  Relaxing the postulates of an axiomatic theory is a natural way to find more
general theories, and historically, the discovery of non-Euclidean geometry is
a famous example of this procedure. Here, we use this way to extend quantum
mechanics by ignoring the heart of Heisenberg's quantum mechanics -- We do not
assume the existence of a position operator that satisfies the Heisenberg
commutation relation, $[\hat x,\hat p]=i\hbar$. The remaining axioms of quantum
theory, besides Galilean symmetry, lead to a more general quantum theory with a
free parameter $l_0$ of length dimension, such that as $l_0 \to 0$ the theory
reduces to standard quantum theory. Perhaps surprisingly, this non-Heisenberg
quantum theory, without a priori assumption of the non-commutation relation,
leads to a modified Heisenberg uncertainty relation, $\Delta x \Delta p\geq
\sqrt{\hbar^2/4+l_0^2(\Delta p)^2}$, which ensures the existence of a minimal
position uncertainty, $l_0$, as expected from various quantum gravity studies.
By comparing the results of this framework with some observed data, which
includes the first longitudinal normal modes of the bar gravitational wave
detector AURIGA and the $1S-2S$ transition in the hydrogen atom, we obtain
upper bounds on the $l_0$.
