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Fujikawa method : ウィキペディア英語版
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\,. The Dirac operator (in Feynman slash notation) is
:D\!\!\!\!/\ \stackrel\ \partial\!\!\!/ + i A\!\!\!/
and the fermionic action is given by
:\int d^dx\, \overlineiD\!\!\!\!/ \psi
The partition function is
:Z()=\int \mathcal\overline\mathcal\psi e^.
The axial symmetry transformation goes as
:\psi\to e^\psi\,
:\overline\to \overlinee^
:S\to S + \int d^dx \,\alpha(x)\partial_\mu\left(\overline\gamma^\mu\gamma^5\psi\right)
Classically, this implies that the chiral current, j_^\mu \equiv \overline\gamma^\mu\gamma^5\psi is conserved, 0 = \partial_\mu j_^\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 = \sum\limits_\psi_ia^i,
:\overline\psi = \sum\limits_\psi_ib^i,
where \ are Grassmann valued coefficients, and \ are eigenvectors of the Dirac operator:
:D\!\!\!\!/ \psi_i = -\lambda_i\psi_i.
The eigenfunctions are taken to be orthonormal with respect to integration in d-dimensional space,
:\delta_i^j = \int\frac\psi^(x)\psi_i(x).
The measure of the path integral is then defined to be:
:\mathcal\psi\mathcal\overline = \prod\limits_i da^idb^i
Under an infinitesimal chiral transformation, write
:\psi \to \psi^\prime = (1+i\alpha\gamma_)\psi = \sum\limits_i \psi_ia^,
:\overline\psi \to \overline^\prime = \overline(1+i\alpha\gamma_) = \sum\limits_i \psi_ib^.
The Jacobian of the transformation can now be calculated, using the orthonormality of the eigenvectors
:C^i_j \equiv \left(\frac\right)^i_j = \int d^dx \,\psi^(x)()\psi_j(x) = \delta^i_j\, - i\int d^dx \,\alpha(x)\psi^(x)\gamma_\psi_j(x).
The transformation of the coefficients \ are calculated in the same manner. Finally, the quantum measure changes as
:\mathcal\psi\mathcal\overline = \prod\limits_i da^i db^i = \prod\limits_i da^db^^(C^i_j),
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:
:\begin^(C^i_j) &= \exp\left(d^dx\, \alpha(x)\psi^(x)\gamma_\psi_i(x)\right )\end
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
:\begin-2\ln C^i_j &= 2i\lim\limits_\alpha\int d^dx \,\psi^(x)\gamma_ e^\psi_i(x)\\
&= 2i\lim\limits_\alpha\int d^dx\, \psi^(x)\gamma_ e^e^\psi_i(k)
:= -\frac)!}(\tfracF)^,
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_\,dx^\mu\wedge dx^\nu\,.
This result is equivalent to (\tfrac)^ Chern class of the \mathfrak-bundle over the d-dimensional base space, and gives the chiral anomaly, responsible for the non-conservation of the chiral current.


抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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