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First part of David Bohm's two-paper account of de Broglie-Bohm theory, published in Physical Review 85, starting at page 166.

Second part of David Bohm's account of de Broglie-Bohm theory, treating measurement and the relation to ordinary predictions. Published in Physical Review 85, starting at page 180.

An empty wave is a nonzero component of the wave function in Bohmian mechanics containing no actual particle configuration. Its persistence is a question about ontology and explanation, not an inconsistent prediction of duplicate detected particles. In de Broglie-Bohm theory, the wave function evolves by the Schrödinger equation while the actual configuration $Q$ evolves by the guidance equation. For spinless particles without a magnetic vector potential,The physical state includes both elements, so a component of $\Psi$ need not contain another set of particles.^[ref]

After a position-recording interaction, write the universal wave function as $\Psi (q,y)=a{\psi }_0(q){\Phi }_0(y)+b{\psi }_1(q){\Phi }_1(y)$, with apparatus configurations $y$ supported in disjoint regions. If the actual $Y$ lies in the first region, ${\Phi }_1(Y)$ and its relevant local derivatives vanish in the ideal disjoint-support model. Evaluating the guidance equation at $(Q,Y)$ therefore gives the velocity field of the occupied component alone. For realistic nearly disjoint packets this is an approximation justified by decoherence. The other component is empty in configuration space, even when the system packets overlap in ordinary SPACE. This is the effective wave function in Bohmian mechanics.

The empty component cannot simply be deleted from the fundamental theory. In a coherent interferometer, separated packets can later overlap and both contribute to the actual probability current. Their relative quantum phase can change the outgoing particle's path. An empty arm is therefore capable of future dynamical significance. When an environment records the alternatives in very many degrees of freedom, restoring their configuration-space overlap is overwhelmingly difficult; that explains effective collapse without a literal change to the universal wave function.

There are two serious objections. First, if every branch has the detailed functional organization of a detector and observer, what additional work do the particles do in making only the occupied branch a world with records? A many-worlds critic can question whether the particle ontology is doing explanatory work rather than merely labelling one branch. The Bohmian response is that ordinary objects and records are constituted by actual configurations, not by every formally similar pattern in $\Psi$. Assessing that response requires an account of material constitution; counting amplitudes alone settles neither side.

Second, the universal wave function affects particles without the usual reciprocal influence of particles on it. This can look unlike a material field. A nomological interpretation of the wave function treats its role more like law than additional matter, although time-dependent effective subsystem waves make that position philosophically demanding.^[ref]

Neither objection licenses assigning ordinary particle energy separately to every empty component as though it contained a second classical apparatus. Such an assignment would require further assumptions. The definite mathematical theory reproduces standard equilibrium statistics while retaining empty components. The problem is the cost and intelligibility of that ontology, especially relative to alternatives, not an unexplained failure of its specified guidance equation.

In de Broglie-Bohm theory, the system has both a wavefunction and an actual configuration of particle positions. The wavefunction obeys the Time-dependent Schrödinger equation; the configuration follows the guidance equation. For spinless particles without electromagnetic vector potentials, write $\Psi =R{e}^{iS/\hslash }$ using a dimensionful action $S$. ThenThe actual configuration selects a trajectory, while the guiding wave generally retains all branches. A measurement correlates system and apparatus configurations; the actual apparatus position lies in one recorded branch. Effective collapse follows when wave packets have separated sufficiently in configuration space, although the full wave continues its Schrödinger evolution.

Substituting the polar form into that equation yields the probability continuity equation and a modified Hamilton-Jacobi equation,The amplitude-dependent term is the quantum potential. Taking gradients gives a second-order force description, but initial velocities remain constrained by the guidance law. Nodes require care because the displayed quotient is singular there.

If the initial configuration distribution is $|\Psi {|}^2$, the same continuity equation preserves that distribution: quantum equilibrium equivariance. It then reproduces ordinary nonrelativistic quantum measurement statistics within the model's applicability. Equivariance preserves a distribution; it is not by itself a proof that every initial distribution equals $|\Psi {|}^2$.

For entangled particles the guiding wave lives on their joint configuration space, so one particle's velocity can depend on the distant actual configuration. This explicit nonlocality avoids the locality premise of a Bell inequality. In quantum equilibrium it does not permit controllable faster-than-light signalling. Spin extensions use a spinor wavefunction and its probability current; they need not assign a classical little spinning vector with simultaneous definite values for every spin component.

David Bohm developed this account in his two 1952 papers. The first sets out the dynamics and the second treats measurement and the relation to ordinary predictions.^[ref][ref] The theory supplies additional dynamical variables and a particular nonlocal account of outcomes, rather than deriving a unique interpretation from the standard wavefunction alone.