Editing De Broglie–Bohm theory (section)

==== Operators as observables ====
In the standard quantum formalism, measuring observables is generally thought of as measuring operators on the Hilbert space. For example, measuring position is considered to be a measurement of the position operator. This relationship between physical measurements and Hilbert space operators is, for standard quantum mechanics, an additional axiom of the theory. The de Broglie–Bohm theory, by contrast, requires no such measurement axioms (and measurement as such is not a dynamically distinct or special sub-category of physical processes in the theory). In particular, the usual operators-as-observables formalism is, for de Broglie–Bohm theory, a theorem.<ref>{{Cite journal|arxiv=quant-ph/0308038|last1=Dürr|first1=Detlef|title=Quantum Equilibrium and the Role of Operators as Observables in Quantum Theory|journal=Journal of Statistical Physics |volume=116|issue=1–4|pages=959|last2=Goldstein|first2=Sheldon|last3=Zanghì|first3=Nino|year=2003|doi=10.1023/B:JOSS.0000037234.80916.d0|bibcode=2004JSP...116..959D|citeseerx=10.1.1.252.1653|s2cid=123303}}</ref> A major point of the analysis is that many of the measurements of the observables do not correspond to properties of the particles; they are (as in the case of spin discussed above) measurements of the wavefunction.

In the history of de Broglie–Bohm theory, the proponents have often had to deal with claims that this theory is impossible. Such arguments are generally based on inappropriate analysis of operators as observables. If one believes that spin measurements are indeed measuring the spin of a particle that existed prior to the measurement, then one does reach contradictions. De Broglie–Bohm theory deals with this by noting that spin is not a feature of the particle, but rather that of the wavefunction. As such, it only has a definite outcome once the experimental apparatus is chosen. Once that is taken into account, the impossibility theorems become irrelevant.
There are also objections to this theory based on what it says about particular situations usually involving eigenstates of an operator. For example, the ground state of hydrogen is a real wavefunction. According to the guiding equation, this means that the electron is at rest when in this state. Nevertheless, it is distributed according to <math>|\psi|^2</math>, and no contradiction to experimental results is possible to detect.

Operators as observables leads many to believe that many operators are equivalent. De Broglie–Bohm theory, from this perspective, chooses the position observable as a favored observable rather than, say, the momentum observable. Again, the link to the position observable is a consequence of the dynamics. The motivation for de Broglie–Bohm theory is to describe a system of particles. This implies that the goal of the theory is to describe the positions of those particles at all times. Other observables do not have this compelling ontological status. Having definite positions explains having definite results such as flashes on a detector screen. Other observables would not lead to that conclusion, but there need not be any problem in defining a mathematical theory for other observables; see Hyman et al.<ref>{{Cite journal|doi=10.1088/0305-4470/37/44/L02|title=Bohmian mechanics with discrete operators|journal=Journal of Physics A: Mathematical and General|volume=37|issue=44|pages=L547|year=2004|last1=Hyman|first1=Ross|last2=Caldwell|first2=Shane A|last3=Dalton|first3=Edward|bibcode=2004JPhA...37L.547H|arxiv=quant-ph/0401008|s2cid=6073288}}</ref> for an exploration of the fact that a probability density and probability current can be defined for any set of commuting operators.