I'm confused whether a symmetrisation/antisymmetrization of a function with respect to its arguments, i.e., $$F(x_1,x_2,...,x_n)=\frac{1}{\sqrt{N!}}\sum_{\pi}\textrm{sgn}(\pi)~f(x_{\pi(1)},...,x_{\pi(n)})$$ has any bearing on its functional derivative. I'd say no and just compute, e.g.,

$$\frac{\delta F(x_1,...,x_n)}{\delta F(y_1,...,y_n)}=\delta(x_1-y_1,...,x_n-y_n). $$

A variation with respect to a particular $f$ gives nothing more than

$$\frac{\delta F(x_1,...,x_n)}{\delta f(y_{\pi'(1)},...,y_{\pi'(n)})}=\delta(x_{\pi'(1)}-y_{\pi'(1)},...,x_{\pi'(n)}-y_{\pi'(n)}). $$

Has anybody come across such a point?


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