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| {{Unreferenced|date=December 2009}}
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| '''Fujikawa's method''' is a way of deriving the [[chiral anomaly]] in [[quantum field theory]].
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| Suppose given a [[Fermionic field#Dirac fields|Dirac field]] ψ which transforms according to a ρ [[representations of Lie groups|representation]] of the [[compact Lie group]] ''G''; and we have a background [[connection form]] of taking values in the [[Lie algebra]] <math>\mathfrak{g}\,.</math> The [[Dirac operator]] (in [[Feynman slash notation]]) is
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| :<math>D\!\!\!\!/\ \stackrel{\mathrm{def}}{=}\ \partial\!\!\!/ + i A\!\!\!/</math> | |
| and the fermionic action is given by
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| :<math>\int d^dx\, \overline{\psi}iD\!\!\!\!/ \psi</math> | |
| The [[partition function (quantum field theory)|partition function]] is
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| :<math>Z[A]=\int \mathcal{D}\overline{\psi}\mathcal{D}\psi e^{-\int d^dx \overline{\psi}iD\!\!\!\!/\psi}.</math>
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| The [[axial symmetry]] transformation goes as
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| :<math>\psi\to e^{i\gamma_{d+1}\alpha(x)}\psi\,</math>
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| :<math>\overline{\psi}\to \overline{\psi}e^{i\gamma_{d+1}\alpha(x)}</math>
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| :<math>S\to S + \int d^dx \,\alpha(x)\partial_\mu\left(\overline{\psi}\gamma^\mu\gamma^5\psi\right)</math>
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| Classically, this implies that the chiral current, <math>j_{d+1}^\mu \equiv \overline{\psi}\gamma^\mu\gamma^5\psi</math> is conserved, <math>0 = \partial_\mu j_{d+1}^\mu</math>.
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| 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]]:
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| :<math>\psi = \sum\limits_{i}\psi_ia^i,</math>
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| :<math>\overline\psi = \sum\limits_{i}\psi_ib^i,</math> | |
| where <math>\{a^i,b^i\}</math> are [[Grassmann]] valued coefficients, and <math>\{\psi_i\}</math> are eigenvectors of the [[Dirac operator]]:
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| :<math>D\!\!\!\!/ \psi_i = -\lambda_i\psi_i.</math>
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| The eigenfunctions are taken to be orthonormal with respect to integration in d-dimensional space,
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| :<math>\delta_i^j = \int\frac{d^dx}{(2\pi)^d}\psi^{\dagger j}(x)\psi_i(x).</math>
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| The measure of the path integral is then defined to be:
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| :<math>\mathcal{D}\psi\mathcal{D}\overline{\psi} = \prod\limits_i da^idb^i</math>
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| Under an infinitesimal chiral transformation, write
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| :<math>\psi \to \psi^\prime = (1+i\alpha\gamma_{d+1})\psi = \sum\limits_i \psi_ia^{\prime i},</math>
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| :<math>\overline\psi \to \overline{\psi}^\prime = \overline{\psi}(1+i\alpha\gamma_{d+1}) = \sum\limits_i \psi_ib^{\prime i}.</math>
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| The [[Jacobian]] of the transformation can now be calculated, using the [[orthonormality]] of the [[eigenvectors]]
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| :<math>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>
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| The transformation of the coefficients <math>\{b_i\}</math> are calculated in the same manner. Finally, the quantum measure changes as
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| :<math>\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>
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| 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:
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| :<math>\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]\\
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| &= \exp\left[2i\int d^dx\, \alpha(x)\psi^{\dagger i}(x)\gamma_{d+1}\psi_i(x)\right]\end{align}</math>
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| to first order in α(x).
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| Specialising to the case where α is a constant, the [[Jacobian]] must be regularised because the integral is ill-defined as written. Fujikawa employed [[Zeta_function_regularization#Heat_kernel_regularization|heat-kernel regularization]], such that
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| :<math>\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)\\
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| &= 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>
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| (<math>{D\!\!\!\!/}^2</math> can be re-written as <math>D^2+\tfrac{1}{4}[\gamma^\mu,\gamma^\nu]F_{\mu\nu}</math>, and the eigenfunctions can be expanded in a plane-wave basis)
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| :<math>= 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>= -\frac{-2\alpha}{(2\pi)^{d/2}(\frac{d}{2})!}(\tfrac{1}{2}F)^{d/2},</math>
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| 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>F \equiv F_{\mu\nu}\,dx^\mu\wedge dx^\nu\,.</math>
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| This result is equivalent to <math>(\tfrac{d}{2})^{\rm th}</math> [[Chern class]] of the <math>\mathfrak{g}</math>-bundle over the d-dimensional base space, and gives the [[chiral anomaly]], responsible for the non-conservation of the chiral current.
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| {{DEFAULTSORT:Fujikawa Method}}
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| [[Category:Anomalies in physics]]
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