Interpreting Bohm quantum potentials in Computing quantum waves exactly from classical action

This technical note extends a previous proof to explicitly include the Bohm quantum potential, demonstrating that while different initializations (Feynman kernel versus standard Madelung) lead to different action and density solutions where the potential may or may not vanish, the resulting overall quantum wave remains independent of this computational choice.

The Gist

The Big Picture: A Misunderstanding About "Ghost Forces"

Imagine you are trying to predict how a wave of water moves across a pond. In the world of quantum physics, there is a famous way to do this called the Madelung approach. It treats the quantum wave like a fluid. However, this fluid has a weird, invisible "ghost force" pushing it around, called the Bohm quantum potential. This force is necessary to make the math work if you start with a specific, complex shape of water.

Recently, someone criticized a paper by Lohmiller and Slotine (let's call them "The MIT Team"). The critic said, "Hey, your proof is missing this ghost force! You can't just ignore it."

The MIT Team's response in this paper is: "We aren't ignoring it. We are starting the race from a different starting line where that ghost force doesn't exist at all. Because of how we set up our initial conditions, the force is mathematically zero, not because we forgot it, but because it's unnecessary for our specific method."

The Two Different Starting Lines

To understand why they say the ghost force is zero, you have to look at how they start their calculations compared to the standard method.

1. The Standard Method (The Madelung Solution)

• The Analogy: Imagine you have a bucket of water and you pour it onto the ground all at once. The water spreads out in a complex, uneven puddle immediately.
• The Math: You start with a known, complex wave shape ($\psi$). When you break this down into a "density" (how much water is where), that density is messy and changes in space.
• The Result: Because the water is uneven, the "ghost force" (Bohm potential) is strong and necessary to explain why the water moves the way it does.

2. The MIT Team's Method (The Feynman Kernel)

• The Analogy: Instead of pouring a bucket, imagine you have a single, tiny drop of water at a specific point. You then imagine thousands of tiny paths radiating out from that single drop.
• The Math: They start with a single point (or a specific momentum) and calculate the path to the destination. Crucially, they initialize the "density" of these paths as a perfectly flat, constant sheet.
• The Result: If your water is a perfectly flat, uniform sheet, there are no bumps or unevenness to create a "ghost force." The math shows that in this specific setup, the Bohm potential is exactly zero.

The "Time Travel" Trick

The paper gets a bit technical in the middle, discussing how to prove this zero-force result even when the paths get complicated (like in a gravitational field or a harmonic oscillator).

• The Problem: Sometimes, as the paths spread out, the "flatness" might seem to get distorted, which would bring the ghost force back.
• The Solution: The authors use a clever mathematical trick involving time. They suggest that instead of using a single clock for the whole universe, every single point in space can have its own "local clock" that ticks at a different speed.
• The Metaphor: Imagine a group of runners on a track. If they all run at the same speed, they stay in a line. If the track curves, they might spread out. But, if you tell each runner to adjust their own watch so that, according to their watch, they are always running in a perfect line, the math stays simple.
• By rescaling time this way (a concept borrowed from d'Alembert), they ensure the density remains "flat" in the eyes of the math, keeping the Bohm potential at zero.

Why This Matters for Their Examples

The paper lists many famous physics examples: the double-slit experiment, the hydrogen atom, tunneling, and the Pauli/Dirac/Maxwell equations.

• The Critic's Fear: "You calculated the hydrogen atom without the ghost force. You must be wrong."
• The Team's Rebuttal: "We calculated the hydrogen atom by starting with a single point and expanding it out (using a Taylor expansion of the kernel). Because we started with that specific 'flat' initialization, the ghost force was never there to begin with. We didn't delete it; we never needed to add it."

They emphasize that they didn't just "import" the known answers from quantum mechanics. They derived them from scratch using classical action, and the math naturally led to the correct quantum results without the extra term.

The Bottom Line

This paper is a technical defense. It says:

1. Yes, the Bohm quantum potential is real in the standard way of doing things (starting with a complex wave).
2. But, in the specific method used in their previous paper (starting with a single point and a constant density), the math naturally results in that potential being zero.
3. Therefore, their previous calculations were correct, and the critic misunderstood the difference between the two starting methods.

It's like someone accusing a chef of forgetting salt in a soup. The chef replies, "I didn't forget it; I used a different recipe that starts with a broth that is already perfectly seasoned, so I didn't need to add any salt."

Technical Summary

Technical Summary: Interpreting Bohm Quantum Potentials in "Computing quantum waves exactly from classical action"

Problem Statement
This technical note addresses a critique raised in a recent arXiv posting [11] regarding the proof of Lemma 3.1 in the authors' previous work [7]. The critique argues that the proof in [7] is incomplete because it fails to explicitly account for the Bohm quantum potential [1, 2] derived from the Madelung partial differential equation (p.d.e.) [9]. The authors acknowledge that while the continuity and Hamilton-Jacobi equations extended by the Bohm potential are undisputed, the specific solutions for action and density depend critically on their initialization at $t=0$. The core issue is a discrepancy between the initialization used in [7] (motivated by the Feynman kernel [4]) and the standard initialization of the Madelung solution [9]. This note aims to extend the proof of Lemma 3.1 to explicitly introduce the Bohm quantum potential and demonstrate why this term vanishes in the specific wave construction used in [7] without loss of generality.

Methodology
The authors employ a derivation in fixed coordinates to make the appearance of the Bohm quantum potential explicit. The methodology involves:

1. Re-evaluating Lemma 3.1: The proof is extended to show that the Schrödinger equation can be solved on each extremal branch $j$ using a Feynman kernel $\psi_j$. This kernel connects a single initial point $(x_o \dashv p_o)$ to any endpoint $x$ via multiple paths.
2. Comparing Initializations: The authors contrast two approaches:
   • Standard Madelung Approach: Derives density $\rho$ and action $\phi$ from a known wavefunction $\psi = \sqrt{\rho}e^{i\phi/\hbar}$. This assumes specific initial distributions $\rho(x,0) = |\psi|^2$ and $\phi(x,0) = \hbar \arg(\psi)$, typically resulting in a non-zero Bohm potential $Q$.
   • Feynman Kernel Approach (Used in [7]): Initializes the multi-paths with a common constant density $\sqrt{\rho_{oj}}$ at $t=0$. The action is initialized such that $\phi(x_o, 0) = 0$ and is undefined elsewhere.
3. Analyzing the Density Evolution: The authors demonstrate that under the Feynman kernel initialization, the density $\rho_j$ for each branch is purely time-dependent. Consequently, the spatial Laplacian of the square root of the density, $\Delta_M \sqrt{\rho_j}$, vanishes.
4. Handling Non-Quadratic Cases: For cases where the Laplacian of the action $\Delta_M \phi_j$ might depend on position (non-quadratic potentials), the authors invoke a time rescaling technique suggested by d'Alembert [3, 5]. By introducing a transformed time variable $t'_j$ via a time transport equation, they ensure that the density remains a function of time only, thereby maintaining the vanishing of the Bohm potential.

Key Contributions

• Explicit Extension of Lemma 3.1: The paper provides a rigorous proof that the Bohm quantum potential $Q_j = -\frac{\hbar^2}{2} \frac{\Delta_M \sqrt{\rho_j}}{\sqrt{\rho_j}}$ is identically zero for the Feynman kernel formulation used in [7].
• Clarification of Initialization Differences: The authors clarify that the absence of the Bohm potential in their results is not an omission but a direct consequence of initializing the system with a constant density along action branches, a condition fundamentally different from the standard Madelung density solution.
• Resolution of Specific Critiques: The note addresses specific claims in [11] regarding the harmonic oscillator (Example 3.9) and the Kepler problem (Example 3.10). It confirms that the purely time-dependent density derived in [7] justifies the removal of the Bohm potential in these cases.
• Correction of Misunderstandings: The authors refute the claim that the Laplacian in the double-slit example was calculated incorrectly, citing the Laplace-Beltrami tensor relation in spherical coordinates to show that $\Delta_M \sqrt{\rho_j} = 0$ holds even at $r=0$.

Results

• Vanishing Bohm Potential: The extended proof confirms that for the specific construction of the wave $\psi_j$ via the Feynman kernel, the Bohm quantum potential is exactly zero. This is because the density $\rho_j$ depends only on time, making its spatial derivatives zero.
• Independence of Overall Wave: While the computational initialization (and thus the presence or absence of the Bohm potential in intermediate steps) differs from the standard Madelung approach, the resulting overall wave $\psi(x, t)$ obtained by integrating over all initial conditions remains independent of this choice.
• Validation of Previous Examples: The results validate the calculations in [7] for various quantum systems (double slit, Aharonov-Bohm, potential well, tunneling, harmonic potential, hydrogen atom, EPR) and the derivations of the Pauli, Dirac, and Maxwell equations. These examples recover known exact quantum results without requiring a non-zero Bohm potential term.
• Systematic Derivation: The paper reiterates that the quantum eigenwaves in the examples are obtained systematically from a Taylor expansion of the kernel, rather than being imported a priori from existing quantum wave solutions.

Significance
The paper serves as a technical clarification rather than a new theoretical framework. Its primary significance lies in defending the mathematical consistency of the method presented in [7]. By explicitly deriving the Bohm potential and showing it vanishes due to the specific initialization of the Feynman kernel, the authors demonstrate that their approach is not missing a critical term but is operating under a different, valid set of initial conditions. The note asserts that the multi-valued action decomposition (Theorem 2.4) in [7] allows for a constant density along each action branch, which is the key mechanism enabling the exact removal of the Bohm quantum potential. This distinction separates their method from standard Madelung calculations where the potential is typically non-zero. The authors conclude that this clarification resolves the concerns raised in [11] and confirms the correctness of the classical action-based computations of quantum waves presented in their prior work.