de Broglie–Bohm Pilot-Wave Theory

De Broglie–Bohm theory supplements the wavefunction with an actual configuration. For nonrelativistic particles, the configuration follows a guiding equation while the wavefunction evolves by Schrödinger’s equation. Measurement outcomes are definite configurations of apparatus records, and quantum equilibrium yields the usual Born statistics.

Required background. Born rule: Translate a specified state and measurement into outcome probabilities.

Wavefunction and configuration

For spinless particles with positions $Q_k$ and masses $m_k$,

$$i\hbar\partial_t\Psi=H\Psi,\qquad \dot Q_k=\frac{\hbar}{m_k} \operatorname{Im}\frac{\nabla_k\Psi}{\Psi}(Q_1,\ldots,Q_N).$$

The formula is stated away from nodes and with the usual scalar-potential Hamiltonian; gauge fields and spin require the appropriate conserved current. The wavefunction lives on configuration space. It is not generally a collection of independent physical-space waves.

Writing $\Psi=Re^{iS/\hbar}$ gives $\dot Q_k=\nabla_kS/m_k$. In a plane wave, the particle velocity is the familiar momentum divided by mass. In a real stationary wavefunction it can vanish, despite a nonzero distribution of momentum-measurement outcomes.

Measurement and effective collapse

Suppose the interaction produces

$$\Psi(x,y)=\sum_a c_a\psi_a(x)\Phi_a(y),$$

where the apparatus packets $\Phi_a$ have effectively disjoint configuration-space supports. The actual apparatus configuration $Y$ lies in one support. The conditional subsystem wavefunction $\Psi(x,Y)$ is then proportional to the corresponding $\psi_a$.

This is effective collapse: other branches persist in the universal wavefunction but cease to guide the occupied configuration under the disjoint-support approximation. Recoherence can invalidate that approximation. A definite position record does not imply that every measured operator had a pre-existing noncontextual value.

Probability and empirical agreement

The equilibrium density $|\Psi|^2$ is preserved by the guiding flow. The probability account distinguishes this equivariance theorem from the initial equilibrium or typicality assumption.

With equilibrium and an appropriate measurement model, the distribution of apparatus configurations reproduces standard quantum statistics. The claim has a domain: a specific nonrelativistic model is not automatically a complete relativistic quantum field theory. Different extensions handle particle creation, fields, or a preferred foliation differently.

Nonlocal dependence

For an entangled wavefunction, $\dot Q_k$ can depend on distant particle positions through $\Psi(Q)$. This is explicit nonlocality of the hidden-variable dynamics. It evades the Bell-local class rather than refuting Bell’s theorem.

Equilibrium predictions remain no-signaling: the measurable marginal agrees with quantum theory. Nonlocal dynamics and controllable faster-than-light communication are distinct claims. Hypothetical nonequilibrium theories need separate analysis.

Strengths, limitations, and caricatures

The theory supplies a clear primitive configuration ontology and a measurement analysis using the same dynamics as other interactions. Its costs include the special role of configuration, nonlocal guiding structure, and the work needed for relativistic extensions.

“Particles follow classical paths” is false: their velocities are guided by the wavefunction and can be strongly nonclassical. “Hidden variables were ruled out by Bell” is also false: Bell excludes a precisely specified local class.

Exercises

For $\psi(x,t)=Ae^{i(kx-\omega t)}$, calculate the guiding velocity. Why does adding a second plane wave generally spoil the interpretation as a classical mixture of two velocities?

Solution

The velocity is $\hbar k/m$. For a coherent superposition, current and density contain interference terms, and the velocity is their ratio $j/|\psi|^2$. It is not obtained by selecting one of the original plane-wave velocities with classical mixture weights.

References

• D. Bohm, “A suggested interpretation of the quantum theory in terms of ‘hidden’ variables. I,” Physical Review 85, 166–179 (1952), doi:10.1103/PhysRev.85.166.
• D. Dürr, S. Goldstein, and N. Zanghì, “Quantum equilibrium and the origin of absolute uncertainty,” Journal of Statistical Physics 67, 843–907 (1992), author manuscript.