Abstract
Necessary and sufficient conditions in order for all components of a local operator (such as a current, for example) to have the same dimension are given and discussed. In addition, we then note that exact scale invariance is in an apparent formal contradiction with a definite scaling behavior for the currents. The solution of this contradiction is then traced back to the fact that in scale-invariant theories, the Schwinger term must be quadratically divergent. We finally conclude by briefly discussing Coleman's theorem for scale and conformal transformations.
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