⚛️ quantum physics

A Bohmian version of a 2-state quantum system

The paper constructs a stochastic Bohmian model for a two-state quantum system where a definite physical state evolves under the guidance of the quantum state, arguing that the resulting probabilities are well-defined and refuting Gillespie's (1994) claims that such Markovian formulations are impossible.

Original authors: Matthew Dickau

Published 2026-07-28

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Matthew Dickau

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand the rules of a game where the pieces don't just sit still or move in a straight line; they seem to be in two places at once, or they change their nature depending on how you look at them. This is the strange world of quantum mechanics, the branch of physics that describes how the tiniest particles in the universe behave. For decades, scientists have debated how to make sense of this "fuzziness." One popular way to think about it is called "Bohmian mechanics." In this view, particles do have a definite position and a definite path, like a car driving down a road. However, there's a catch: the car's path is guided by a mysterious, invisible "wave" that tells it where to go. This wave isn't just a map; it's a force that can make the car jump around in ways that look random, even though the wave itself follows strict, predictable rules.

The big question this paper tackles is whether this "guided jumping" idea actually works without breaking the rules of math. Some critics have argued that if you try to make a particle jump between two states (like a light switch being on or off) based on these quantum rules, the math eventually explodes. They say the "jump rates"—how likely the particle is to switch—would shoot up to infinity at certain moments, making the whole theory impossible to use. If the math breaks, the theory breaks. But what if the critics were looking at the math wrong? What if the "infinity" isn't a crash, but just a moment where the rules need a tiny bit of special handling?

This paper, written by Matthew Dickau, steps into that debate to defend the Bohmian version of a simple two-state system. The author builds a mathematical model where a particle has a definite state (either 0 or 1) but jumps between them stochastically (randomly) under the guidance of a quantum wave. The paper argues that, contrary to the claims of critics like Gillespie, the probabilities in this model are actually well-defined and do not break down. The author shows that while the "jump rates" might look like they go to infinity at specific moments when the particle is definitely in one state, this is a natural feature of the math, not a fatal error. By carefully tracking how the probabilities evolve over time, the paper demonstrates that the system remains consistent and that the "infinite" moments are just points where the particle is certain to be in one state, making the rate of leaving that state momentarily undefined but harmless to the overall picture.

The Story of the Two-State Dancer

To understand what Dickau is doing, let's imagine a dancer on a stage with only two spots: Spot 0 and Spot 1. In the standard quantum view, this dancer is a ghostly blur, standing in both spots at the same time with varying degrees of "ghostliness." But in Dickau's Bohmian version, the dancer is a real person who is always standing firmly on either Spot 0 or Spot 1. The mystery is: how does the dancer decide to jump from one spot to the other?

The answer lies in the "quantum wave," which acts like a choreographer. This choreographer doesn't just tell the dancer where to stand; it dictates the rate at which the dancer should jump. If the choreographer is shouting "Jump now!" the dancer might leap. If they whisper, the dancer stays put. The paper sets up a specific scenario where the choreographer is a simple, oscillating rhythm (a Hamiltonian that makes the system swing back and forth). The goal is to make sure that if you watch the dancer over a long time, the percentage of time they spend on Spot 0 matches exactly what standard quantum mechanics predicts.

The Problem of the "Infinite Jump"

Here is where the trouble started. A critic named Gillespie looked at this setup and said, "Wait a minute! At certain times, the choreographer's instructions become impossible to follow."

Gillespie argued that to match the quantum predictions, the "jump rate" (the speed at which the dancer is told to switch spots) would have to become infinite at specific moments. In the world of standard probability, if a rate is infinite, the math breaks. It's like a traffic light that turns green for an infinite number of cars per second; the road would instantly clog, and the system would collapse. Gillespie claimed this proved that you couldn't have a Bohmian version of a two-state system because the probabilities wouldn't be well-defined.

The Paper's Solution: It's Not a Crash, It's a Pause

Dickau's paper says, "Hold on. Gillespie is looking at the math too strictly."

The author explains that the "infinite rate" only happens at the exact moment when the dancer is certain to be in one spot (say, Spot 0) and the quantum wave says, "Okay, now you must be in Spot 1." If the dancer is 100% sure they are on Spot 0, the probability of them jumping away from Spot 0 is technically undefined because they haven't left yet. It's like asking, "How fast is the car moving the exact instant it is parked?" The answer isn't a broken number; it's just a moment where the concept of "speed" needs to be handled carefully.

Dickau shows that if you look at the actual probability of the dancer being in Spot 1 over a tiny slice of time, it works out perfectly fine. The "infinite rate" is just a mathematical artifact that appears when you try to divide by zero (the probability of being in the starting spot is zero). But the paper proves that if you calculate the total probability of the jump happening over a small interval, it comes out to a sensible, finite number.

Think of it like a video game character who is stuck in a wall. If you ask, "How fast is the character moving through the wall?" the answer might seem infinite or broken. But if you look at the game's code, you see that the character simply doesn't move until the wall disappears. The "infinite speed" is just the code's way of saying, "Wait here until the next frame." Dickau demonstrates that the Bohmian model handles these "stuck" moments correctly. The probabilities remain well-defined, the dancer eventually makes the jump, and the quantum predictions are perfectly matched.

The Verdict

The paper concludes that the critics were wrong to say the model is broken. The "divergence" (the infinite numbers) that Gillespie pointed out is a known, manageable feature of the math, not a fatal flaw. The author rigorously derives the equations for how the particle jumps and shows that even at the tricky moments where the particle is certain to be in one state, the system behaves logically.

While this proof is specific to a simple two-state system (like a single qubit or a light switch), the author suggests there is no reason to believe that more complex systems would behave differently. The paper doesn't claim to have solved all of quantum mechanics, but it successfully defends the idea that a Bohmian version of a two-state system is mathematically sound and that the probabilities are, indeed, well-defined. The "ghostly" quantum predictions can be matched by a "real" particle jumping around, provided you know how to read the choreographer's instructions correctly.

Technical Summary: Bohmian Version of a 2-State Quantum System

Problem Statement
The paper addresses a specific criticism leveled against stochastic extensions of Bohmian mechanics, particularly regarding two-state quantum systems. In a prior exchange, Gillespie [2, 4] argued that models proposing a definite physical state (such as a "clock variable" or a Bohmian particle) guided stochastically by a quantum wavefunction suffer from a fundamental flaw: the probability jump rates required to reproduce quantum predictions diverge to infinity at certain times (specifically when the quantum probability of a state vanishes). Gillespie contended that such divergences render the probabilities ill-defined and violate the axioms of probability, thereby suggesting that quantum mechanics cannot be formulated as a Markov process. This critique has been cited as an obstacle to extending Bohmian mechanics to quantum field theory and other stochastic physical state models.

Methodology
The author constructs a rigorous Bohmian model for a simple two-state system (a qubit) with basis states $|0\rangle$ and $|1\rangle$ evolving under a specific Hamiltonian $H = \frac{\omega}{2}(|0\rangle\langle 1| + |1\rangle\langle 0|)$ . The system is characterized by:

A deterministic quantum state $|\psi(t)\rangle$ evolving via the Schrödinger equation.
A definite physical state $S(t) \in \{0, 1\}$ that jumps stochastically between the two states.

The core of the methodology involves deriving the transition rates $W_{a \to b}(t)$ required to ensure the physical state distribution matches the quantum probabilities $y_a(t) = |\langle a | \psi(t) \rangle|^2$ . The author employs a rigorous probabilistic framework rather than a heuristic one:

Conditional Probabilities: The analysis defines forward-directed conditional probabilities $p_{a \to b}(t, t')$ and utilizes the Markov condition, where future probabilities depend only on the most recent past state.
Rate Derivation: By taking the limit of infinitesimal time steps ( $\delta t \to 0$ ), the author derives differential equations for these conditional probabilities. This leads to the definition of jump rates $W_{a \to b}(t)$ as the time derivative of the conditional probability.
Handling Singularities: The paper explicitly investigates the behavior of these rates when the single-time probability $p_a(t)$ approaches zero. It acknowledges that the conditional probability $p_{a \to b}(t, t')$ is technically undefined when $p_a(t)=0$ , leading to a divergence in the formal definition of the rate $W_{a \to b}(t)$ .
Integration: The author solves the resulting ordinary differential equations (ODEs) for the conditional probabilities using an integrating factor method, demonstrating that the cumulative probabilities remain well-behaved even when instantaneous rates diverge.

Key Contributions and Results

Reinterpretation of Divergent Rates: The paper demonstrates that the divergence of jump rates $W_{a \to b}(t)$ at times when $p_a(t) = 0$ is a mathematically consistent consequence of the rigorous definition of conditional probability, not a violation of probability axioms. The author argues that Gillespie's criticism stems from treating the infinitesimal time step as a fixed positive real number rather than a limit, and from failing to account for the undefined nature of conditional probabilities when the conditioning event has zero probability.
Well-Defined Probabilities: Through the explicit solution of the probability rate equations, the author shows that the integrated conditional probabilities (e.g., $p_{0 \to 1}(0, t)$ ) remain finite, bounded between 0 and 1, and perfectly match the quantum mechanical predictions ( $y_a(t)$ ) throughout the evolution, including at points where the instantaneous jump rate diverges.
Resolution of the "Quadratic" Objection: The paper addresses a secondary criticism regarding the behavior of conditional probabilities being quadratic in $\delta t$ at specific instants (e.g., $t=0$ ). The author shows that this quadratic behavior is instantaneous and does not persist over an interval, thereby avoiding the contradiction Gillespie derived in his Theorem 2. The author provides a corrected expansion of the conditional probability that resolves the alleged inconsistency.
Generalization of Jump Rates: The model allows for a class of jump rates defined by an arbitrary non-negative "excess flux" $J'(t)$ , showing that the well-defined nature of the probabilities holds for the minimal rates (where $J'=0$ ) as well as for the broader class of rates discussed in previous literature [1].

Significance and Claims
The paper claims to refute the argument that stochastic Bohmian models for two-state systems are mathematically invalid due to divergent jump rates. By providing a rigorous derivation that yields well-defined probabilities despite instantaneous divergences in the rate functions, the author argues that the objection raised by Gillespie does not hold.

The author modestly concludes that while this analysis specifically addresses a two-state system, there is no apparent reason to expect qualitatively different behavior in systems with more than two states. Consequently, the paper asserts that the argument against extending Bohmian mechanics to include stochastic evolution of physical states (such as in quantum field theory) is removed, at least regarding the specific issue of ill-defined probabilities in simple systems. The work serves as a defense of the mathematical consistency of such models against the specific critique of divergent rates.

The Story of the Two-State Dancer

The Problem of the "Infinite Jump"

The Paper's Solution: It's Not a Crash, It's a Pause

The Verdict

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