Quantum Mechanics – Does a Pilot Wave Have Mass in Bohmian Mechanics?

Question

In Bohmian mechanics does a pilot wave have a mass and what is the mass?

Answer 1

Pilot waves have no more and no less mass than Schroedinger wavefuction waves do, since their mathematics is identical.

To bypass ghastly ontological quandaries, recall that the math of the waves cares not about what the particles they guide ("pilot") do or don't. The math of the waves is just the math of the Schroedinger equation in polar (Madelung) language.

So, for a free particle, of mass m, you have $$\psi = R \; \exp \left( \frac{i \, S}{\hbar} \right), ~~~~~ R^2 = |\psi|^2, ~~~ \leadsto $$
$$ \frac{\, \partial R^2 \,}{\, \partial t \,} + \nabla \cdot \left( R^2 \vec{v} \right) = 0 ~ , \tag {1} $$ where the particle velocity field is determined by the “guidance equation”, $\vec{v}\left(\,\vec{r},\,t\,\right) = \frac{1}{\,m\,} \, \nabla S\left(\, \vec{r},\, t \,\right) $, and also a further equation, a modified Hamilton–Jacobi equation,
$$- \frac{\partial S}{\partial t} = \frac{\;\left|\, \nabla S \,\right|^2\,}{\,2m\,} + Q ~ ,\tag {2} $$ where Q is the quantum potential, an additional piece lacking in the classical equation,
$$ Q = - \frac{\hbar^2}{\,2m\,} \frac{\nabla^2 R }{ R } ~, $$ responsible for all the "mysterious" effects of quantum mechanics.

The above guidance equation then amounts to
$$ m \, \frac{d}{dt} \, \vec{v} = - \nabla Q ~ , ~~~~\hbox{where} ~~~\frac{d}{dt} \equiv \frac{ \partial }{\, \partial t \,} + \vec{v} \cdot \nabla ~ . $$

But notice that, interpretations aside, except for the fact we used the velocity field as a mere shorthand for the gradient of S in (1), the probability continuity equation, the two PDEs (1), (2), don't know or care about the particle: they are really a transcription of the two components (real and imaginary) of Schroedinger's equation describing probability amplitude waves.

• As such, the waves they describe obviously don't have a mass, unless you bought the destructive interpretational blather about the particle being the same as its probability wave. They just dictate how a particle moves as they pilot it.

They are also dispersive, i.e., they have a group velocity different than the phase velocity, but this is not thought of as a mass.