A Remark on Gauge Invariance in Wavelet-Based Quantum Field Theory

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1 New Avnces in Physics Vol. 5 No. Jnury-June 0 pp. -8 A Remrk on Gue Invrince in Wvelet-Bse Quntum Fiel Theory S. Albeverio & M. V. Altisky b c Wvelet trnsform hs been ttrctin ttention s tool for reulriztion of ue theories since the first pper of Feerbush [] where the interl representtion of the fiels by mens of the wvelet trnsform ws sueste: x b b C + A(x) = A(b) with A (b) bein unerstoo s the fiels A mesure t point b with resolution +. In present pper we consier wvelet-bse theory of ue fiels provie counterprt of the ue trnsform for the scle-epenent fiels: A (x) A (x) + f (x) n erive the Wr-Tkhshi ientities for them. PACS:.5.-q Keywors: Gue invrince Wvelets Nonlocl fiel theory. Troubles with ultrviolet iverences tken toether with the fct tht strict loclisbility of quntum events is just n pproximtion tht cnnot be reche experimentlly stimulte the eorts to construct self-consistent nonlocl fiel theory t the possible lck of strict microcuslity [ 4 ]. This is specilly importnt for ue fiel theories incluin quntum electroynmics n quntum chromoynmics. In locl belin ue fiel theory the locl phse trnsformtion of the fermionic mtter fiels (x) e ief (x) (x) (x) e ief (x) (x) (). Interisziplinäre Zentrum für Komplexe Systeme University of Bonn Bonn D-55 Germny E-mil: serio.lbeverio@yhoo.com b. Joint Institute for Nucler Reserch Joliot-Curie 6 Dubn 4980 Russi E-mil: ltisky@mx.iki.rssi.ru c. Spce Reserch Institute RAS Profsoyuzny 84/ Moscow 7997 Russi.

2 S. Albeverio & M. V. Altisky (where f is rel-vlue ue function of the spce-time vrible x ( x ) =... e is the chre of the mtter fiels ) is ccompnie by the substitution of spce-time erivtives by covrint erivtives D + iea x which mkes the theory invrint with respect to the locl phse trnsformtion () if the ue fiel A trnsforms ccorinly: A A + f. () The heuristic enertin functionl of such theory is invrint uner trnsformtions () () if the source terms n the ue fixin terms re invrint heuristiclly it is thus iven by Z [J η η] = A exp ( i Leff x) exp ( i Leff x) () L eff = ν i γ D m Fν F () A + J A + η + η (4) 4 α where L eff is the effective Lrnin is normlistion constnt α is ue fixin prmeter γ re the mm mtrices J η η re test-functions for A respectively F is the curvture ssocite with A. The invrince of the enertin functionl () uner the trnsformtions () () is ensure by so-clle Wr-Tkhshi ientities [5 6]. The im of present pper is to formulte theory of the ue fiels A (x) tht epen on both the position x n the resolution. As in previous ppers [8 7 ] this is one by substitutin the fiels in the eective Lrnin (4) in terms of their continuous wvelet trnsforms: x b = b A ()() x 0 A b x b > (5) + C where C is positive normlistion constnt of the bsic wvelet stisfyin the missibility conition C = () k < (6) 0 n where mens the continuous Fourier trnsform of ; see e.. [9 0] for reviews on the continuous wvelet trnsform.

3 A Remrk on Gue Invrince in Wvelet-Bse Quntum Fiel Theory The substitution (5) mkes the effective Lrnin (4) into n effective Lrnin for nonlocl fiel theory. Tht is why the locl ue invrince principle () shoul be reconsiere for such theory. Usin the ies from nonlocl ue fiel theory [] we ssume tht the locl phse invrince of the mtter fiels shoul be preserve uner the substitution (5) n the ue trnsformtions of the scle-epenent ue fiels x b A ()() b = A x x (where is the complex conjution of the bsic wvelet ) shoul be chosen ccorinly to keep tht invrince. This implies the trnsformtion conitions: (x) () x e ( ) ie x b f () b b C + f() x A (x) A + x (7) where f (b) = x b f () x x is the continuous wvelet trnsform of the ue function f of the oriinl locl theory (). In the infinitesiml form ( i.e. up to orer two in the power expnsion of the exponent) this les to the trnsformtion lw e x b b ()() x x () ı f b C. Becuse of the linerity of the wvelet trnsform the eqution (7) urntees the ue trnsform () for orinry locl ue fiels. Let us now specify the ue theory n the Wr-Tkhshi ientities [5] for the theory of scle-epenent fiels A. The effective Lrnin itself is ue invrint by construction n only the source term cquires multipliction by the fctor exp i ()() A f + J f ief η η x α

4 4 S. Albeverio & M. V. Altisky which cn pproximtely (up to the secon orer term in heuristic expnsion in power series) be represente by first orer term for smll f tht is + iδ with δ x ()() A J ie η η α f () x. Let us replce the fiels in the eqution bove by their interl representtions in terms of wvelet trnsform (5). Intertin by prts we put the Lplcin prmeter f. Heuristiclly: x onto the ue fixin δ = x b x α A () b C x b ( )()( b ) f b x b [()]()( J ) b f b b C ie [()()()()] η η b b b b C x b x b f () b () x b x ( )( b )( ) b b (8) = x b where is the rient of the bsic wvelet function ( ) b = cn be looke upon s the mesure on ffine roup [ ] written in L norm [4] n the curly brckets... enote the functionl verin men vlue obtine by Feynmn functionl intertion (). Introucin the mtrix elements of opertors between wvelet bsic functions

5 A Remrk on Gue Invrince in Wvelet-Bse Quntum Fiel Theory 5 T ( ) x b x b x () T () ( ) x b x b x () M ( ) x b x b x b x () we heuristiclly erive the Wr-Tkhshi ientities for the scle-epenent fiels A. In terms of the bove efine opertors T T () M the vrition term (8) cn be written in the form δ = T ( )()()( b A ) b f b b α C ( )()( b ) J b b C ıe [()()()()] η η b b b b C f ()( b M )( )( )( b) b b. To obtin the heuristic vrition of the enertin functionl the fiels shoul be substitute by corresponin functionl erivtives: δ δ δ A ι δη ι δη ι δj with ll vritions tken with respect to the mesure on the ne roup ( b). Assumin tht the full vrition of the enertin functionl with respect to ue trnsformtions is zero this ives the functionl eqution

6 6 S. Albeverio & M. V. Altisky i δ T ( )() J α C δ J () C δ η() δ η e () M ( )( )( b) b C δ η() δ η () Z [ η η J] = 0. To erive thewr-tkhshi equtions for connecte Green functions we heuristiclly substitute Z = exp (iw). This ives the followin eqution in functionl erivtives i δw α ( )( b )() T J δ () C J C ie δw δw η()() η δη δη C ()() M ( )( )( b) 0. b = To et the equtions for the vertex functions Γ[ A ] we pply the functionl Leenre trnsform Γ [ A ] = W [η η J] η + η + JA to the ltter equtions. Doin so we rrive heuristiclly to the followin eqution in functionl erivtives for the vertex function δγ + α C C A () A ()( T ) δ ie δγ δγ ()() M ( ) = 0. C δ ()() δ where the short-hn nottion A () A (b ) () (b ) is use n the intertion over ll repete inices is ssume. The Wr-Tkhshi equtions re heuristiclly erive by tkin the secon erivtives of the eqution (9) t zero fiels (A = = = 0). This ives

7 A Remrk on Gue Invrince in Wvelet-Bse Quntum Fiel Theory 7 δ Γ[0] ie δ δ )()() δ δ δ x )() δ C x y A C δγ δγ ()() M ( ) = 0. δ ()() δ y Performin the heuristic functionl ierentition n usin the symmetry uner the permuttion fter the intertion we hve the Wr-Tkhshi ientities δ Γ[0] δ Γ[0] δ Γ[0] = ie ie M( ) = 0. C δ x )()() δ y δa x δ )())() y δ x δ (0) where x x y re rbitrry position-resolution ruments x ( x b x ) +. Followin e.. [5] we efine vertex functions n inverse proptors in the Fourier spce δ Γ[0] b x b x b y exp (i (p b x pb y qb x )) δ x )()() δ y δa x = ie (π) δ (p p q) Γ x y x (p q p ) () b x b y exp ( (p b x pb y )) δ Γ[0] δ x )() δ y x y = ()()() π δ p p S p. () Usin the efinitions () we multiply eqution (0) by e (p b x pb y qb x ) n interte over b x b x b y. This ives q Γ ( p q) p + q 4 = S p M p + q q p M p + q q p S p 4 ( )( )( )() 4 () where M ( k k )()()()()() k = π δ k k k k k k is the Fourier ime of the vertex opertor M.

8 8 S. Albeverio & M. V. Altisky The eqution () is non-locl nlo of the orinry (locl)wr-tkhshi eqution in Fourier spce. Acknowleement The pper ws supporte by DFG Project 46 RUS /95. References [] P. Feerbush A New Formultion n Reulriztion of Gue Theories Usin Non-Liner Wvelet Expnsion Pror. Theor. Phys (995). [] M. Altisky Quntum Fiel Theory without Diverences Physicl Review D 8() 500 (00). [] V. A. Alebstrov n G. V. Efimov Cuslity in Quntum Fiel Theory with Nonlocl Interction Communictions in Mthemticl Physics 8() -8 (974). [4] G. Bttle Wvelets n Renormliztion Group Worl Scientific (999). [5] J. Wr An Ientity in Quntum Electroynmics Phys. Rev (950). [6] Y. Tkhshi On the Generlize Wr Ientity Nuovo Cimento 6() 7-75 (957). [7] M. Altisky Wvelet-Bse Quntum Fiel Theory Symmetry Interbility n Geometry: Methos n Applictions 05 (007). [8] M. V. Altisky Scle-Depenent Functions Stochstic Quntiz tion n Renormliztion Symmetry Interbility n Geometry: Methos n Applictions 046 (006). [9] I. Dubechies Ten Lectures on Wvelets S.I.A.M. Philelphie (99). [0] C. Chui An Introuction to Wvelets Acemic Press Inc. (99). [] G. Efimov Problems in Quntum Theory of Nonlocl Interctions Nuk Moscow In Russin (985). [] A. L. Crey Squre-Interble Representtions of Non-Unimoulr Groups Bull. Austr. Mth. Soc. 5 - (976). [] M. Duflo n C. C. Moore On Reulr Representtions of Nonuni-moulr Loclly Compct Group J. Func. Anl (976). [4] C. M. Hny n R. Murenzi Moment-Wvelet Quntiztion: A First Principles Anlysis of Quntum Mechnics Throuh Continuous Wvelet Trnsform Theory Phys. Lett. A (998). [5] L. Ryer Quntum Fiel Theory Cmbrie University Press (985).