Such a situation was already anticipated by Einstein [14] when he wrote
....perhaps the success of the Heisenberg method points to a purely algebraic description of nature, that is, to the elimination of the continuous functions from physics. Then, however, we must give up, in principle, the space-time continuum.
Source:
Foundations of Quantum Theory in the Light of Bohmian Non-commutative Dynamics
B. J. Hiley, 2013
Einstein here attributes the non-commutativity to Heisenberg. The von Neumann-Moyal algebra builds Heisenberg’s non-commutativity into a noncommutative symplectic phase space.
When we consider a non-commutative geometry, we do not have a unique underlying manifold, but we can construct shadow manifolds....
On the philosophical side, this non-commutative algebra is actually a
mathematical expression of Bohm’s implicate order [3]. The algebra is a
mathematical description of what Bohm calls the implicate order. The
shadow manifolds are examples of what Bohm calls explicate orders aris-
ing from the participation of ourselves or our measuring instruments in the
process itself.
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