Reply to "Interpreting Bohm quantum potentials in `Computing quantum waves exactly from classical action'"

This paper refutes the claims made by Lohmiller and Slotine regarding the exact equivalence between classical action and the Schrödinger equation, demonstrating that their proposed position-dependent time transformation violates the multivariable chain rule and that their framework remains limited to the well-known semiclassical Van Vleck propagator, which is exact only for quadratic potentials.

The Gist

Imagine you are trying to predict how a ripple moves across a pond. In the world of quantum physics, this ripple is described by a complex equation called the Schrödinger equation. For a long time, scientists have known that for simple, smooth ponds (like a perfect bowl of water), you can predict the ripple's path using simple, straight-line rules from classical physics. But for ponds with weird, bumpy bottoms (complex forces), those simple rules usually fail.

Recently, a team of researchers (Lohmiller and Slotine) claimed they had found a "magic trick" to make those simple rules work for any pond, even the bumpy ones. They argued that the mysterious "quantum force" (called the Bohm quantum potential) that usually messes up the simple predictions actually disappears if you just change how you measure time.

Gábor Vattay, the author of this paper, is here to say: "That magic trick doesn't work."

Here is a simple breakdown of his argument:

1. The "Magic Clock" Trick

The researchers claimed that if you give every single spot in space its own personal clock (a "local time"), you could make the math work out so that the messy quantum force vanishes.

Vattay's Analogy: Imagine you are driving a car through a city with traffic. The researchers say, "If we just slow down the clock in the heavy traffic and speed it up in the empty streets, the car will feel like it's driving on a perfect, empty highway with no traffic at all."

2. The Math Mistake (The Chain Rule)

Vattay points out that this logic fails because of a basic rule of calculus (the multivariable chain rule).

The Analogy: Even if you change the speed of the clock in different parts of the city, the road itself hasn't changed. The traffic jams (the bumps in the potential) are still physically there.

• If you try to calculate how fast the car is moving relative to the actual road (the physical space), you have to account for the fact that the clock is ticking at different rates in different places.
• Vattay shows that when you do the math correctly, the "traffic" (the spatial changes in the wave) doesn't disappear just because you changed the clock. The "bumps" in the road still create a "bump" in the wave.

3. The "Special Case" Confusion

The researchers tested their idea on a few specific examples, like a harmonic oscillator (a perfect spring). In these specific cases, their math did work.

The Analogy: It's like someone claiming they invented a new way to fly that works for any vehicle. They prove it works for a bicycle (which is easy) and a unicycle (also easy), and then they say, "See? It works for everything!"
Vattay explains that their method is actually just the old, well-known "Van Vleck" method. This method is famous for working perfectly only for simple, smooth shapes (like springs or free-falling objects). It has never worked for complex, bumpy shapes (like the electric pull of an atom).

4. The Conclusion

Vattay concludes that:

• You cannot erase the "quantum force" just by changing your clock.
• The "magic trick" is actually just a re-discovery of an old approximation that only works for simple, smooth situations.
• For complex, real-world quantum systems, their method is not an exact solution; it's just a rough guess (the WKB approximation) that happens to look exact only in very specific, simple cases.

In short: The researchers tried to solve a complex puzzle by changing the rules of time, but Vattay shows that the puzzle pieces still don't fit unless the picture is already simple. The "quantum weirdness" is still there, and you can't mathematically wave it away.

Technical Summary

Technical Summary: Reply to "Interpreting Bohm quantum potentials in 'Computing quantum waves exactly from classical action'"

Problem Statement
This paper addresses a recent claim by Lohmiller and Slotine (arXiv:2605.20443) asserting an exact equivalence between classical action and the Schrödinger equation for arbitrary potentials, specifically arguing that the Bohm quantum potential ($Q$) vanishes in their formulation. The authors of this reply contend that Lohmiller and Slotine's attempt to resolve the omission of the quantum potential for non-quadratic potentials relies on a mathematically flawed position-dependent time transformation. The core problem is the assertion that a local time coordinate $t'_j(x, t)$ can eliminate the spatial derivatives of the probability density amplitude, thereby nullifying the quantum potential in the physical reference frame.

Methodology
The paper employs a rigorous application of multivariable calculus to re-evaluate the mathematical framework proposed by Lohmiller and Slotine. The analysis focuses on two primary areas:

1. Chain Rule Verification: The authors examine the proposed transformation where the probability density amplitude is parameterized solely by a local clock $t'_j(x, t)$. They apply the multivariable chain rule to compute the spatial gradient ($\nabla_x$) and Laplacian ($\nabla^2_x$) of the density in the physical laboratory frame.
2. Comparison with Established Theory: The paper contrasts the proposed method with the well-established Van Vleck-Pauli-Morette semiclassical propagator. It analyzes the conditions under which semiclassical approximations yield exact results, specifically testing the claim against non-quadratic potentials (e.g., Coulomb potential) versus quadratic potentials (e.g., harmonic oscillator).

Key Contributions and Results

• Refutation of the Time Transformation: The analysis demonstrates that the authors' reasoning is analytically invalid. Because the proposed time transformation $t'_j(x, t)$ explicitly depends on spatial coordinates to compensate for non-uniform potentials, the spatial gradient of the time coordinate is non-zero ($\nabla_x t'_j \neq 0$). Consequently, the chain rule dictates that the spatial gradient of the density in the physical frame is finite:
  $$\nabla_x R(t'_j) = \frac{dR}{dt'_j} \nabla_x t'_j$$
  Furthermore, the spatial Laplacian does not vanish:
  $$\nabla^2_x R(t'_j) = \frac{d^2R}{dt'^2_j} (\nabla_x t'_j)^2 + \frac{dR}{dt'_j} \nabla^2_x t'_j$$
• Persistence of the Quantum Potential: The results show that substituting these non-zero derivatives back into the Schrödinger equation restores the Bohmian quantum potential ($Q = -\frac{\hbar^2}{2M} \frac{\nabla^2_x \sqrt{\rho}}{\sqrt{\rho}}$) in its entirety. The mathematical maneuver of shifting to a spatially dependent clock fails to erase the spatial curvature of the probability amplitude in the physical frame.
• Identification of the Framework: The paper concludes that the mathematical framework presented by Lohmiller and Slotine is indistinguishable from the standard semiclassical Van Vleck propagator. It is a known theorem that this propagator is exact if and only if the Hamiltonian is at most quadratic in position and momentum.
• Limitation to Quadratic Potentials: The analysis confirms that the examples provided by Lohmiller and Slotine (such as the harmonic oscillator) succeed only because they fall into the specific category of quadratic potentials where the semiclassical approximation is naturally exact. For non-quadratic systems, the methodology re-derives the WKB approximation rather than providing an exact solution, as it restricts the formulation to classical local extrema and ignores the integration over non-classical paths required for exact quantum propagation.

Significance and Claims
The paper claims to provide a definitive mathematical correction to the interpretation of the relationship between classical action and quantum waves. Its significance lies in:

1. Correcting a Misinterpretation: It clarifies that the Bohm quantum potential cannot be mathematically eliminated by coordinate transformations that depend on spatial position.
2. Reaffirming Established Limits: It reinforces the rigorous boundary of semiclassical methods, confirming that exact equivalence between classical action and the Schrödinger equation is restricted to quadratic potentials.
3. Methodological Rigor: It asserts that the proposed "exact" solution for arbitrary potentials is, in fact, a restatement of the standard semiclassical approximation, which is inherently approximate for non-linear potentials.

The paper does not propose new applications or experimental tests; rather, it serves as a technical rebuttal to ensure the mathematical consistency of semiclassical wave propagation theories.