The Anomaly And Soliton Solutions In Noncommutative Field Theory

The idea that the space-time coordinates do not commute is quite old. but Noncommutative geometry can not attract the physist's interest for a long time. In the past several years, noncommutative geometry has played more and more roles with the development of string theory. Indeed, noncommutative geometry has arisen in at least three distinct but closely related contexts in string theory. Witten's open string field theory formulate the interaction of bosonic open strings in the language of noncommutative geometry. Compactification of matrix theory on the noncommutative torus was argued to correspond to supergravity with constant background three form tensor field. More generally, it has been realised that noncommutative gauge theory arises in the worldvolume theory on D-brane in the presence of a constant background B field in string theory.Noncommutative space means the space that their coordinate do not commu-tute. Noncommutative field theory is a quantum field theory on noncommutative space and means that fields are thoughts of as function over noncommutative space. Noncommutative gauge theory are discribed via two equivalent descriptions. In the first way, the coordinate become operaotrs acting on a Hilbert space, which provides for a respresentation space of the noncommutative space, on the other hand, starting with the standard commutative gauge theory action and then reinterpreting any products of fields appearing in terms of the moyal product. The weyl-Moyal transformation takes operator multiplication into the Moyal product of functions on the phase space.One would like to know to which extent properties of quantum field theories on commutative spaces also arise in quantum field theories on noncommutative space.This worth us further studying, we should to do so in noncommutative field theory. In this paper, we mainly take two dimensional non-commutative chiral QCD2 model as an exameple to discuss the chiral anomaly in noncommutative field theory. Chiral anomaly in noncommutative field theory renewed the physist's interest in the past years. We mainly study the chiral anomaly of two dimensional non-commutative chiral QCD2 model using Fujikawa's path integral approach. We commute the Jacobian phase of the integral measure under the chiral rotation, we also discuss the fermion determent of the non-commutative chiral QCD2 model and provide the effective action of the non-commutative chiral QCD2 model. It is shown that vector boson have a mass generation and the effective Lagrangian contains a term corresponding to the a Wess-Zumino-Witten like term, which was consistant with noncommutative chiral Schwinger model.Although Derrick's theorem forbids solitons in more than 1 + 1 dimensions, Gopakumar, Minwalla and Strominger found that soliton solution in (2+1) noncommutative scalar field theory can be exactly given in terms of projection operators. Martinec and Moore discussed how D-branes on orbifolds fit naturally into the algebraic framework as described by projection operators. Thus the study of projection operators in various noncommutative spaces are important in string theory. Rieffel has presented a general formula for the projection operators on noncommutative torus. Boca further described the projection operators on orbifold T2/G and discussed the relation between their trace and the commutator q of the operators U and V on noncommutative torus(UV = Vt/e2). He proved the existence of nontrivial projection operators and explicitly presented an example with trace 1/q for q an integer in Z4 case. Konechny, Schwartz and Walters have also given some Z2, Z4 invariant projection operators. Martinec and Moore pointed out that no explicit expressions for the projection operators in Z3, Z6 case have been found. Gopakumar et al succeeded in constructing the projection operator on noncommutative integral torus with generic r. We find that if the vacuum state |0 > in their paper is replaced by any state vector their construction still works. We notice that if the statevector has some s...