On the operator origins of classical and quantum wave functions

Abstract

Abstract We investigate operator algebraic origins of the classical Koopman–von Neumann wave function $$\psi _{KvN}$$ ψ KvN as well as the quantum-mechanical one $$\psi _{QM}$$ 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, $$\psi _{KvN}$$ ψ KvN and $$\psi _{QM}$$ 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 $${{\mathcal {E}}}$$ E follows from this. $$\psi _{KvN}$$ ψ KvN and $$\psi _{QM}$$ ψ QM are shown to be sections in $${{\mathcal {E}}}$$ E . The difference between $$\psi _{KvN}$$ ψ KvN and $$\psi _{QM}$$ ψ QM originates from a quantization map interpreted as “twisting” of sections over $${{\mathcal {E}}}$$ 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.