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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.

Coauthor of work on quantum equilibrium equivariance and Bohmian mechanics.

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.

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.

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.