  A representation theorem for non-semibounded Hermitian quadratic forms in
terms of a (non-semibounded) self-adjoint operator is proven. The main
assumptions are closability of the Hermitian quadratic form, the direct
integral structure of the underlying Hilbert space and orthogonal additivity.
We apply this result to several examples, including the position operator in
quantum mechanics and quadratic forms invariant under a unitary representation
of a separable locally compact group. The case of invariance under a compact
group is also discussed in detail.
