# Quasiparticles in Bohmian mechanics

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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?

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I think that your inherent suggestion that the Bohmian mechanics suffers from an inconsistency because of quasiparticles is right. QM shows that it doesn't matter whether one says that something is a particle or quasiparticle: the Hilbert spaces and dynamics and probabilistic predictions are isomorphic. In the Bohmian mechanics, it always matters because particles' degrees of freedom are "beable" while other e.g. quasiparticles' degrees of freedom are not. For the same reason, quantum computers can't simulate the reality in Bohmian mechanics. It just doesn't work.

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@Luboš Motl: Electron (a particle) in Hydrogen atom is always in a mixed state - it does not have an electron wave function $\psi (\vec{r}_e)$. Instead, it is a quasi-particle that may be in a pure state $\psi (\vec{r})$ where $\vec{r}=\vec{r}_e-\vec{r}_p$. As well, proton is also in a mixed state but the center of mass may be in a pure state like $exp\left ( i(\vec{P}\vec{R}-\frac{P^2}{2M_A\hbar}t)\right)$.There are no such real particles with $\mu$ and $M_A=m_e+m_p$; the corresponding variables describe quasi-particles. It is namely quasi-particle energy in atom that is quantized.

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If we lock an atom in a box, its center of mass (a quasi-particle) motion will be quantized too. In this respect we observe quasi-particle properties: proper frequencies, total masses, etc. They are beable but belong to compound systems.

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The answer to your question, @Vladimir, is obviously No: every quantum system (whether it is a particle or quasiparticle) may always be found in a pure state. That's why any quantum system may exhibit 100% destructive interference etc., something that would indeed be impossible if an object were "forced" to be partly mixed, and this is one reason why any picture of the quantum phenomena that tries to be more classical than QM (including Bohmian mechanics) is inevitable inconsistent with the tests of interference and other empirically established features of the quantum world.

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I did not ask any question for you to answer "No", Lubosh. I just told something apparently unknown to you, unfortunately.

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What do you mean by "meaning"? How can you say that quasiparticles have any meaning in standard quantum mechanics? They're just a very good way of describing an emergent phenomenon; they don't really exist in the usual sense of the word.

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Can I also say that after thinking it over, I suspect there may be a very good question lurking in here. Namely: do the trajectories of quasiparticles in Bohmian mechanics obey the same rules as the trajectories of particles in Bohmian mechanics? And if not, what rules do they obey?

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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

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answered Dec 24, 2011 by (1,725 points)

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