How does CSD differ from Bohmian mechanics?

The short answer

Bohmian mechanics is the standing proof that deterministic quantum theories exist. In it, particles always have definite positions, the wavefunction acts as a real field that guides their motion, and the resulting statistics reproduce those of quantum mechanics, so any other deterministic programme owes an account of how it differs, and the differences here are structural rather than cosmetic.

The first difference is the arena, the space of possible states of the world. Bohm's arena is the space of particle configurations, with the wavefunction as an extra field on top, and CSD's arena isn't a configuration space at all. It's a compact geometric space (finite in extent, so nothing can escape to infinity) that projects onto the space of quantum states, so every point of the arena maps to a quantum state, with many points mapping to each one, and the quantum state is the output of that projection and not an additional field with dynamics of its own.

The second difference is the measure, the rule that assigns sizes, and hence probabilities, to regions of the arena. Bohm's statistics require quantum equilibrium, the assumption that particle positions are distributed according to the squared amplitude of the wavefunction, and CSD's counterpart is a theorem, as its measure is the only one the symmetries of the state space leave unchanged, so a distributional assumption is replaced by a uniqueness proof.

The third difference is the record of an outcome. In Bohmian mechanics an outcome is a particle position, and in CSD an outcome is membership of a basin, a region of the arena whose trajectories all flow to the same result, and that membership is written into the fibre, the extra directions of the arena that lie above each quantum state. The records come with their own stability theorem.

The longer answer

The programme's respect for Bohm is on record, as Bohmian mechanics is the standing counterexample to the claim that hidden variables are dead, and so is the divergence. CSD keeps the quantum state epistemic in function, a description of what's known about the underlying arena rather than a physical agent, where Bohm makes it a dynamical agent that guides the particles. Contextuality, the dependence of an outcome on what else is measured alongside it, lives in CSD in outcome maps fixed by the apparatus, where in Bohmian mechanics it lives in the contextual responses of the guidance dynamics. And the equilibrium question, why the right measure is the right one, is answered by symmetry-forced uniqueness rather than by hypothesis or by arguments that the distribution relaxes towards equilibrium over time.

No experiment currently separates the two theories, and neither side claims one. The separation lies in what must be assumed, and in what has been machine-checked, where the comparison is one-sided.

What is actually proved

The contrast is carried by the corpus's uniqueness and reconstruction theorems rather than by any formalisation of Bohmian mechanics itself, so it's fsMeasure_unique versus the quantum equilibrium hypothesis, the projection postulate carried as a hypothesis versus the guidance equation carried as dynamics, and the record layer's basin machinery versus position as the universal pointer.

Related entries

• Bohmian mechanicsBohmian mechanics is CSD's nearest neighbour, deterministic, with definite states, and openly non-local, and the dispute between them is over where the probabilities come from.
• Is CSD a hidden-variable theory?Yes, in the technical sense, as something definite underlies the quantum state in CSD, and the programme openly pays the two prices the no-go theorems demand, dependence on measurement context and correlations beyond what locality permits.
• Single trajectoryThere's one world and it follows one path, so probability lives in how experiments are repeated and not in how the world evolves, and theorems rather than wording keep the two apart.
• DeterminismThe substrate has no dice, so the spread in the data is variation in what your preparation couldn't pin down.
• Hidden variablesThe no-go theorems don't kill hidden variables, as they price them, and CSD reads the price tag and pays it.
• Nelson stochastic mechanicsNelson derived the Schrodinger equation from a diffusion, so the same continuity equation that lets Bohm read a velocity off the wavefunction lets Nelson read a drift with noise against it, and the Born density solves both.