Editing De Broglie–Bohm theory (section)

=== The conditional wavefunction of a subsystem ===
In the formulation of the de Broglie–Bohm theory, there is only a wavefunction for the entire universe (which always evolves by the Schrödinger equation). Here, the "universe" is simply the system limited by the same boundary conditions used to solve the Schrödinger equation. However, once the theory is formulated, it is convenient to introduce a notion of wavefunction also for subsystems of the universe. Let us write the wavefunction of the universe as <math>\psi(t, q^\text{I}, q^\text{II})</math>, where <math>q^\text{I}</math> denotes the configuration variables associated to some subsystem (I) of the universe, and <math>q^\text{II}</math> denotes the remaining configuration variables. Denote respectively by <math>Q^\text{I}(t)</math> and <math>Q^\text{II}(t)</math> the actual configuration of subsystem (I) and of the rest of the universe. For simplicity, we consider here only the spinless case. The ''conditional wavefunction'' of subsystem (I) is defined by

:<math>\psi^\text{I}(t, q^\text{I}) = \psi(t, q^\text{I}, Q^\text{II}(t)).</math>

It follows immediately from the fact that <math>Q(t) = (Q^\text{I}(t), Q^\text{II}(t))</math> satisfies the guiding equation that also the configuration <math>Q^\text{I}(t)</math> satisfies a guiding equation identical to the one presented in the formulation of the theory, with the universal wavefunction <math>\psi</math> replaced with the conditional wavefunction <math>\psi^\text{I}</math>. Also, the fact that <math>Q(t)</math> is random with [[Probability density function|probability density]] given by the [[square modulus]] of <math>\psi(t,\cdot)</math> implies that the [[Conditional probability density function|conditional probability density]] of <math>Q^\text{I}(t)</math> given <math>Q^\text{II}(t)</math> is given by the square modulus of the (normalized) conditional wavefunction <math>\psi^\text{I}(t,\cdot)</math> (in the terminology of Dürr et al.<ref>{{Cite journal|arxiv=quant-ph/0308039 |last1=Dürr |first1=Detlef |title=Quantum Equilibrium and the Origin of Absolute Uncertainty |journal=Journal of Statistical Physics |volume=67 |issue=5–6 |pages=843–907 |last2=Goldstein |first2=Sheldon |last3=Zanghí |first3=Nino |year=2003 |doi=10.1007/BF01049004 |bibcode=1992JSP....67..843D|s2cid=15749334 }}</ref> this fact is called the ''fundamental conditional probability formula'').

Unlike the universal wavefunction, the conditional wavefunction of a subsystem does not always evolve by the Schrödinger equation, but in many situations it does. For instance, if the universal wavefunction factors as

:<math>\psi(t, q^\text{I} ,q^\text{II}) = \psi^\text{I}(t, q^\text{I}) \psi^\text{II}(t, q^\text{II}),</math>

then the conditional wavefunction of subsystem (I) is (up to an irrelevant scalar factor) equal to <math>\psi^\text{I}</math> (this is what standard quantum theory would regard as the wavefunction of subsystem (I)). If, in addition, the Hamiltonian does not contain an interaction term between subsystems (I) and (II), then <math>\psi^\text{I}</math> does satisfy a Schrödinger equation. More generally, assume that the universal wave function <math>\psi</math> can be written in the form

:<math>\psi(t, q^\text{I}, q^\text{II}) = \psi^\text{I}(t, q^\text{I}) \psi^\text{II}(t, q^\text{II}) + \phi(t, q^\text{I}, q^\text{II}),</math>

where <math>\phi</math> solves Schrödinger equation and, <math>\phi(t, q^\text{I}, Q^\text{II}(t)) = 0</math> for all <math>t</math> and <math>q^\text{I}</math>. Then, again, the conditional wavefunction of subsystem (I) is (up to an irrelevant scalar factor) equal to <math>\psi^\text{I}</math>, and if the Hamiltonian does not contain an interaction term between subsystems (I) and (II), then <math>\psi^\text{I}</math> satisfies a Schrödinger equation.

The fact that the conditional wavefunction of a subsystem does not always evolve by the Schrödinger equation is related to the fact that the usual collapse rule of standard quantum theory emerges from the Bohmian formalism when one considers conditional wavefunctions of subsystems.