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I am reading Efetov's article on Anderson Localization, where some kind of supersymmetric formalism is used, and I am currently trying to make sense of the definitions. The most useful reference is this long review article, around equation 2.20.

Here are the useful definitions. I will only use 2x2 supermatrices, which is enough for my question, and I will write complex numbers with latin letters, and Grassmann numbers with greek letters.

The complex conjugate of a Grassmann number $\theta$ is $\theta^*$, and we use the definition $$(\theta^*)^*=-\theta.$$

A supermatrix $F$ takes the form

$$F=\begin{pmatrix}a & \theta \\ \eta & b\end{pmatrix},$$ and its transpose is defined as $$F^T=\begin{pmatrix}a & -\eta \\ \theta & b\end{pmatrix},$$ which implies the nice property that if we denote the hermitian conjugate by $F^\dagger=(F^T)^*$, then $(F^\dagger)^\dagger=F$. (NB: it is important that the Grassmann elements of $F$ are complex to have this property).

A supermatrix $U$ is said to be unitary if $$U U^\dagger = U^\dagger U =1.$$

Now here's the question. Writing $U$ in the form of $F$, I am trying to find the most general form of $U$ such that it is unitary. The problem is that I cannot find a consistent way to do that.

The relevant equations, using $U U^\dagger = U^\dagger U =1$, are $$ |a|^2-\theta^*\theta=1,\\ |b|^2+\eta^*\eta=1,\\ |a|^2-\eta^*\eta=1,\\ |b|^2+\theta^*\theta=1,\\ a\theta^*+b^*\eta=0,\\ a^*\eta+b\theta^*=0,\\ $$ as well as the complex conjugate of the last two equations.

Playing with this, one finds (without any assumptions) that $|a|^2+|b|^2=2$ and $\eta^*\eta=\theta^*\theta$. Now assuming that $a\neq 0$, one gets $\eta=-b/a^*\theta^*$, which implies that (if $\theta^*\neq 0$) $|a|^2=|b|^2=1$.

Here comes the problem : we also have that $|a|^2-\theta^*\theta=1$ which implies that $\theta^*\theta=0$, which is not possible, unless $\theta^*=\theta$, but this breaks our (important) assumption that $\theta$ is complex (to have $(U^\dagger)^\dagger=U$).

If instead we have $a=0$, this implies $\theta^*\theta=-1$, and I am not sure how to interpret this... (Yes, $\theta^*\theta$ is bosonic, but I don't think this equality makes sense, since if we integrate with respect to $\theta$ and $\theta^*$ both side, we get $1=0$...).

Otherwise, one could just impose $\theta=\eta=0$, but in that case, there is not really a point to define unitary supermatrices, since their effect is kind of trivial (i.e. change the phase of the bosons and fermions independently).

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  • $\begingroup$ D0es multiplication of supermatrices differ from ordinary matrix multiplication? I am not versed in the subject, but just going from what you have written and using matrix multiplication I get different equations for being unitary. Namely, I get some (but not all) of your minus signs to be plus signs and vice versa. $\endgroup$ Oct 4, 2016 at 13:43
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    $\begingroup$ Yes, matrix multiplication is the same. But $\theta^*\theta=-\theta\theta^*$, which might explain some signs. $\endgroup$
    – Adam
    Oct 4, 2016 at 13:45

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The problem was in assuming that because one has $$ (|b|^2-|a|^2)\theta^*=0,$$ then $|b|^2=|a|^2$. Indeed, the only thing it tells you is that $$|b|^2=|a|^2+\sigma \theta^*,$$ where $\sigma$ can be another Grassmanian (or a complex number). In fact, one can check that the two independent equations $|b|^2=1-\theta^*\theta$ and $|a|^2=1+\theta^*\theta$ are compatible with $ (|b|^2-|a|^2)\theta^*=0$.

The best way to solve the above equations is to solve the equations for $a$ and $b$ in terms of $\theta^*\theta$. One gets $$a=e^{i\alpha}\left(1+\frac{\theta^*\theta}{2}\right),\\ b=e^{i\beta}\left(1-\frac{\theta^*\theta}{2}\right), $$ with $\alpha$ and $\beta$ two arbitrary phases. Then the equation linking $\eta$ and $\theta^*$ is used to get $$\eta=-e^{i(\alpha+\beta)}\theta^*.$$ One then checks that these results are compatible with all the equations in the question.

Thus, a general 2x2,unitary supermatrix is parametrized by only three numbers : two real numbers $\alpha$ and $\beta$, and a Grassman number $\theta$, and it reads $$U=\begin{pmatrix} e^{i\alpha}\left(1+\frac{\theta^*\theta}{2}\right) && \theta \\ -e^{i(\alpha+\beta)}\theta^* && e^{i\beta}\left(1-\frac{\theta^*\theta}{2}\right). \end{pmatrix}$$

One then checks that it is indeed superunitary.

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