Fujikawa's method is a way of deriving the chiral anomaly in quantum field theory.
Suppose given a Dirac field ψ which transforms according to a ρ representation of the compact Lie group G; and we have a background connection form of taking values in the Lie algebra
and the fermionic action is given by
The partition function is
The axial symmetry transformation goes as
Classically, this implies that the chiral current,
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
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
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,
This result is equivalent to