Quantum Electrodynamics with arbitrary charge on noncommutative space Wan-Ping Zhou1,2,*, Shao-hong Cai2,3, Zheng-Wen Long1,2. 1Department of Physics, GuiZhou University,GuiYang 550025,PR China 2Laboratory for Photoelectric Technology and Application, GuiZhou University,GuiYang 550025,PR China 3College of Finance and Economics GuiZhou University,GuiYang 550025,PR China Using the Seiberg-Witten Map. We obtain a quantum electrodynamics on nonocommutative space, which has arbitrary charge and the gauge invariant is also kept at the leading order of theta. The one-loop divergence and Compton Scattering are reinvestigated. The noncommutative effect is much more remarkable then the ordinary nonocommutative quantum electrodynamics PACS numbers: 11.10.Nx, 12.20.-m Keywords: noncommutative,Quantum Electrodynamics,charge *Electronic address: zzzwp917sohu.com Electronic address: aa.shcaigzu.edu.cn Electronic address: sci.zwlonggzu.edu.cn http:/www.paper.edu.cn 1.INTRODUCTION The idea that the coordinates are noncommutative(NC) has a long history 1. It has been thought as a way to eliminate the infinite in quantum field by providing natural cut off, but the success of renormalization makes the idea obscure. In the past few years, noncommutativity has attracted much attention 2. It is widely accepted that open strings end is attach on the D-brane2,3, where the background Neveu-Schwarz B field exist. It makes the strings coordinates noncommutative2,4-6, and gives an extra factor of phase to the scattering amplitude2. The open string theory indicates that we must replace the ordinary product in effective actions by star product 0,)()()2exp()()(=+ =xgxfixgxfjiij, (1) Where ij is the fuction of B field. Using the star product, we can obtain the commutator of coordinates ixxxxxx=*,*. (2) Replacing the ordinary product in the gauge field actions by star product, we would get noncommutative gauge theory. The simplest noncommutative gauge theory is the NC U (1) theory 7-15, It has two important propertys14-15. One is the theory isnt Lorentz invariance, the other is the new vertex such as three and four photon self-interaction are introduced. It is similar to the non-abelian theory and renormalizable at single loop level. Unfortunately, there are some problems in NC gauge theory. First the ordinary SU (N) theory, which is the foundation of the standard model cant expend to NC SU (N) theory because SU (N) groups NC counterpart destroy the closure condition 16-17. The only group which admits a simple noncommutative extension is U (N). In order to get NC SU (N) theory, the authors of the Ref 18 use the Seiberg-Witten map to the get low energy effective theory. Second the time components ofbring the unitarity problem. Another problem is the no-go theorem 16: the matter fields can transform nontrivially under at most two NC group factors. In other words, the matter fields cannot carry more than two NC gauge group charge. Additionally only NC U (1) charge 1,0, -1 is allowed, because the gauge transform is charge dependent , +=aiga. (3) All of that are conflicts with the observation. In the standard model the matter fields can couple with three gauge fields, and the U (1) charge is more then three. We must introduce additional fields for the new charge in the theory, and there are too many degrees of freedom. It cant coincide with ordinary QED in the commutative limite 0.The later problem has been partly dealt with in the Ref 19. Calmet require the U (1) field depends explicitly on charge Q, and found all of the NC U (1) field could be expressed as function of the ordinary U (1) field via the Seiberg-Witten map ,4,4 afgaagaa+=(4) ,4 ag+=(5) . They got an effective theory of standard model. In the effective theory all fields are ordinary fields. But there is a flaw. The original NC U (1) action isnt gauge invariant, the reason is the same as (3)- The new covariant http:/www.paper.edu.cn 2derivative cant transform covariantly. This paper is aim to constuct a NC U (1) theory with arbitrary charge in noncommutative space. To avoid problem with unitarity we assume that only the space-space components of are nonzero, namely00=. It is organized as follows. In section 2, we construct the generalize form of NC U (1) theory. In section 3, we discuss the one-loop divergence of minimal deviations from the ordinary quantum electrodynamics, the function. In section 4, we calculate noncommutative correct of the Compton Scattering amplitude. Section 5 is the conclusion. II.NONCOMMUTATIVE QED In order to make different NC U (1) field degenerate to the same ordinary U (1) field in the commutative limite, we should ask all the NC U (1) field are the functions of the classical gauge field a, as Calmet did as in the Ref