Measurement of Position in Bohmian Mechanics

Question

In many formulations of Bohmian mechanics, researchers seem to claim that 1) measurements of observables such as spin are just measurements of the position of a pointer variable, such as the Stern-Gerlach experiment apparatus and 2) measurements of position reveal the position of particles. This effectively removes the 'measurement problem', where superpositions of classical pointers can be explained.

However, this explanation seems to be insufficient. There exists the (philosophical) question of why the position is such an important variable - in classical mechanics, the position and the momentum are considered the state of a system. Even worse, position-measuring apparatus, such as the screen in the double slit experiment, obviously perturb physical quantities such as momentum. How is it that we can say measurement reveals the exact position, but perturbs other variables? Isn't it more natural to assume that position measurements are also performed by entangling the system with a classical pointer object (for instance, the screen and the particles become entangled in a double-slit experiment)? For example, weak measurements are performed similarly, by entangling the system with a Gaussian pointer distribution with a high spread. Therefore, position measurement also perturbs particles and therefore cannot reveal the exact position values. A paper by N. Gisin supports this claim, that position measurements in Bohmian mechanics do not reveal exact positions: https://doi.org/10.3390/e20020105.

Answer 1

In Bohmian mechanics, the entire trajectory of all particles through space as a function of time is well-defined. So they have both a position and a momentum at all times.

Yes, you are correct that measurements never reveal the exact position or momentum. So what?

Answer 2

Actually, Bohmian mechanics does not talk about measurement at all.

The central objects of Bohmian mechanics are particles that move according to a certain equation (involving the wave function). It is as simple as that. There are no additional postulates about measurements.

However, just because Bohmian mechanics does not involve measurements in its formulation doesn't mean Bohmian mechanics has nothing to say about measurements. The formulation also doesn't involve pianos, but we can analyse the behaviour of particles that constitute a piano. In the same way, Bohmian mechanics can be used to analyse physical situations that correspond to experiments that are called measurements. It is actually a strength of the theory, that it explains measurements without additional axioms. Everything can be derived from its dynamics.

Now, concerning an experimental setup that yields some numerical result, one can analyse (this has been done to a great extent) whether the numbers on the apparatus or the pointer position correspond to the true particle position. Whether this is the case depends on the experiment.

It turns out that such genuine measurements (i.e., measures that reveal a pre-existing property) of position are indeed possible. E.g., the position of the flash on the screen does indeed correspond to the position of the electron (this is a consequence of the dynamics).

However, not all measurements of "position operators" correspond to measurements of the true particle position. Sometimes (e.g., in the context of weak measurements), experiments are called position measurements if they have the same Born distribution as the particle's position. But of course, just because two things are identically distributed, they are not necessarily the same.

The Bohmian moral would be that the notion of measuring an observable should be taken with a grain of salt.