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It seems OP's question (v4) is related to the proper handling of derivatives of Dirac delta distributions. Reductions are performed with the help of (the appropriate 3D generalizations of) the following formulas: $$\tag{A} \{\partial_x+\partial_y\}\delta (x-y)~=~ 0,$$ $$\tag{B} \{f(x)-f(y)\}~\delta (x-y)~=~ 0,$$ $$\tag{C} \{f(y)-f(x)\}~\partial_x \delta ...


No, the gauge current need not be gauge invariant, since it carries a group index in non-abelian theories. You should recall that both sides of the Yang-Mills equation (and therefore the current itself) are Lie-algebra valued and therefore transform in the adjoint representation. Not even the field strength $F^a_{\mu\nu}$ is gauge invariant, but transforms ...

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