Quasiparticles in Bohmian mechanics

Question

My questions are about de Broglie-Bohm "pilot wave" interpretation of quantum mechanics (a.k.a. Bohmian mechanics).

1. Do quasiparticles have any meaning in Bohmian mechanics, or not? Specifically, is it possible to trace the motion of a quasiparticle (e.g. a phonon, or a hole) by watching Bohmian trajectories?

2. Bohmian mechanics provides some explanation of difficulties related to quantum measurement process. But imagine that, in some kind of "theory of everything", all known elementary particles (leptons, quarks, gluons etc.) are actually quasiparticles (real particles are always confined). Would the explanation, provided by Bohmian mechanics, survive in this case?

Answer 1

Nobody wrote an answer, so I'll give it a try

1. Indeed in Bohmian QM quasiparticles are not on the same ground as ordinary particles. For example consider phonons in a crystal lattice. From a Bohmian POV, the atom position observables have fundamental significance. But observables natural from the phonon POV, such as phonon number are not functions of the atom positions. Of course these observables are still functions of atom positions + momenta, so in principle they can be assigned values along a Bohmian trajectory. However, this approach has serious issues. For one thing the "phononic" observables are still not going to play a symmetric role with the atomic positions. For another, the phonon number will not be integer*! This shows it hardly makes sense to talk about phonon trajectories
2. Actually, the elementary particles are "quasiparticles". They are excitations of the quantum fields. This indeed means the Bohmian approach runs into trouble, since in Bohmian QFT the fields become the fundamental observables, making it inconsistent with nonrelativistic Bohmian QM. Actually it's not the only problem of Bohmian QFT: it also fails to be Lorentz invariant

*In the simplest harmonic model, the occupation number of each mode is a linear function of its energy i.e. something quadratic in both positions and momenta

Answer 2

1. Given that Bohmian mechanics (better named de Broglie-Bohm or dBB theory) contains the Schrödinger equation, it contains quantum theory completely, and therefore all things which make sense in quantum theory remain meaningful. Of course, for quasiparticles nobody plans to construct trajectories.

And it is also not possible to simply "watch" Bohmian trajectories.

2. dBB theory is usually presented as a theory for particles, but this is far too restrictive. The better way is to present it as a theory for the configuration space: It defines a trajectory for the configuration $q(t)$.

So, the configuration may be, as well, a field $f(x)$ on the space. Then the "trajectory" is $f(x,t)$.

The mathematics of dBB theory works if the Hamiltonian has the form $H=p^2 + V(q)$. This works nicely with relativistic field theories $L = (d_tf)^2-(d_xf)^2$ + interaction terms gives momentum $p(x)=d_t$ $f(x)$ and gives a Hamiltonian quadratic in $p(x)$.

All the advantages provided by dBB theory do not depend on the question what is the particular configuration space. Thus, they would survive whatever the choice of Q is. All one has to care for is a way to obtain a choice of a configuration space so that $H=p^2 + V(q)$. For bosonic field theories no problem, for fermionic field theories proposals exist too. In the worst case, one can consider partial realizations (like defining trajectories only for bosons). This is less beautiful but preserves the major advantages (realism, causal explanation, no measurement problem).