On the Operator Origins of Classical and Quantum Wave Functions

We investigate operator algebraic origins of the classical Koopman-von Neumann wave function ψ_KvN as well as the quantum mechanical one ψ_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, ψ_KvN and ψ_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 {cal E} follows from this. ψ_KvN and ψ_QM are shown to be sections in {cal E}. The difference between ψ_KvN and ψ_QM originates from a quantization map interpreted as "twisting" of sections over {cal E}. We also show that the Schrödinger equation is obtained from the Koopman-von Neumann equation. What this suggests is that neither the Schrödinger 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.