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A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In 2D, the infinite-dimensional algebra of local conformal transformations normally permits exact solution or classification of such theories. Further use for CFT applications to string theory, statistical mechanics, and condensed matter physics.

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Is it valid to change the order of tensors by changing their sign?

Perhaps you're looking for integration by parts. If you introduce some scalar field $\phi(x)$ (vanishing as $x \to \infty$) that you act on with your derivatives, you can move a derivative $\partial_\ …
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