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For the separation of a two-dimensional central-potential eigenstate, the guidance equation gives zero radial velocity and constant-radius circular motion, with circulation $2\pi \hslash k/m$. A real superposition of the degenerate $k$ and $-k$ eigenstates has zero current away from nodes instead. The speed depends on the actual wavefunction, not just its energy.

A deterministic interpretation with a definite particle configuration guided by the wavefunction. The latter obeys the Time-dependent Schrödinger equation, while the particle follows the guidance equation. A Born rule distribution of initial positions is preserved by quantum equilibrium equivariance. The guidance law restricts admissible initial velocities, even when trajectories are rewritten as a second-order force equation.

If $\Psi (q,y)=\psi (q)\Phi (y)+{\Psi }^{\perp }(q,y)$ and the actual environmental configuration $Y$ lies in the support of $\Phi$ disjoint from the environmental support of ${\Psi }^{\perp }$, then evaluating the guidance equation at $(Q,Y)$ gives the velocity generated by $\psi$. This is effective collapse without deleting a universal empty wave.

A component of the wave function whose configuration-space support does not contain the actual configuration. It can affect later guidance equation evolution if branches overlap again; enduring environmental separation makes it dynamically irrelevant to the occupied branch to excellent approximation.

In Bohmian mechanics the particle has a definite position $\mathbf{X}(t)$ at every time. Its wavefunction obeys the usual autonomous wave equation, while its actual position follows the guidance equationHere $\mathbf{j}$ is the probability current. The Born rule is the quantum-equilibrium choice of initial position distribution $\rho =|\psi {|}^2$; quantum equilibrium equivariance ensures that this distribution persists because it obeys the same probability continuity equation as the wave amplitude. The guidance equation fixes the initial velocity as well as subsequent velocities: the second-order equation below does not permit an independent arbitrary initial velocity.

Define the quantum potentialTaking the gradient of the real Madelung equations givesOn any smooth phase patch $\nabla \times \mathbf{v}=0$, so $\nabla (|\mathbf{v}{|}^2/2)=(\mathbf{v}\cdot \nabla )\mathbf{v}$. Along the actual path, differentiation is the material derivative $D/Dt={\partial }_t+\mathbf{v}\cdot \nabla$. ThereforeThis is the Bohmian mechanics Newton form: the classical force is supplemented by the amplitude-dependent quantum potential. Neither division by $R$ nor a smooth phase is justified at a wavefunction node, so the derivation applies on nonzero-amplitude regions. A nonzero circulation around a node is compatible with the locally curl-free guidance equation.

For the separated stationary state, restore its time factor $e^{-iEt/\hslash }$. Away from radial nodes, the quantum phase is $S=k\varphi -Et/\hslash$, up to a constant $0$ or $\pi$ where the real radial function has fixed sign. The guidance equation in plane polar coordinates therefore gives the Bohmian circulation of an angular-momentum eigenstateThe direction is $\hat{\mathbf{\varphi }}$ for $k>0$ and $-\hat{\mathbf{\varphi }}$ for $k<0$; the velocity is zero for $k=0$. Each admissible trajectory is a circle:The orbital angular momentum along the trajectory is $m{r}_0{v}_{\varphi }=\hslash k$. The origin or any zero-amplitude circle is excluded from this local formula. For the real degenerate superposition constructed above, the spatial quantum phase is constant on each nodal sector, so its Bohmian mechanics velocity is instead zero. Thus the circular motion is a conclusion about the separated angular-momentum eigenstate, not an arbitrary energy eigenstate.

The Born rule assigns $p_i=⟨\psi |P_i|\psi ⟩$ to a projective measurement with projection operators $P_i$. Its numerical form can be shared by theories while the source of the probabilities differs. Comparing Bohmian mechanics, the GRW model and the many-worlds interpretation exposes three different roles: uncertainty over actual configurations, fundamental stochastic events, and weights for branch-relative expectation.

In Bohmian mechanics, the Schrödinger equation implies a probability continuity equation for $|\Psi {|}^2$ with current $j$, while the guidance equation transports an ensemble density $\rho$ with velocity $v=j/|\Psi {|}^2$:Thus $\rho =|\Psi {|}^2$ is preserved: quantum equilibrium equivariance. If an apparatus interaction produces disjoint configuration regions ${\Gamma }_i$ with occupied-branch norm $|c_i|$, integrating the equilibrium distribution gives $\mathrm{Pr}(Q\in {\Gamma }_i)=|c_i{|}^2$. The actual outcome is fixed by initial configuration; the distribution describes the typical or uncertain configuration, not a fundamental random collapse. Equivariance alone does not prove equilibrium initially: every admissible density is transported by the velocity field. A typicality account privileges the equivariant $|\Psi {|}^2$ measure and then explains subsystem statistics conditionally; that explanatory premise must not be hidden in the continuity equation.^[ref]

In the GRW model, the centre law $p(X)=\Vert L_i(X)\Psi {\Vert }^2$ is part of the stochastic dynamics. Since $\int L_i(X{)}^{†}{L}_i(X)dX=I$, it is normalized. For two well-separated correlated pointer packets with coefficients $a,b$, integration over centres around their respective occupied regions gives approximately $|a{|}^2,|b{|}^2$; cross-contributions are suppressed at separations large compared with $r_C$. This explains why macroscopic records follow the Born rule approximately, without a special postulate applying only when an observer measures. However, the squared-norm centre law already carries the relevant mathematical probability structure. The theory relocates the rule into a universal physical law; it does not derive all probabilities from a deterministic equation. There are also small predicted deviations from standard unitary quantum mechanics.^[ref]

In the many-worlds interpretation, every nonzero decohering component persists, so the question is not which single global branch will survive. It is why a rational observer should use branch weight $|c_i{|}^2$ for expectation or caring about successor records. Equal-amplitude symmetry and decision theory motivate equal weights for physically equivalent alternatives. For rational $|c_i{|}^2=m_i/M$, an ancillary refinement into $m_i$ equal-norm alternatives can motivate weight $m_i/M$, followed by continuity for other amplitudes. This argument requires assumptions that refining irrelevant details does not change value, that preferences respect the relevant quantum symmetries, and that weight tracks the intended notion of expectation. Those are substantive premises, not consequences of branch existence alone.^[ref]

Naive branch counting is unsuitable: decoherence yields approximate, refinement-dependent branches, and splitting one record into more environmental records should not change its weight. Nor is diagonality of a density matrix already an account of probability. The comparison therefore has no common derivation from bare formalism. Bohmian mechanics needs a justified equilibrium or typicality measure, GRW model needs a justified stochastic law, and an Everettian account needs a justified connection from branch structure and rationality to experienced statistics. Each offers a coherent location for the Born rule; the remaining explanatory costs differ.

Quantum mechanics is compatible with quantum no-signalling, but its full correlations are incompatible with Bell local causality under the usual additional assumptions. The answer depends on which locality condition is meant. A local operational prohibition on controllable faster-than-light communication is weaker than a causal explanation of correlations by a complete shared past.

Let $a,b\in \{-1,1\}$ be outcomes, $x,y$ local settings and $\lambda$ a complete relevant past specification. Bell factorization and measurement independence demandNeither assumes deterministic response. Put $A_x(\lambda )={\sum }_{a}ap(a|x,\lambda )$ and similarly $B_y$, so $|A_x|,|B_y|\le 1$. Pointwise,Averaging with the same $\rho$ for all settings gives the CHSH inequality $|E_{00}+E_{01}+E_{10}-E_{11}|\le 2$. For a spin singlet state, $E(\mathbf{a},\mathbf{b})=-\mathbf{a}\cdot \mathbf{b}$. Choose ${\mathbf{a}}_0=\hat{z}$, ${\mathbf{a}}_1=\hat{x}$, ${\mathbf{b}}_0=(\hat{z}+\hat{x})/\sqrt{2}$, ${\mathbf{b}}_1=(\hat{z}-\hat{x})/\sqrt{2}$; the magnitude becomes $2\sqrt{2}$. Thus the assumptions cannot all be maintained.^[ref]

There is nevertheless no signalling contradiction. If a local unread measurement has Kraus operators $M_a$ with ${\sum }_a{M}_a^{†}{M}_a=I$, thenThe equality follows by testing both sides against any distant observable and using cyclicity of the trace. Distant conditional states can differ after selecting $a$, but communicating that selection needs an ordinary message. The microcausality condition in quantum field theory similarly ensures commutation of spacelike-separated local observables; it does not make the vacuum or other entangled quantum states Bell-factorizable.

One can preserve different notions of locality by changing premises. Bohmian mechanics retains unique outcomes but its guidance equation is nonlocal for entangled systems; quantum equilibrium equivariance preserves the equilibrium distribution which masks that dependence from signalling. The many-worlds interpretation can combine local interaction equations with no unique global outcome, but a claim of locality must explain its branch-relative correlations; it has not thereby constructed the above Bell-local single-outcome model. The measurement-independence loophole, including suitable retrocausal or correlated-setting models, drops the shared setting-independent $\rho$ premise. Such models are logical possibilities, not demonstrations that the ordinary locality assumptions survive unchanged.

The justified answer is therefore conditional rather than an unqualified yes or no. Operational locality is retained by the quantum formalism. A theory reproducing all its correlations cannot retain Bell factorization, independent settings and the ordinary joint-outcome framework together. Neither denial of hidden variables nor an appeal to relativity alone removes the probabilistic contradiction.

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.

Both the ensemble position density transported by the guidance equation and the squared wave amplitude satisfy the same probability continuity equation. Subject to existence and uniqueness of that transport, an initial Born rule distribution stays a Born rule distribution. This is a preservation statement, not a derivation that every initial ensemble is already in equilibrium.

The amplitude-dependent extra potential in the Madelung equations. Taking the gradient of their phase equation and using the guidance equation yields $mD\mathbf{v}/Dt=-\nabla (V+Q)$, where $D/Dt$ is the material derivative. At wavefunction nodes this local expression may be singular.