On the operator origins of classical and quantum wave functions

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.

Qualia and the Formal Structure of Meaning

“…In fact, sense and feeling are sometimes used interchangeably in this literature. 4 The Formal Structure of Meaning So far, we have put forth, mostly on phenomenological grounds, that meaning, generated intrinsically by the experiencing self, is an integral and inseparable characteristic of qualia, across the board. Furthermore, the subjectivity, non-representational and as well, the transparency of, what has been referred to as subjective meaning, suggests why it may be directly linked to the phenomenal contents of conscious experience.…”

“…In recent years, what one means by language and grammar has vastly generalized beyond traditional forms of written language. Examples include graph grammars in graphical languages [57,83], pregroup grammars in diagrammatic languages involving string diagrams [40,77], visual grammars corresponding to compositions involving visual scenes and actions [96,139], and hypergraph / operator algebras that extend graph grammars [4,136,137], just to name a few. In our case here, the collections of objects in both, Obj P and Obj M are not arbitrary collections.…”

“…A feature of these self-models is that they not only constitute a representation of the agent, but they also co-represent the representational relation (between the agent and the world) itself [89,91]. In other words, they are equipped with a phenomenal model of the intentionality relation 4 . According to Metzinger, phenomenal self-models are characterized by phenomenal properties pertaining to ownership, perspectivalness and selfhood; all of which are essential for creating a rst-person perspective [89,91].…”