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

Some recent experiments claim to show that any model in which a quantum state represents mere information about an underlying physical reality of the system must make predictions which contradict those of quantum theory. The present work undertakes to investigate the issue of reality, treading a more fundamental route from the Hamilton-Jacobi equation of classical mechanics to the Schrodinger equation of quantum mechanics. Motivation for this is a similar approach from the eikonal equation in geometrical optics to the wave equation in electromagnetic theory. We rewrite the classical Hamilton-Jacobi equation as a wave equation and seek to generalise de Broglie’s wave particle duality by demanding that both particle and light waves have the freedom of being described by any square-integrable function. This generalisation, which allows superposition also for matter wave functions, helps us to obtain the Schrodinger equation, whose solution can be seen to be as much objective as the classical mechanics wave function. Several other equations which one writes in quantum mechanics, including the eigenvalue equations for observables, series expansion of energy states in terms of eigenstates of observables other than energy, etc., can be written in the classical case too. Absence of any collapse of the wave function, entanglement, etc. in the classical realm have their origin in the nonlinearity of the classical wave equation. These considerations indicate that many of the puzzles in quantum mechanics are present also in classical mechanics in a dormant form, which fact shall help to demystify quantum mechanics to a great extent.

💡 Research Summary

The paper tackles the long‑standing debate on whether the quantum state (the wave function) represents an element of reality or merely an observer’s information. The author adopts a “bottom‑up” strategy, starting from classical mechanics and progressing to quantum mechanics by drawing an analogy with the transition from geometrical optics (eikonal equation) to full electromagnetic wave optics.

In the first part the author revisits the eikonal equation, showing that it can be derived as the short‑wavelength (λ̄→0) limit of the linear scalar wave equation that follows from Maxwell’s equations. Because the full wave equation is linear, any superposition of solutions is again a solution; the eikonal limit, being nonlinear, does not admit superposition. This illustrates how a richer set of possible fields exists in the wave description, while the ray description is a restricted subset.

The second part mirrors this construction in mechanics. The Hamilton‑Jacobi (HJ) equation, S(q,t), is a first‑order nonlinear partial differential equation that encodes classical trajectories. By introducing a complex exponential ansatz ψ = exp(iS/ħ) the author claims that the HJ equation can be rewritten as a linear wave‑type equation. If ψ is allowed to be any square‑integrable (L²) function, the superposition principle becomes available, and the resulting linear equation is identified with the Schrödinger equation. The author further argues that in the limit ħ → 0 the Schrödinger equation reduces back to the classical HJ equation, completing the circle.

From this construction the author draws several philosophical and interpretational conclusions:

1. The wave function can be regarded as an “objective” field, just as the classical “wave function” (the HJ solution) is.
2. Phenomena usually considered uniquely quantum—wave‑function collapse, entanglement, measurement paradoxes—are in fact dormant manifestations of the non‑linearity of the underlying classical wave equation; they become explicit only after the linearisation that yields the Schrödinger equation.
3. Spin is interpreted as a consequence of the multi‑component nature of the wave function, rather than as a fundamentally non‑classical degree of freedom.
4. The superposition principle, rather than the quantisation of action, is the essential feature that separates quantum mechanics from classical mechanics.

The paper is written in a pedagogical style, with many historical remarks and a clear intent to make quantum mechanics appear as a natural extension of classical mechanics, thereby demystifying it for students.

Critical assessment:

Mathematical rigor: The transition from the HJ equation to a linear wave equation hinges on the insertion of Planck’s constant ħ and the exponential substitution ψ = exp(iS/ħ). The author does not provide a systematic derivation of this step, nor does he discuss the conditions under which the resulting ψ satisfies the Schrödinger equation (e.g., the need for a potential term, ordering ambiguities, and the role of the Laplacian). Consequently, the claim that the Schrödinger equation “emerges” from a merely reformulated HJ equation is not fully substantiated.

Physical constraints: Allowing any L² function as a valid wave function ignores the boundary and continuity conditions that arise from the underlying differential equations (e.g., square‑integrability plus differentiability, matching conditions at interfaces). In quantum mechanics, the Hilbert space structure also encodes the inner‑product that yields probabilities; the paper does not address how these probabilistic interpretations survive the classical limit.

Non‑linearity argument: While it is true that the eikonal and HJ equations are nonlinear, the statement that non‑linearity “forbids” superposition is oversimplified. Certain nonlinear equations admit linear combinations of particular solutions (e.g., soliton superposition in integrable systems). Moreover, the classical limit of quantum mechanics is usually obtained via the WKB approximation, which retains the linear Schrödinger equation but expands the phase in powers of ħ; the paper’s approach bypasses this well‑established route.

Spin and internal degrees of freedom: Interpreting spin solely as a multi‑component wave function neglects the algebraic structure of SU(2) and the necessity of Pauli matrices or spinor representations. These features have no direct analogue in the scalar HJ framework, so the claim that spin is “not wholly strange” is not convincingly demonstrated.

Overall contribution: The manuscript offers an interesting pedagogical narrative that highlights analogies between optics and mechanics and attempts to place quantum mechanics on a continuum with classical theory. However, the lack of rigorous derivations, insufficient treatment of boundary conditions, and oversimplified philosophical conclusions limit its impact. Future work would benefit from a more detailed mathematical bridge (e.g., explicit WKB derivation, inclusion of potentials, and a careful discussion of the role of ħ) and from addressing how measurement theory and entanglement arise—or fail to arise—in the proposed classical‑wave picture.

In summary, the paper succeeds in presenting a coherent story that many “quantum puzzles” have classical precursors, but it falls short of providing the quantitative and conceptual depth required to substantiate the claim that the reality of the quantum state can be settled by this classical‑to‑quantum wave analogy.