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**ppyvabw** Say you have a scalar field $\displaystyle \Phi(x)$ defined on a manifold M. Then imagine evaluating the field at a point an infitesimal distance away from x at $\displaystyle x'=x+\epsilon X$ where X is a illing vector of the manifold. Then

$\displaystyle \Phi(x')=\Phi(x)+\epsilon X^{\mu}\partial_{\mu}\Phi(x)$ to first order. Now the second term on the rhs is the Lie derivative of a scalar, so

$\displaystyle \Phi(x')=\Phi(x)+\epsilon L_{X}\Phi(x)$

Say now that $\displaystyle \Phi$ is not a scalar, say maybe a vector of spinor or tensor. Does the same thing follow, that

$\displaystyle \Phi^{\nu}(x')=\Phi^{\nu}(x)+\epsilon L_{X}\Phi^{\nu}(x)$ Where now $\displaystyle L_{X}$ is not the appropriate Lie derivative of whatever object $\displaystyle \Phi$ is.