I'm confused by something super simple. When taking functional variation (e.g. of the action) in the context of field theory, I often see $$ {\delta \phi(x) \over \delta \phi(y)} = \delta(x-y) \ .$$ How does this make dimensional sense when the LHS naively appears dimensionless, while the RHS has mass dimension 1?
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1Does this answer your question? Functional derivative and units. See also the links therein. – Tobias Fünke May 21 '23 at 14:15
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Because functional derivatives are implicitly computed inside the integral of the functional, which will kill the delta function and the associated units. – Abezhiko May 21 '23 at 15:58
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Possible duplicate: https://physics.stackexchange.com/q/605472/2451 – Qmechanic May 21 '23 at 16:48