Reply To: A problem of Bohmian mechanics

AvatarRoderich Tumulka

Hi Rainer, here are three remarks on the problem you raised.

First, the situation is in a certain respect the same in Bohmian mechanics and in standard quantum mechanics. That is because if we consider the wave function of object and apparatus together (or of the whole solar system) then Bohmian mechanics and standard quantum mechanics both use the Schroedinger equation, so they use the same wave function.

Second, your question is ultimately one about the form of that wave function. Considering two systems, the x-system and the y-system (the environment of the x-system), for the x-system to be an “effective system,” i.e., to have an effective wave function, we need that Psi(x,y) is locally a product, i.e., Psi(x,y) = psi(x)phi(y) + Phi(x,y) where Phi has y-support macroscopically disjoint from the actual (Bohmian) Y. So the question is, why should the wave function of object and apparatus before a quantum experiment be of such local product form? Why should the object be initially unentangled with the apparatus? In fact, random wave functions of two systems typically have a high degree of entanglement. For example, if the x-system and the y-system are jointly in thermal equilibrium then they are highly entangled.

A relevant mechanism is the stretching of wave functions under the free evolution. It is known from scattering theory that the wave function of a single particle, when evolving freely (zero potential), becomes locally a plane wave. Let us apply this to, for example, a double-slit experiment, in which the electron used for the experiment gets “boiled off” from a piece of metal; initially, that electron is in thermal equilibrium with the metal, and thus highly entangled with other electrons in the metal. But once it leaves the metal, it propagates freely, so its wave function approaches a spherical shape, locally a plane wave. A collimation slit allows only the part of the wave propagating in the right direction to enter the double-slit setup. If the electron (the x-system) actually passes through the collimation slit, and gets used in the experiment, then the wave function Psi(x,y) of electron and apparatus is, in a neighborhood of (X,Y) in configuration space, a plane wave in the x-variable (and complicated in y). Since plane waves are not entangled (in the spinless case), Psi is locally a product psi(x)phi(y). (In the presence of spin, the spin of the electron in the double-slit setup may well be entangled with the spin of electrons in the metal piece, but that does not matter for the experiment.)

Third, the measurement rule can be formulated in Bohmian mechanics as follows: *IF* the object is initially unentangled with the apparatus *THEN* the probability distribution of the outcome is given by the standard formula. However, Bohmian mechanics is still meaningful, and still makes unambiguous predictions, in case the object *IS* initially entangled (or not perfectly unentangled) with the apparatus. The prediction is just whatever follows from the Schroedinger equation and Bohm’s equation of motion for object and apparatus together. For standard quantum mechanics, in contrast, this situation would be problematical because it is not covered by the standard axioms.

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