On the operator origins of classical and quantum wave functions. (English) Zbl 07886856
Summary: We investigate operator algebraic origins of the classical Koopman-von Neumann wave function \(\psi_{KvN}\) 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, \(\psi_{KvN}\) 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 \(\mathcal{E}\) follows from this. \(\psi_{KvN}\) and \(\psi_{QM}\) are shown to be sections in \(\mathcal{E}\). The difference between \(\psi_{KvN}\) and \(\psi_{QM}\) originates from a quantization map interpreted as “twisting” of sections over \(\mathcal{E}\). 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.
MSC:
| 81-XX | Quantum theory |
Keywords:
operator algebras; non-commutative differential calculus; Koopman-von Neumann mechanics; pre-Hilbert spaces; operator entanglementReferences:
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