In more classical language, " z " is a differential operator, constructed from the infinitesimal transformations which are induced on " V " by the Lie algebra of " G ".
22.
The Lagrangian density \ mathcal { L } is a scalar, and so will infinitesimally transform as \ delta \ mathcal { L } = \ alpha ^ \ mu \ partial _ \ mu \ mathcal { L } under the infinitesimal transformation.
23.
In 1940, L�on Rosenfeld and before him Wolfgang Pauli introduced what he called a local variation \ delta ^ { \ ast } A of a geometric object A \, induced by an infinitesimal transformation of coordinates generated by a vector field X \,.
24.
The derivative of any smooth curve with " I " } } in the group depending on some group parameter with respect to that group parameter, evaluated at 0 } }, serves as a definition of a corresponding group generator, and this reflects an infinitesimal transformation away from the identity.
25.
It is then possible to extend Lichnerowicz's definition to all vector fields ( generic infinitesimal transformations ) by retaining Lichnerowicz's local expression for a " generic " vector field X, but explicitly taking the antisymmetric part of \ nabla _ { a } X _ { b } only.
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