On the Operator Origins of Classical and Quantum Wave Functions
- URL: http://arxiv.org/abs/2211.01838v2
- Date: Sun, 21 May 2023 14:31:16 GMT
- Title: On the Operator Origins of Classical and Quantum Wave Functions
- Authors: Xerxes D. Arsiwalla, David Chester, Louis H. Kauffman
- Abstract summary: We introduce a formalism of Operator Mechanics based on a noncommutative Poisson, symplectic and noncommutative differential structures.
We show that the Schr"odinger equation is obtained from the Koopman-von Neumann equation.
What this suggests is that neither the Schr"odinger equation nor the quantum wave function are fundamental structures.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: 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 ${\cal E}$
follows from this. $\psi_{KvN}$ and $\psi_{QM}$ are shown to be sections in
${\cal E}$. The difference between $\psi_{KvN}$ and $\psi_{QM}$ originates from
a quantization map interpreted as "twisting" of sections over ${\cal E}$. We
also show that the Schr\"{o}dinger equation is obtained from the Koopman-von
Neumann equation. What this suggests is that neither the Schr\"{o}dinger
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.
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