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- Fujikawa_method abstract "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 and the fermionic action is given byThe partition function is The axial symmetry transformation goes asClassically, 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:where are Grassmann valued coefficients, and are eigenvectors of the Dirac operator:The eigenfunctions are taken to be orthonormal with respect to integration in d-dimensional space,The measure of the path integral is then defined to be:Under an infinitesimal chiral transformation, writeThe Jacobian of the transformation can now be calculated, using the orthonormality of the eigenvectorsThe transformation of the coefficients are calculated in the same manner. Finally, the quantum measure changes aswhere 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: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(can be re-written as , and the eigenfunctions can be expanded in a plane-wave basis)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.".
- Fujikawa_method wikiPageID "2703614".
- Fujikawa_method wikiPageRevisionID "457369799".
- Fujikawa_method hasPhotoCollection Fujikawa_method.
- Fujikawa_method subject Category:Anomalies_in_physics.
- Fujikawa_method type Abnormality114501726.
- Fujikawa_method type Abstraction100002137.
- Fujikawa_method type AnomaliesInPhysics.
- Fujikawa_method type Anomaly114505821.
- Fujikawa_method type Attribute100024264.
- Fujikawa_method type Condition113920835.
- Fujikawa_method type PhysicalCondition114034177.
- Fujikawa_method type State100024720.
- Fujikawa_method comment "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 and the fermionic action is given byThe partition function is The axial symmetry transformation goes asClassically, 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_method label "Fujikawa method".
- Fujikawa_method sameAs m.07y_z7.
- Fujikawa_method sameAs Q5507421.
- Fujikawa_method sameAs Q5507421.
- Fujikawa_method sameAs Fujikawa_method.
- Fujikawa_method wasDerivedFrom Fujikawa_method?oldid=457369799.
- Fujikawa_method isPrimaryTopicOf Fujikawa_method.