrl F(s) eists for some big s, if we dd d substrct powers of the form k s s for iteger k d ( ), we c split F(s) ito coverget itegrl I (s) vlid for s d some diverget m s itegrls of the form d s, usig formule (3) (4) (5) d (9) we c epress these itegrls i terms of the series s d s, m which will be coverget for Re( s m) > d Re( s ) >, ow usig the Fuctiol equtio for the urwitz d Riem Zet fuctio we c mke the lytic cotiutio of both series to s voidig the pole t s= by the use of Riem Zet fuctio t egtive itegers ζ ( ) plus some correctios ivolvig Γ ' of course the rules for chge of vrible d still vlid so Γ s s f ( ) = duf ( u) u this c be used to void some IR divergeces t = by splittig the itegrl ito IR diverget prt d UV diverget prt β α du = du du. For other types of diverget itegrls like log for positive α d β oe could differetite with respect to m or s iside formul () β i order to obti recurrece equtio for the itegrls log equtio is fiite (pproimtely) sice for Re ( p ) > eed to be regulrized provided >. Other useful idetities c be α, this recurrece β log p is fiite d do ot 8

9 / or the epsio of the logrithm vlid for y >.4 ( ) log = to mke logrithms more trctble, lso we could use Luret epsios to hdle complicte o-polyomil epressios like ( µ by epdig it for big ito symptotic (iverse) power series. ) k Aother method to del with coverget Luret series vlid for > so s, would be to epd the itegrd ito = ( ) d use the equtios (-5) term by term to obti fiite results for the itegrls m=,,- however this is bit trickier th (). 3 m s with o Puli Villrs regulriztio emple for logrithmic diverget itegrls: To ed with this sectio we will give other emple of how c itegrl with logrithmic divergece be regulrized, for emple we set ( s, Λ ) = d k ( k s iε ) Λ k Λ 4 Λ ε (3) ere the regultor for smll k is d s is physicl prmeter, this emples ppers i Newcomb [8] pge 33, i the reormliztio process we hve tht s s so, we must tke ito ccout the differece betwee the bre vlue of the prmeter d the mesured oe ( s, Λ) ( s, Λ ) = δ, i this cse fter reormliztio, the itegrl iside (3) hs the vlue δ s = iπ log s REGULARIZATION FOR IR DIVERGENT INTEGRALS: If the itegrd hs pole t = so is IR diverget itegrl, we c mke the Tylor substrctio λ f b ( ) λ = λ b ( λ ) ( ) b = ( f )! (4) For the cse λ = we hve 9

10 ( ) = = ( ) ( ) ( ) ( ) ( ) (5) To void the pole t = we isert smll comple qutity Epsilo so iε, the formul (4) c be grite = ( iε ) ( iε ) ( iε ) (6) The iside (5) we hve oly UV logrithmic divergece, for λ > there re oly IR divergeces, if we use the Biomil theorem ( ) λ ( iε ) λ k λ k = λ k k λ = ( ( iε ) ) (7) Itegrls iside (6) d (7) c be evluted with the epressio m m r m r Γ ( ) π ( i ) = = I(, m, r) r ( ( iε ) ) mπ π m si ( r )! Γ 3 r ε (8) m o Regulriztio of itegrls m : There re cses, for m= we hve ( ) = = ( u) Where we hve mde the chge of vrible u = iside the lst itegrl to tur the IR divergece ito Logrithmic UV divergece. For m > we itroduce regultor s to mke the IR diverget itegrl to coverge = = m s m s m s m s / duu m s u = (9)

11 The lst itegrl iside (8) is diverget i the limit s d c be regulrizd with λ λ λ formul (3). If my IR divergeces pper ( ).( )...( ) m m we must perform the Tylor substrctio (4) o ech poit ( i )i REGULARIZATION OF MULTIPLE INTEGRALS: I geerl i QFT there will be lso multi-loop (multiple) itegrls so we will lso hve to regulrize itegrls i the form. I( s) = d q d q... d q F( q, q,..., q ) () i= ( qi ) ere we hve itroduced regultor depedig o eterl prmeter s i order the itegrl () to coverge for big s d the use the lytic regulriztio to tke the limit s, this regultor must be chose with cre i order ot to spoil y symmetries of the Physicl system. Aother method is to cosider the multiple itegrl s iterte itegrl d the mke the substrctio for every vrible for emple k k i, i i i= i= i q F( q q,..., q ) ( q,..., q )( q ) q ( q,..., q )( q ) () The symbol q mes tht the itegrl is mde over the vrible q keepig the other vribles costt, the umber k is chose so the first itegrl is fiite, this itegrl will deped o I( q,..., q ), the diverget itegrls (eve for the logrithmic cse i=-) c be regulrized. Now we hve regulrized the first itegrl, we hve reduced i oe vrible the multiple itegrl, repetig the itertive process for the fuctios i ( q,..., q ) k k i q i ( q..., q) b ( q,..., q)( q) q bi( q,..., q )( q ) = = () Usig () d () o every vrible we c reduce the dimesio of the itegrl util we rech to the oe dimesiol cse, which is esier to hdle. If the itegrd F( q, q,..., q ) hd o sigulrities for every q >, we my epd this itegrd ito multiple Luret series of severl vribles, d the

12 s, s,..., s m m m m, m,..., m i m, m,..., m perform the substrctio C ( q b ) ( q b )...( q b ) order to defie fiite prt of the itegrl d q d q d q F C q b q b q b (3) s, s,..., s m m m... m, m,..., m ( ) ( )...( ) m, m,..., m Plus some correctios due to diverget itegrls ( q ) m i bi dqi m=-,,,.... I my cses lthough the itegrls give i () d (3) re fiite they will hve o ect epressio or the ect epressio will be too complicte, i this cse we c use the Guss-Lguerre Qudrture formul (i cse the itervl is [, ) ) to pproimte the itegrl by sum over the zeros of Lguerre Polyomils wi f ( q, q,..., q, i ) with the weigth epressed i terms of Lguerre Polyomils d their roots i wi =, L ( i ) = (4) ( ) L ( ) i uas emple we will regulrize the simples -loop itegrl l dldk k ( k p) ( k l) s s We itroduce the zet regultors k l to get fiite results d void divergeces, the regultor will be elimited fter clcultios. Itegrtio over l gives (here p is costt) i= s s s l l dl l dl π dl = ( k) ( k l) ( k l)( l ) l (5) To perform itegrtio over k we will eed to regulrize the itegrls k dk A k k s ( k) dk p / s dl ( k p) k ( k l)( l ) A = π (6) The first itegrl becomes logrithmic UV diverget itegrl u du s with the ( up ) chge of vrible u = ( s lwys the regultor s teds to ), this c be regulrized by dditio d substrctio to this itegrl of the fctor, this logrithmiclly diverget itegrl c be regulrized with the Euler Mcluri formul d the idetity = γ. reg

13 To evlute the secod itegrl we will use the Lguerre qudrture formul i order to simplify clcultios ( k) dk dl w ( k) dk w dke ( k p) k ( k l)( l ) (7) ( k p) k ( k ) ( ) Epressio (7) is legitimte sice the itegrl over l is coverget, with chge of vrible u = ech itegrl of the sum iside (7) becomes logrithmic diverget wu( u ) du itegrl dk, this is gi logrithmic diverget itegrl d c be ( up) u ( ) evluted by ddig /substrctig term of the form. Usig the equtios (4-7) we hve show emple of how -loop itegrl c be ζ regulrizd, by pplyig the zet regulriztio method of our pper o ech vrible, first over l keepig k costt d the over k to get fiite results, ech UV diverget itegrl c be regulrized with the formule (-5) o A esier emple of -loop itegrl d zet regulriztio: l dldk k ( k p) ( k l) If the itegrl seems too complicte we will give esier model of how Zet regulriztio Works beyod the -loop itegrl, let be the diverget itegrl dy, this is diverget, d hs lso overlppig divergece i y d y, itegrtio d regulriztio over the sub-divergece o the vrible gives (regultor ( y) s ssumed implicitly). y dy = dy dy y ( y )( ) (8) y The first itegrl iside (8) is coverget dy so we c use the y y Lguerre Qudrture formul to evlute it ω e, we hve ow ( y )( ) sum of fiite term, ech term will be fuctio of y, regulriztio i the vrible y gives filly (for ech term of the Lguerre qudrture) y ( ) dy ( y )( ) ( y )( ) y = dy dy ( ) (9) Where, we must pply the regulriztio iside (9) to every term of the Lguerre y Qudrture formul ω e to obti fiite result ( y )( ) 3

14 ( ) Now itegrl iside (9) dy is coverget (itegrtio over y ) so ( y )( ) it c be evluted without y regulriztio reormliztio. The three diverget itegrls dy d ydy c be regulrized usig the equtios () (3) d (4) plus the idetity = γ. Of course we could hve used the Euler-Mcluri summtio formul to replce the y coverget itegrl dy y y will deped o the vrible y, the we tructe the series d ( y )( ) pply zet regulriztio (9) to ech term of the series. by fiite coverget series whose terms We would like to put much emphsis ito this method of zet regulriztio for multiple loop itegrl ( loop) with overlppig divergeces, becuse we hve hd our pper reected severl times becuse of the chep ecuse tht zet regulriztio would ot be vlid for higher loops, with these emples we prove them wrog, we show how we c regulrize multiple itegrl usig zet regulriztio d iterted itegrtio o ech of vribles, first over (subdivergece) d the over y, of course this process c be geerlized to -loop multiple itegrls too, so referee s ecuse is ust wrog d zet regulriztio is vlid lso for multiple itegrls CONCLUSIONS AND FINAL REMARKS: We hve eteded the defiitio of the zet regulriztio of series to pply it to the m Zet regulriztio of diverget itegrl > m > by usig the Zet regulriztio techique combied with the Euler Mcluri summtio formul. For good itroductio to the Zet regulriztio techiques, there is the book by Elizlde [4] or the Book by Bredt bsed o the mthemticl discoveries of Rmu d its method of summtio equivlet to the Zet regulriztio lgorithm [], other good referece (but bit more dvced) is Zeidler [], for the cse of Zetregulrized determits [7] is good olie referece describig lso the process of Zet regulriztio vi lytic cotiutio d how it c be pplied to prove the idetity ( ) = log π. Appretly there is cotrdictio, sice the Riem Zet fucito hs pole t s= so the rmoic series could ot be regulrized, however usig the defiitio of 4

15 fuctiol determit E µ E = oe gets the fiite result for the rmoic Γ ' (geerlized) series =, with the id of the Euler-mcluri summtio Γ formul this result for the rmoic series c be used to give pproimte regulrized vlue of the logrithmic itegrl, for the cse of other types of diverget itegrls ( ) m we c use gi Euler-Mcluri summtio formul to epress this diverget itegrls i terms of the egtive vlues of the urwitz or Riem Zet fuctio ζ ( s,) = ζ ( s ) ζ ( m,) (UV) m=,,,3,4,...d the vlue of the derivtive of urwitz zet fuctio log s = ζ (, ) (logrithmic UV), these vlues ecode the UV divergeces []. s For the cse of the IR (ifrred ) divergeces i the form chge of vrible to re-iterprette these itegrls s q m s oe could mke m s q dq, for the cse m= we hve logrithmic divergece both t = s so we must split the / itegrl ito IR d UV diverget prt = fter few simple clcultios this itegrl will be equl to / forml UV d IR regultor so = log µ, sice we c simply itroduce Λ lim log UV Λ = Λ Λ itroduced to esure tht the itegrl will be coverget., UV regultor is We lso believe tht similr procedure c be pplied to eted our Zet regulriztio lgorithm to multiple (multi-loop) itegrls d q d q... d q F( q, q,..., q ) The dvtges of zet regulriztio re Zet regulriztio gives oly fiite results without couterterms if we use the lytic cotiutio of Riem zet m s = ζ ( s m) d the regulriztio Γ ' of the rmoic series = reg Γ Zet regulriztio does ot lter the dimesio of spce so we c work with Mtrices γ 5 = iγ γγ γ 3 d Tesors ε ik, ulike dimesiol regulriztio. Zet regulriztio respect my of the symmetries of the system 5

16 Zet regulriztio is comptible with my other regulriztio methods: Puli- Villrs, Cut-off regulriztio... Zet regulriztio is vlid lso i curved spcetimes. For the cse of IR divergeces we c mke the chge of vrible = u so the IR diverget itegrl m s m s turs ito u du, for the logrithmic itegrl we get = i the sese of zet regulriztio. I the Zet regulriztio method these equtios re equivlet Γ ' = reg log d = this my be proved usig the reg Γ Euler-Mcluri formul If we use Numericl methods we c eted this zet regulriztio lgorithm to multiple loop itegrls, see eqs. (-9) Elizlde d others hve proved tht Zet regulriztio is well defied [5], lso through history, zet regulriztio hs give correct vlues i severl brches of mth d physics, for emple i the Csimir effect. [] I both cses zet regulriztio d dimesiol regulriztio re bsed o the priciple of lytic cotiutio, however some people still hve preudices gist zet regulriztio d it is ot commoly used, ecept for the evlutio of fuctiol determits. Dimesiol regulriztio is ust Zet regulriztio i disguise, i fct whe fter clcultios they tke the limit d = 4 ε s ε they fid the epressio γ O( ε ), the Euler-Mscheroi costt ppers i both regulriztio lthough i ε zet regulriztio we do ot hve the pole / ε t the ed of the clcultios. The impositio i formul () tht must be turl umber is i order to void oddities i the process of Zet regulriztio with the Zet d urwitz Zet fuctio, sice uless is positive iteger the equlity ζ (, ) = ( ) = (3) does ot hold. Equtio (3) must be uderstood s ( ) = = ζ (, ) so zet regulriztio is cosistet d yields to correct results. Aother dvtge is tht we get oly fiite qutities, wheres i dimesiol regulriztio you will lwys fid poles of the Gmm fuctio Γ ( z) = γ O( z) i z the limit z this epressio blows up, d eed couterterm to tur it fiite. The mi dvtge of our Zet regulriztio method is tht due to formul (3) d the Γ ' regulrized idetity for the Digmm fuctio = ( ) reg, the reltioship Γ 6

17 betwee dimesiol regulriztio d dimesiol regulriztio is recovered if we use the followig defiitio for the logrithmic diverget itegrl = Ψ () log ( 4π ) log µ s s (3) s Where we hve itroduced the Eergy scle µ. APPENDIX A: OW TO OVERCOME TE POLE ζ () = I this pper we hve see how due to the pole of the Riem zet t the poit s = we could ot regulrize the itegrl uless we use the result for the rmoic series = reg ( ) fro > d fiite, the if we itroduce this result iside the Γ ' ( ) Γ Euler-Mcluri summtio formul we c get fiite results for Aother ltertive is to use the idetity. ( ) ( α ) = e = log!! α = (A.) I this cse we c evlute the itegrls iside (A.) by the Euler Mcluri formul f f B f f f f ( k)! k = k = (k ) (k ) (A.) ere The Beroulli umbers o the lst side of (A.) re the coefficiets of the Tylor log f = epsio of e if we isert iside (A.) the epresió α i log log r log B r log ( ) = ζ ( α) α α α r s i= i r= ( r)! u ( ) = (A.3) ( ere α is smll o iteger, so the zet fuctio d its derivtives ζ ) ( α) re FINITE Aother ltertive is to look for Pde or Rtiol pproimtio for the squre root of for emple. 7

18 P Q ( )! = > (A.4) ( )(!) 4 P I this cse (A.4) we hve the pproimtio 3/ Q formul P c i i 3/ 3/ ci ci c 5/ Q i i (A.5), ow if we pply the Iside (A.5) ow there re o logrithmic-diverget itegrls, so the pole ζ () will ot ow pper Refereces [] Abrmowitz, M. d Stegu, I. A. (Eds.). "Riem Zet Fuctio d Other Sums of Reciprocl Powers." 3. i dbook of Mthemticl Fuctios. New York: Dover, pp , 97. [] Berdt B. Rmu's Notebooks: Prt II Spriger; editio (993) ISBN-: [3] Dowlig Joth P, "The Mthemtics of the Csimir Effect", Mth. Mg. 6, (989). [4] Elizlde E. ; Zet-fuctio regulriztio is well-defied, Jourl of Physics 7 (994), L [5] Elizlde E. ; Zet regulriztio techiques with pplictios Ed. World Scietific Press ISBN [6] Mizuo, Y. "Geerlized Lerch Formuls: Emples of Zet-Regulrized Products." J. Number Th. 8, 55-7, 6. [7] Muñoz Grcí, E. d Pérez Mrco, R. "The Product Over All Primes is." Preprit IES/M/3/34. My 3. [8] Newcomb W. Reormliztio Methods: A Guide For Begiers Oford Uiversity Press, USA; editio (Jury 6, 8) ISBN-: [9] Pvel E "Note o dimesiol regulriztio", Qutum fields d strigs: course for mthemticis, Vol.,(Priceto, NJ, 996/997), Providece, R.I.: Amer. Mth. Soc., pp , ISBN (999) 8

19 [] Prellberg Thoms, The mthemtics of the Csimir Effect tlk vlible t e-prit t [] Voros A. Zet fuctios over zeros of zet fuctios Ed. SPRINGER () ISBN: [] Zee, A. (3). Qutum Field Theory i Nutshell. Priceto Uiversity ISBN ISBN [3] Zeidler E. Qutum Field Theory I : Bsis i Physics d Mthemtics Ed : Spriger Verlg ISBN: