On the Efetov–Wegner terms by diagonalizing a Hermitian supermatrix

The diagonalization of Hermitian supermatrices is studied. Such a change of coordinates is inevitable if one wishes to find certain structures in random matrix theory. However, it still poses serious problems since until now the calculation of all Rothstein contributions, known as Efetov–Wegner terms in physics, was quite cumbersome. We derive the supermatrix Bessel function with all Efetov–Wegner terms for an arbitrary rotation invariant probability density function. As applications we consider representations of generating functions for Hermitian random matrices with and without an external field as integrals over eigenvalues of Hermitian supermatrices. All results are obtained with all Efetov–Wegner terms which were previously unknown in such an explicit and compact representation.