De Broglie–Bohm Pilot Wave Theory

Bohmian mechanics answers a question the orthodox account refuses to ask: where is the particle? In de Broglie's 1927 pilot-wave proposal, revived and completed by David Bohm in 1952, a system of $N$ particles is described by two things at once — a wavefunction $\psi(\mathbf q,t)$ living on configuration space, and an actual configuration $\mathbf Q(t)$ of definite particle positions. The wave guides; the particles move on trajectories. Determinism is restored: fix $\psi$ and the initial $\mathbf Q$ , and the entire future is settled. The dynamics has exactly two ingredients, and the first is the ordinary many-body Schrödinger equation — nothing is altered there. The second is the guidance equation, a first-order law whose velocity field is the quantum probability current divided by the density, $\mathbf v = \mathbf j/|\psi|^2 = (\nabla S)/m$ . Because that velocity is built from the same current that satisfies the continuity equation — recalled by name from Probability Current in One Dimension — an initial $|\psi|^2$ distribution of configurations is carried along so as to stay $|\psi|^2$ . This single structural fact, equivariance, is why the theory reproduces every statistical prediction of standard quantum mechanics without a separate Born postulate. Two payoffs and one price are owned here. Writing $\psi = R\,e^{iS/\hbar}$ splits Schrödinger's equation into a continuity equation and a quantum Hamilton–Jacobi equation carrying an extra quantum potential $Q = -\tfrac{\hbar^2}{2m}\nabla^2R/R$ — all interference, tunnelling, and entanglement live in $Q$ . Measurement needs no axiom: the pointer is itself a Bohmian particle that lands in one wavepacket, an effective collapse with nothing added. The price, made unavoidable by Bell's theorem, is manifest nonlocality — the guidance velocity of one particle depends instantaneously on the positions of distant entangled partners.

Key takeaways

Bohmian mechanics adds definite particle positions $\mathbf Q(t)$ to $\psi$ : the wave obeys the usual Schrödinger equation, the particles follow the guidance equation $\dot{\mathbf Q} = \mathbf j/|\psi|^2 = \nabla S/m$ .
Polar decomposition (owned): $\psi = R\,e^{iS/\hbar}$ turns Schrödinger's equation into a continuity equation plus a quantum Hamilton–Jacobi equation with the quantum potential $Q = -\tfrac{\hbar^2}{2m}\nabla^2R/R$ ; all non-classical physics lives in $Q$ .
Equivariance (owned): if positions are $|\psi|^2$ -distributed at $t=0$ , the continuity equation keeps them $|\psi|^2$ forever — quantum equilibrium — so the theory reproduces the Born rule exactly, no extra postulate.
Manifest nonlocality (owned): for entangled particles the velocity of one depends instantaneously on the other's position; this is exactly the nonlocality Bell's theorem forces on any hidden-variable theory that matches quantum mechanics (developed in Bell's Theorem and the CHSH Inequality by name).
No-signalling survives: quantum equilibrium hides the nonlocality statistically (marginals are untouched by a distant operation), recalled from The Reduced Density Matrix and Partial Trace.
Worked: a moving Gaussian gives $v=\hbar k/m$ (group velocity) and $Q(0)=+\hbar^2/2ma^2$ ; an entangled $\psi=e^{ix_1x_2}$ gives $v_1=x_2$ — the velocity of particle 1 literally is the position of particle 2.
Key idea: Bohmian mechanics recovers all of quantum mechanics as the statistics of definite trajectories guided by $\psi$ — deterministic and realist, at the mandatory cost of explicit nonlocality.

The core insight

De Broglie–Bohm theory keeps the Schrödinger equation untouched and adds actual particle positions guided by $\mathbf v = \nabla S/m$ . Equivariance makes an initial $|\psi|^2$ ensemble stay $|\psi|^2$ , so the Born rule is a theorem, not an axiom. The polar form exposes a quantum Hamilton–Jacobi equation with a quantum potential $Q$ ; the classical limit is $Q\to0$ . The theory is deterministic, realist, and — as Bell's theorem requires of any empirically adequate hidden-variable theory — explicitly nonlocal.

Where this lives today

Pilot-wave theory is far from a museum piece. As a computational tool, Bohmian trajectories are used in quantum chemistry and molecular dynamics to give an intuitive, trajectory-based picture of tunnelling, nonadiabatic transitions, and reaction dynamics — the quantum potential becomes a practical bookkeeping device. In hydrodynamic quantum analogues, millimetric oil droplets bouncing on a vibrating bath (Couder–Fort "walkers") were shown to self-propel guided by their own wave field, reproducing single- and double-slit statistics and quantized orbits — a macroscopic, classical caricature of pilot-wave dynamics that reignited interest in the picture (though the analogy is not exact). And in foundations, the modern Dürr–Goldstein–Zanghì formulation put quantum equilibrium and the origin of the Born rule on a rigorous "typicality" footing, and Bohmian mechanics remains the sharpest existence proof in the debate over whether quantum mechanics can be given a realist, observer-free ontology — the benchmark against which spontaneous-collapse models (developed in Objective Collapse Theories by name) and the no-collapse alternative (The Many-Worlds Interpretation) are measured.

The physical picture

Picture a single electron heading toward a double slit. In the orthodox account it is a wave that mysteriously "becomes" a particle at the screen. In Bohmian mechanics it is, always, a particle at a definite place, plus a wave that fills both slits and beyond. The wave is not a probability cloud that the particle is somewhere inside; it is a real field, on configuration space, that pushes the particle around. The particle rides the wave like a surfer rides a swell — its velocity at each instant is read directly off the local flow of the wave, and where the wave has a crest of current the particle goes fast, where the wave interferes to a standstill the particle is steered aside. Two features fall out of this image and organize the whole theory. First, the trajectories are choreographed by the phase of $\psi$ : the velocity is the gradient of the phase $S$ divided by the mass, so the particle always moves perpendicular to surfaces of constant phase, exactly along the probability current. Second, the amplitude $R$ of $\psi$ exerts a force too — but a strange one, encoded in the quantum potential $Q$ , which depends on the shape of $R$ rather than its size and so can reach undiminished across a laboratory. That reach is the seed of nonlocality: squeeze the wave far away, change its curvature, and $Q$ changes here. The classical world is the corner of this picture where $Q$ is negligible and the trajectories collapse onto Newton's.

Figure

The pilot wave guides a definite particle: velocity is the local probability flow

The dynamics: two laws, one wavefunction

The wavefunction obeys the ordinary many-body Schrödinger equation — that half is unchanged from textbook quantum mechanics and is recalled by name from Time-Dependent Schrödinger Equation. What Bohmian mechanics adds is a law of motion for the particles, and the natural candidate is the one whose consistency with the wave equation is automatic: the velocity equal to the probability current per unit density. The guidance equation, for particle $k$ in an $N$ -body system, is

\frac{d\mathbf Q_k}{dt} = \frac{\mathbf j_k}{|\psi|^2}\bigg|_{\mathbf q=\mathbf Q} = \frac{\hbar}{m_k}\,\mathrm{Im}\frac{\nabla_k\psi}{\psi}\bigg|_{\mathbf q=\mathbf Q} = \frac{\nabla_k S}{m_k}\bigg|_{\mathbf q=\mathbf Q} \qquad (1)

where: $\mathbf j_k = (\hbar/m_k)\,\mathrm{Im}(\psi^\ast\nabla_k\psi)$ is the quantum probability current for particle $k$ , $|\psi|^2$ is the configuration-space density, and the third form uses the phase $S$ of $\psi = R\,e^{iS/\hbar}$ . Read it physically: the particle's velocity is the local flow of probability, evaluated at the particle's actual position $\mathbf Q$ . It is first-order — velocity, not acceleration, is fixed by the state — which is why an initial position (not position and momentum) determines the trajectory. This is the whole extra content of the theory; everything else is a consequence.

Derivation AThe polar decomposition: Schrödinger's equation splits into continuity and quantum Hamilton–Jacobi

We substitute $\psi = R\,e^{iS/\hbar}$ (with $R,S$ real) into the one-particle Schrödinger equation and separate real and imaginary parts. The imaginary part is the continuity equation for $|\psi|^2 = R^2$ ; the real part is a Hamilton–Jacobi equation identical to the classical one except for a single extra term, the quantum potential. The strategy exposes exactly where quantum mechanics deviates from classical mechanics — it is all in that one term.

Step 1. Insert the polar form. With $\psi = R\,e^{iS/\hbar}$ the time derivative is $\partial_t\psi = (R_t + iRS_t/\hbar)e^{iS/\hbar}$ and the second space derivative is $\partial_x^2\psi = (R_{xx} + 2iR_xS_x/\hbar + iRS_{xx}/\hbar - RS_x^2/\hbar^2)e^{iS/\hbar}$ . Substituting into $i\hbar\,\partial_t\psi = -\tfrac{\hbar^2}{2m}\partial_x^2\psi + V\psi$ and cancelling the common phase $e^{iS/\hbar}$ leaves a single complex equation in $R$ and $S$ .

Step 2. Take the imaginary part. Collecting the coefficient of $i$ gives $\hbar\big(R_t + \tfrac{1}{2m}(2R_xS_x + RS_{xx})\big)=0$ , which is exactly $\partial_tR^2 + \partial_x\!\big(R^2\,S_x/m\big) = 0$ — the continuity equation for $\rho = R^2 = |\psi|^2$ with the current $j = R^2S_x/m$ . Why this is the same object owned upstream: $j = (\hbar/m)\,\mathrm{Im}(\psi^\ast\psi_x) = R^2S_x/m$ , and the continuity equation $\partial_t|\psi|^2 + \partial_xj = 0$ is recalled by name from Probability Current in One Dimension; the guidance velocity is $v = j/|\psi|^2 = S_x/m$ .

Step 3. Take the real part. The coefficient of $1$ gives, after dividing by $-R$ ,

\partial_tS + \frac{(\partial_xS)^2}{2m} + V + Q = 0, \qquad Q \equiv -\frac{\hbar^2}{2m}\frac{\partial_x^2R}{R} \qquad (2)

the quantum Hamilton–Jacobi equation. Why this is illuminating: without the last term this is the classical Hamilton–Jacobi equation with $S$ the classical action and $p = \partial_xS$ the momentum. The lone quantum term is the <strong>quantum potential</strong> $Q$ , and it depends on the curvature of $R$ relative to $R$ — on the shape of the amplitude, not its magnitude.

Step 4. Read off the classical limit. When $Q$ is negligible next to the kinetic and potential energies, (2) is Newtonian: the trajectories from (1) reduce to classical ones, and quantum mechanics becomes classical mechanics. All interference, tunnelling, zero-point energy, and entanglement are carried by $Q$ . Verified symbolically: substituting $\psi = R\,e^{iS/\hbar}$ into the TDSE and separating gives the imaginary part $\hbar(R_t + (2R_xS_x + RS_{xx})/2m)$ (continuity) and the real part $-R(S_t + S_x^2/2m + V + Q)$ with $Q = -(\hbar^2/2m)R_{xx}/R$ , matched term-by-term.

Go deeper · Derivation B · 1 minEquivariance: why an initial Born distribution stays Born forever

Derivation BEquivariance: why an initial Born distribution stays Born forever

We show that the guidance law is tuned to the continuity equation so that the $|\psi|^2$ distribution is dynamically preserved. This is the load-bearing result: it is why a deterministic theory of trajectories reproduces the probabilistic Born rule with no extra assumption. The engine is that the velocity field of the particles is exactly the velocity field of the probability fluid.

Step 1. Write the transport law for the particle ensemble. If we have a cloud of configurations distributed with density $\rho_{\rm part}(\mathbf q,t)$ , each moving with the guidance velocity $\mathbf v = \nabla S/m$ , then $\rho_{\rm part}$ obeys its own continuity equation $\partial_t\rho_{\rm part} + \nabla\!\cdot(\rho_{\rm part}\,\mathbf v) = 0$ — the standard conservation of a fluid carried by a velocity field.

Step 2. Compare with the wave's continuity equation. From Derivation A (Step 2), the quantum density $|\psi|^2$ obeys $\partial_t|\psi|^2 + \nabla\!\cdot(|\psi|^2\,\mathbf v) = 0$ with the same velocity field $\mathbf v = \nabla S/m$ . The two transport equations are identical in form and share the velocity field.

Step 3. Conclude equivariance. Two densities obeying the same first-order transport equation with the same velocity field, and agreeing at $t=0$ , agree for all $t$ . So if $\rho_{\rm part}(\mathbf q,0) = |\psi(\mathbf q,0)|^2$ then $\rho_{\rm part}(\mathbf q,t) = |\psi(\mathbf q,t)|^2$ for all $t$ . Why it matters: this "quantum equilibrium" hypothesis, once true, stays true, and it makes the Born rule a consequence of the dynamics rather than a postulate — measurement statistics come out exactly as in orthodox quantum mechanics. *Verified symbolically: $\partial_t|\psi|^2 + \partial_x(|\psi|^2 v)$ with $v = S_x/m$ equals the continuity residual of the Schrödinger equation, which vanishes on solutions — the particle-transport and probability-transport equations are the same object.*

Limiting cases

case	result	reading
$Q \ll V, T$	quantum HJ $\to$ classical HJ	trajectories become Newtonian; classical limit
free plane wave $\psi\propto e^{ikx}$	$v = \hbar k/m$ , $Q=0$	particle moves at the group velocity, no quantum force
real $\psi$ (e.g. a bound state)	$v = \nabla S/m = 0$	particle sits still; all energy is quantum-potential energy
product $\psi=\phi_1(\mathbf q_1)\phi_2(\mathbf q_2)$	$v_1$ independent of $\mathbf q_2$	no entanglement $\Rightarrow$ no nonlocality
entangled $\psi(\mathbf q_1,\mathbf q_2)$	$v_1$ depends on $\mathbf q_2$	manifest nonlocality (Bell-required)
ensemble $=\|\psi\|^2$ at $t=0$	stays $\|\psi\|^2$ (equivariance)	Born rule reproduced exactly

Common trapBohmian mechanics is NOT a local hidden-variable theory that "sneaks past" Bell

A common misreading treats pilot-wave theory as the local, deterministic completion of quantum mechanics that Einstein wanted — and therefore as something Bell's theorem should have killed. It is deterministic and it has hidden variables (the positions), but it is emphatically not local: the guidance velocity of one particle depends instantaneously on the configuration of its entangled partners, through the joint phase $S(\mathbf q_1,\mathbf q_2,t)$ . This is not a defect to be patched; it is required. Bell's theorem (developed in Bell's Theorem and the CHSH Inequality by name) proves that any hidden-variable theory reproducing quantum statistics must be nonlocal, and pilot-wave theory is the clean existence proof that such a theory can be built — at exactly the mandated cost. The reassurance is that the nonlocality is not usable for signalling: in quantum equilibrium the observable marginals are untouched by a distant operation (no-signalling, recalled from The Reduced Density Matrix and Partial Trace), so relativity's causal order is respected at the statistical level even though the underlying trajectories are nonlocally coordinated.

Worked examples

Example 1 — a moving Gaussian packet: velocity and quantum potential (confidence builder). Take a one-dimensional packet $\psi = R(x)\,e^{ikx}$ with a Gaussian amplitude $R = e^{-x^2/2a^2}$ and a linear phase $S = \hbar k x$ . Find the guidance velocity and the quantum potential.

Step 1. Guidance velocity. From (1), $v = \partial_xS/m = \hbar k/m$ — a constant. The particle moves at the group velocity $\hbar k/m$ , exactly what a classical particle of momentum $\hbar k$ would do, so the packet's centroid travels ballistically (machine-verified).

Step 2. Quantum potential. With $R = e^{-x^2/2a^2}$ , $\partial_x^2R/R = (x^2 - a^2)/a^4$ , so $Q = -\tfrac{\hbar^2}{2m}(x^2-a^2)/a^4$ . At the packet center $x=0$ , $Q(0) = +\dfrac{\hbar^2}{2ma^2}$ (machine-verified) — a positive quantum potential, a "pressure" that pushes probability outward and is the Bohmian face of a Gaussian's tendency to spread.

Step 3. Read the physics. The linear phase drives uniform motion (the classical part); the curved amplitude supplies the quantum potential that will spread the packet. The narrower the packet (smaller $a$ ), the larger $Q(0)\propto 1/a^2$ — sharper localization costs more quantum-potential energy, the trajectory-level echo of the uncertainty principle.

Example 2 — entanglement makes the velocity nonlocal (exam level). Consider the (unnormalized) two-particle wavefunction $\psi(x_1,x_2) = e^{i x_1 x_2}$ — a state that does not factorize into a product $\phi_1(x_1)\phi_2(x_2)$ , so the two particles are entangled. Compute the guidance velocity of particle 1 and show it depends on where particle 2 is.

Step 1. Velocity of particle 1. With $\hbar=m=1$ , $v_1 = \mathrm{Im}(\partial_{x_1}\psi/\psi) = \mathrm{Im}(i x_2 e^{ix_1x_2}/e^{ix_1x_2}) = \mathrm{Im}(i x_2) = x_2$ . The velocity of particle 1 is literally the position of particle 2.

Step 2. Put in numbers. If particle 2 sits at $x_2 = 0.5$ , then $v_1 = 0.5$ ; if instead $x_2 = 1.7$ , then $v_1 = 1.7$ (both machine-verified). Nothing about particle 1's own position entered — moving the distant partner from $0.5$ to $1.7$ changes how fast particle 1 moves right now.

Step 3. Contrast a product state. For $\psi = e^{ix_1}e^{ix_2^2}$ (a product, hence unentangled), $v_1 = \mathrm{Im}(i e^{ix_1}\cdots/e^{ix_1}\cdots) = 1$ regardless of $x_2$ (machine-verified). So the nonlocality is switched on only by entanglement — exactly the structural link between entangled correlations and Bell nonlocality. The statistical marginals still hide it: averaged over a quantum-equilibrium ensemble, a distant operation on particle 2 leaves particle 1's observable distribution unchanged (no-signalling), and a full CHSH treatment — the bound $2$ for local models versus $2\sqrt2$ for quantum mechanics — is developed in Bell's Theorem and the CHSH Inequality by name.

Figure

The nonlocality switch: entanglement makes particle 1's speed depend on particle 2's position

A wave that thought it was a guide

Origins · 1 minbohmian mechanics and how this was discovered

bohmian mechanics

1927–1952 · Brussels, Princeton, Birkbeck

The pilot wave has an unusual biography: proposed, abandoned by its own author, and revived a quarter-century later by a physicist under political siege. Louis de Broglie presented a pilot-wave dynamics at the 1927 Solvay Congress, but under sharp criticism — Wolfgang Pauli's objection about inelastic scattering chief among it — he withdrew it, and the orthodox Copenhagen view carried the day. In 1952, working at the University of São Paulo after being driven from Princeton during the McCarthy era, David Bohm rediscovered and completed the theory in two Physical Review papers, supplying the quantum potential and, crucially, a treatment of measurement as ordinary interaction that dissolved the need for a collapse postulate. The theory found its most influential champion in John Bell, who called it the disproof of the claim that hidden variables are impossible and whose own theorem was, in part, an attempt to understand the nonlocality that Bohm's construction made unavoidable. The rigorous modern form is due to Dürr, Goldstein and Zanghì, who clarified equivariance and the typicality account of quantum equilibrium.

History

1927

— de Broglie presents the pilot wave at the Solvay Congress, then abandons it

1932

— von Neumann's "impossibility proof" is read as ruling out hidden variables (later shown flawed)

1952

— Bohm's two Physical Review papers revive and complete the theory with the quantum potential

1966

— Bell cites Bohm's theory as the explicit counterexample to von Neumann's no-go

1992

— Dürr, Goldstein and Zanghì formalize quantum equilibrium and the Born rule via typicality

2005

— Couder and Fort's bouncing-droplet "walkers" realize a macroscopic pilot-wave analogue

Going further

References. D. Bohm, Phys. Rev. 85, 166 and 180 (1952) · L. de Broglie, Solvay 1927 (in Electrons et Photons, 1928) · J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, 2nd ed. · D. Dürr, S. Goldstein & N. Zanghì, J. Stat. Phys. 67, 843 (1992) (quantum equilibrium and typicality); Quantum Physics Without Quantum Philosophy (2013) · Holland, The Quantum Theory of Motion (1993) · Couder & Fort, Phys. Rev. Lett. 97, 154101 (2006).

Provenance. All results are standard. The many-body Schrödinger equation is owned by Time-Dependent Schrödinger Equation; the probability current $\mathbf j$ and the continuity equation $\partial_t|\psi|^2 + \nabla\!\cdot\mathbf j = 0$ by Probability Current in One Dimension; the no-signalling property of local operations by The Reduced Density Matrix and Partial Trace — all recalled by name. Owned and derived here: (1) the polar decomposition of the Schrödinger equation into continuity plus the quantum Hamilton–Jacobi equation with $Q = -\tfrac{\hbar^2}{2m}\nabla^2R/R$ , verified term-by-term symbolically; (2) the guidance law $v = \nabla S/m = j/|\psi|^2$ ; (3) equivariance, that the particle-transport and probability-transport equations coincide, verified symbolically. Worked numbers recomputed: a moving Gaussian gives $v = \hbar k/m$ and $Q(0) = +\hbar^2/2ma^2$ ; the entangled $\psi = e^{ix_1x_2}$ gives $v_1 = x_2$ ( $=0.5$ and $1.7$ for $x_2 = 0.5, 1.7$ ), while a product state gives $v_1 = 1$ independent of $x_2$ . The CHSH bounds ( $2$ classical, $2\sqrt2$ quantum) are stated with the derivation credited to Bell's Theorem and the CHSH Inequality; the hidden-variable program and the origin of the nonlocality requirement to Hidden Variable Theories and The EPR Paradox and Local Realism; spontaneous collapse and many-worlds are contrasted by name.

Builds on: Probability Current and the Continuity Equation · Hidden Variable Theories · The EPR Paradox and Local Realism · Bell's Theorem and the CHSH Inequality

Leads to: Objective Collapse Theories · The Many-Worlds Interpretation · The Quantum Measurement Problem

If the wavefunction still obeys the Schrödinger equation, what does adding trajectories actually buy you?

It buys a definite answer to "where is the particle?" at all times, and with it a dissolution of the measurement problem. In orthodox quantum mechanics the wavefunction is the whole story, so a measurement needs a separate collapse rule to produce a definite outcome. In Bohmian mechanics the outcome is just which wavepacket the actual configuration is in after an ordinary interaction — no collapse axiom, no observer, no classical–quantum cut. The trajectories also give intuition (double-slit paths, tunnelling times) and a clean realist ontology. The cost is nonlocality; the benefit is a complete, deterministic theory with the same predictions.

Does the quantum potential do work on the particle like an ordinary force?

It enters the quantum Hamilton–Jacobi equation exactly where a potential energy would, and its gradient $-\nabla Q$ acts like a force in the second-order (Newtonian) form of the trajectory equation. But it is unusual in two ways: it depends on the shape of the amplitude $R$ , not its magnitude, so it does not weaken with distance the way an ordinary potential does; and it is generated by the wavefunction itself, so it responds to the global configuration. That scale-independence is precisely why entangled quantum potentials can couple distant particles — the root of Bohmian nonlocality.

Bohmian mechanics is deterministic — so where does quantum randomness come from?

From ignorance of the initial configuration, exactly as in classical statistical mechanics. The trajectories are determined, but we do not know the precise starting position; we know only that it is drawn from the $|\psi|^2$ distribution (quantum equilibrium). Equivariance then guarantees the statistics stay $|\psi|^2$ , reproducing the Born rule. So quantum "randomness" is epistemic — like a coin flip in Newtonian mechanics — not a fundamental indeterminism. Whether quantum equilibrium itself needs justification is answered by the typicality analysis of Dürr, Goldstein and Zanghì.

If it is nonlocal, doesn't it violate relativity?

The underlying dynamics is nonlocal — distant entangled particles are coordinated instantaneously through the joint wavefunction — and reconciling this with Lorentz invariance is genuinely subtle and an active research area (a preferred foliation of spacetime is usually invoked). But at the level of observable statistics, no relativity-violating signalling is possible: in quantum equilibrium the marginal distribution of any local measurement is unaffected by distant operations (no-signalling). So the empirical predictions never conflict with relativistic causality, even though the hidden trajectories are nonlocally choreographed. This mirrors standard quantum mechanics, which is also nonlocal in its correlations but non-signalling.

How is this different from the many-worlds interpretation, which also keeps the Schrödinger equation?

Both keep the Schrödinger equation and reject a collapse postulate, but they differ on ontology. Many-worlds says the wavefunction is everything and all branches are real; the appearance of one outcome is a fact about which branch "you" are in. Bohmian mechanics adds a single actual configuration that is always in exactly one branch — the others are "empty waves" that no longer guide the particle. So Bohm has a definite, single history; many-worlds has all histories. Both reproduce quantum statistics; the choice is metaphysical, and it is drawn by name in The Many-Worlds Interpretation.

What happened to spin and relativistic particles — is Bohmian mechanics only for positions?

The basic theory is built for particle positions in nonrelativistic quantum mechanics, and spin is handled by letting the (spinor) wavefunction guide the positional trajectories — spin is a property of the wave, not an extra hidden variable, so a Stern–Gerlach measurement still produces a definite deflection via the pointer position. Relativistic and field-theoretic extensions (Bohmian field ontologies, Bell-type "beable" QFT) exist but are more involved and less settled, largely because of the tension between the theory's nonlocality and Lorentz invariance. These extensions are beyond this article's scope and belong to the graduate research literature.