Topological Singularity of Fermion Current in Abelian Gauge Theory

Since calculations in a quantized theory are often plagued by divergencies , we have to impose a regularization scheme in order to eliminate the singularity from ill-field functions such as the infrared-divergence problem in quantum electrodynamics[1,2]. Among these divergence phenomena, a type of divergence appears in product of quantum operators taken at the same time. The product of the local operators is often singular, which can destroy symmetries of conservation currents and classical equations of motions. For instance, in derivation of the conservation equation for the axial vector current, the differential equation of motion for fermion fields can not be used to reduce fermion-photon vertex in quantum electrodynamics in the most straightforward way[3]. Through standard perturbative treatments of singularity in the operator product of fermion current, a closer look at the manipulations reveals some subtleties. These unavoidable divergences reflect the feature that the symmetry of the classical theory may be ruined in quantum theory by quantum anomaly. Usually, we deal with the divergence of the anomaly by examining the “triangle graph”, which consist of an internal fermion loop connected to two vector fields and to one axial vector field. These triangle graphs give rise to anomaly in orders of perturbation theory, which arise from momentum-space integrals. In consequence, the anomaly is affected by interactions of gauge bosons attached to the fermion loop[4,5].


Introduction
Since calculations in a quantized theory are often plagued by divergencies , we have to impose a regularization scheme in order to eliminate the singularity from ill-field functions such as the infrared-divergence problem in quantum electrodynamics [1,2]. Among these divergence phenomena, a type of divergence appears in product of quantum operators taken at the same time. The product of the local operators is often singular, which can destroy symmetries of conservation currents and classical equations of motions. For instance, in derivation of the conservation equation for the axial vector current, the differential equation of motion for fermion fields can not be used to reduce fermion-photon vertex in quantum electrodynamics in the most straightforward way [3]. Through standard perturbative treatments of singularity in the operator product of fermion current, a closer look at the manipulations reveals some subtleties. These unavoidable divergences reflect the feature that the symmetry of the classical theory may be ruined in quantum theory by quantum anomaly. Usually, we deal with the divergence of the anomaly by examining the "triangle graph", which consist of an internal fermion loop connected to two vector fields and to one axial vector field. These triangle graphs give rise to anomaly in orders of perturbation theory, which arise from momentum-space integrals. In consequence, the anomaly is affected by interactions of gauge bosons attached to the fermion loop [4,5].
From the path-integral viewpoint, the anomaly term associated with the axial-vector current can be understood in the path-integral formulation of quantum gauge theory as a consequence of the fact that the functional measure is not invariant with respect to the relevant local group transformations on the fields [6]. The chiral anomaly responds to local group transformations but vanishes for a class of space-independent transformations. One discovers that the anomaly has a topological character given by the index of a Dirac operator in a gauge background [1]. The mathematical explanation of such a anomaly is directly related to Atiyah-Singer index theorem. The theorem relates the number of chirality zero modes of Dirac operator to the topological charge of the gauge field. Furthermore, a number of papers have addressed the relationships among chiral anomalies,the geometry of the space of vector potentials and families of Dirac operators [7,8,9]. The authors showed that the first characteristic class of the index bundle for the Dirac operator is related to anomalies. In particular, in the gravitation case,the

Transformation property of fermion current
Now we are in position to specify a transformation of operator product of fermion current in Abelian quantum field to determine the property of the current. According to the form of fermion current coupling with gauge field ( ) Bx  in Dirac equation, the fermion current coupling with gauge field is defined as where    indicates the corresponding Dirac matrix, which is also an element of Dirac algebra,  x  and   x  is Dirac spinors.
One easily verifies that these fermion currents possess bilinear covariant property under Lorentz transformation. Some of the "products of currents" itself have quantum field theoretical meaning, which is subject to infinitesimal change of varibles along the symmetry direction. To do this, in the broad sense, a set of the changing variables of matter field in quantum theory can be transformed as [3]       where the matrices     , Hh    are the functions composed of both Dirac matrix and field variables, the group parameter    x   is a real function.
In the light of Eq,(2), the infinitesimal local transformation of fermion fields in Abelian gauge theory is taken tentatively as [12]    remains the same. This is because changing variables in an integral never affects its value. In this sense, we can consider a gauge transformation to be more general type of field redefinition.This is the transfornation Eq.(2) as a guiding rule.
Thus, to discuss the property of the fermion current, we need to choose a comparatively simple transformation on the fields for fermion currents. This is transformation Eq.(3), in which the term coupling with the spacetime derivative of    In order to understand the meaning of the operator product composed of ' n c and ' n c in the , we now recall the algebraic property of functional measure in the path integral formulation. Under the transformation Eq. (3), the corresponding functional measure ,  J  is the Jacobian determinant of the corresponding transformation of the fermion variables, which is a key quantity for anomaly current. Then we see that the Jacobian of differential operator d connects with the transformation of the operator function  , which is defined as the operator product of the anticommuting coefficient , After the transformation Eq.
where  J   is the Jacobian determinant of the transformations Eq.(3).
From the properties of the Grassmann algebra we find the identity It shows that the Jacobian  J   corresponding to the operator product  connects with the inverse of the Jacobian of the corresponding transformation of the integral measure.
Having clarified the relation between the operator product of fermion current and the integral measure, we therefore see that the power function of fermion current Eq. (8) This is the desired result, which is just the transformation identity of fermion current. The expression illustrates that the property of fermion current is presented in its functional Jacobian through the operator product function  . In other words, the mathematical property of fermion current can be characterized by the nature of Jacobian of functional integral measure due to transformation of fermion field. For instance, the process of regularization of Jacobian in the functional measure gives rise to the anomaly for the axial vector current. Its topological property of the anomaly term is described by Atiyah-Singer index theorem for Dirac operator.

Topological property of fermion current
From the topological viewpoint, the non-perturbative effect of the Abelian anomaly associated with Ward-Takahashi type's identity for axial vector current is related to the topological character in the presence of topologically non-trivial field configuration [14,15]. The topological exposition of the quantum anomaly for the current is addressed by Atiyah-Singer index theorem in a gauge background .The Abelian anomaly term can be derived by using path integral formulation in Euclidean spacetime E 4 . Of particular interest is applications of the index theorem for Dirac operator, because it can be used to discuss the topological property of fermion current.  (16) with "trace" here denoting a trace only over indices labeling the various fermion species, which could consider as a regularization of the trace in function space. F  is the field strength, and   5 Ax is the anomaly function.
This formula shows that the index of D is given in terms of the restriction to the diagonal of the kernel of 2 D at arbitrarily large masses, which we know by the asymptotic expansion of the kernel to be given by local formula in the curvature of the connection. In the above equation, the following identity is used The fact reveals the important information that the integral of the local anomaly function   5 Ax can not change smoothly under variations in the gauge field. Since the anomaly function in Eq. (16) can be written as a total derivative, the space integral of the anomaly function depends only the behavior of the gauge field at the boundaries. That is, there exists a number of singularities of the E vector bundle on the manifold.
Since D is an elliptic operator, the Atiyah-Singer local index theorem gives a formula for the topological index t Ind D of D in terms of the Chern character of ChE and the Â -genus of 4 E [18,19] at Ind D Ind D  So that [11]     Obviously, the quantity on the right of the Eq. (22) is kown as Chern-Pontrjagin term.
Since the anomaly function    5 Ax does not vanish in the Jacobian determinant J , we have the explicit identity The gauge group parameter    x   is independent of the topological index.
The Abelian anomaly arisen from axial vector current is a local quantity, because it is a consequence of short distance singularities, which does violate the chiral symmetry. That is, the topological singularity of operator product of fermion current Eq.(13) is presented by Jacobian of the measure due to the transformation of fermion fields.  The computation shows that anomaly functions for many fermion currents vanish. In other words, there is no topological singularity for these currents.

An Example of Abelian gauge theory
As one see from the following quantum electrodynamics case, the above argument provides a approach to examine the topological singularity in the operator products of the fermion current. Let us consider an Abelian gauge field to show our argument. The Lagrangian density based on the minimal coupling ansatz for quantum electrodynamics is the following [4]   where g and m denote, respectively, the charge and mass of the electron. In this case, the gauge field is just the photon field   This means that the fact holds irrespecttive whether these re-naming field variables take the form of symmetry of action, or not. Therefore, this implicitly expect that a anomaly might take place at the quantum measure, which is the failure of a class symmetry to survive the process of quantization and regularization. Normally, under gauge transformation, the determinant in the measure transformation is descarded because it appears to be a constant. However, closer analysis of this term shows that it is actually divergent and here requires regularization.
Following Fujikawa's prescription [6,20], the regularization procedure for the variation of the integral measure can provide access to a wider class of such anomaly objects. In order to analyze the topology property of various fermion current, the anomaly functions   iv. v. vi.
The above computation shows that the anomaly function in Jacobian vanishes for many fermion currents,(in other words, they cancel each other in regularization).
For simplicity, we take a two-dimensional eigenspace (2 ) d  of the regulation operator

Conclusion
We have presented that the topological singularity in operator product of various fermion currents coupling to a gauge field is characterized by the topological properties of anomaly function in a quantum gauge background in terms of Atiyah-Singer index theorem for Dirac operator. The anomaly functions corresponding to various fermion currents have been evaluated through the calculus of the kernel of Dirac operator.
As the above illustration, the topological singularity of various fermion currents coupling gauge field is indeed understand on Atiyah-Singer index theorem in quantum field theory as a consequence of the fact that the Jacobian of integration measure possesses anomaly terms. That is, the kernel of the Dirac operator may have short distance singularities.
No doubt, the singularity of the fermion current has to be considered when dealing with reduction of the interaction vertex by using the Dirac differential equation of motion in the Dyson-schwinger equation. Also in the non-abelian gauge case, the relation of anommalies in conservation o general axial currents to the indes of the Dirac operator in a gauge background need to discussed further. In addition, the property of quantum anomaly associated with Ward-Takahashi relation plays an important role in the nonperturbative study of gauge theories, such as the dynamical chiral symmetry breaking. In terms of the trace over Dirac indices, when we expand the regularization operator   22 (/ fDM  in D , the terms with less than four gamma matrices evaluate to zero. In the light of the limit over M (mass), terms with more than four D s will also drop out.

Appendix
In the last step, we have used the operator identities