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 [math]\displaystyle{ \psi }[/math] which transforms according to a representation [math]\displaystyle{ \rho }[/math] of the compact Lie group G; and we have a background connection form of taking values in the Lie algebra [math]\displaystyle{ \mathfrak{g}\,. }[/math] The Dirac operator (in Feynman slash notation) is

[math]\displaystyle{ D\\!\\!\\!\\!/\ \stackrel{\mathrm{def}}{=}\ \partial\\!\\!\\!/ + i A\\!\\!\\!/ }[/math]

and the fermionic action is given by

[math]\displaystyle{ \int d^dx\, \overline{\psi}iD\\!\\!\\!\\!/ \psi }[/math]

The partition function is

[math]\displaystyle{ Z[A]=\int \mathcal{D}\overline{\psi}\mathcal{D}\psi\,e^{-\int d^dx\,\overline{\psi}iD\\!\\!\\!/\,\psi}. }[/math]

The axial symmetry transformation goes as

[math]\displaystyle{ \psi\to e^{i\gamma_{d+1}\alpha(x)}\psi\, }[/math]
[math]\displaystyle{ \overline{\psi}\to \overline{\psi}e^{i\gamma_{d+1}\alpha(x)} }[/math]
[math]\displaystyle{ S\to S + \int d^dx \,\alpha(x)\partial_\mu\left(\overline{\psi}\gamma^\mu\gamma_{d+1}\psi\right) }[/math]

Classically, this implies that the chiral current, [math]\displaystyle{ j_{d+1}^\mu \equiv \overline{\psi}\gamma^\mu\gamma_{d+1}\psi }[/math] is conserved, [math]\displaystyle{ 0 = \partial_\mu j_{d+1}^\mu }[/math].

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:

[math]\displaystyle{ \psi = \sum\limits_{i}\psi_ia^i, }[/math]
[math]\displaystyle{ \overline\psi = \sum\limits_{i}\psi_ib^i, }[/math]

where [math]\displaystyle{ \\{a^i,b^i\\} }[/math] are Grassmann valued coefficients, and [math]\displaystyle{ \\{\psi_i\\} }[/math] are eigenvectors of the Dirac operator:

[math]\displaystyle{ D\\!\\!\\!\\!/ \psi_i = -\lambda_i\psi_i. }[/math]

The eigenfunctions are taken to be orthonormal with respect to integration in d-dimensional space,

[math]\displaystyle{ \delta_i^j = \int\frac{d^dx}{(2\pi)^d}\psi^{\dagger j}(x)\psi_i(x). }[/math]

The measure of the path integral is then defined to be:

[math]\displaystyle{ \mathcal{D}\psi\mathcal{D}\overline{\psi} = \prod\limits_i da^idb^i }[/math]

Under an infinitesimal chiral transformation, write

[math]\displaystyle{ \psi \to \psi^\prime = (1+i\alpha\gamma_{d+1})\psi = \sum\limits_i \psi_ia^{\prime i}, }[/math]
[math]\displaystyle{ \overline\psi \to \overline{\psi}^\prime = \overline{\psi}(1+i\alpha\gamma_{d+1}) = \sum\limits_i \psi_ib^{\prime i}. }[/math]

The Jacobian of the transformation can now be calculated, using the orthonormality of the eigenvectors

[math]\displaystyle{ C^i_j \equiv \left(\frac{\delta a}{\delta a^\prime}\right)^i_j = \int d^dx \,\psi^{\dagger i}(x)[1-i\alpha(x)\gamma_{d+1}]\psi_j(x) = \delta^i_j\, - i\int d^dx \,\alpha(x)\psi^{\dagger i}(x)\gamma_{d+1}\psi_j(x). }[/math]

The transformation of the coefficients [math]\displaystyle{ \\{b_i\\} }[/math] are calculated in the same manner. Finally, the quantum measure changes as

[math]\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^i_j), }[/math]

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:

[math]\displaystyle{ \begin{align}{\det}^{-2}(C^i_j) &= \exp\left[-2{\rm tr}\ln(\delta^i_j-i\int d^dx\, \alpha(x)\psi^{\dagger i}(x)\gamma_{d+1}\psi_j(x))\right]\\\ &= \exp\left[2i\int d^dx\, \alpha(x)\psi^{\dagger i}(x)\gamma_{d+1}\psi_i(x)\right]\end{align} }[/math]

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

[math]\displaystyle{ \begin{align}-2{\rm tr}\ln C^i_j &= 2i\lim\limits_{M\to\infty}\alpha\int d^dx \,\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^dx\, \psi^{\dagger i}(x)\gamma_{d+1} e^{{D\\!\\!\\!/\,}^2/M^2}\psi_i(x)\end{align} }[/math]

([math]\displaystyle{ {D\\!\\!\\!\\!/}^2 }[/math] can be re-written as [math]\displaystyle{ D^2+\tfrac{1}{4}[\gamma^\mu,\gamma^\nu]F_{\mu\nu} }[/math], and the eigenfunctions can be expanded in a plane-wave basis)

[math]\displaystyle{ = 2i\lim\limits_{M\to\infty}\alpha\int d^dx\int\frac{d^dk}{(2\pi)^d}\int\frac{d^dk^\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) }[/math]
[math]\displaystyle{ = -\frac{-2\alpha}{(2\pi)^{d/2}(\frac{d}{2})!}(\tfrac{1}{2}F)^{d/2}, }[/math]

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, [math]\displaystyle{ F \equiv F_{\mu\nu}\,dx^\mu\wedge dx^\nu\,. }[/math]

This result is equivalent to [math]\displaystyle{ (\tfrac{d}{2})^{\rm th} }[/math] Chern class of the [math]\displaystyle{ \mathfrak{g} }[/math]-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.

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