I am trying to study Susy from Patrick Labelle's book. While I see that we want the Susy variation to take \(\phi\) into \(\chi\) and vice versa, I don't see why I cannot argue that since \(\delta\phi=\zeta.\chi\) is itself a scalar, another variation by a Susy parameter (say, \(\beta\)) won't just produce \(\delta_\beta \delta_\zeta \phi=(\beta.\zeta)(\zeta.\chi)\), or something like that, since \[\delta_\beta \delta_\zeta \phi=\delta_\beta(\zeta.\chi)=\delta_\beta(\phi')=\beta.\chi\]again? Here I have defined \[\phi'=\zeta.\chi\] to be the new scalar field. Is it right to think of \(\zeta.\chi\) as a scalar field?
Put differently, what stops me from thinking of \(\zeta.\chi\) as another scalar, the Susy variation of which should again give something similar to the Susy variation of \(\phi\)?
I have a "feeling" that I am perhaps not clear about the fact that \(\phi\) is a scalar under Lorentz transformation. I am also not clear about whether it is the scalar or the bosonic nature of \(\phi\) that I should be thinking about.
Any help would be much appreciated!