On the reality of quantum states: A pedagogic survey from classical to quantum mechanics

Technical Summary: On the Reality of Quantum States: A Pedagogic Survey from Classical to Quantum Mechanics

Problem Statement
The paper addresses the long-standing debate regarding the ontological status of the quantum wave function ($\Psi$). Specifically, it investigates whether $\Psi$ represents an objective physical reality independent of the observer or merely a representation of the observer's state of knowledge (information) about an underlying system. Recent experimental claims suggest that models treating quantum states as mere information contradict quantum theory, rekindling interest in the reality of the wave function. The author posits that the "bizarre" nature of quantum mechanics and the confusion surrounding the reality of $\Psi$ stem from presenting quantum mechanics as a disparate discipline disconnected from classical mechanics.

Methodology
The author employs a pedagogic and theoretical approach, tracing a direct route from classical mechanics to quantum mechanics, analogous to the transition from geometrical optics to wave optics.

1. Optical Analogy: The paper first reviews the relationship between the nonlinear eikonal equation of geometrical optics (a limiting case) and the linear electromagnetic wave equation of Maxwell (the general case). It highlights that the eikonal equation does not permit the superposition principle, whereas the wave equation does, allowing electromagnetic signals to take any square-integrable functional form.
2. Classical Mechanics Reformulation: The author rewrites the classical Hamilton-Jacobi (HJ) equation, which is nonlinear and prohibits superposition, into a "classical mechanics wave equation" form by defining a function $X(r,t) = \exp(iS(r,t)/\hbar)$, where $S$ is Hamilton's principal function.
3. Generalization of Wave-Particle Duality: The core methodological step involves generalizing de Broglie's wave-particle duality. The author argues that just as photon wave functions can be any square-integrable solution to Maxwell's equations, matter wave functions should also be allowed to be any square-integrable solution to a fundamental equation. This necessitates that the matter wave equation must obey the superposition principle.
4. Derivation of the Schrödinger Equation: To satisfy the superposition requirement, the author modifies the nonlinear classical mechanics wave equation by adding a specific term (of order $\hbar$) to the kinetic energy term. This linearization transforms the classical equation into the linear Schrödinger equation.
5. Comparative Analysis: The paper systematically compares classical and quantum formalisms, including eigenvalue equations for observables (momentum, position, energy), series expansions, and probability assignments, demonstrating that these structures exist in both realms but function differently due to the presence or absence of superposition.

Key Contributions and Results

• Derivation of Schrödinger's Equation: The paper presents an alternative derivation of the Schrödinger equation not as a postulate, but as a necessary generalization of the classical HJ equation to allow for the superposition of matter waves, mirroring the transition from the eikonal equation to Maxwell's equations.
• Reinterpretation of Classical "States": The author introduces the concept of a "classical mechanics wave function" ($X$) representing a classical energy state. It is shown that while classical mechanics allows for the mathematical definition of eigenfunctions for observables (e.g., momentum eigenfunctions), these do not constitute valid physical states of the system because they cannot be superposed to form other solutions of the nonlinear classical wave equation.
• Explanation of Quantum Puzzles: The paper argues that phenomena such as wave function collapse, entanglement, and the assignment of probabilities to non-energy observables are direct consequences of the superposition principle.
  • Collapse: In classical mechanics, a system is restricted to specific energy states (solutions of the nonlinear equation). A measurement cannot "collapse" a classical wave function into an eigenstate of a different observable because such eigenstates are not valid solutions to the classical wave equation. In quantum mechanics, the linearity allows any superposition, making collapse possible.
  • Entanglement: The absence of entanglement in the classical realm is attributed to the lack of wave function collapse and the inability to form superpositions of distinct energy states.
  • Spin: The paper suggests that intrinsic spin arises from the multi-component nature of wave functions. In the classical limit ($\hbar \to 0$), multi-component functions effectively reduce to a single component, explaining the absence of classical spin analogues.
• Observable Fields: The author defines "observable fields" (e.g., momentum field, energy field) as the ratio of the operator acting on the wave function to the wave function itself ($A(r,t) = A_{op}\Psi / \Psi$). These fields are shown to be real, position-dependent quantities that, when averaged over the probability density $|\Psi|^2$, yield the standard expectation values.

Significance and Claims
The paper claims that the "puzzles" of quantum mechanics are not unique to the quantum realm but exist in classical mechanics in a "dormant form." By demonstrating that the mathematical structures of eigenvalues, expansions, and probability assignments are present in classical mechanics, the author argues that the distinction between the two theories lies solely in the validity of the superposition principle.

The significance of this work is twofold:

1. Pedagogical: It aims to demystify quantum mechanics by presenting it as a natural, linear generalization of classical mechanics, rather than an esoteric discipline based on abstract Hilbert spaces disconnected from reality.
2. Ontological: The author concludes that the wave function is real (objective) rather than merely a representation of knowledge. This conclusion is supported by the argument that wave function collapse is a physical phenomenon requiring a change in the system's state, which is only possible in a linear theory where eigenstates of different observables can coexist as valid states. The paper asserts that the reality of the wave function and the associated observable fields is endorsed by the fact that expectation values can be computed as averages of these physically existing fields over the probability density.

The work does not propose new experiments but seeks to resolve conceptual issues by re-examining the foundational route from classical to quantum theory, suggesting that the "dormant" features in classical mechanics, once activated by the superposition principle, fully explain the behavior of quantum systems.