Fujikawa method

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 $mathfrak\{g\}$. The Dirac operator (in Feynman slash notation) is :$D!!!!/\; stackrel\{mathrm\{def\{=\}\; partial!!!/\; +\; i\; A!!!/$and the fermionic action is given by:$int\; d^dx\; overline\{psi\}iD!!!!/\; psi$The partition function is :$Z\; [A]\; =int\; mathcal\{D\}overline\{psi\}mathcal\{D\}psi\; e^\{-int\; d^dx\; overline\{psi\}iD!!!!/psi\}.$

The axial symmetry transformation goes as:$psi\; o\; e^\{igamma\_\{d+1\}alpha(x)\}psi,$:$overline\{psi\}\; o\; overline\{psi\}e^\{igamma\_\{d+1\}alpha(x)\}$:$S\; o\; S\; +\; int\; d^dx\; alpha(x)partial\_muleft(overline\{psi\}gamma^mugamma^5psi\; ight)$Classically, this implies that the chiral current, $j\_\{d+1\}^mu\; equiv\; overline\{psi\}gamma^mugamma^5psi$ is conserved, $0\; =\; partial\_mu\; j\_\{d+1\}^mu$.

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::$psi\; =\; sumlimits\_\{i\}psi\_ia^i,$:$overlinepsi\; =\; sumlimits\_\{i\}psi\_ib^i,$where $\{a^i,b^i\}$ are Grassmann valued coefficients, and $\{psi\_i\}$ are eigenvectors of the Dirac operator::$D!!!!/\; psi\_i\; =\; -lambda\_ipsi\_i.$The eigenfunctions are taken to be orthonormal with respect to integration in d-dimensional space,:$delta\_i^j\; =\; intfrac\{d^dx\}\{(2pi)^d\}psi^\{dagger\; j\}(x)psi\_i(x).$The measure of the path integral is then defined to be::$mathcal\{D\}psimathcal\{D\}overline\{psi\}\; =\; prodlimits\_i\; da^idb^i$

Under an infinitesimal chiral transformation, write:$psi\; o\; psi^prime\; =\; (1+ialphagamma\_\{d+1\})psi\; =\; sumlimits\_i\; psi\_ia^\{prime\; i\},$:$overlinepsi\; o\; overline\{psi\}^prime\; =\; overline\{psi\}(1+ialphagamma\_\{d+1\})\; =\; sumlimits\_i\; psi\_ib^\{prime\; i\}.$The Jacobian of the transformation can now be calculated, using the orthonormality of the eigenvectors:$C^i\_j\; equiv\; left(frac\{delta\; a\}\{delta\; a^prime\}\; ight)^i\_j\; =\; int\; d^dx\; psi^\{dagger\; i\}(x)\; [1-ialpha(x)gamma\_\{d+1\}]\; psi\_j(x)\; =\; delta^i\_j\; -\; iint\; d^dx\; alpha(x)psi^\{dagger\; i\}(x)gamma\_\{d+1\}psi\_j(x).$The transformation of the coefficients $\{b\_i\}$ are calculated in the same manner. Finally, the quantum measure changes as:$mathcal\{D\}psimathcal\{D\}overline\{psi\}\; =\; prodlimits\_i\; da^i\; db^i\; =\; prodlimits\_i\; da^\{prime\; i\}db^\{prime\; i\}\{det\}^\{-2\}(C^i\_j),$where the Jacobian is the reciprocal of the determinant because the integration variables are Grassmanian, 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:D!!!!/}^2/M^2}psi_i(x)end{align}($\{D!!!!/\}^2$ can be re-written as $D^2+\; frac\{1\}\{4\}\; [gamma^mu,gamma^\; u]\; F\_\{mu\; u\}$, and the eigenfunctions can be expanded in a plane wave basis):$=\; 2ilimlimits\_\{M\; oinfty\}alphaint\; d^dxintfrac\{d^dk\}\{(2pi)^d\}intfrac\{d^dk^prime\}\{(2pi)^d\}\; psi^\{dagger\; i\}(k^prime)e^\{ik^prime\; x\}gamma\_\{d+1\}\; e^\{-k^2/M^2+1/(4M^2)\; [gamma^mu,gamma^\; u]\; F\_\{mu\; u$e^{-ikx}psi_i(k):$=\; -frac\{-2alpha\}\{(2pi)^\{d/2\}(frac\{d\}\{2\})!\}(\; frac\{1\}\{2\}F)^\{d/2\},$

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, $F\; equiv\; F\_\{mu\; u\}dx^muwedge\; dx^\; u$

This result is equivalent to $(\; frac\{d\}\{2\})^\{\; m\; th\}$ Chern class of the $mathfrak\{g\}$-bundle over the d-dimensional base space, and gives the chiral anomaly, responsible for the non-conservation of the chiral current.