Towards a fnte conformal QED A D Alhadar Saud Center for Theoretcal Physcs P O Box 3741 Jeddah 143 Saud Araba In 196 whle at UCLA workng wth C Fronsdal and M Flato I proposed a model for conformal QED that I clamed to be dvergence-free and nontrval The results for one loop calculaton were gven However a debate about untarty and nontrvalty of the model caused the wthholdng of publcaton of that work Now and more than 30 years students and colleagues suggested that wth recent results there s a mert to the publcaton of the orgnal study so that the problem could be revsted ndependently nvestgated and the calculatons be repeated Consequently I present here the work exactly as t appeared then n a UCLA preprnt of the Theoretcal Elementary Partcle physcs group wth preprnt number UCLA/6/TEP/31 UCLA/6/TEP/31 August 196 TOWARD A FINITE CONFORMAL QED A D Hadar Department of Physcs Unversty of Calforna Los Angeles CA 9004 ABSTRACT: A fnte (dvergence-free) nontrval conformal QED s formulated and the results of on-loop calculatons are presented Recently [1] t was dscovered that the conformal spnor feld s a gauge feld whose physcal subspace s the massless electron The representaton of the conformal group carred by the spnor s non-decomposable and forms a symmetrc trplet (the Gupta- Bleuler trplet) An ndefnte-metrc quantzaton whch s conformally covarant was carred out n the same paper Upon quantzaton of charged conformal felds an assocated dffculty arses [1-3]: No conformally covarant two-pont functon can be found that solves the free wave equaton The suggeston was made [3] that the scalar part of the trplet the cyclc for the whole representaton space may resolve ths problem when taken as an ndependent feld component In ths paper we study some of the mplcatons of the fndngs made n Ref [1] and wrte conformal QED n terms of the three components of the spnor gauge multplet besdes the usual vector potental and dpole ghost The resultng theory elmnates the above-mentoned dffculty but more dramatcally produces cancellaton of dvergent dagrams verfed here up to one loop We start by formulatng the theory n Drac s sx-cone [4] notaton whch has the advantage of manfest conformal nvarance The sx-cone s Mnkowsk space 1
6 compactfed and embedded n as the surface y yy y0 y1 y y3 4yy 0 wth the projecton y y for 0 The spnor acton s [4] S 1 0( ) ( dy) y y c 6 ( dy) ( y ) d y (1) where s a homogeneous -component spnor wth degree of homogenety and c s a dmensonless real parameter y y where { } s a set of matrces satsfyng the Clfford algebra { } and as a bass we choose ({ } and are respectvely the Drac and Paul matrces): 3 1 0 where and In Mnkowsk notaton [14] forms the spnor multplet ˆ ( x) ˆ ( x) usng the transformaton ˆ( x) ( y ) 1 x ( y) The most general homogeneous -pont functon s 3 ( y) ( y) yy cyy y y c s arbtrary It carres the representaton [15]: D 5 1 0 D 3 01 D 5 1 0 D5 01 D3 1 0 D5 01 The arrows are sem-drect sums ndcatng leaks among the rreducble parts under the acton of the group of conformal transformatons The left /3 of the trplet the scalar and physcal s contaned n ˆ whle ˆ s pure gauge and of the form y Asde from the trval soluton no -pont functon solves the free wave equaton obtaned from (1) for all c and c To try to resolve ths problem we splt ˆ nto ts physcal and scalar parts n the followng way: c1 ˆ ( x) c 1 c3 ˆ( x) 4 where c are dmensonless constants and To ths end we wrte the theory n terms of a homogenous spnor of degree 1 and make the dentfcaton:
( y) 1 ( y) (mod y ) () Now s not ntrnsc on the cone therefore an extenson off the cone [136] s needed for modulo y y In Mnkowsk notaton ths extenson takes the form of the multplet ( y ) and defnton () gves c1 c c3 1 The acton of specal conformal transformaton ( K ) on these spnors s K 0 x 1 K x K x n and x x where x x x n The requrements of conformal covarance and the exstence of a non-sngular kernel (nvertble propagator) gve the followng unque acton and -pont functon: S 1 1 1 0( ) ( dy) y y y y (3) 1 1 1 3 ( y y) ( y) ( y) yy yy yy y y yy (4) satsfyng the followng equatons (mod y ): 0 y 0 (5) Asde from the last term whose orgn wll become clear shortly ths acton s the same as (1) wth c 1 and n x-notaton t reads: 4 dx (6) The non-vanshng covarant -pont functons correspondng to (4) are: ( x) ( x) rr ( x) ( x) rr 4 ( x) ( x) r ( x) ( x) rr 4 (7) where r x x and r r These functons satsfy the free wave equatons 3
whch also follow from (5) or (6) Introducng mnmal couplng ( 0 0 0 ea ) n the orgnal free acton (1) gves S( ) ( dy) 1 y y c e y a () and a s the 6-vector potental whch decomposes n x-notaton nto ( A A ) the followng acton [7] and has 1 4 Saj ( ) ( dy) a a ( ya) ( ya) a j where s a dmensonless gauge parameter In the present model y j 0 whch can be used to elmnate A and gve the electromagnetc acton [] 1 1 1 4 4 S( A J) d x F F A A A A A J AJ The free propagators are [7] 1 TA ( x) A ( x ) r ( 1) r r r 4 (9) 1 1 TA ( x) A ( x ) r r TA ( x) A ( x ) 4 The full spnor acton () s nvarant under the gauge transformaton: e e ; alternatvely e e and n Mnkowsk notaton t s equvalent to the followng set of local gauge transformatons e e e e e e e The unconventonal affne gauge transformaton of matter feld s also present n conformal scalar QED [1] In a gauge nvarant perturbatve quantum feld theory defned by ts n-pont functons one needs to defne free propagators If gauge transformaton alters ths defnton one fxes the gauge Covarant gauge fxng s usually accomplshed by addng a gauge-dependent term that vanshes on the physcal subspace e the Lorentz condton term For example n QED we add ( A) and n conformal QED A A correspondng to the Lorentz condton A 0 and A 0 [7] respectvely For the present case the Lorentz condton whch removes the scalar mode from the spnor gauge feld s y 0 and can only be mposed as an ntal condton on the 4
physcal subspace Therefore we propose to fx the gauge n () by addng a term proportonal to 0 whch leaves only rgd U(1) nvarance Ths exactly what has been done n a unque way under the requrements that lead to (3) actually the last term s nothng but 0 The nteracton part of the Lagrangan densty s e 1 1 y a y a ea e 1 1 1 1 ea ea The equaton 0 s stll mantaned sgnfyng that s a generalzed free feld and preservng the Lorentz condton Arrangng the three spnors n the trplet and the vertces are: 1 4 q p q 0 e 1 p 0 0 0 0 nm 0 1 4 qp q 1 e 4 qp ( p q) 1 p 0 nm Usng these vertces and the free propagators (7) and (9) one can demonstrate the fnteness of the theory by explct calculatons [9] In ths report we state the results for one loop: ( p) ( pq ) 4 p p 0 p e 0 p 3 p 0 nm 1 3 4 q p q 0 e 1 p 0 3 0 0 0 nm 5
( pq ) 3 e 3 1 1 ( q p p q) 4 q q 4 p 0 1 p 0 nm () r 0 () r 0 () r 4 4 4 e [ ( r)] e ( ) d q The last dstrbuton s defned as follows: Let Fr () and Gr () be less sngular dstrbutons then the nserton of s gven by 4 4 dxdxfr 3 ( 1) ( r3) Gr ( 34) efr ( 14) Gr ( 14) whch also leads to consstency wth covarance The Ward-Takahash dentty ( p q) ( p q) e ( p) ( q) s satsfed on the physcal subspace ( phys 0 ) The frst order contrbuton ( p) to the free propagators s 0 0 0 e ( p) 0 0 1 3 0 1 4 p e ( p) p p p p 4 e 4 ( p) p p 4 e 4 ( p) ( p) 4 The author s ndebted to C Fronsdal for advce and helpful dscussons and grateful to M Flato for very stmulatng and frutful conversatons Ths work was supported n part by the Saud Mnstry of Hgher Educaton and the Unversty of Petroleum and Mnerals 6
REFERENCES [1] F Bayen M Flato C Fronsdal and A Hadar Phys Rev D 3 (195) 673 [] S Adler Phys Rev D 6 (197) 3445; D (1973) 400 [3] A D Hadar J Math Phys 7 (196) 409 [4] P A M Drac Ann Math 37 (1936) 49; H A Kastrup Phys Rev 150 (1966) 113; G Mack and A Salam Ann Phys (NY) 53 (1969) 174 [5] W Hedenrech Nuovo Cmento A 0 (194) 0 [6] S Ichnose Lett Math Phys 11 (196) 113 [7] B Bnegar C Fronsdal and W Hedenrech J Math Phys 4 (193) [] R P Zakov Report No JINR E-3- Dubna 193 (unpublshed); P Furlan V Petkova G Sotkov and I Todorov Revsta Nuovo Cmento (195) 1 [9] To be publshed 7