Some authors would write the following: σ is a permutation of the numbers 1, ..., n and

(σf)(x_1, ..., x_n) = f(x_{σ(1)}, ..., x_{σ(n)});

nothing truly different, but possibly more comfortable. Another thing to say is that a field automorphism can't move the base field Q: all we can do and all we have done is move around these formal variables. So with the rest of your notation,

(σf)(x_1, x_2) = f(x_{σ(1)}, x_{σ(2)}) = f(x_2, x_1) = x_2 - x_1