# Origin of position-momentum asymmetry in quantum mechanics

Elementary quantum theory teaches there exists a symmetry between position space and momentum space - you are free to switch by Fourier transform between position eigenvectors or momentum eigenvectors to express the wave function of a particle.

In articles on decoherence I have read, but don't understand, that this symmetry is broken by environmental interaction and position becomes the preferred basis (or the preferred observable is the position of the system).

Have I understood this correctly? If so is there a simple way to understand the origin of this asymmetry between position and momentum (or an SE post where this has already been discussed at a simple level)?

• you could ask the same thing about Hamiltonian mechanics... Aug 18, 2016 at 7:52
• So to help orientate my question further: as I understand it in Hamiltonian mechanics a particle has a definite position and momentum and during evolution that "symmetry" between position and momentum remains. In discussions of quantum decoherence you read phrases like "decoherence leads to mixture of narrow position-space wave packets". It is this asymmetry I am trying to understand. Aug 18, 2016 at 9:55
• OK that makes sense. So is it that typical environmental interactions measure position? Can you give examples of an environmental interaction which measures momentum (or other observable not position)? Aug 18, 2016 at 10:06
• Your sources lied. The environment measures both the position nor the momentum. It certainly doesn't measure the position to infinite accuracy, and it certainly measures the momentum to some degree. Aug 18, 2016 at 14:22
• I've now found this statement: "since the force law describing system-environment interaction typically depends on some power of distance (e.g. Coulomb law) the interaction Hamiltonian will commute with the position operator." This would answer my question if it is correct. Aug 18, 2016 at 15:02

The asymmetry arises from the measurement apparatus. Which basis is chosen depends on what kind of environmental interaction you have.

In general this is formalised using the von-Neumann measurement scheme. It describes how a pointer gets entangled with the state variable, where the pointer is an approximately classical object.

The Hamiltonian that causes this entanglement to occur then determines which basis of the quantum system can be distinguished using the measurement apparatus consisting of the pointer and the interaction (e.g. the screen and the magnet respectively, in the Stern-Gerlach experiment, where the spin states become entangled with the position states on the screen).

Note that this only describes how the system and the pointer entangle, i.e. how correlations between them occur. From the comments below this answer I conclude that the OP's question is actually about how decoherence happens dynamically. The compulsory reference on this is Nieuwenhuizen et al., where they solve models that can describe real measurement processes. Please note that this does not have to yield the position basis as the preferred one, in fact in the particular Curie-Weiss model that is solved in the paper it is the spin basis again (simply because spin is easy to deal with).

• Yes ok, I am familiar with the von-Neumann measurement scheme. But I am not a working physicist - I don't have many Stern-Gerlach apparatus in the environment of my living room. Yet quantum measurement through decohorence is playing out all around me. The claim is that here position measurements are preferred over other types of measurement. Aug 18, 2016 at 13:16
• @BruceGreetham I am not sure this is still a physics question. are you asking if humans preferably "do" position measurements? well yes, because you can see where something is, but you will have a hard time seeing what momentum something has... I may misunderstand your follow-up question. Aug 18, 2016 at 13:27
• No I'm definitely not asking that, but I see how you got to that thought. What I am discovering here is that there are two meanings of environmental interaction among physicists (and probably two meanings of measurement). So to be clear I am trying to study the motion of a particle (treated quantum mechanically) in the presence of an environment of atoms and photons hitting it. This interaction is regarded in decoherence literature as a form of "measurement" in itself. i.e. no measurement apparatus required. That was the meaning of my original question. Aug 18, 2016 at 14:06
• @BruceGreetham see edit (v2). Hope that helps. Aug 18, 2016 at 14:15
• Thanks that's more what I'm after - you've given me something to think about. Aug 18, 2016 at 14:19