de Broglie-Bohm theory

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.

The ensemble density of actual configurations equals the squared norm of the guiding wave function. For spinless particles with $v_k=j_k/|\Psi {|}^2$, the Schrödinger continuity equation is ${\partial }_t|\Psi {|}^2+{\sum }_k{\nabla }_k\cdot (|\Psi {|}^2{v}_k)=0$. A density transported by the same guiding field satisfies the same equation, so initial equality is preserved. This equivariance is a consistency property, not a proof that every possible initial ensemble is in equilibrium. The physical account of the equilibrium assumption and subsystem statistics requires an additional argument.

In the spinless version of Bohmian mechanics, insert the actual environment configuration $Y(t)$ into a total wave function $\Psi (x,y,t)$ to obtain a subsystem's conditional wave function, when that slice is nonzero. It encodes the actual environment relevant to the subsystem's guiding dynamics. It need not satisfy an autonomous subsystem Schrödinger equation; further separation conditions are needed for an effective wave function.^[ref]

If total-wave branches become disjoint in the relevant environment configuration variables and the actual environment lies within one branch, the subsystem's conditional wave function is proportional to that branch's subsystem factor. Persistent separation makes its autonomous use an effective description. No term of the universal wave function has literally been deleted. Possible recombination prevents treating every temporary packet separation as irreversible collapse.^[ref]

An interpretation treating the universal wave function as part of the law governing actual configurations rather than an additional material field. The interpretation is motivated partly by the absence of reciprocal configuration-to-wave influence in Bohmian mechanics.^[ref]

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.

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.

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.

For a spinless particle without a magnetic vector potential, Bohmian mechanics sets $\dot{\mathbf{X}}=\hslash \nabla S/m$, where $S$ is the dimensionless quantum phase. Equivalently the velocity is the probability current divided by the probability density. This formula is local to nonzero wavefunction regions.

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.