Unsharp degrees of freedom and the generating of symmetries

While real degrees of freedom are usually described by operators which are self-adjoint, there are exceptions described by merely symmetric operators. It has been shown that such exceptional degrees of freedom generally display a form of “unsharpness.” Various studies in quantum gravity indicate that the widely expected unsharpness of space-time at very short distances can be described by such operators. It is also known, however, that unlike self-adjoint operators, merely symmetric operators do not generate unitary transformations, at least not straightforwardly. This raises the question of whether merely symmetric operators are able to play the important double role which self-adjoint operators often play, namely, both to represent a real degree of freedom and also to act as a symmetry generator. Here, we answer this question for a large class of symmetric non-self-adjoint operators X. We show that operators which coincide with such an X on the physical domain are even able to generate the entire unitary group of the Hilbert space.