Fujikawa method
In physics, Fujikawa's method is a way of deriving the chiral anomaly in quantum field theory. It uses the correspondence between functional determinants and the partition function, effectively making use of the Atiyah–Singer index theorem. Derivation. Suppose given a Dirac field formula_1 which transforms according to a representation formula_2 of the compact Lie group "G"; and we have a background connection form of taking values in the Lie algebra formula_3 The Dirac operator (in Feynman slash notation) is and the fermionic action is given by The partition function is The axial symmetry transformation goes as Classically , this implies that the chiral current, formula_10 is conserved, formula_11. Quantum mechanically, the chiral current is not conserved: Jackiw discovered this due to the non-vanishing of a triangle diagram. Fujikawa reinterpreted this as a change in the partition function measure under a chiral transformation. To calculate a change in the measure under a chiral transformation, first consider the Dirac fermions in a basis of eigenvectors of the Dirac operator: where formula_14 are Grassmann valued coefficients, and formula_15 are eigenvectors of the Dirac operator: The eigenfunctions are taken to be orthonormal with respect to integration in d-dimensional space, The  measure of the path integral is then defined to be: Under an infinitesimal chiral transformation, write The Jacobian of the transformation can now be calculated, using the orthonormality of the eigenvectors The transformation of the coefficients formula_22 are calculated in the same manner. Finally, the quantum measure changes as where the Jacobian is the reciprocal of the determinant because the integration variables are Grassmannian, and the 2 appears because the a's and b's contribute equally. We can calculate the determinant by standard techniques: to first order in α(x). Specialising to the case where α is a constant, the  Jacobian must be regularised because the integral is ill-defined as written. Fujikawa employed heat-kernel regularization, such that after applying the completeness relation for the eigenvectors, performing the trace over γ-matrices, and taking the limit in M. The result is expressed in terms of the field strength 2-form, formula_27 This result is equivalent to formula_28 Chern class of the formula_29-bundle over the d-dimensional base space, and gives the chiral anomaly, responsible for the non-conservation of the chiral current.