THE THEORY OF DISTRIBUTIONS APPLIED TO DIVERGENT INTEGRALS OF THE FORM

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1 THE THEOY OF DISTIBUTIONS APPLIED TO DIVEGENT INTEGALS OF THE FOM ( ) u ( b) Jose Jvier Gri Moret Grdute studet of Physis t the UPV/EHU (Uiversity of Bsque outry) I Solid Stte Physis Address: Address: Prtites Ad y Grijlb 5 G P.O Portuglete Vizy (Spi) Phoe: () E-mil: josegr@yhoo.es MSC: 6E5, 6E99, 35Q4 ABSTACT: I this pper we review some results o the regulriztio of diverget itegrls of ( ) the form d b i the otet of distributio theory Keywords: egulriztio, distributio theory, frtiol lulus egulriztio of diverget itegrls: I QFT (Qutum field theroy) there re mily two ids of diverget itegrl, the UV (ultrviolet) divergee, tht hppes wheever itegrl is diverget whe d the I (ifrred) oe tht ours if the itegrl is diverget s, few emples of these itegrls re 3 ( ) 3 ( ) or Z () The first two itegrls hve UV divergee, wheres the lst oes hve I divergee,the mes I d UV divergees ome from the ft tht if we use the h Wve-prtile Dulity i QM settig =p we fid tht UV divergee is p

2 equivlet to hve smll wvelegths (ulr violet) d the I divergee hppes for big wvelegths (ifrred) so the me is ot sul, for more detils [5] d [9] d To del with UV divergees o itegrl of the form d F( ) o d d we simply me hge of vrible to d- dimesiol polr oordites, to rewrite the itegrl d d d i ( ) ( ) (, ) i ( ) ( ) i r ( ) d F d drr F r d dr r d With the ostts () dzf( z, ) ( ) i, here mes tht we must itegrte z over the gulr vribles, i se F is ivrit uder rottios o the d-dimesiol d / ple the gulr prt of the itegrl be lulted etly s d ( d / ), the ide of epdig our futio ito Luret series, is to isolte the UV divergeies of the form drr so they be ured usig the zet regulriztio lgorithm. Choosig y >, sie the itegrl hs UV divergee, fter epdig the itegrd F( r, ) r d ito overget Luret series for r > there will be oly fiite umber of diverget itegrls of the form drr plus logrithmi diverget dr itegrl (this is other emple of UV divergee), ow to get fiite results r from the diverget itegrls we will pply the reurree dedued i our previous pper [4 ] whe disussig the zet regulriztio pplied to itegrls m m m B rm!( m r ) mr ( m) (3) ( r)!( m r )! r Equtio (3) is regulriztio i the sese tht if we hd the upper limit N- isted N N of the epressio (3) would give the well-ow result > or =, for the se, the series vlue i the spirit of Zet futio regulriztio [ 4] so is o loger overget d must be give fiite ( ), this is why the term ( m) ( m) ppers iside (3), ote tht (3) is reurree formul with iitil term () / d llows to lulte or ssig

3 fiite vlue to every diverget itegrl of the form drr for positive iteger, (the mi problem for the logrithmi divergee is the pole of the Zet fuito t s=) the first terms re I(, ) () / I(, ) I(, ) ( ) B I(, ) I(, ) ( ) I(, ) (4) 3 B 3 I(3, ) I(, ) ( ) I(, ) ( 3) B3I (, ) Nottio I ( m, ) physil meig. stds for lim, beig Lmbd ut-off with erti This emple is vlid for UV divergees but wht would hppe to the se of dr sigulr itegrls <b< or to the logrithmi divergee b? r o egulriztio of diverget itegrls usig Distributio theory: As first emple, let us suppose we wt to lulte the itegrl f ( ) D ( ), for = we ow tht i spite of the pole of the delt futio t = we hve tht f ( ) ( ) f ( ) for y test futio f(), the we use two method We perform itegrtio by prts voidig the sigulr poit t = so ( D) f ( ) ( ) f ( ) D ( ) ( D) f ( ) (5) We itegrte -times (o mtter if is o-iteger) with respet to (6) I( ) f ( ) D ( ) D I( ) f ( )( ) ( ) ( ) f ( ) 3

4 Both methods re equivlet sie if we rell the idetity D D I( ) I( ) we yield to the sme result, o mtter if we itegrte/differetite with respet to the prmeter or if we itegrte by prts. A similr ide be pplied to itegrls of the form d b first we defie s ( b) for << the distributio T ( b, ), for regulr eough futio ( ) we otherwise osider (i the sese of distributio) our diverget itegrl to be the lier opertor T[ ] ( ) for y rel s, the ide is to perform differ-itegrtio s ( b) -times so s /. First we must itrodue the oept of differ-itegrl or frtiol derivtive of y order, there re mily [6] 3 defiitios ( ) ( ) ( ) D f dt t f t dt ( ) q m q D f ( ) lim ( ) f ( ( q m) h) h q h m m (7) The first defiitio iside (7) is the epressio for frtiol itegrl (ot vlid for egtive ), the seod oe is the Gruwld-Letiov differitegrl vlid for positive q, the third ltertive for the derivtive omes from the defiitio of q ( q ) dz Cuhy s itegrl formul D f ( ) f ( z) q i for y retifible urve ( z ) o the omple ple tht iludes the poit z= Emple: Let be the sigulr itegrl I ( b) ( ) s ( b), i order to give it fiite vlue first we differetite -times with respet to b so s / hee ( s) Db I( b) ( ) ( s ), we me the the hge of vrible b our itegrl beomes b ( s) b ( ) ( ) D I b du b u ( s ), the we defie the b b u so df futio F so ( b u ) this implies du ( s) Db I( b) F( b) F( b) filly tig the iverse opertor we ( s ) ( s) set I ( b) Db F( b) Db F( b) ( b, ) (8) ( s ) d f Here mu is rel umber d Db f eists i the sese of frtiol derivtive/itegrl d D ( b, ). We set the oditio s / beuse with b hge of vrible we void the pole b / t =b. 4

5 The sme strtegy be pplied to sigulr itegrl equtio, for emple if we wished to solve the followig equtio dt f ( ) g( ) f ( t) so t ( ) dt D f ( ) D g( ) f ( t) ( ) (9) t Where /, mig the hge of vrible t u so (9) beomes ( ) ( ) ( ) ( ) D f D g du f u ( ) Gmm futio () Now, equtio () hs NO poles t t=, to solve this itegrl equtio without sigulrity we ould use itertive proess ( ) D f D g du f u f ( ) g( ) ( ) ( ) ( ) ( ) () Where we hve supposed tht D (, ) (, ) i order to simplify the lultios of the solutio of the itegrl. The oe ould s, wht hppes i the limit b, i this se the qutity b beomes zero for every, so the I(b) must ted to for b big hee we should hoose the futio ( b, ) with the oditios D ( b, ) d b ( s) lim Db F( b) Db F( b) ( b, ) b ( s ) () o Logrithmi divergees The se of the logrithmi itegrl is bit differet, sie formul (4) ot hdle it,due to the pole of zet futio t s=, oe of the ides to pply [4] is just to reple the diverget itegrl by diverget series with h beig step (we / h use etgle method ),this series is still diverget but be ssiged fiite vlue vi muj resummtio equl to the logrithmi derivtive of Gmm futio ',however this id of method depeds o the vlue of the step h give. h 5

6 From Fourier lysis oe iterprette, the itegrl s the followig H ( ) H ( ) ovolutio H * with H() the Heviside step futio, usig the property of the Fourier trsform d the ovolutio theorem H ( ) i iu H * d u ( u) e I( ) i u (3) Here is the bsolute vlue futio tht tes the vlue or depedig o if is either positive or egtive, solvig Fourier trsform (3) we solve the logrithmi UV divergee. If we solve (3) for fied the for other vlue b so b is b differet from or iifiite we hve log I( ) b Aother simpler method is tht if we hve I( ) diverget itegrl, we differetite with respet to I '( ) so itegrtig gi with ( ) respet to I( ) log( ) ( ), with uiversl diverget ( ) D I( ) ( ) ( ) ( ) ( ) (4) So I ( ) ( ) ( ) D (, ), with d D (, ), oe of the problems is the ppret bsurdity sie due to the term there is omple otributio to itegrl with rel-vlued itegrd. A ltertive formultio bsed o Hurwitz Zet futio is the followig log( ) d H (, ) log ( ) log (5) For the sum lo g( ) (, ), this is the Zet-regulrized defiitio for the s H Determit of opertor, ombitio of the epressios iside (5) gives the ' regulrized vlue for the Hrmoi sum ( ) (i se = we get the Euler-Msheroi ostt), this is the logue result to simply usig muj resummtio for the series s s >, d s=-, stds for the emider term r B r iside Euler-Mluri summtio formul d is r r ( r)! 6

7 Appedi: Covolutio theorem I this pper we hve itrodued d used the Covolutio theorem, if we hve the ovolutio of futios or distributios f() d g() defied s h( z) f ( ) g( z) the H ( u) F( u) G( u) with F, G d H the fourier trsform respetively of h(z) f() d g(). iu e f ( ) F( u) Proof:= if we defie iuz iuz dze h( z) dze f ( ) g( z) iu d e g( ) G( u), we me the hge of vrible y z so = -dz ito the first itegrl so we hve the followig idetities iuz iu( y) iu iuy dze f ( ) g( z) dye f ( ) g( y) dye f ( ) g( y) e The first itegrl o the left is just H(u) d usig Fubii s theorem to iterhge the order of itegrtio we hve bee ble to proof tht H ( u) F( u) G( u). If we me g( ) H ( ) d f ( ), the H ( ) pplyig ovolutio theorem d the Fourier trsform iu i i du ( u) D ( u) e u > (7), hee Ufortutely there re some oddities with defiig produt of distributios D (6) D,, withi distributio theory, however if the itegrl f ( ) mes sese s iem itegrl, if we defie F(u) s the Fourier trsform of f() the the F() Covolutio theorem gives the regulrized result dtf ( t) f ( ) for some d el, for the se of f ( ) with Fourier trsform the itegrl is d diverget i the iem sese but be tthed fiite vlue log( ), sie is eve its derivtive will be odd so the me vlue of the derivtive er = 7

8 will be d we hve Mluri summtio formul to get log( ). The lst method would be to use the Euler- r B r d r r ( r )! (8) The problem is tht (, ) H is still diverget, lthough usig mujsummtio we tth this series the fiite vlue (, ) ( ) H d plug this result ito (7), I both ses the pproimtio of the itegrl by sum ( ) d the result log( ) re equivlet for (big ) sie usig the Stirlig s pproimtio for log ( ) d tig the derivtive we get '( ) / ( ) the symptoti result lim log( ) eferees: [] Estrd. Kwl. A distributiol pproh to symptotis Bosto Birhäuser Birhäuser () ISBN: [] Elizlde E. ; Zet-futio regulriztio is well-defied, Jourl of Physis A 7 (994), L [3] Gri J.J Chebyshev Sttistil Prtitio futio : A oetio betwee Sttistil Mehis d iem Hypothesis Ed. Geerl Siee Jourl (GSJ) 7 (ISSN ) [4] Gri J.J A ew pproh to reormliztio of UV divergees usig Zet regulriztio tehiques Ed. Geerl Siee Jourl (GSJ) 8 (ISSN ) [5] Griffiths, Dvid J. (4). Itrodutio to Qutum Mehis (d ed.). Pretie Hll. ISBN OCLC A stdrd udergrdute tet. [6] Keeth S. Miller & Bertrm oss A Itrodutio to the Frtiol Clulus d Frtiol Differetil Equtios Publisher: Joh Wiley & Sos; (993). ISBN [7] Lighthill M.J Itrodutio to Fourier Alysis d geerlized futio Cmbridge Uiversity Press (978) (ISBN ) 8

9 [8] Shwrtz L (954): Sur l'impossibilité de l multiplitios des distributios, C..Ad. Si. Pris 39, pp [9] Yduri, F.J. (996). eltivisti Qutum Mehis d Itrodutio to Field Theory (st ed.). ISBN