On the operator origins of classical and quantum wave functions

We investigate operator algebraic origins of the classical Koopman-von Neumann wave function psi(K upsilon N) as well as the quantum-mechanical one psi(QM). We introduce a formalism of Operator Mechanics (OM) based on a noncommutative Poisson, symplectic, and noncommutative differential structures. OM serves as a pre-quantum algebra from which algebraic structures relevant to real-world classical and quantum mechanics follow. In particular,psi(K upsilon N) and psi(QM) are both consequences of this pre-quantum formalism. No a priori Hilbert space is needed. OM admits an algebraic notion of operator expectation values without invoking states. A phase space bundle epsilon follows from this. psi(KvN) and psi(QM) are shown to be sections in epsilon. The difference between psi(K upsilon N) and psi(QM) originates from a quantization map interpreted as "twisting" of sections over epsilon. We also show that the Schrodinger equation is obtained from the Koopman-von Neumann equation. What this suggests is that neither the Schrodinger equation nor the quantum wave function are fundamental structures. Rather, they both originate from a pre-quantum operator algebra. Finally, we comment on how entanglement between these operators suggests emergence of space; and possible extensions of this formalism to field theories.