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
The Dirac operator (in Feynman slash notation) is
![{\displaystyle D\!\!\!\!/\ {\stackrel {\mathrm {def} }{=}}\ \partial \!\!\!/+iA\!\!\!/}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b265048a4358e7cb71813cefabc01e17f4137cce)
and the fermionic action is given by
![{\displaystyle \int d^{d}x\,{\overline {\psi }}iD\!\!\!\!/\psi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf96b41952b8b7b8fa3eb4c1443fc30e1c437c6b)
The partition function is
![{\displaystyle Z[A]=\int {\mathcal {D}}{\overline {\psi }}{\mathcal {D}}\psi e^{-\int d^{d}x{\overline {\psi }}iD\!\!\!\!/\psi }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85e0111a1411161eb7ba69e0e6ae09c1043cfdfa)
The axial symmetry transformation goes as
![{\displaystyle \psi \to e^{i\gamma _{d+1}\alpha (x)}\psi \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e010affb9fa1f49ea2e5ceee364a53f268d5e20d)
![{\displaystyle {\overline {\psi }}\to {\overline {\psi }}e^{i\gamma _{d+1}\alpha (x)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b0a0c0eaac9f0e9ca4b69afb317b577ba7c881e)
![{\displaystyle S\to S+\int d^{d}x\,\alpha (x)\partial _{\mu }\left({\overline {\psi }}\gamma ^{\mu }\gamma _{d+1}\psi \right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/086567a3f25a5d0a3007762c57561a066d93c596)
Classically, this implies that the chiral current,
is conserved,
.
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:
![{\displaystyle \psi =\sum \limits _{i}\psi _{i}a^{i},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af9a3fd36522aebf2afbb8e7b1081e615be3107f)
![{\displaystyle {\overline {\psi }}=\sum \limits _{i}\psi _{i}b^{i},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8db019bc6da2fd1d410d30af67a678e644149822)
where
are Grassmann valued coefficients, and
are eigenvectors of the Dirac operator:
![{\displaystyle D\!\!\!\!/\psi _{i}=-\lambda _{i}\psi _{i}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96eea8acf0e24f9cd47097306fb4dece55d9b660)
The eigenfunctions are taken to be orthonormal with respect to integration in d-dimensional space,
![{\displaystyle \delta _{i}^{j}=\int {\frac {d^{d}x}{(2\pi )^{d}}}\psi ^{\dagger j}(x)\psi _{i}(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbf0f9417cd6d929891f70e52182add4fc9feaa9)
The measure of the path integral is then defined to be:
![{\displaystyle {\mathcal {D}}\psi {\mathcal {D}}{\overline {\psi }}=\prod \limits _{i}da^{i}db^{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2df6d282f6594bca41bf644844c3e9485cad277)
Under an infinitesimal chiral transformation, write
![{\displaystyle \psi \to \psi ^{\prime }=(1+i\alpha \gamma _{d+1})\psi =\sum \limits _{i}\psi _{i}a^{\prime i},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfd3960b08224075cb2392f5b71d56cc89caa0c)
![{\displaystyle {\overline {\psi }}\to {\overline {\psi }}^{\prime }={\overline {\psi }}(1+i\alpha \gamma _{d+1})=\sum \limits _{i}\psi _{i}b^{\prime i}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a72b0aa44a5671db768011c6494dfd316df6ee1f)
The Jacobian of the transformation can now be calculated, using the orthonormality of the eigenvectors
![{\displaystyle C_{j}^{i}\equiv \left({\frac {\delta a}{\delta a^{\prime }}}\right)_{j}^{i}=\int d^{d}x\,\psi ^{\dagger i}(x)[1-i\alpha (x)\gamma _{d+1}]\psi _{j}(x)=\delta _{j}^{i}\,-i\int d^{d}x\,\alpha (x)\psi ^{\dagger i}(x)\gamma _{d+1}\psi _{j}(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0ff4a9b91e03ec9e3a18e5f57eaadd03b10b1ec)
The transformation of the coefficients
are calculated in the same manner. Finally, the quantum measure changes as
![{\displaystyle {\mathcal {D}}\psi {\mathcal {D}}{\overline {\psi }}=\prod \limits _{i}da^{i}db^{i}=\prod \limits _{i}da^{\prime i}db^{\prime i}{\det }^{-2}(C_{j}^{i}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6146eb49f3aa902b67a60db4a458962ae21de4fa)
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:
![{\displaystyle {\begin{aligned}{\det }^{-2}(C_{j}^{i})&=\exp \left[-2{\rm {tr}}\ln(\delta _{j}^{i}-i\int d^{d}x\,\alpha (x)\psi ^{\dagger i}(x)\gamma _{d+1}\psi _{j}(x))\right]\\&=\exp \left[2i\int d^{d}x\,\alpha (x)\psi ^{\dagger i}(x)\gamma _{d+1}\psi _{i}(x)\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26a547accf4438f355faec9182c227ad3d642ad3)
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
![{\displaystyle {\begin{aligned}-2{\rm {tr}}\ln C_{j}^{i}&=2i\lim \limits _{M\to \infty }\alpha \int d^{d}x\,\psi ^{\dagger i}(x)\gamma _{d+1}e^{-\lambda _{i}^{2}/M^{2}}\psi _{i}(x)\\&=2i\lim \limits _{M\to \infty }\alpha \int d^{d}x\,\psi ^{\dagger i}(x)\gamma _{d+1}e^{{D\!\!\!\!/}^{2}/M^{2}}\psi _{i}(x)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2442cab92806facfaea1fc68548559d6becf95d)
(
can be re-written as
, and the eigenfunctions can be expanded in a plane-wave basis)
![{\displaystyle =2i\lim \limits _{M\to \infty }\alpha \int d^{d}x\int {\frac {d^{d}k}{(2\pi )^{d}}}\int {\frac {d^{d}k^{\prime }}{(2\pi )^{d}}}\psi ^{\dagger i}(k^{\prime })e^{ik^{\prime }x}\gamma _{d+1}e^{-k^{2}/M^{2}+1/(4M^{2})[\gamma ^{\mu },\gamma ^{\nu }]F_{\mu \nu }}e^{-ikx}\psi _{i}(k)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7950abb404ba18c112133787793ba0fe6d8c870)
![{\displaystyle =-{\frac {-2\alpha }{(2\pi )^{d/2}({\frac {d}{2}})!}}({\tfrac {1}{2}}F)^{d/2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81307f20b3056e84d0d137c490657aec80d2b7c4)
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
Chern class of the
-bundle over the d-dimensional base space, and gives the chiral anomaly, responsible for the non-conservation of the chiral current.
References
- K. Fujikawa and H. Suzuki (May 2004). Path Integrals and Quantum Anomalies. Clarendon Press. ISBN 0-19-852913-9.
- S. Weinberg (2001). The Quantum Theory of Fields. Volume II: Modern Applications.. Cambridge University Press. ISBN 0-521-55002-5.