I have found in the literature at least two different definitions of $\bf{super}$ Hilbert spaces:
- Definition 1: A super Hilbert space is a complex super-vector space $\mathcal{H}=\mathcal{H}_0\oplus \mathcal{H}_1$ with a inner product such that $$\overline{\langle v,v'\rangle}=(-1)^{|v||v'|}\langle v',v\rangle.$$ Here $v$ and $v'$ are homogeneous elements. It implies that $\mathcal{H}_0$ and $\mathcal{H}_1$ are perpendicular and that the norm square of an even vector is real and for an odd one is pure imaginary.
- Definition 2: A super Hilbert space is a complex super-vector space $\mathcal{H}=\mathcal{H}_0\oplus \mathcal{H}_1$ with an inner product such that $\mathcal{H}$ is a usual Hilbert space and $\mathcal{H}_0$ is perpendicular to $\mathcal{H}_1$.
It is clear that there is a bijective correspondence between this two kind of super Hilbert spaces. If $\mathcal{H}$ satisfies definition 1, then only changing the inner product in the odd elements as $\langle v',v''\rangle' = \overline{\langle v',v''\rangle}$ you get a super Hilbert space of definition 2.
My question is: Which is the most used definition in physics and why?
