Unfortunately, this important property is sensitive to the environment that the electromagnetic and gravitational fields live in, i e what they interact with. In the case of electromagnetic fields, and the related (Yang-Mills) fields responsible for the known short range fundamental forces in nature, the requirements are not that stringent. On the other hand, the gravitational field interacts with itself in a way that appears to be in conflict with quantum mechanics. Ordinarily, we should not worry too much, since we cannot yet measure any quantum mechanical effects of gravity, but we should realize that gravity by itself is not understood microscopically.
It could help if there was something more in nature that compensated for the problems caused by gravity. At present there is only one known theory of gravity, string theory, that contains fields and matter that manages to give calculable non-negative quantum mechanical probabilities. String theory contains a multitude of forces and particles that only are important at extremely short distances, but also others determined by a symmetry called supersymmetry, which relates forces and matter. The effects of supersymmetry may well be directly observable.
Recently, string theory has been argued to be just a particular way
of looking at a theory called M-theory. If this is is correct one can use
string theory in defining M-theory, which is not yet precisely defined,
and one can use an M-theory perspective to get new results in string theory.
Both approaches have of course been attempted with a good deal of success,
but there are many more things to understand... One perspective on M-theory
called matrix theory implies that it is very closely related to Yang-Mills
theories. Then the reason that gravity works so well in string theory could
be that it is not a fundamental force, only a effective force which one
can see at larger distances.